A viability theorem for set-valued states in a Hilbert space

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

A viability theorem for set-valued states in a Hilbert space Thomas Lorenz Applied Mathematics, RheinMain University of Applied Sciences, 65197 Wiesbaden, Germany

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 19 October 2016 Available online xxxx Submitted by H. Frankowska

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H. © 2017 Elsevier Inc. All rights reserved.

Keywords: Evolution inclusion Reachable set Viability condition Set differential inclusion Scalar topology Bounded scalar convergence

1. Introduction 1.1. Motivating set-valued states by robust control problems Currently modeling uncertainty is catching more and more interest in the mathematical community. The focus of this article is originally motivated by the following notion of “robust” control problems with state constraints: We extend the standard form of control equations with state constraints by a further timedependent parameter v(·), i.e., we consider the following autonomous control equation for a vector-valued   state x : [0, T ] −→ Rn with two time-dependent parameters u, v ∈ L1 [0, T ], Rm ⎧ ⎪ x (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u(t) v(t) ⎪ ⎪ ⎪ x(0) ⎪ ⎪ ⎪ ⎩ x(t)

  = f x(t), u(t), v(t) for a.e. t ∈ [0, T ], for a.e. t ∈ [0, T ], ∈ U ⊂ Rm ˜ ⊂ Rm for a.e. t ∈ [0, T ], ∈ U ∈ K0 ⊂ Rn , ∈ V

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2017.08.011 0022-247X/© 2017 Elsevier Inc. All rights reserved.

⊂ Rn

for every t ∈ [0, T ].

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Differential games represent a popular class of examples which can be reformulated in this way (see, e.g., [3,8,49,61] and references therein). But now we are free to choose only one of these parameters, say u(·), ˜ . This whereas there is no opportunity to specify the other parameter v(t) more precisely than just v(t) ∈ U situation occurs in differential games, for examples, whenever the first player has no detailed information about the strategy of the second player. His goal, however, remains to preserve the state constraint x(t) ∈ V – no matter how his antagonist decides to proceed. This situation is often considered a starting point for stochastic formulations. Probabilities play the role of weights and, the choice of the underlying probability space has an immediate consequence on the class of events which cannot be taken into consideration explicitly (due to their probability being 0). There are many concrete applications in which it is not recommendable at all to “neglect” any events. Collision avoidance of cars or planes are just two typical examples both dealing with securing human life. Hence, we prefer focusing on a completely deterministic formulation instead. Let us return to the situation in equation (1) for a moment, but without the state constraint x(t) ∈ V . The open-loop control u ∈ ˜ ), L1 ([0, T ], U ) can be chosen by the (first) player. In regard to the other control function v ∈ L1 ([0, T ], U n however, all possibilities should be taken into consideration. Then the state vector x(t) ∈ R at time t ∈ [0, T ] can attain each element of the set    ˜ ) : x(0) ∈ K0 , K t; u(·) = x(t) ∃ x ∈ W 1,1 ([0, t], Rn ), v ∈ L1 ([0, t], U  

x (τ ) = f x(τ ), u(τ ), v(τ ) for a.e. τ ∈ [0, t]  = x(t) ∃ x ∈ W 1,1 ([0, t], Rn ) : x(0) ∈ K0 ,  

˜ for a.e. τ ∈ [0, t] x (τ ) ∈ f x(τ ), u(τ ), U due to Filippov’s selection theorem (see, e.g., [10, Theorem 8.2.10] or [47, Proposition II.2.25]). The latter   ˜ . In comparison with stochasis called the reachable set of the (ordinary) differential inclusion x ∈ f x, u, U tic approaches, it does not consider any form of weighting (like a probability) for distinguishing between attainable vectors x(t) and thus, no “rare events” will be neglected in the very end. So far, u(·) plays the role of a time-dependent control parameter which determines the set-valued map ˜ −→ Rn for the K(·, u) : [0, T ] ; Rn and, the literature provides sufficient conditions on f : Rn × U × U   compactness of each reachable set K t, u(·) ⊂ Rn . The condition that all attainable states x(t) satisfy the   state constraint is now equivalent to the inclusion K t, u(·) ⊂ V for every t ∈ [0, T ]. This simple observation leads us to the following question: Does there exist an open-loop control u ∈   1 ˜ ) and the initial set K0 is contained L ([0, T ], U ) such that the reachable set K t, u(·) ⊂ Rn of x ∈ f (x, u, U in V for every t ∈ [0, T ]? From a more conceptual point of view, this question represents a viability problem – in a generalized sense: Here, the reachable set K(t; u(·)) ⊂ Rn is the time-dependent state variable. u ∈ L1 ([0, T ], U ) is interpreted as an open-loop control and, the state constraint has the form of a set inclusion, namely   K t, u(·) ⊂ V . ˜ ) does not occur any longer. It is “hidden” in the In this setting, the other control function v ∈ L1 ([0, T ], U ˜) construction of the set-valued state by means of the non-autonomous differential inclusion x ∈ f (x, u(t), U instead.   We are looking for (at least) one open-loop control u(·) such that the state constraint K t; u(·) ⊂ V is satisfied for every t ∈ [0, T ]. This reflects the gist of a viability problem (a.k.a. weak invariance problem, e.g., [3,8,9,21,24,25]). The key analytical challenge is based on the fact that the state variable is now a compact subset of Rn , not a vector in Rn . The set K(Rn ) of non-empty compact subsets of Rn , however, does not have an obvious linear structure. It is usually regarded as a metric space – due to the popular Pompeiu–Hausdorff distance dlPH .

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1.2. Extending analytical approaches from vectors to set-valued states The preceding example of robust control problems with state constraints has led us to the following ˜ −→ Rn , each open-loop control u ∈ analytical situation: Under suitable assumptions about f : Rn × U × U   1 n L ([0, T ], U ) and initial set K0 ⊂ R is related to a unique set-valued map K · , u(·) : [0, T ] ; Rn by means ˜ ) (Lebesgue-almost of the reachable set of K0 and the non-autonomous differential inclusion x ∈ f (x, u, U everywhere). This construction can be regarded as a generalization of the Aumann integral because in the special case   ˜ −→ Rn , the composition f u(·), U ˜ : [0, T ] ; Rn that f does not depend on x explicitly, i.e., f : U × U depends just on “time” t and so,   Def.  K t; u(·) = x(t) ∈ Rn ∃ x ∈ W 1,1 ([0, t], Rn ) : x(0) ∈ K0 ,  

˜ for a.e. τ ∈ [0, t] x (τ ) ∈ f u(τ ), U    ˜ dτ. = K0 + f u(τ ), U [0,t]

The essential advantage of reachable sets over Aumann integrals is that convexity (of the initial set K0 ⊂ Rn , for example) need not be preserved for all times. For this reason, we prefer approaches on the basis of reachable sets to the many concepts of set integral equations which start with the Aumann integral and are thus restricted to convex sets (see, e.g., [54,62,63,69–71,98] and references therein). Having specified a general form of set “integration” (namely reachable sets of differential inclusions), we can follow essentially the same steps as in calculus for introducing set integral equations which we are free to consider set differential equations: So far, the reachable set has been determined basically by the right-hand side of the underlying differential     ˜ : [0, T ] × Rn ; Rn , (t, x) → f x, u(t), U ˜ . inclusion, i.e., the set-valued map f ·, u(·), U   The basic notion of the next generalizing step is to let it also depend on the current set K t; u(·) ⊂ Rn . In other words, we take “feedback” into consideration, i.e., the time-dependent set K(t) ⊂ Rn has an explicit influence on its own evolution. This dependence on K(t) might be expressed in terms of non-local features like its diameter or its distance from a given fixed set, for example. The feedback can be formulated in terms of a set-valued map f˜ : [0, T ] × Rn × K(Rn ) ; Rn inducing the differential inclusion x (t) ∈   f˜ t, x(t), K(t) (e.g., [28–30]). In this article, we prefer re-arranging the arguments and relate each tuple of time t and compact set K ⊂ Rn to a set-valued map Rn ; Rn of “admissible” velocities depending on space x ∈ Rn . In the notation introduced below, this reads as a single-valued function [0, T ] × K(Rn ) −→ LIPco (Rn , Rn ). This perspective is mathematically equivalent (under suitable assumptions) and goes back to Aubin [4–6]. It is the starting point of so-called morphological equations and, they were originally introduced by Aubin for non-empty compact subsets of Rn supplied with the Pompeiu–Hausdorff metric dlPH . Meanwhile the underlying concept for infinitesimal evolutions in non-linear spaces has proved to be suitable for various other examples like (non-convex) random closed sets, closed sets in Banach spaces, fuzzy differential equations and stochastic differential equations with non-local sample dependence (see, e.g., [11,35,36,41,55–60,64–68, 73,83] and the references therein). There are several further approaches, of course, how to introduce integral or (equivalently) differential equations for non-convex closed sets in a vector space. A very popular alternative is based on describing reachable sets as so-called integral funnel solutions a.k.a. R-solution (see, e.g., [75–77,94,95,98]). Then the step to differential equations uses the so-called quasi-differential equations in a metric space as introduced by Panasyuk (e.g., [78–80]).

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The equivalence between these concepts for compact sets in Rn has already been mentioned in [28,30,67]. Sufficient conditions on the set-valued maps involved are formulated even for closed subsets of a Banach space in [68] and, we quote these results in Proposition 2.3 as well as Corollary 2.7 below. 1.3. The focus: sufficient conditions for viability of set-valued states In the so-called viability theory (e.g., [3,8]), the following well-known characterization is the core: Proposition 1.1 (Viability theorem in Rn ). Let the set-valued map F : Rn ; Rn be upper semicontinuous with non-empty, convex and compact values. Then these statements are equivalent for every non-empty closed subset V ⊂ Rn : (i) F (x0 ) ∩ TV (x0 ) = ∅ for every x0 ∈ V with TV (x0 ) ⊂ Rn denoting the so-called contingent cone of V at x0 in the sense of Bouligand, i.e., for every x0 ∈ V , there is some w ∈ F (x0 ) with lim inf h1 · h↓0   dist x0 + h · w, V = 0. (ii) V ⊂ Rn is viable (a.k.a. weakly invariant) w.r.t. F , i.e., in every element x0 ∈ V , there starts (at least) one absolutely continuous solution x : [0, T ] −→ Rn of the differential inclusion x ∈ F (x) with all its values in V . Extensions to vector-valued solutions of evolution inclusions in a Hilbert space can be found in [3,21, 48,86–88], for example. The main goal of this article is to provide a corresponding sufficient condition of viability for set-valued states rather than vectors. It is the starting point for generalizing the well-established viability theory to set-valued states. First results in this direction can be found for compact subsets of Rn in [64,65,67]. Now several new aspects are taken into consideration: • We consider compact subsets of a separable Hilbert space H (instead of a fixed finite-dimensional vector space like Rn ). K(H) abbreviates the set of all non-empty compact subsets of H. • The constraints on the set-valued state are not restricted to inclusions of the form K(t) ⊂ V with a fixed set V ⊂ H (see, e.g., [41] for H = Rn ). A non-empty subset V ⊂ K(H) is given instead which we assume to be closed w.r.t. the Pompeiu–Hausdorff metric dlPH . As a consequence, the counterpart of the contingent cone of Bouligand has to be specified for the   metric space K(H), dlPH . • Reachable sets serve as a generalized approach to “integrating” closed sets w.r.t. time again, but we ˜ x) as before. Evolution inclusions are preferred do not use ordinary differential inclusions x ∈ G(t,   instead, i.e., a strongly continuous semigroup S(t) t≥0 with generator A is given and, the reachable ˜ x). (Hence, we seize set is constructed by means of the mild solutions x : [0, T ] −→ H of x ∈ A x + G(t, the definition and results in [68].) The strong continuity of the semigroup implies directly that the Lipschitz continuity of reachable sets in time w.r.t. the Pompeiu–Hausdorff metric is not obvious any more. Hence whenever considering regularity in time, we use another distance function ΔCl (for non-empty closed subsets of H) which takes the semigroup into consideration appropriately as suggested in [56,67,68]. The additional aspect of mild solutions extends the field of applications significantly since the recent results cover various examples of compact sets generated by partial differential equations. • The main result, i.e., Theorem 3.3 below, will be proved by means of several arguments which have already been useful for compact subsets of Rn in [64,65,67]. Indeed, we start with adapting Haddad’s notion of approximate solutions. The idea of “improving     the given accuracy” leads to essentially two sequences Kk (·) k∈N , fk (·) k∈N . Each member Kk (·) is

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a compact-valued map of time representing the current value of the approximate solution and, fk (·) can be interpreted as a form of its (generalized weak) derivative (more strictly speaking, it specifies the right-hand side of the underlying evolution inclusion). In the finite-dimensional case, we continued by parametrizing each set-valued map fk (t) : H ; H and then used weak compactness criteria for vector-valued L1 spaces. The standard theorems about Lipschitz continuous parametrizations, however, are restricted to finite-dimensional vector spaces as they rely on constructions like the Steiner point. But the Hilbert space H here might be of infinite dimension. For this reason, a supplementary step is needed. Motivated by Galerkin’s method, we use orthogonal projections on finite-dimensional subspaces Hd ⊂ H which are spanned by eigenvectors of the generator A. Assuming A to be normal with a compact resolvent, there is always an orthonormal system of eigenvectors spanning a subspace dense in H and being invariant with respect to each linear operator of the semigroup. The challenging question is how the “partial approximate” solutions in each of the finite-dimensional subspaces Hd (d ∈ N) can be induced by one and the same reachable set in H. We obtain these results by means of the so-called scalar topology on uniformly bounded convex subsets [12,13,90] and earlier results about trajectories of ordinary differential inclusions in finite dimensions by Stassinopoulos and Vinter [92,93]. • Many proofs of similar viability theorems rely on a notion of approximate solutions (as the one by Haddad presented in, e.g., [43,45] and [3, § 3.4]). Then the Arzelà–Ascoli theorem is the key tool for extracting a uniformly converging subsequence of continuous curves in the state space and, its limit proves to be a solution satisfying the state constraint in addition. Even in a metric space, however, the Arzelà–Ascoli theorem requires the relative compactness of all values at one and the same point of a dense subset of the domain (see, e.g., [42]). In the case of an infinite-dimensional Hilbert space, uniform boundedness of the continuous functions is not sufficient any more. Hence, we follow a different strategy, which can be regarded as similar to the so-called Euler compactness (see, e.g., [60,67]): We focus on an appropriate form of compactness of the curves representing the derivatives and, this is to lay the basis for concluding the pointwise convergence of the state curves. In the special case of vector-valued states, the gist is rather obvious: Consider a sequence (yk )k∈N of absolutely continuous functions [0, T ] −→ Rn with joint initial point ξ0 ∈ Rn . If the corresponding sequence of weak derivatives (yk )k∈N converges to some v ∈ L1 (0, T ; Rn ) weakly, then we can conclude from the representation  yk (t) = ξ0 + yk (s) dL1 s [0,t]

  for each index k ∈ N that for every t ∈ [0, T ], the limit of yk (t) k∈N exists and satisfies  v(s) dL1 s,

lim yk (t) = ξ0 +

k→∞

[0,t]





i.e., yk (·) k∈N converges pointwise to an absolutely continuous function y : [0, T ] −→ Rn with y(0) = ξ0 and y  = v Lebesgue-almost everywhere in [0, T ]. (This gist has already proved to be useful for stochastic differential equation with non-local sample dependence in [58].) Now we adapt this notion to compact reachable sets in a separable Hilbert space (as states) and, the set-valued maps occurring in the underlying evolution inclusions play the role of the weak derivatives.

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In particular, we do not need the Arzelà–Ascoli theorem for the sequence of “approximate solutions”   [0, T ] −→ K(H), dlPH . • These conclusions about converging states are based on the appropriate topology of the “weak derivatives”. In the recent example of states in Rn , the weak convergence in L1 (0, T ; Rn ) proves to be appropriate. For compact sets as states, however, we require the counterpart for the set-valued co˜ : [0, T ] × H ; H in the evolution inclusion x ∈ A x + G(t, ˜ x). efficient map G Most results about reachable sets supply the coefficient maps with the supremum Pompeiu–Hausdorff metric since they result from Filippov-like theorems about existence of solutions (see, e.g., [4,6,67]). As one of the new results here, the compact reachable sets are proved to depend continuously on the ˜ w.r.t. the pointwise scalar topology. coefficients G These new aspects have the inevitable consequence that the section of proofs has become quite technical. Furthermore, this article is written with the objective of a self-contained presentation, i.e., all definitions and tools are formulated and, we state a reason for each step of the proofs. The rest is structured as follows: First, we list the notation concerning sets, metrics and hypotheses used below. Section 2 provides all the definitions and earlier results used here on the way to differential equations for sets (in § 2.3). In § 2.4, Proposition 2.14 represents our new results about the continuous dependence of reachable sets on the set-valued coefficient maps. Solutions to set differential inclusions follow in § 2.5. They are defined in terms of an infinitesimal condition at Lebesgue-almost every time. Proposition 2.17 formulates sufficient conditions for an equivalent characterization by means of a measurable selection (instead). The latter plays the role of the (weak) time derivative of the compact-valued solution. The main result about set differential inclusions with state constraints is presented in section 3. Indeed, Theorem 3.3 specifies sufficient conditions on the set of constraint V ⊂ K(H) for being viable (a.k.a. weakly invariant). Section 4 provides all the proofs of the new results. Theorem 3.3 is verified in several steps presented in the six subsections of § 4.3. The proof of Proposition 2.17 follows in § 4.4. It is based on the same method as for Theorem 3.3, but requires some adaptation because the underlying set differential inclusion is non-autonomous and less regular. We usually collect the relevant statements in the beginning of each subsection and then, their proofs are presented in detail. For the sake of a self-contained presentation, the appendix provides various earlier results about evolution inclusions in a Banach space and several known characterizations of (weak) compactness used in the proofs here. Notation   H, ·, · abbreviates a Hilbert space and, we use the following notations: 

B the open unit ball in H, i.e., B := x ∈ H x < 1 ,

 the open ball with radius R > 0, i.e., BR := x ∈ H x < R , BR Cl(H) the set of all non-empty closed subsets of H, Clco (H) the set of all non-empty closed and convex subsets of H, Clbd,co (H) the set of all non-empty closed bounded and convex subsets of H, K(H) the set of all non-empty compact subsets of H, Kco (H) the set of all non-empty compact and convex subsets of H, the Lebesgue measure on R (and its Borel subsets). L1 dlPH : Cl(H) × Cl(H) −→ [0, ∞] denotes the Pompeiu–Hausdorff distance between non-empty closed subsets of H dlPH (A, B) := max

sup dist(x, B), sup dist(y, A)

x∈A

y∈B

∈ [0, ∞]

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where we admit the value +∞ just to avoid restricting all our further considerations to bounded subsets of H (as, e.g., in [13,56]). In the literature, it is often just called Hausdorff metric. σ : Furthermore, 

Cl(H) × H −→ R ∪ {+∞} abbreviates the support function, i.e., σ(A, p) := sup x, p x ∈ A . As in [3,5–8,10,40,67], for example, the arrow ; indicates a set-valued (or multivalued) mapping. General hypotheses   (H1) H, ·, · is a real Hilbert space. · := ·, · : H −→ R denotes the corresponding norm on H. (H2) (H, · ) is separable.   (H3) S(t) t≥0 is a strongly continuous semigroup of bounded linear operators H −→ H and, its generator is called A : D(A) −→ H. Let CS ≥ 1, ωS ∈ R denote constants such that S(t) Lin(H,H) ≤ CS · eωS ·t holds for every t ≥ 0 due to [37, Proposition I.5.5].   (H4) Let S(t) t≥0 be a strongly continuous semigroup of bounded linear operators H −→ H with infinitesi  mal generator A. In addition, suppose S(t) t≥0 to be quasi-contractive, i.e., there is a constant ωS ∈ R with S(t) Lin(H,H) ≤ eωS ·t for every t ≥ 0 [37, Ch. II condition (3.10)]. (H5) A is normal with compact resolvent. 2. Solutions to set differential equations and inclusions We apply the concepts and results of [68, §§ 2,3] to the special case of closed subsets in a Hilbert space, which are not necessarily bounded (or even compact). 2.1. Infinitesimal characterizations of set evolutions in time ˜ : Definition 2.1. In addition to hypotheses (H1), (H3), suppose 0 ≤ t0 < T and a set-valued map G [t0 , T ] × H ; H to have non-empty values. A continuous function x : [t0 , T ] −→ H is called a mild solution of the non-autonomous evolution inclusion ˜ x) x (t) ∈ A x(t) + G(t,

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if there exist a vector x0 ∈ H and a (Lebesgue–) Bochner integrable function g : [t0 , T ] −→ H with ⎧ t ⎪ ⎪ ⎨ x(t) = S(t − t ) x + S(t − s) g(s) ds for every t ∈ [t0 , T ] 0 0 ⎪ t 0 ⎪   ⎩ ˜ t, x(t) g(t) ∈ G for Lebesgue-a.e. t ∈ [t0 , T ]. Furthermore, ϑG˜ (t, M ) :=



x(t) ∈ H ∃ x ∈ C 0 ([0, t], H) such that x(0) ∈ M and

˜ x) x is a mild solution of x ∈ A x + G(·,

˜ at time t ∈ [0, T ]. is called the reachable set of M ⊂ H and G The next proposition suggests three approaches to characterizing the infinitesimal evolution of closed ˜ : [0, T ] × H ; H for their equivalence sets in time. It specifies sufficient conditions on the set-valued map G and was proved for the more general case of closed (not necessarily bounded) sets in a Banach space in [68, § 2.4, Theorem 1]:

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Definition 2.2. Under the assumptions (H1), (H3) define     ΔCl : R × Cl(H) × R × Cl(H) −→ [0, ∞] by   ΔCl (s, M ), (t, N ) :=



  |s − t| + dlPH S(t − s) M, N   |s − t| + dlPH M, S(s − t) N

if s ≤ t if s > t

A function K : [0, T ] −→ Cl(H) is called sequentially continuous w.r.t. ΔCl if for every time instant t ∈ [0, T ] and any sequence (tn )n∈N in [0, T ] tending to t, ΔCl



   t, K(t) , tn , K(tn ) −→ 0

(n → ∞).

˜ : [0, T ] × H ; H Proposition 2.3. In addition to hypotheses (H1), (H3), suppose for the set-valued map G the following conditions: ˜ is a non-empty closed and convex subset of H, (i) each value of G ˜ is integrably bounded, i.e., there exists k ∈ L1 ([0, T ]), k ≥ 0, with G(t, ˜ ξ) ⊂ Bk(t) ⊂ H for every (ii) G t ∈ [0, T ], ξ ∈ H, (iii) for every ε > 0, there exist a compact subset Jε ⊂ [0, T ] and a function ωε : Jε × [0, 1] −→ [0, ∞) satisfying the following three conditions   • L1 [0, T ] \ Jε < ε, • for every t ∈ Jε , lim ωε (t, h) = 0 and h↓0

• for every t ∈ Jε and h ∈ [0, 1], sup s ∈ Jε : |s−t| ≤ h

  ˜ ξ), G(s, ˜ ξ) ≤ ωε (t, h), sup dlPH G(t,

ξ∈H

˜ ·) : H ; H is (iv) there is λ ∈ L1 ([0, T ]) such that for Lebesgue-almost every t ∈ [0, T ], the map G(t, λ(t)-Lipschitz continuous w.r.t. the Pompeiu–Hausdorff distance dlPH . Then the following statements are equivalent for every set-valued map K : [0, T ] ; H with non-empty closed values in H: (1.) K(t) = ϑG˜ (t, K(0)) ⊂ H for every t ∈ [0, T ], ˜ in the sense that K(·) is sequentially continuous w.r.t. (2.) K(·) is a so-called R-solution generated by G ΔCl and the equation

lim

h↓0

 1 dlPH K(t+h), h

 

h S(h) ξ +

ξ ∈ K(t)

   ˜ t, S(σ) ξ dσ S(h−σ) G

 = 0

0

holds for Lebesgue-almost every t ∈ [0, T ]. ˜ in the sense that K(·) is sequentially continuous w.r.t. (3.) K(·) is a so-called morphological primitive of G ΔCl and the equation lim

h↓0

  1 · dlPH K(t + h), ϑG(t,·) (h, K(t)) = 0 ˜ h

is satisfied for Lebesgue-almost every t ∈ [0, T ].

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Remark 2.4. Strictly speaking, [68, § 2.4, Theorem 1] concerns a quasi-contractive semigroup on a real Banach space. This assumption is stronger than hypothesis (H3) here, but we are still free to replace the norm · (related to the inner product ·, ·) on H by an equivalent norm | · |.   Indeed, for every C0 semigroup S(t) t≥0 of bounded linear operators H −→ H, there exist constants CS ≥ 1, ωS ∈ R satisfying S(t) Lin(H,H) ≤ CS · eωS ·t for every t ≥ 0 due to [37, Proposition I.5.5]. The ˜ := e−ωS · t S(t) : H −→ H, t ≥ 0, are uniformly first step is based on rescaling: The linear operators S(t) bounded. Next, [37, Lemma II.3.10] states that | x | := sup

  ˜ x t ≥ 0 S(t)

for x ∈ H

  ˜ is equivalent to the original norm on H, and S(t) becomes a contraction semigroup on the Banach t≥0     space H, | · | . Hence, S(t) t≥0 is quasi-contractive w.r.t. | · | and satisfies the assumptions of [68, § 2.4, Theorem 1] (in its original form). Remark 2.5. ΔCl proves to be an appropriate substitute for the Pompeiu–Hausdorff metric between closed sets whenever the two sets are related to different time instants. In particular, the strong continuity   of S(t) t≥0 prevents us from standard conclusions about the Lipschitz continuity of reachable sets in time. We have already suggested counterparts of ΔCl in [56,67,68], for example, and now summarize several features (as in [68, § 2.2 Remark 2]): (1.) ΔCl is defined for tuples in R×Cl(H) whose first component can be interpreted as time instant whereas the second component represents a set state. When “comparing” two closed subsets in H quantitatively, the additional real component helps us to “distinguish” which subset is the “earlier” one. (2.) The quantitative “comparison” of two closed subsets in H is based on a form of “distance function”, not necessarily a metric in the classical sense. Whenever two closed subsets do not refer to the same time   instant we suggest to take the flow along the semigroup S(t) t≥0 into consideration first and then to apply the well-established Pompeiu–Hausdorff metric dlPH on Cl(H). This supplementary step proves to be very useful for conclusions about continuity in time since all “effects”   of the strong continuity of S(t) t≥0 are covered. Proposition A.1.4 citing [68, § 2.2, Lemma 3], for example, ensures that under the assumptions of Proposition 2.3, the set-valued map [0, T ] ; H, t → ϑG˜ (t, M ) with all values in Cl(H) is sequentially continuous w.r.t. ΔCl . (3.) The price to pay for this rather technical advantage is that ΔCl is not a metric on R × Cl(H) in general. Obviously, ΔCl is positive definite and symmetric. Furthermore, it fulfills the following time-oriented modification of the triangle inequality     ΔCl (t1 , M1 ), (t3 , M3 ) ≤ CS · e|ωS | (t3 −t2 ) · ΔCl (t1 , M1 ), (t2 , M2 ) +   ΔCl (t2 , M2 ), (t3 , M3 ) for any (t1 , M1 ), (t2 , M2 ), (t3 , M3 ) ∈ R × Cl(H) with t1 ≤ t2 ≤ t3 because   ΔCl (t1 , M1 ), (t3 , M3 )   + dlPH S(t3 − t1 ) M1 , M3 = t 3 − t1   ≤ t3 − t2 + t2 − t1 + dlPH S(t3 − t1 ) M1 , S(t3 − t2 ) M2   + dl  PH S(t3 −t2 ) M2 , M3   ≤ t3 − t2 + t2 − t1 + S(t3 − t2 )Lin(H,H) · dlPH S(t2 − t1 ) M1 , M2   + dlPH S(t3 − t2 ) M2 , M3     ≤ CS · e|ωS | (t3 −t2 ) · ΔCl (t1 , M1 ), (t2 , M2 ) + ΔCl (t2 , M2 ), (t3 , M3 ) .

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2.2. Differential equations for closed sets in a real Hilbert space Definition 2.6. In addition to hypotheses (H1), (H3), let M be a non-empty subset of H. Define LIPco (M, H) as the set of all set-valued maps G : M ; H satisfying (1.) each value of G is a non-empty closed convex subset of H, (2.) the set values of G are uniformly bounded in H, (3.) G is Lipschitz continuous w.r.t. the Pompeiu–Hausdorff metric dlPH , i.e., there is a non-negative constant LipG < ∞ with   dlPH G(ξ), G(ζ) ≤ LipG · ξ − ζ

for all ξ, ζ ∈ M.

Moreover, set for G, G1 , G2 ∈ LIPco (M, H) G ∞ := sup

sup

ξ ∈ M v ∈ G(ξ)

v ,

  dl∞ (G1 , G2 ) := sup dlPH G1 (ξ), G2 (ξ) . ξ∈M

The subset LIPco,cp (M, H) consists of all set-valued maps G ∈ LIPco (M, H) satisfying the stronger condition (1. ) each value of G is a non-empty compact convex subset of H. This definition extends Aubin’s original specification [6, Definition 3.7.1] and [67, Definitions 1.49, 5.9] to a real Hilbert space H instead of Rn . Recently, [68, § 3.1, Definition 8] suggested essentially the same generalization of LIPco (Rn , Rn ) to a separable Banach space, the condition of compact values, however, was not required in that reference. Proposition 2.3 in combination with feedback lays the foundations for differential equations instantaneously: Corollary 2.7. In addition to hypotheses (H1), (H3), assume for the single-valued function F : [0, T ] × Cl(H) −→ LIPco (H, H): (i) There is k ∈ L1 ([0, T ]) satisfying for Lebesgue-almost every t ∈ [0, T ], 

 sup F(t, M )∞ M ∈ Cl(H) ≤ k(t). (ii) There is λ ∈ L1 ([0, T ]) such that for Lebesgue-almost every t ∈ [0, T ], 

sup LipF(t, M ) M ∈ Cl(H) ≤ λ(t).   (iii) F is sequentially continuous w.r.t. ΔCl and dl∞ respectively, i.e., any sequence (tk , Mk ) k∈N in [0, T ] × Cl(H) and every (t, M ) ∈ [0, T ] × Cl(H) with lim

k→∞

  ΔCl (tk , Mk ), (t, M ) = 0

satisfy lim

k→∞

  dl∞ F(tk , Mk ), F(t, M ) = 0.

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Then the following three statements are equivalent for any curve K : [0, T ] −→ Cl(H) and the related set-valued map   F˜K : [0, T ] × H ; H, (t, x) → F t, K(t) (x) :

(1.) At each time t ∈ [0, T ], the set K(t) ⊂ H coincides with the closed reachable set ϑF˜K (t, K(0)) ⊂ H of the non-autonomous evolution inclusion x ∈ A x + F˜K (·, x). (2.) K is an R-solution generated by F˜K , i.e., K is sequentially continuous w.r.t. ΔCl and satisfies for Lebesgue-almost every t ∈ [0, T ] and h ↓ 0  dlPH K(t + h),

  ξ ∈ K(t)

h S(h) ξ +

   S(h−σ) F˜K t, S(σ) ξ dσ

 = o(h).

0

(3.) K is a morphological primitive of F˜K , i.e., it is sequentially continuous w.r.t. ΔCl and fulfills for Lebesgue-almost every t ∈ [0, T ] and h ↓ 0    dlPH K(t + h), ϑF˜K (t,·) h, K(t) = o(h). Definition 2.8. Suppose the assumptions of Corollary 2.7 for the function F : [0, T ] ×Cl(H) −→ LIPco (H, H). A curve K : [0, T ] −→ Cl(H) is called a solution to the set differential equation   ˚ K(t) = F t, K(t) if it fulfills one of the equivalent criteria (1.)–(3.) in Corollary 2.7. 2.3. Sufficient conditions for compact-valued solutions [68, Corollary A.3] specifies sufficient conditions on the right-hand side of an evolution inclusion for its solution set being compact with respect to the supremum norm. Applying this result to a Hilbert space H, we obtain directly: ˜ : [0, T ] × H ; H: Proposition 2.9. In addition to hypotheses (H1)–(H3), suppose for the set-valued map G ˜ is a non-empty closed and convex subset of H, (i) each value of G ˜ ˜ x) ⊂ Bk(t) for any x ∈ H and Lebesgue(ii) G is integrably bounded, i.e., there is k ∈ L1 ([0, T ]) s.t. G(t, almost every t ∈ [0, T ], ˜ x) : [0, T ] ; H is (strongly) Lebesgue measurable, (iii) for every x ∈ H, the set-valued map G(·, 1 ˜ ·) : H ; H is (iv) there is λ ∈ L ([0, T ]) such that for Lebesgue-almost every t ∈ [0, T ], the map G(t, λ(t)-Lipschitz continuous w.r.t. the Pompeiu–Hausdorff distance dlPH , (v) at least one of the following conditions is satisfied:   ∗ S(t) t≥0 is immediately compact, ˜ ∗ there is a compact subset K ⊂ H containing all set values of G. Then for every compact subset M0 ⊂ H, the set-valued solution map   M0 ; C 0 ([0, T ], H), · sup ,

˜ x), x(0) = ξ ξ → x(·) x is a mild solution of x ∈ A x + G(t, has a compact graph.

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Corollary 2.10. Under the assumptions of preceding Proposition 2.9, the reachable set ϑG˜ (t, M0 ) of each initial set M0 ∈ K(H) is compact in H. Remark 2.11. Among the assumptions of Proposition 2.9, hypothesis (v) makes the most restrictive impression. Immediately compact semigroups are induced by uniformly parabolic operators, for example (e.g., [38,82,85]). ˜ is mainly to provide an alThe second condition demanding a compact superset K ⊂ H of all values of G ternative for other types of evolution inclusions. In a word, it is not clear how to construct limiting solutions (as required for compact graph or compact reachable sets) without aspects of compactness. Whenever we   want to avoid such assumptions about the strongly continuous semigroup S(t) t≥0 the integral representation of mild solutions motivates the focus on the set-valued map related to the integrated inhomogeneous part. It is regarded as what could be called “compact perturbations”. In connection with partial differential equations, this condition is satisfied in several problems with non-local operators like convolutions as the following example shows. Example 2.12. Consider the wave equation problem with Dirichlet zero boundary conditions ⎧ ⎪ ∂t2 u − Δx u ⎪ ⎪ ⎨ u(t, ·) ⎪ u(0, ·) ⎪ ⎪ ⎩ ∂t u(0, ·)

= = = =

0 0 u0 u1

in on in in

]0, T [ ×Ω ∂Ω Ω Ω

on a bounded open domain Ω ⊂ Rn with smooth boundary. By means of the supplementary component v = ∂t u, the tuple w := (u, v) ∈ W01,2 (Ω) × L2 (Ω) leads to an abstract evolution equation of first order ∂t w = A w on the separable Hilbert space H := W01,2 (Ω) × L2 (Ω) with the operator 

 0 idL2 (Ω) Δx 0

A :=

  : W 2,2 (Ω) ∩ W01,2 (Ω) × W01,2 (Ω) −→ H

as specified in [85, § 3.8.4]. According to [85, Lemma 38.7], A is skew-adjoint, has compact resolvent   and generates a unitary group S(t) t∈R of bounded linear operators on H. The latter property implies   S(t) Lin(H,H) = 1 for every t ≥ 0 and so, the semigroup S(t) t≥0 is even contractive. Hence, hypotheses (H1)–(H5) are satisfied.   S(t) t≥0 is not immediately compact, however. Indeed, if S(t) : H −→ H was a compact linear operator for some t > 0 then so would be the composition with the bounded linear operator S(−t) : H −→ H, namely the identity idH = S(−t) ◦ S(t) : H −→ H (similarly to the conclusions in [20, Remark 5.6.1]). Now we give an example for a set-valued map G : H ; H whose closed convex values are all contained in a Def. joined compact set K ⊂ H: Fix (μ, η) ∈ H = W01,2 (Ω)×L2 (Ω) arbitrarily and choose some ζ1 , ζ2 ∈ C ∞ (Rn ) with compact support in B1 (0) ⊂ Rn and non-negative values (but not necessarily symmetric w.r.t. 0 ∈ Rn ). For the non-linear operator N : H −→ H defined by  N (u, v) := 0, ζ1 ∗ 

 =

u v  + ζ2 ∗ 1 + |u| 1 + |v|

ζ1 (y)

0, Rn

u(· − y) dy + 1 + |u(· − y)|

 ζ2 (y) Rn

 v(· − y) dy , 1 + |v(· − y)|

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we consider the trivial extensions of u, v, respectively, to Rn in the convolutions. In particular, the convolution w.r.t. “space” covers the non-local and weighted influence of neighboring states (similarly to the gist in various recent traffic flow models like [15,27,60]). The fractions exemplify a Holling-type 2 functional dependence. Mild solutions to the abstract evolution equation ∂t w = A w + N (w) are related to the non-local initial–boundary value problem ⎧ ⎪ ∂t2 u − Δx u ⎪ ⎪ ⎨ u(t, ·) ⎪ u(0, ·) ⎪ ⎪ ⎩ ∂t u(0, ·)

= = = =

ζ1 ∗ 0 u0 u1

u(t,·) 1+|u(t,·)|

+ ζ2 ∗

∂t u(t,·) 1+|∂t u(t,·)|

in on in in

]0, T [ ×Ω ∂Ω Ω Ω.

For taking uncertainties into consideration quantitatively, the step to an evolution inclusion is based on the set-valued map G : H ; H defined as the L2 (Ω) closed convex hull   2 G(u, v) := coL N B1 (u) × B1 (v)

      2 ˜ − uL2 (Ω) , v˜ − v L2 (Ω) ≤ 1 . = coL N u ˜, v˜ u ˜, v˜ ∈ L2 (Ω), u Def.

It has non-empty bounded convex and closed values in H = W01,2 (Ω) × L2 (Ω). Furthermore, the convolu  ˜, v˜ ∈ {0} × L2 (Ω). tions with the same functions ζ1 , ζ2 ∈ Cc∞ (Rn ) imply the equi-integrability of all N u The theorem of Kolmogorov, Riesz and Fréchet (formulated in, e.g., [1, Theorem 4.16] or [19, Theorem 4.26]) ensures the relative compactness of the image set G(H) ⊂ {0} × L2 (Ω) ⊂ H. 2.4. Continuous dependence of compact reachable sets on coefficients Shortly speaking, most of the proofs below have the notion in common that reachable sets ϑG˜ (t, M0 ) ⊂ H of non-autonomous evolution inclusions are approximated. Hence, we need an appropriate form of their  : [0, T ] × continuous dependence on both the initial set M0 ∈ K(H) and the set-valued coefficient map G H ; H. In the literature, these conclusions usually rely on estimates of the Pompeiu–Hausdorff distance between the reachable sets and, they are based on a counterpart of Filippov’s theorem for “comparing” appropriate vector-valued solutions (see, e.g., [5,6,28,56,67,68] and the results cited here in § A.1.2). As a consequence, the resulting estimates depend on the Pompeiu–Hausdorff distance between the values  frequently in terms of the metric dl∞ for coefficient functions. of G, In this article, we need similar (at least, qualitative) results with respect to a different topology on LIPco (H, H) instead. Motivated by earlier results about solution sets in finite-dimensional vector spaces by Stassinopoulos and Vinter [92,93], we consider an alternative based on the “pointwise” convergence of the support functions with respect to each “direction” p ∈ H. This topology for bounded convex subsets of a Banach space is called the “weak topology” by De Blasi and Myjak in [32]. Sonntag, Zălinescu and Beer, for example, prefer the name “scalar topology” instead (in, e.g., [12,90] and [13, § 4.3] respectively). Definition 2.13. Let H be a real Hilbert space. A sequence (Ck )k∈N of bounded, closed and convex subsets of H is said to converge scalarly to a set C ∈ Clbd,co (H) if lim σ(Ck , p) = σ(C, p) holds for every vector p ∈ H.

k→∞

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One of our new results is the following statement in a separable Hilbert space:  : [0, T ] × H ; H Proposition 2.14. In addition to hypotheses (H1)–(H3), (H5), let the set-valued maps G  and F : [0, T ] × H ; H ( ∈ N) satisfy the assumptions of Proposition 2.9 with joint bound functions λ, k ∈ L1 ([0, T ]). Suppose for all x ∈ H, p ∈ H and Lebesgue-almost every t ∈ [0, T ]     ˜ x), p σ F˜ (t, x), p −→ σ G(t,

( → ∞).

Then, for every initial set M0 ∈ K(H), the following convergence holds sup t ∈ [0,T ]

  dlPH ϑF˜ (t, M0 ), ϑG˜ (t, M0 ) −→ 0

( → ∞).

2.5. The step to set differential inclusions Similarly to the standard situation in vector spaces, the step from a differential equation to a differential inclusion is based on replacing the single-valued function on the right-hand side by a set-valued map. This notion motivates the following adaptation of [67, Definition 5.1]: Definition 2.15. In addition to hypotheses (H1), (H3), let F : [0, T ] × Cl(H) ; LIPco (H, H) be a set-valued map with non-empty values. A set-valued map K : [0, T ] ; H is called a solution to the set differential inclusion   ˚ K(t) ∈ F t, K(t)

in [0, T ]

if it satisfies the following conditions: (a) K has non-empty closed values in H, (b) K is sequentially continuous w.r.t. ΔCl (in the sense of Definition 2.2),   (c) for Lebesgue-almost every t ∈ [0, T ), there exists at least one set-valued map G ∈ F t, K(t) ⊂ LIPco (H, H) with lim

h↓0

   1 · dlPH K(t + h), ϑG h, K(t) = 0. h

Criterion (c) considers just one out of three criteria which we used for solutions to set differential equations in Definition 2.8. The next proposition provides a connection with the “integral characterization”, i.e., the reachable set of a related non-autonomous evolution inclusion as in Corollary 2.7 (1.). The underlying aspects of measurability and separability require an appropriate metric on LIPco (H, H). In particular, the set-valued maps in LIPco (H, H) serve as right-hand sides of evolution inclusions and, so we are not obliged to consider the standard metric dl∞ , i.e., the supremum metric w.r.t. dlPH . Proposition 2.14 inspires an alternative: 

Definition 2.16. Let (H, ·, ·) be a separable Hilbert space and, pn n ∈ N denotes a dense subset of H. We define dlco : Clbd,co (H) × Clbd,co (H) −→ R,

dlco (C1 , C2 ) :=

∞  n=1

2−n

|σ(C1 , pn ) − σ(C2 , pn )| 1 + |σ(C1 , pn ) − σ(C2 , pn )|

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and the corresponding supremum metric on LIPco (M, H) (for any non-empty set M ⊂ H) dlco,∞ : LIPco (M, H) × LIPco (M, H) −→ R,

  (F1 , F2 ) −→ sup dlco F1 (x), F2 (x) . x∈M

dlco induces the scalar topology on the class of uniformly bounded and closed convex subsets – as formu  lated in Definition 2.13 and Lemma 4.1 below. Whenever H is separable, the metric space Clbd,co (H), dlco is both separable and complete (as a consequence of [32, Theorems 3.1 and 3.2], which even consider separable reflexive Banach spaces). Proposition 2.17. Under hypotheses (H1)–(H2), (H4), (H5), let F : [0, T ] × K(H) ; LIPco (H, H) be a set-valued map satisfying the following conditions • every set value F(t, M ) ⊂ LIPco (H, H) is non-empty and convex, i.e., for any G1 , G2 ∈ F(t, M ) ⊂ LIPco (H, H) and λ ∈ [0, 1], the set-valued map H ; H, x → λ · G1 (x) + (1 − λ) · G2 (x) also belongs to F(t, M ),     • for Lebesgue-almost every t ∈ [0, T ], the mapping F(t, ·): K(H), dlPH ; LIPco (H, H), dlco,∞ has closed graph, • F is globally bounded in the sense that sup

sup

t ∈ [0,T ] M ∈ K(H)

G ∈ F(t,M )

  Lip G + G ∞ < ∞,

• at least one of the following conditions is satisfied:   ∗ S(t) t≥0 is immediately compact,  ∗ there is a compact subset K0 ⊂ H with

G(H) ⊂ K0 .

G ∈ F([0,T ]×K(H))

  ˚ Every compact-valued solution K : [0, T ] ; H to the set differential inclusion K(t) ∈ F t, K(t) in [0, T ] satisfies the following conditions: (i) K : [0, T ] −→ K(H) is continuous w.r.t. dlPH and, ˜ : [0, T ] × H ; H such that (ii) there exists a convex-valued Carathéodory map G   ˜ • G(t, ·) ∈ F t, K(t) ⊂ LIPco (H, H) for Lebesgue-almost every t ∈ [0, T ] and   • for every t ∈ [0, T ], the set K(t) ⊂ H coincides with the closed reachable set ϑG˜ t, K(0) ⊂ H of ˜ x). the non-autonomous evolution inclusion x ∈ A x + G(·, If we assume in addition that all values of F belong to LIPco,cp (H, H) then the inverse implication also holds for a curve K : [0, T ] −→ K(H). 3. Set differential inclusions with state constraints: main results We adapt [6, Definition 1.5.2] to closed subsets of a real Hilbert space: Definition 3.1. Let V be a non-empty subset of Cl(H) and the set M ∈ V. The so-called contingent transition set TV (M ) consists of all set-valued maps G ∈ LIPco (H, H) satisfying (with respect to the Pompeiu–Hausdorff metric on Cl(H))

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lim inf h↓0

  1 · dist ϑG (h, M ), V = 0. h

Remark 3.2. The equivalence in Proposition 2.3 applied to the autonomous evolution inclusion x ∈ A x + G(x) leads to a further infinitesimal condition characterizing G ∈ TV (M ) ⊂ LIPco (H, H), i.e.,     h    1 · dist lim inf S(h) ξ + S(h − σ) G S(σ) ξ dσ , V = 0. h↓0 h ξ∈M

0

In comparison with the reachable set mentioned in Definition 3.1, it might be regarded as closer to the standard definition of “tangents” or “contingent vector” in the special case of state vectors in a Hilbert or a Banach space and their evolution inclusions. Their viability conditions have already been investigated thoroughly in [16,17,21,48,74,86–88], for example. The essential difference between these established results about viability in a Banach space and our   considerations here is that (from now on) our state space is K(H), dlPH and so, it does not provide an obvious linear structure. Theorem 3.3. In addition to hypotheses (H1), (H2), (H4), (H5), suppose for the set-valued map F : K(H) ; LIPco (H, H): (i) every set value F(M ) ⊂ LIPco (H, H) is non-empty and convex, i.e., for any G1 , G2 ∈ F(M ) ⊂ LIPco (H, H) and λ ∈ [0, 1], the set-valued map H ; H, x → λ · G1 (x) + (1 − λ) · G2 (x) also belongs to F(M ), (ii) α  := sup sup Lip G < ∞, β :=

M ∈ K(H) G ∈ F(M )

sup

sup

M ∈ K(H) G ∈ F(M )

G ∞ < ∞,

(iii) the graph of F is closed (w.r.t. locally uniform convergence in LIPco (H, H) as specified in Remark 3.4 below), (iv) at least one of the following conditions is satisfied:   ∗ S(t) t≥0 is immediately compact,  ∗ there is a compact subset K0 ⊂ H satisfying G(H) ⊂ K0 . G ∈ F(K(H))

If the non-empty closed subset V ⊂ K(H) satisfies F(M ) ∩ TV (M ) = ∅

for every set M ∈ V

then V is viable (a.k.a. weakly invariant) with respect to F in the following sense: For every initial set M0 ∈ V, there exist at least one compact-valued map K : [0, T ] ; H and a convex-valued Carathéodory ˜ : [0, T ] × H ; H satisfying map G   • K(t) = ϑG˜ t, M0 ∈ V for every t ∈ [0, T ] and   ˜ ·) ∈ F t, K(t) ⊂ LIPco (H, H) for Lebesgue-almost every t ∈ [0, T ]. • G(t, If all set-valued maps in the range of F have compact values in H in addition, then K(·) is a solution to the   ˚ ∈ F K(t) in [0, T ] (in the sense of Definition 2.15). set differential inclusion K(t)

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Remark 3.4. (1.) Hypothesis (iii) can be formulated as the following sequential criterion: Choose M ∈ K(H), G ∈ LIPco (H, H). Let (Mk )k∈N and (Gk )k∈N be arbitrary sequences in K(H) and LIPco (H, H) respectively satisfying for every compact set K ⊂ H and each vector p ∈ H ⎧ ⎨ Gk ∈ F(Mk ) for each k ∈ N,   ⎩ lim dlPH (Mk , M ) + sup σ(Gk (x), p) − σ(G(x), p) = 0. k→∞

x∈K

Then, the tuple (M, G) belongs to the graph of F or, equivalently, G ∈ F(M ). (2.) The proof of Theorem 3.3 is based on suitable forms of sequential compactness and so, we use approximations by means of orthogonal projections on finite-dimensional invariant subspaces Hd ⊂ H (d ∈ N). All reachable sets are bounded and closed. Thus, their counterparts in Hd are even compact due to Heine–Borel’s theorem. While the dimension of Hd tends to infinity, we can approximate only compact sets in H w.r.t. the Pompeiu–Hausdorff metric. In other words, the method of proof requires   assumptions which guarantee the compactness of all “state” sets, i.e., the state space is K(H), dlPH .   This restriction, however, does not concern the values in LIPco (H, H) if the C0 semigroup S(t) t≥0 is immediately compact. 4. Proofs 4.1. Auxiliary results about scalarly converging sequences of uniformly bounded convex and closed sets In this subsection, we collect several results about sequences of closed and convex subsets converging w.r.t. the metric dlco . In particular, the related scalar convergence in a separable Hilbert space proves to be a good choice for drawing conclusions from converging projections on finite-dimensional subspaces. 

Lemma 4.1. Let (H, ·, ·) be a separable real Hilbert space and, the set pd d ∈ N is supposed to be dense in H. For each index d ∈ N, set Hd := span{p1 , . . . , pd } ⊂ H and, Πd : H −→ Hd denotes the orthogonal projection on Hd . (d ∈ N indicates the maximal finite dimension of Hd .) Then the function dlco : Clbd,co (H) × Clbd,co (H) −→ R introduced in Definition 2.16 is a metric on Clbd,co (H) and, the following equivalence holds for each C ∈ Clbd,co (H) and every sequence (Cm )m∈N in Clbd,co (H) with supm Cm ∞ < ∞ lim dlco (Cm , C) = 0

m→∞

⇐⇒

∀p∈H :

lim σ(Cm , p) = σ(C, p).

m→∞

Lemma 4.2 (Corollary of [32, Proposition 3.2]). In addition to the hypotheses of Lemma 4.1, let (Cm )m∈N be a sequence in Clbd,co (H) with supm Cm ∞ < ∞ and converging to C ∈ Clco (H) w.r.t. dlco . Def.

Then, C ∞ = sup x ≤ lim inf Cm ∞ . x∈C

m→∞

Lemma 4.3 (Corollary of [32, Theorem 3.2]). Under the assumptions of Lemma 4.1, the metric space   Clbd,co (BR ), dlco is compact for every R > 0. The two recent lemmata imply directly   Corollary 4.4. For every closed ball BR in a separable Hilbert space H, the metric space Clbd,co (BR ), dlco is complete and separable.

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Next, various connections with the set convergence of projections on finite-dimensional subspaces Hd (d ∈ N) are summarized. Lemma 4.5. In addition to the assumptions of Lemma 4.1, let (Cm )m∈N be any sequence in Clbd,co (H) and C ∈ Clbd,co (H) satisfying supm Cm ∞ < ∞ and lim dlco (Cm , C) = 0. m   →∞  Then, for every index d ∈ N, the sequence dlPH Πd Cm , Πd C m∈N tends to 0. Lemma 4.6. In addition to the assumptions of Lemma 4.1, let (Cm )m∈N be any sequence in Clbd,co (H) and   C ∈ Clbd,co (H) satisfying supm Cm ∞ < ∞ and lim dlPH Πd Cm , Πd C = 0 for every dimension index m→∞ d ∈ N. Then, (Cm )m∈N converges to C w.r.t. dlco . Lemma 4.7. In addition to the assumptions of Lemma 4.1, let (Cm )m∈N be any sequence in Clbd,co (H) satisfying supm Cm ∞ < ∞ and suppose that for every dimension index d ∈ N, there exists a set Pd ∈   Clbd,co (Hd ) = Kco (Hd ) with lim dlPH Πd Cm , Pd = 0. m→∞

Then (Cm )m∈N converges to a set C ∈ Clbd,co (H) w.r.t. dlco . In particular, Πd C = Pd holds for every d ∈ N. Finally, we prove those lemmata which do not result immediately from statements by De Blasi and Myjak [32]: Proof of Lemma 4.1. Obviously, dlco is symmetric. It is positive definite as a consequence of the separation theorem for convex sets (see, e.g., [13, § 1.4]). Furthermore, the triangle inequality of dlco results essentially from the triangle inequality of the absolute value | · | and the following monotonicity property for all real r, s1 , s2 ≥ 0 (indicated in, e.g., [1, Example 2.8], [31, § 15]) r ≤ s1 + s2

=⇒

s1 + s2 r s1 s2 ≤ ≤ + . 1+r 1 + s1 + s2 1 + s1 1 + s2

Thus, dlco is a metric on Clbd,co (H). Now choose C ∈ Clbd,co (H) and let (Cm )m∈N be any sequence in Clbd,co (H) with R := supm Cm ∞ < ∞. The definition of dlco implies the equivalence (see, e.g., [31, § 21]) lim dlco (Cm , C) = 0

m→∞

⇐⇒

∀d∈N:

lim σ(Cm , pd ) = σ(C, pd ).

m→∞

Hence, the claimed implication “⇐=” is obvious, i.e., whenever σ(Cm , p) −→ σ(C, p) (m → ∞) holds for each p ∈ H, then (Cm )m∈N converges to C w.r.t. dlco . In regard to the remaining claim “=⇒”, we now assume limm→∞  dlco (Cm ,εC) = 0 and fix p ∈ H, σ(Cm , pn0 ) − ε > 0 arbitrarily. Then there exist indices n0 , m0 ∈ N with pn0 − p < 2 (1+R+C and ∞) σ(C, pn0 ) < 2ε for all m ≥ m0 . For every m ≥ m0 , the triangle inequality of the absolute value and Cauchy–Schwarz’ inequality lead to σ(Cm , p) − σ(C, p) = sup x, p − sup y, p x ∈ Cm

y∈C

≤ sup x, pn0  − sup y, p + sup x, p − pn0  x ∈ Cm

y∈C

x ∈ Cm

y∈C

x ∈ Cm

  ≤ sup x, pn0  − sup y, pn0  + R p − pn0  + sup y, p − pn0  y∈C

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        ≤ σ Cm , pn0 − σ C, pn0 + R p − pn0  + C ∞ p − pn0    ε < 2ε + R + C ∞ 2 (1+R+C < ε, ∞) i.e.,

lim σ(Cm , p) = σ(C, p).

m→∞

2 d Def.

Proof of Lemma 4.5. Set R := supm Cm ∞ and fix ε > 0 arbitrarily. B1 = Hd ∩ B1 is compact due to   η d Heine–Borel’s theorem. Hence, there exists an index η = η(ε, d) ∈ N with B1 ⊂ n = 1 pn +B 1 + R +ε C∞ . Then we conclude from [13, Corollary 3.2.8] (representing the Pompeiu–Hausdorff distance between bounded closed convex sets as a supremum of support functions) and a transformation rule for support functions (e.g., [10, Table 2.1])   dlPH Πd Cm , Πd C = sup σ(Πd Cm , p) − σ(Πd C, p) d

p ∈ B1

≤ sup d

p ∈ B1

≤ = =

inf

1≤n≤η

  σ(Πd Cm , pn ) − σ(Πd C, pn ) + (R + C ∞ ) · p − pn

sup

σ(Πd Cm , pn ) − σ(Πd C, pn ) + ε

sup

σ(Cm , Π∗d pn ) − σ(C, Π∗d pn ) + ε

sup

σ(Cm , Πd pn ) − σ(C, Πd pn ) + ε

1≤n≤η

1≤n≤η

1≤n≤η

since the adjoint Π∗d coincides with the orthogonal projection Πd : H −→ Hd . Finally, it yields   lim sup dlPH Πd Cm , Πd C ≤ 0 + ε for any ε > 0. 2 m→∞

 Proof of Lemma 4.6. Choose p ∈ H and ε > 0 arbitrarily and set R := supm Cm ∞ . Due to H = d∈N Hd , ε there exist an index d ∈ N and a related vector q ∈ Hd with p − q < 1+R+C . The same arguments as ∞ before yield the estimate   σ(Cm , p) − σ(C, p) ≤ σ(Cm , q) − σ(C, q) + R + C ∞ p − q < σ(Cm , Πd q) − σ(C, Πd q) + ε = σ(Cm , Π∗d q) − σ(C, Π∗d q) + ε = σ(Πd Cm , q) − σ(Πd C, q) + ε   

≤ max 1, q · dlPH Πd Cm , Πd C + ε   and so, lim sup σ(Cm , p) − σ(C, p) ≤ 0. This convergence of σ(Cm , p) m∈N for all p ∈ H is equivalent m→∞

to the convergence of (Cm )m∈N w.r.t. dlco according to Lemma 4.1.

2

  Proof of Lemma 4.7. Set R := supm Cm ∞ again. Due to Lemma 4.3, the metric space Clbd,co (BR ), dlco is compact and so, (Cm )m∈N has a subsequence converging to a set C ∈ Clbd,co (H) (with C ⊂ BR ⊂ H) w.r.t. dlco . As a consequence of Lemma 4.5, Πd C = Pd is fulfilled for every d ∈ N.

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It remains to prove that the “whole” sequence (Cm )m∈N tends to this set C for m → ∞. Otherwise,     there exist some ε > 0 and a subsequence Cm ∈N satisfying dlco Cm , C ≥ ε for all ∈ N. According     to Lemma 4.3 again, Cm ∈N has a converging subsequence which is also denoted by Cm ∈N . Its limit D ∈ Clbd,co (H) w.r.t. dlco fulfills dlco (D, C) ≥ ε,

  dlPH Πd Cm , Πd D −→ 0 ( → ∞)

  due to Lemma 4.5. This implies dlPH Πd C, Πd D = 0 for every d ∈ N. The limit for d → ∞ reveals indirectly C = D – a contradiction. 2 4.2. Continuous dependence of compact reachable sets on the coefficients Now we focus on the proof of Proposition 2.14. On our way to approximations in finite-dimensional subspaces of H, we benefit from hypothesis (H5):   The generator A of the strongly continuous semigroup S(t) t≥0 is supposed to be normal with compact resolvent. Def. The latter property means that for some (and thus, for all) λ > ωS , the resolvent operator R(λ; A) = −1  : H −→ H is compact. Furthermore, R(λ; A) proves to be normal as well due to [51, § I.6.6]. λ IdH − A Thus, the Hilbert space H has an orthonormal basis {ˆ e1 , eˆ2 , . . . } consisting of eigenvectors of R(λ; A) as a consequence of the spectral theorem for compact normal operators (see, e.g., [51, § V.2.3 Theorem 2.10], [1, Proposition 12.12]). Each eˆd ∈ D(A) ⊂ H (d ∈ N) is also an eigenvector of the generator A and, the corresponding eigenvalue is abbreviated as κd ∈ R. Then at each time instant t ≥ 0, the bounded linear operator S(t) : H −→ H also has the eigenvector eˆd , but related to the eigenvalue eκd t (due to [37, Lemma I.1.9], for example). From now on, let Hd denote the d-dimensional real subspace of H spanned by the orthonormal vectors eˆ1 , . . . , eˆd for each d ∈ N. In particular, we observe Hd ⊂ Hd+1 ⊂ D(A)

for every d ∈ N,

H =



Hd

d∈N

and, the orthogonal projection Πd : H −→ Hd is easy to specify by means of the inner product ·, · on H. d As an abbreviation, set Bρ := Bρ (0) ∩ Hd for the closed ball of radius ρ ≥ 0 in Hd . Then each subspace Hd ⊂ H is invariant w.r.t. both A and S(t) for every t ≥ 0, i.e., A Hd ⊂ Hd and S(t) Hd ⊂ Hd as we can also conclude from the more general results about invariant subspaces in  Propositions A.3.1 and A.3.2. In particular, the restriction S(t) H t≥0 is a norm continuous semigroup d with the bounded generator A D(A)∩H = A H : Hd −→ Hd . d d The proof of Proposition 2.14 is based on approximations in Hd with the dimension d ∈ N being chosen sufficiently large. In regard to the finite-dimensional case, the key tool is the following result by Stassinopoulos and Vinter [92,93]: Proposition 4.8 ([93, Theorem 7.1]). Let D : [0, 1] × Rn ; Rn and each D : [0, 1] × Rn ; Rn ( ∈ N) satisfy the following assumptions: (1.) D and D have non-empty convex compact values, (2.) D(·, x), D (·, x) : [0, 1] ; Rn are Lebesgue measurable for every x ∈ Rn , (3.) there exists k ∈ L1 ([0, 1]) such that D(t, · ), D (t, · ) : Rn ; Rn are k(t)-Lipschitz continuous for L1 -almost every t ∈ [0, 1], sup |v| ≤ h(t) for every x ∈ Rn and L1 -almost every (4.) there exists h ∈ L1 ([0, 1]) such that t ∈ [0, 1].

v∈D(t,x) ∪ D (t,x)

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Fixing the initial point a ∈ Rn arbitrarily, the absolutely continuous solutions of   y  (·) ∈ D (·, y(·)) a.e. y  (·) ∈ D(·, y(·)) a.e. and y(0) = a y(0) = a   respectively form compact subsets of C 0 ([0, 1], Rn ), · ∞ denoted by D ( ∈ N) and D. Then, D converges to D (w.r.t. the Pompeiu–Hausdorff metric on compact subsets of C 0 ([0, 1], Rn )) if and only if for every solution d(·) ∈ D, the set-valued maps D (·, d(·)) : [0, 1] ; Rn ( ∈ N) converge to D(·, d(·)) : [0, 1] ; Rn weakly – in the sense of Artstein [2] that 

 dlPH

D (s, d(s)) ds, J

 →∞ D(s, d(s)) ds −→ 0

J

holds for every Lebesgue measurable subset J ⊂ [0, 1]. This statement lays the foundations for the subsequent introducing steps: Lemma 4.9. For any d ∈ N, u ∈ C 0 ([0, T ], Hd ) and Lebesgue measurable set J ⊂ [0, T ] given, the assumptions of Proposition 2.14 imply        ˜ s, u(s) ds −→ 0 dlPH Πd F˜ s, u(s) ds, Πd G ( → ∞) . J

J

Corollary 4.10. For d ∈ N and initial set M0 ∈ K(H), let D (M0 ) ( ∈ N) and D(M0 ) denote the sets of all Carathéodory solutions y ∈ C 0 ([0, T ], Hd ) to the non-autonomous differential inclusions 

y  ∈ A y + Πd F˜ ( · , y) a.e. y(0) ∈ Πd M0 ⊂ Hd

 and

˜ · , y) a.e. y  ∈ A y + Πd G( y(0) ∈ Πd M0 ⊂ Hd

respectively.   Then each set D (M0 ) ( ∈ N) and D(M0 ) are compact in the Banach space C 0 ([0, T ], Hd ), · sup (due to the closedness and the Arzelà–Ascoli theorem, e.g., [42]). Moreover, D (M0 ) converges to D(M0 ) w.r.t. the related Pompeiu–Hausdorff metric dlPH,C 0 for → ∞. Corollary 4.11. Under the hypotheses of Proposition 2.14, the following convergence holds for every M0 ∈ K(H) and d ∈ N sup t ∈ [0,T ]

  dlPH ϑΠd F˜ (t, Πd M0 ), ϑΠd G˜ (t, Πd M0 ) −→ 0

( → ∞).

Their missing proofs follow: Proof of Lemma 4.9. Choose the dimension index d ∈ N, u ∈ C 0 ([0, T ], Hd ) and the Lebesgue measurable set J ⊂ [0, T ] arbitrarily. As a consequence of the assumptions of Proposition 2.9, both Πd F˜ ( ∈ N) and ˜ : [0, T )×H ; Hd are Carathéodory maps with compact and convex values. Hence, the compositions Πd G     ˜ ·, u(·) : [0, T ) ; Hd Πd F˜ ·, u(·) , Πd G are Lebesgue measurable with non-empty closed and compact values due to [10, Theorem 8.2.8]. Furthermore, they have a Lebesgue integrable bound in common. Now fix any p ∈ H. The underlying convergence assumption of Proposition 2.14 implies for Lebesgue-almost every t ∈ [0, T )

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        ˜ t, u(t) , Π∗d p lim σ F˜ t, u(t) , Π∗d p = σ G         ˜ t, u(t) , p . lim σ Πd F˜ t, u(t) , p = σ Πd G

→∞

=⇒

→∞

In regard to the corresponding Aumann integrals, we conclude from [10, Proposition 8.6.2 (3.)] and Lebesgue’s theorem of dominated convergence          lim σ Πd F˜ t, u(t) dt, p = lim σ Πd F˜ t, u(t) , p dt →∞

→∞

J

J





   ˜ t, u(t) , p dt σ Πd G

= J



=σ

   ˜ t, u(t) dt, p . Πd G

J

These Aumann integrals are closed, convex and uniformly bounded subsets of Hd and so, Lemma 4.1 leads to        ˜ t, u(t) dt = 0. lim dlco Πd F˜ t, u(t) dt, Πd G →∞

J

J

Finally, they coincide with their respective projections on Hd ⊂ H and thus, we conclude the claim from Lemma 4.5:        ˜ ˜ t, u(t) dt = 0. 2 lim dlPH Πd F t, u(t) dt, Πd G →∞

J

J

Proof of Corollary 4.10. In a word, it results from Proposition 4.8 by Stassinopoulos and Vinter [92,93] and, preceding Lemma 4.9 provides the sufficient conditions needed there. First, we consider the special case M0 = {a} ⊂ Hd . Lemma 4.9 implies   lim dlPH

→∞

J

  ds, A u(s) + Πd F˜ s, u(s)

 

   ˜ ds = 0 A u(s) + Πd G s, u(s)

J

for every function u ∈ C 0 ([0, T ], Hd ) and each Lebesgue measurable set J ⊂ [0, T ] and so, it provides the sufficient condition for concluding from Proposition 4.8 that D ({a}) converges to D({a}) for → ∞ w.r.t. dlPH,C 0 . Next, we extend this observation to arbitrary M0 ∈ K(H). Fix ε > 0 arbitrarily. We conclude from Filippov’s existence theorem for differential inclusions in a finite-dimensional vector space (see, e.g., [9, § 2.4 Theorem 1], [33, § 8.5] or [101, Theorem 2.4.3]) that the following estimate holds for any x0 , y0 ∈ Hd and ∈ N   dlPH,C 0 D ({x0 }), D ({y0 }) ≤ x0 − y0 · e(λL1 ([0,T ]) +ALin(Hd ,Hd )

T)

and, this leads to     dlPH,C 0 D (M1 ), D (M2 ) ≤ dlPH M1 , M2 · e(λL1 ([0,T ]) +ALin(Hd ,Hd )

T)

for any non-empty compact initial sets M1 , M2 ⊂ Hd . Similarly, the same upper bound holds for   dlPH,C 0 D(M1 ), D(M2 ) .

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The compact set Πd M0 − (λL1 ([0,T ]) +ALin(Hd ,Hd ) T )

e

23

⊂ Hd can be covered by finitely many open balls of radius and with centers x1 , . . . , xn ∈ Πd M0 . Then we obtain

ε 2

·

  dlPH,C 0 D (M0 ), D(M0 )    ≤ dlPH,C 0 D (M0 ), D {x1 , . . . , xn }      + dlPH,C 0 D {x1 , . . . , xn } , D {x1 , . . . , xn }     + dlPH,C 0 D {x1 , . . . , xn } , D(M0 )   ≤ 2 · dlPH Πd M0 , {x1 , . . . , xn } · e(λL1 ([0,T ]) +ALin(Hd ,Hd )      + sup dlPH,C 0 D {xj } , D {xj }

T)

1≤j≤n

≤ε +

sup

1≤j≤n

     dlPH,C 0 D {xj } , D {xj } .

The special case about single-point initial sets discussed first leads to   lim sup dlPH,C 0 D (M0 ), D(M0 ) ≤ ε + 0 →∞

with ε > 0 having been chosen arbitrarily small. 2 Proof of Corollary 4.11. This is a consequence of Corollary 4.10 since  ϑΠd F˜ (t, Πd M0 ) = y(t) ∈ Hd  ϑΠd G˜ (t, Πd M0 ) = y(t) ∈ Hd



y ∈ D (M0 ) ,

y ∈ D(M0 )

imply sup t ∈ [0,T ]

    dlPH ϑΠd F˜ (t, Πd M0 ), ϑΠd G˜ (t, Πd M0 ) ≤ dlPH,C 0 D (M0 ), D(M0 ) .

2

Finally, we focus on the step from reachable sets in finite-dimensional subspaces Hd (d ∈ N) to compact sets in the infinite-dimensional Hilbert space H. ˜ Lemma 4.12. In addition to the general assumptions of this subsection 4.2 and Proposition 2.14 about G, ˜ F : [0, T ] × H ; H ( ∈ N), choose M0 ∈ K(H) and ε > 0 arbitrarily. Then there exist a dimension index d ∈ N and a radius ρ > 0 satisfying  t ∈ [0,T ]

  ∈N

ϑF˜ (t, M0 ) ∪



 ϑF˜ (·, Πd · ) (t, M0 ) ∪ ϑG˜ (t, M0 )

d

⊂ Bρ +

ε 4

B.

∈N

Proof of Lemma 4.12. According to assumption (v) of Proposition 2.9, at least one of the following conditions is satisfied:   ∗ the strongly continuous semigroup S(t) t≥0 is immediately compact, ˜ F˜ ( ∈ N). ∗ there is a compact subset K0 ⊂ H containing all set values of G,

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  We first focus on the case supposing that S(t) t≥0 is immediately compact. For arbitrary ε > 0, the bound k ∈ L1 ([0, T ]) (specified in assumption (ii) of Proposition 2.9) leads to τˆ1 ∈ (0, T ) such that ε CS e|ωS | T k L1 ([a, b]) < 16 holds for any a, b ∈ [0, T ] with 0 < b − a ≤ τˆ1 (see, e.g., [1, Lemma A3.17]).   Moreover, the strong continuity of S(τ ) τ ≥0 and the compactness of M0 ⊂ H guarantee the uniform continuity of [0, T ] × M0 −→ H, (τ, x) −→ S(τ ) x due to [37, Lemma I.5.2]. Hence, there exists some τˆ2 ∈ (0, τˆ1 ) ε B. For each ∈ N, the reachable set of F˜ is defined as sufficiently small with S(t) M0 ⊂ M0 + 16 x(t) ∈ H ∃ x ∈ C 0 ([0, t], H) such that x(0) ∈ M0 and

x is a mild solution of x ∈ A x + F˜ (·, x) = x(t) ∈ H ∃ x ∈ C 0 ([0, t], H), g ∈ L1 (0, T ; H) : x(0) ∈ M0 , τ S(τ − s) g(s) ds ∀ τ, x(τ ) = S(τ ) x(0) +

ϑF˜ (t, M0 ) :=



  0 g(τ ) ∈ F˜ τ, x(τ ) for Lebesgue-a.e. τ and so, we conclude from F˜ (s, y) ⊂ k(s) B ⊂ H for all s ∈ [0, T ], y ∈ H that 

  S(t − s) k(s) B ds

ϑF˜ (t, M0 ) ⊂ S(t) M0 + [0,t]

for all t ∈ [0, T ]. First, we obtain for every t ∈ [0, τˆ2 ], ϑF˜ (t, M0 ) ⊂ M0 + ⊂ M0 +

ε 16 ε 16

B + CS e|ωS | T k L1 ([0, t]) B ε B + 16 B.

Next, choose any t ∈ (ˆ τ2 , T ]. Then the reachable set ϑF˜ (t, M0 ) is contained in 

 k(s) S(t − s) B ds +

S(t) M0 + [0,t−ˆ τ2 ]





= S(ˆ τ2 ) S(t − τˆ2 ) M0 +

k(s) S(t − s) B ds [t−ˆ τ2 ,t]

 k(s) S(t − τˆ2 − s) B ds +

[0,t−ˆ τ2 ]



k(s) S(t − s) B ds [t−ˆ τ2 ,t]

  ⊂ S(ˆ τ2 ) S(t − τˆ2 ) M0 + CS e|ωS | T k L1 B + CS e|ωS | T k L1 ([t−ˆτ2 , t]) B ⊂ S(ˆ τ2 ) BR + with the radius R := CS e|ωS | T

ε 16

·B

  M0 ∞ + k L1 ([0,T ]) ≥ 0 and, for the same reasons, we have

    ϑF˜ (·, Πd · ) t, M0 ∪ ϑG˜ t, M0 ⊂ S(ˆ τ2 ) BR +

ε 16

· B.

S(ˆ τ2 ) : H −→ H is a compact linear operator by assumption and so, the subset S(ˆ τ2) BR is relatively compact in H. Now the dimension index d ∈ N and the radius ρ > 0 can be selected sufficiently large such that 

 M0 ∪ S(ˆ τ2 ) BR +

ε 8

d

B ⊂ Bρ +

ε 4

B ⊂ Hd +

ε 4

B.

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Thus,  

 t ∈ [0,T ]

ϑF˜ (t, M0 ) ∪

∈N



 ϑF˜ (·, Πd · ) (t, M0 ) ∪ ϑG˜ (t, M0 )

d

⊂ Bρ +

ε 4

B.

∈N

˜ F˜ ( ∈ N). Then, each Finally, we suppose that a compact set K0 ⊂ H contains all the set values of G, reachable set at time t ∈ [0, T ] satisfies  ϑF˜ (t, M0 ) ⊂ S(t) M0 +

S(t − s) K0 ds [0,t]



⊂ S(t) M0 +

S(t − s) co K0 ds [0,t]



= S(t) M0 +

 S(t − s) ds · co K0 .

[0,t]

  Due to the strong continuity of S(τ ) τ ≥0 , [0, T ] × K −→ H, (τ, x) −→ S(τ ) x is uniformly continuous for every compact subset K ⊂ H (according to [37, Lemma I.5.2]). We conclude from the compactness of M0 and co K0 ⊂ H that MT :=

 t ∈ [0,T ]



 S(t) M0 +

  ⊂ H S(t − s) ds · co K0

[0,t]

is relatively compact (a similar proof by means of a Riemann sum is presented in [67, Lemma 3.133], for example). Now the dimension index d ∈ N and the radius ρ > 0 can be chosen sufficiently large with d MT ⊂ Bρ + 4ε B ⊂ Hd + 4ε B. This completes the proof of Lemma 4.12. 2 ˜ Lemma 4.13. In addition to the general assumptions of this subsection 4.2 and Proposition 2.14 about G, ˜ F : [0, T ] × H ; H ( ∈ N), choose M0 ∈ K(H) and ε > 0 arbitrarily. Then there exist a dimension index d ∈ N and a constant c = const( M0 ∞ , λ L1 ([0,T ]) , CS , ωS , T ) ≥ 0 satisfying for all ∈ N ⎧   ⎨ dlPH Πd ϑ ˜ (t, M0 ), ϑ ˜ (t, Πd M0 ) ≤ c · ε Π d F F   ⎩ dlPH Πd ϑ ˜ (t, M0 ), ϑ ˜ (t, Πd M0 ) ≤ c · ε. Πd G G Proof of Lemma 4.13. It is essentially based on the observation that     ϑΠd F˜ t, Πd M0 = Πd ϑF˜ (·,Πd · ) t, M0 and the estimate of the Pompeiu–Hausdorff distance between reachable sets in Proposition A.1.2 below. Let d ∈ N and ρ > 0 be given as in the preceding Lemma 4.12. The d-dimensional subspace Hd ⊂ H was constructed by means of an orthonormal sequence of eigenvectors of both A and each S(τ ) (τ ≥ 0). Hence, Hd is invariant w.r.t. every S(τ ) (τ ≥ 0) and, the related orthogonal projection Πd : H −→ Hd commutes with both each linear operator S(τ ) (τ ≥ 0) and the Lebesgue–Bochner integral in H. Thus, we obtain for every t ∈ [0, T ], d, ∈ N

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  Πd ϑF˜ (·,Πd · ) t, M0 = Πd x(t) ∃ x ∈ C 0 ([0, t], H), g ∈ L1 (0, T ; H) : x(0) ∈ M0 , τ S(τ − s) g(s) ds ∀ τ, x(τ ) = S(τ ) x(0) +

 0 g(τ ) ∈ F˜ τ, Πd x(τ ) for Lebesgue-a.e. τ

⊂



y(t) ∈ Hd ∃ y ∈ C 0 ([0, t], Hd ), g ∈ L1 (0, T ; H) : y(0) ∈ Πd M0 , τ S(τ − s) Πd g(s) ds ∀ τ, y(τ ) = S(τ ) y(0) +

  0 g(τ ) ∈ F˜ τ, y(τ ) for Lebesgue-a.e. τ

⊂



y(t) ∈ Hd ∃ y ∈ C 0 ([0, t], Hd ), g˜ ∈ L1 (0, T ; Hd ) : y(0) ∈ Πd M0 , τ S(τ − s) g˜(s) ds ∀ τ, y(τ ) = S(τ ) y(0) +

 0 g˜(τ ) ∈ Πd F˜ τ, y(τ ) for Lebesgue-a.e. τ   = ϑΠd F˜ t, Πd M0 . For proving the opposite inclusion, we start from Definition 2.1 of reachable sets   ϑΠd F˜ t, Πd M0 = y(t) ∃ y ∈ C 0 ([0, t], H), g˜ ∈ L1 (0, T ; H) : y(0) ∈ Πd M0 , τ S(τ − s) g˜(s) ds ∀ τ, y(τ ) = S(τ ) y(0) +

 0 g˜(τ ) ∈ Πd F˜ τ, y(τ ) for Lebesgue-a.e. τ . As a consequence of Filippov’s selection theorem (see, e.g., [10, Theorem 8.2.10]), each g˜ ∈ L1 (0, T ; H) is related to a function g ∈ L1 (0, T ; H) satisfying g˜(τ ) = Πd g(τ ),

  g(τ ) ∈ F˜ τ, y(τ )

for Lebesgue-almost every τ ∈ [0, T ]. In the next step, we relate each function y ∈ C 0 ([0, t], Hd ) (mentioned in the last characterization) to some x ∈ C 0 ([0, t], H) by means of an element x(0) ∈ M0 with y(0) = Πd x(0) and τ S(τ − s) g(s) ds

x(τ ) = S(τ ) x(0) +

(τ ∈ [0, t]).

0

In particular, the invariance of the subspace Hd w.r.t. each linear operator S(τ ) (τ ≥ 0) implies y(τ ) = Πd x(τ ) for every τ ∈ [0, t) and so,

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  ϑΠd F˜ t, Πd M0 ⊂ y(t) ∃ y ∈ C 0 ([0, t], Hd ), g ∈ L1 (0, T ; H) : y(0) ∈ Πd M0 , τ S(τ − s) Πd g(s) ds ∀ τ, y(τ ) = S(τ ) y(0) +

  0 g(τ ) ∈ F˜ τ, y(τ ) for Lebesgue-a.e. τ

⊂



Πd x(t) ∃ x ∈ C 0 ([0, t], H), g ∈ L1 (0, T ; H) : x(0) ∈ M0 , τ S(τ − s) g(s) ds ∀ τ, x(τ ) = S(τ ) x(0) +

 0 g(τ ) ∈ F˜ τ, Πd x(τ ) for Lebesgue-a.e. τ   ⊂ Πd ϑF˜ (·,Πd · ) t, M0     (4.12) d completing the proof of ϑΠd F˜ t, Πd M0 = Πd ϑF˜ (·,Πd · ) t, M0 ⊂ Bρ + 4ε B. The orthogonal projection Πd : H −→ Hd on the finite-dimensional subspace Hd ⊂ H is a bounded     linear operator with Πd Lin(H,H) ≤ 1. Thus, Proposition A.1.2 and the λ(τ )-Lipschitz continuity of each F˜ (τ, ·): H ; H yield   dlPH Πd ϑF˜ (t, M0 ), ϑΠd F˜ (t, Πd M0 )   = dlPH Πd ϑF˜ (t, M0 ), Πd ϑF˜ (·,Πd · ) (t, M0 )   ≤ dlPH ϑF˜ (t, M0 ), ϑF˜ (·,Πd · ) (t, M0 ) t ≤

sup

  dlPH F˜ (s, ξ), F˜ (s, Πd ξ) ds · eγT ,S · (1 + λL1 ([0,t]) )

d

0

ξ ∈ Bρ + ε4 B

t ≤

sup

  λ(s) ξ − Πd ξ ds · eγT ,S · (1 + λL1 ([0,t]) )

d

0

ξ ∈ Bρ + ε4 B

≤ λ L1 ([0,T ])

ε 4

· eγT ,S · (1 + λL1 ([0,T ]) )

for every ∈ N with a constant γT,S = const(CS , ωS , T ). The same arguments lead to the corresponding   estimate for dlPH Πd ϑG˜ (t, M0 ), ϑΠd G˜ (t, Πd M0 ) . This completes the proof of Lemma 4.13. 2 Proof of Proposition 2.14. Fix ε > 0 arbitrarily small. Then Lemma 4.12 provides a dimension index d ∈ N and a radius ρ > 0 with  t ∈ [0,T ]

  ∈N

ϑF˜ (t, M0 ) ∪



 ϑF˜ (·, Πd · ) (t, M0 ) ∪ ϑG˜ (t, M0 )

d

⊂ Bρ +

ε 4

B.

∈N

In particular, it implies for every t ∈ [0, T ] and ∈ N     dlPH ϑF˜ (t, M0 ), Πd ϑF˜ (t, M0 ) , dlPH ϑG˜ (t, M0 ), Πd ϑG˜ (t, M0 ) ≤ Now we conclude from Corollary 4.11 and Lemma 4.13

ε 4

.

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  lim sup dlPH ϑF˜ (t, M0 ), ϑG˜ (t, M0 ) →∞

≤ lim sup →∞

≤2·

ε 4



  dlPH ϑF˜ (t, M0 ), Πd ϑF˜ (t, M0 ) +

  dlPH Πd ϑF˜ (t, M0 ), ϑΠd F˜ (t, Πd M0 ) +   dlPH ϑΠd F˜ (t, Πd M0 ), ϑΠd G˜ (t, Πd M0 ) +   dlPH ϑΠd G˜ (t, Πd M0 ), Πd ϑG˜ (t, M0 ) +   dlPH Πd ϑG˜ (t, M0 ), ϑG˜ (t, M0 )  + const( M0 ∞ , λ L1 ([0,T ]) , CS , ωS , T ) · ε + 0

≤ const( M0 ∞ , λ L1 ([0,T ]) , CS , ωS , T ) · ε . 2 4.3. The sufficient condition for viability of V ⊂ K(H) in Theorem 3.3 Throughout this subsection 4.3, we assume the hypotheses about F : K(H) ; LIPco (H, H) formulated in   Theorem 3.3. S(t) t≥0 denotes a quasi-contractive C0 semigroup of bounded linear operators and, γT,S = const(γS , T ) ≥ 1 abbreviates the constant mentioned in Corollary A.1.3 (about Pompeiu–Hausdorff   distances between reachable sets). Moreover, let V be a non-empty closed subset of K(H), dlPH satisfying F(M ) ∩ TV (M ) = ∅

for every M ∈ V.

  ˚ ∈ F K(t) For M0 ∈ V fixed arbitrarily, it is to prove that a compact-valued solution K: [0, T ] ; H of K(t) starts in M0 and has all its values in V ⊂ K(H). We are going to adapt the approximation method that was introduced for functional differential inclusions by Haddad [43,44] and can be found for “standard” differential inclusions in Rn in [3, § 3.4], for example. Its gist has already proved to be appropriate for compact subsets of Rn as presented in [65,66] and [67,   § 5.2]. Now the challenge is to take the strongly continuous semigroup S(t) t≥0 and the missing local   compactness of H and so of K(H), dlPH into consideration properly. 4.3.1. The approximate solutions Lemma 4.14. Choose any ε > 0.   Under the assumptions of Theorem 3.3, there exist a continuous function Kε : [0, T ] −→ K(H), dlPH  T and a function fε : [0, T ) −→ LIPco (H, H) satisfying with Rε (T ) := ε eγT ,S (1+α) T >0 (a) Kε (0) = M0 ,   (b) dist Kε (t), V ≤ Rε (T ) for all t ∈ [0, T ],   (c) for all t ∈ [0, T ), the map fε (t) ∈ LIPco (H, H) belongs to F B2 Rε (T ) (Kε (t)) and satisfies lim h1 · h↓0   dlPH K(t + h), ϑfε (t) (h, K(t)) = 0. (d) fε is piecewise constant “to the right”, i.e., in the following sense: For each t ∈ [0, T ), there exists some δ > 0 such that fε [t, t+δ) is constant. In a word, the proof of Lemma 4.14 is based on arguments adapting the proof of [67, Lemma 5.16] to     the situation in K(H), dlPH and LIPco (H, H), dl∞ respectively, which do not provide local compactness. This topological gap is bridged by a supplementary component approximating the respective projection on V similarly to what Bothe used in [18].

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  For ε > 0 fixed, let Aε (M0 ) denote the set of all tuples τK , K(·), P (·), f (·) consisting of a scalar τK ∈ [0, T ], two compact-valued maps K, P : [0, τK ] ; H and a function f : [0, τK ) −→ LIPco (H, H) satisfying the following conditions: (a) K(0) = M0 .    t t > 0, (b’) 1.) dlPH K(τK ), P (τK ) ≤ Rε (τK ) with Rε (t) := ε eγT ,S (1+α)   2.) dlPH K(t), P (t) ≤ Rε (T ) for all t ∈ [0, τK ].   (c’) For all t ∈ [0, τK ), the map f (t) ∈ LIPco (H, H) belongs to F BRε (T ) (P (t)) and satisfies lim h1 · h↓0   dlPH K(t + h), ϑf (t) (h, K(t)) = 0. (d) f is piecewise constant “to the right”, i.e., in the following sense: For each t ∈ [0, τK ), there is some δ > 0 such that f [t, t+δ) is constant.  K is Λ-Lipschitz continuous w.r.t. ΔCl i.e., for any s, t ∈ [0, τK ] (s ≤ t) (e) Setting Λ := e|ωS | T β,   Def.   ΔCl (s, K(s)), (t, K(t)) = |t − s| + dlPH S(t − s) K(s), K(t) ≤ Λ |t − s|. (f) P is piecewise constant “to the left” in (0, τK ) – in the following sense: For each t ∈ (0, τK ), there is some δ > 0 such that P (t−δ, t] is constant.   (g) There exists an (at most) countable family [aj , bj ) j∈J of pairwise disjoint intervals whose union is P [0, τK ) and which satisfies inf

s∈[t,bj ]

        ΔCl aj , P (aj ) , s, P (t) ≤ Λ + ε · bj − aj ,         ΔCl aj , P (aj ) , t˜, P (t˜) ≤ Λ + ε · bj − aj ,         ΔCl t˜, P (t˜) , bj , P (bj ) ≤ Λ + ε (1 + e|ωS | T ) · bj − aj

for all j ∈ JP , t ∈ [aj , bj ] and at every point t˜ of discontinuity of P in (aj , bj ]. (h) P has all its values in V. Obviously, Aε (M0 ) = ∅ since it contains (0, K(·) ≡ M0 , P (·) ≡ M0 , f (·) ≡ f0 ) with arbitrary f0 ∈ LIPco (H, H). Moreover, an order relation  on Aε (M0 ) is specified by (τ1 , K1 , P1 , f1 )  (τ2 , K2 , P2 , f2 ) :⇐⇒ τ1 ≤ τ2 , K2 [0,τ1 ] = K1 , P2 [0,τ1 ] = P1 and f2 [0,τ1 ) = f1 .   Corollary 4.17 below shows that Aε (M0 ),  satisfies the assumptions of Zorn’s Lemma and so, it has a   maximal element τ, Kε (·), Pε (·), fε (·) ∈ Aε (M0 ). Finally, an indirect conclusion from Lemma 4.15 reveals τ = T . We formulate these steps as separate lemmata – for the sake of a (more) transparent presentation.   Lemma 4.15. Under the assumptions of Lemma 4.14, let Aε (M0 ),  be defined as before. For every tuple (τ, K, P, f ) ∈ Aε (M0 ) with τ < T , there exist some ρ ∈ (0, T − τ ) and a related tuple     ˜ P˜ , f˜ ∈ Aε (M0 ) with (τ, K, P, f )  τ + ρ, K, ˜ P˜ , f˜ . τ + ρ, K,     Proof. By assumption, there exists a set-valued map G ∈ TV P (τ ) ∩ F P (τ ) ⊂ LIPco (H, H). According     to Corollary 2.10, the reachable sets ϑG h, K(τ ) and ϑG h, P (τ ) are compact in H for every h ≥ 0.   Due to Definition 3.1 of the contingent transition set TV P (τ ) , there is a sequence hm ↓ 0 in (0, T − τ )     ˜ P˜ : [0, τ + h1 ] −→ K(H) and with dist ϑG hm , P (τ ) , V < 2ε hm for all m ∈ N. Now we define K, ˜ f : [0, τ + h1 ] −→ LIPco (H, H) as

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 ˜ K(t) =  f˜(t) =

for t ∈ [0, τ ] for t ∈ (τ, τ + h1 ]

K(t)   ϑG t − τ, K(τ ) f (t)   G ∈ F P (τ )

for t ∈ [0, τ ) for t ∈ [τ, τ + h1 )

P˜ (t) = P (t) for t ∈ [0, τ ] P˜ remains to be defined on (τ, τ + h1 ] so far. For each index m ∈ N, we select a set P˜ (τ + hm ) ∈ V with         dlPH ϑG hm , P˜ (τ ) , P˜ (τ + hm ) ≤ dist ϑG hm , P˜ (τ ) , V +

ε 2

hm < ε hm .

Then P˜ is extended as a constant function to every interval (τ + hm+1 , τ + hm ], m ∈ N, implying that ˜ P (τ,τ +h ] is piecewise constant “to the left”. 1 The adequate choice of ρ > 0 is still open, i.e., we are to choose the index m0 ∈ N sufficiently large so ˜ P˜ , f˜ to [0, τ + ρ] with ρ := hm satisfy all conditions on Aε (M0 ). that the restrictions of K, 0   According to hypothesis (H4), S(t) t≥0 is quasi-contractive and so, for any index m ∈ N, we conclude from Corollary A.1.3   ˜ + hm ), P˜ (τ + hm ) dlPH K(τ     ˜ )), ϑG (hm , P˜ (τ )) + dlPH ϑG (hm , P˜ (τ )), P˜ (τ + hm ) ≤ dlPH ϑG (hm , K(τ    hm ˜ ), P˜ (τ ) · eγT ,S (1+α) + ε · hm < dlPH K(τ γT ,S (1+α)  τ γT ,S (1+α)  hm ≤ ε e τ · e + ε · hm ≤ Rε (τ + hm ), i.e., condition (b’)(1.) is also satisfied at time t = τ + hm for any m ∈ N. Next, we verify condition (b’)(2.) for the interval [τ, τ + hm ] with m ∈ N sufficiently large. The functions [0, T − τ ) −→ R × K(H) defined as    ˜ ) h −→ t + h, ϑG h, K(τ

   and h −→ t + h, ϑG h, P˜ (τ )

respectively are sequentially continuous w.r.t. ΔCl due to Proposition A.1.4 and so, they are continuous w.r.t. dlPH according to Lemma A.2.3. Thus, there exists δ ∈ (0, T − τ ) such that all h ∈ [0, δ) satisfy 

   ˜ ), ϑG h, K(τ ˜ ) < dlPH K(τ    dlPH P˜ (τ ), ϑG h, P˜ (τ ) <

1 3 1 3

  · Rε (T ) − Rε (τ + h1 )   · Rε (T ) − Rε (τ + h1 ) .

Due to hm ↓ 0, we can always find m0 ∈ N with 0 < hm < δ for all m ≥ m0 . Then for each t ∈ (τ, τ + hm0 ], a unique index m ≥ m0 satisfies t − τ ∈ (hm+1 , hm ] and, the triangle inequality of dlPH guarantees       ˜ ˜ ˜ ) + dlPH K(τ ˜ ), P˜ (τ ) + dlPH K(t), P˜ (t) ≤ dlPH K(t), K(τ     dlPH P˜ (τ ), P˜ (τ + hm ) + dlPH P˜ (τ + hm ), P˜ (t)     ˜ ) , K(τ ˜ ) + Rε (τ ) + ≤ dlPH ϑG t − τ, K(τ    dlPH P˜ (τ ), ϑG hm , P˜ (τ ) + 0 <

Rε (T )−Rε (τ +h1 ) 3

+ Rε (τ ) +

<

Rε (T )−Rε (τ +h1 ) 3

+ Rε (τ + h1 ) +

< Rε (T ).

Rε (T )−Rε (τ +h1 ) 3 Rε (T )−Rε (τ +h1 ) 3

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In regard to property (c’), the infinitesimal condition 1 h↓0 h

lim

   · dlPH K(t + h), ϑf (t) h, K(t) = lim

1 h↓0 h

   · dlPH K(t + h), ϑG h, K(t) = 0

results directly from Proposition 2.3 for every t ∈ [τ, τ + h1 ). Moreover, for every t ∈ (τ, τ + hm0 ], we obtain the following estimate with the unique index m ≥ m0 satisfying t − τ ∈ (hm+1 , hm ]   dlPH P˜ (τ ), P˜ (t)   = dlPH P˜ (τ ), P˜ (τ + hm )        ≤ dlPH P˜ (τ ), ϑG hm , P˜ (τ ) + dlPH ϑG hm , P˜ (τ ) , P˜ (τ + hm )   ≤ 13 · Rε (T ) − Rε (τ + h1 ) + ε hm   ≤ 13 · Rε (T ) − Rε (τ + h1 ) + Rε (τ + hm ) < Rε (T ),     i.e., f˜(t) = G ∈ F P˜ (τ ) ⊂ F BRε (T ) (P˜ (t)) . This completes the proof of condition (c’). The conditions (d), (f) and (h) are obvious in [τ, τ + h1 ). Property (e) results directly from Proposi  tion A.1.4 since S(t) t≥0 is assumed to be a quasi-contractive semigroup (i.e., CS = 1). Last, but not least, we focus on condition (g). The index set JP related to P : [0, τ ] −→ K(H) is extended by a new element j and, we set aj := τ , bj := τ + hm0 with the index m0 ∈ N specified before. Then, the   triangle-type inequality for ΔCl in Remark 2.5 (3.) and Proposition A.1.4 imply for every t ∈ τ, τ + hm0 and m ≥ m0 with t − τ ∈ (hm+1 , hm ] inf

s∈[t,bj ]

    ΔCl (aj , P˜ (aj )), (s, P˜ (t)) ≤ ΔCl (τ, P˜ (τ )), (τ + hm , P˜ (t))

  = ΔCl (τ, P˜ (τ )), (τ + hm , P˜ (τ + hm ))   ≤ ΔCl (τ, P˜ (τ )), (τ + hm , ϑG (hm , P˜ (τ ))) +   ΔCl (τ + hm , ϑG (hm , P˜ (τ ))), (τ + hm , P˜ (τ + hm ))   ≤ Λ · hm + dlPH ϑG (hm , P˜ (τ )), P˜ (τ + hm ) ≤ Λ · hm + ε · hm   ≤ Λ + ε · hm0 .   Every point t˜ of discontinuity of P˜ in (aj , bj ] = τ, τ + hm0 has the form t˜ = τ + hm with m ≥ m0 and so, the preceding arguments show     ΔCl (aj , P˜ (aj )), (t˜, P˜ (t˜)) = ΔCl (τ, P˜ (τ )), (τ + hm , P˜ (τ + hm ))   ≤ Λ + ε · hm0 . In essentially the same way, we obtain   ΔCl (t˜, P˜ (t˜)), (bj , P˜ (bj ))   = ΔCl (τ + hm , P˜ (τ + hm )), (τ + hm0 , P˜ (τ + hm0 ))   ≤ ΔCl (τ + hm , P˜ (τ + hm )), (τ + hm0 , ϑG (hm0 , P˜ (τ ))) +

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  ΔCl (τ + hm0 , ϑG (hm0 , P˜ (τ ))), (τ + hm0 , P˜ (τ + hm0 ))   ≤ e|ωS | (hm0 −hm ) · ΔCl (τ + hm , P˜ (τ + hm )), (τ + hm , ϑG (hm , P˜ (τ ))) +   ΔCl (τ + hm , ϑG (hm , P˜ (τ ))), (τ + hm0 , ϑG (hm0 , P˜ (τ ))) +   ΔCl (τ + hm0 , ϑG (hm0 , P˜ (τ ))), (τ + hm0 , P˜ (τ + hm0 ))   ≤ e|ωS | hm0 · dlPH P˜ (τ + hm ), ϑG (hm , P˜ (τ )) + Λ · hm0 − hm +   dlPH ϑG (hm0 , P˜ (τ )), P˜ (τ + hm0 ) ≤ e|ωS | hm0 · ε hm + Λ · hm0 + ε hm0 .   ˜ P˜ , f˜ with ρ := hm ∈ (0, T − τ ) satisfies conditions (a)–(h) on tuples in Aε (M0 ). Thus, τ + ρ, K, 0

2

  Lemma 4.16. Under the assumptions of Lemma 4.14, let Aε (M0 ),  be defined as before.   For every sequence (τk , Kk , Pk , fk ) k∈N in Aε (M0 ) satisfying (τk , Kk , Pk , fk )  (τk+1 , Kk+1 , Pk+1 , fk+1 ) for each k ∈ N, there exists a tuple (τ, K, P, f ) ∈ Aε (M0 ) with (τk , Kk , Pk , fk )  (τ, K, P, f )

for every k ∈ N.

In regard to the assumptions of Zorn’s lemma (see, e.g., [102, § 0.I]), the gist of Lemma 4.16 can be reformulated in the following way:   Corollary 4.17. Under the assumptions of Lemma 4.14, let Aε (M0 ),  be defined as before. Then every totally ordered subset of Aε (M0 ) has an upper bound w.r.t. .   Proof of Lemma 4.16. Assuming that (τk , Kk , Pk , fk ) k∈N is monotone w.r.t. , the sequence (τk )k∈N is non-decreasing in [0, T ] and so, it converges to some τ ∈ [0, T ]. Then we can construct candidates for the wanted functions K, P, f in [0, τ ) by choosing the index k sufficiently large: For every t ∈ [0, τ ), there exists an index k0 ∈ N with t < τk ≤ τ for every k ≥ k0 and, set K(t) := Kk (t), P (t) := Pk (t), f (t) := fk (t) with any index k ≥ k0 . In particular, the values do not depend on the choice of k (just provided τk ≥ t) due to the assumed monotonicity w.r.t. . The sets K(τ ), P (τ ) ∈ K(H) remain to be specified such that (τ, K, P, f ) belongs to Aε (M0 ).   As a consequence of condition (e), the sequence (τk , K(τk )) = (τk , Kk (τk )) k∈N is a Cauchy sequence w.r.t. the time-oriented distance function ΔCl – in the sense that for any δ > 0, there exists an index kδ ∈ N satisfying   ΔCl (τk , K(τk )), (τ , K(τ )) < δ

whenever k, ≥ kδ .

    Hence, S(τ − τk ) K(τk ) k∈N is a Cauchy sequence in the complete metric space K(H), dlPH because for any indices k ≤ ,   dlPH S(τ − τk ) K(τk ), S(τ − τ ) K(τ )   = dlPH S(τ − τ ) S(τ − τk ) K(τk ), S(τ − τ ) K(τ )   ≤ eωS (τ −τ ) · dlPH S(τ − τk ) K(τk ), K(τ )   ≤ e|ωS | T · ΔCl (τk , K(τk ), (τ , K(τ )) .   Now we define K(τ ) ∈ K(H) as the limit of S(τ − τk ) K(τk ) k∈N w.r.t. dlPH . Then K : [0, τ ] −→ K(H) is sequentially continuous w.r.t. ΔCl at time instant t = τ and so, it is continuous w.r.t. dlPH due to Lemma A.2.3.

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Similarly, we specify P (τ ) ∈ V ⊂ K(H): Each function Pk : [0, τk ] −→ K(H) is related to an (at most) countable family of pairwise disjoint subintervals [ak,j , bk,j ), j ∈ JPk , as specified in condition (g). Due to the assumed monotonicity w.r.t. , we can suppose without loss of generality that 





[ak,j , bk,j ) j ∈ JPk ⊂ [ak+1,j , bk+1,j ) ∩ [0, τk ) j ∈ JPk+1

holds for every k ∈ N. Then the family [ak,j , bk,j ) for any k ∈ N and j ∈ JPk is countable and has the property that any two subintervals are either disjoint or their left boundary coincide. This induces a countable family of pairwise disjoint subintervals [aj , bj ), j ∈ JP , whose union is [0, τ ) and which satisfy for any j ∈ JP , t ∈ [aj , bj ] ∩ [0, τ ) and each point t˜ of discontinuity in (aj , bj ] ∩ (0, τ ) inf

s∈[t,bj ]

        ΔCl aj , P (aj ) , s, P (t) ≤ Λ + ε · bj − aj ,         ΔCl aj , P (aj ) , t˜, P (t˜) ≤ Λ + ε · bj − aj ,         ΔCl t˜, P (t˜) , bj , P (bj ) ≤ Λ + ε (1 + e|ωS | T ) · bj − aj .

These inequalities result from property (g) of Pk with any index k = k(j) ∈ N sufficiently large such that either τk = τ (if bj = τ ) or bj < τk ≤ τ (if bj < τ ). There are two cases excluding each other: In the first case, there is no sequence (t˜ )∈N in [0, τ ) which converges to τ and consists of points of discontinuity of P . There exist some δ > 0 and an index k0 = k0 (δ) ∈ N instead such that for all k ≥ k0 , τk > τ − δ is satisfied and the restriction P (τ −δ,τ ] is constant. Due to the assumed monotonicity w.r.t. , k these restrictions have a unique value P0 ∈ V ⊂ K(H) in common and, we define P (τ ) := P0 ∈ V. In the second case, there exists a sequence t˜ ↑ τ consisting of points of discontinuity of P . Then,   S(τ − t˜ ) P (t˜ ) ∈N proves to be a Cauchy sequence w.r.t. the Pompeiu–Hausdorff metric dlPH whose limit is chosen as P (τ ) ∈ V ⊂ K(H). (In particular, this limit does not depend on the sequence (t˜ )∈N as an indirect standard conclusion reveals.) Indeed, the last two estimates and the triangle-type inequality for the time-oriented distance function ΔCl in Remark 2.5 (3.) imply for all , m ∈ N ( ≤ m)      ˜ ˜  ΔCl (t˜ , P (t˜ )), (t˜m , P (t˜m )) ≤ e|ωS | (tm −t ) Λ + ε (1 + e|ωS | T ) · t˜m − t˜ due to a piecewise argument (to the left) in combination with suprema and thus,   dlPH S(τ − t˜ ) P (t˜ ), S(τ − t˜m ) P (t˜m )     ≤ S(τ − t˜m )Lin(H,H) · dlPH S(t˜m − t˜ ) P (t˜ ), P (t˜m )   ˜ ≤ e|ωS | (τ −tm ) · ΔCl (t˜ , P (t˜ )), (t˜m , P (t˜m ))     ≤ e|ωS | T Λ + ε (1 + e|ωS | T ) · t˜m − t˜ . This construction of K(τ ), P (τ ) ∈ K(H) preserves conditions (b’), (d)–(h) because V is assumed to be   closed in K(H), dlPH . In particular, P : [0, τ ] −→ K(H) is continuous w.r.t. dlPH at time instant t = τ . 2 Remark 4.18. The conclusions of this subsection can be extended to a weaker form of contingent condition which opens the door to corresponding results about non-autonomous set differential inclusions. We mention this detail here for preparing the proof of Proposition 2.17 in § 4.4 below.  : [0, T ] × K(H) ; LIPco (H, H) with non-empty values For a moment, consider the set-valued map F   and global bounds (as in Theorem 3.3 (ii)). Let V still be a non-empty closed subset of K(H), dlPH , but

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assume an (at most) countable set S = {t1 , t2 , . . . } ⊂ [0, T ] such all t ∈ [0, T ] \ S and every M ∈ V satisfy  M ) ∩ TV (M ) = ∅. Then the statement of Lemma 4.14 holds for arbitrary ε > 0 with the modified radius F(t,   Def.  T  T eγT ,S (1+α) ε + Rε (T ) = ε eγT ,S (1+α) 1 + T > 0. Indeed, the only change required concerns the proof of Lemma 4.15. We need an alternative choice for      τ, P (τ ) ∩ TV P (τ ) = ∅, i.e., if τ = tj ∈ S for some index j ∈ N. The G ∈ LIPco (H, H) in the case F function [0, T ] × P (τ ) −→ H, (t, x) −→ S(t) x is uniformly continuous (due to [37, Lemma I.5.2]) and so, there exists some δ > 0 such that all (s, x), (t, y) ∈ [0, T ] × P (τ ) fulfill |t − s| < δ and x − y < δ

  S(s) x − S(t) y  <

=⇒

ε 2j+1

.

ε Choose ρ˜ ∈ (0, δ) sufficiently small with ρ˜ · e|ωS | T β < 2j+1 and τ + ρ˜ ∈ [0, T ) \ S. G ∈ LIPco (H, H) is now    τ, P (τ ) = ∅ arbitrarily and, P : [0, τ ] −→ K(H) is extended to P˜ : [0, τ + ρ˜] −→ K(H) by selected in F means of the constant value P (τ ) ∈ V. This implies for all h ∈ [0, ρ˜]

    dlPH P˜ (τ + h), ϑG (h, P˜ (τ )) = dlPH P (τ ), ϑG (h, P (τ )) <

ε 2j

due to the standard representation of mild solutions. Finally, Lemma A.2.3 about the continuity w.r.t. dlPH provides some ρ ∈ (0, ρ˜] with τ + ρ ∈ [0, T ) \ S and 

   ˜ ), ϑG h, K(τ ˜ ) < dlPH K(τ    dlPH P˜ (τ ), ϑG h, P˜ (τ ) <

1 3 1 3

  · Rε (T ) − Rε (τ + ρ˜)   · Rε (T ) − Rε (τ + ρ˜)

for all h ∈ [0, ρ]. Then we obtain essentially all relevant estimates in the proofs of Lemma 4.15 and 4.16 with the modified radius      t γT ,S (1+α)  t ε Def. 1 ε (t) := Rε (t) + eγT ,S (1+α) = ε e R t + . k k 2 2 k ∈ N: tk < t

k ∈ N: tk < t

4.3.2. Aspects of compactness for subproblems in finite dimensions From now on, we again benefit from hypothesis (H5). Indeed, it lays the foundations for constructing an expanding sequence of finite-dimensional invariant subspaces Hd ⊂ H (d ∈ N) as described at the beginning of § 4.2.   Moreover for every index k ∈ N, Lemma 4.14 provides a continuous function Kk : [0, T ] −→ K(H), dlPH and a function fk : [0, T ) −→ LIPco (H, H) satisfying (a) Kk (0) = M0 ,   (b) dist Kk (t), V ≤ k1 for all t ∈ [0, T ],   (c) for all t ∈ [0, T ), the map fk (t) ∈ LIPco (H, H) belongs to F B1/k (Kk (t)) and satisfies lim h1 · h↓0    dlPH Kk (t + h), ϑfk (t) h, Kk (t) = 0. (d) fk is piecewise constant “to the right”, i.e., in the following sense: For each t ∈ [0, T ), there exists some δ > 0 such that fk [t, t+δ) is constant. Now we continue this construction by means of finite-dimensional parametrizations: Lemma 4.19. For every set-valued map G ∈ LIPco (H, H) and each index d ∈ N, the composition Πd ◦ G H : d

d

Hd ; Hd belongs to LIPco (Hd , Hd ) and so, there is a single-valued function gd : Hd × B1 −→ Hd satisfying (i) for every x ∈ Hd , the image set of gd (x, · ) is Πd ◦ G(x) ⊂ BG∞ ∩ Hd and thus compact,

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d

(ii) for every p ∈ B1 , gd ( · , p) : Hd −→ Hd is Lipschitz continuous with a Lipschitz constant ≤ const(d) · LipG, d (iii) for every x ∈ Hd , gd (x, · ) : B1 −→ Hd is Lipschitz continuous with a Lipschitz constant ≤ const(d) · G ∞ , (iv) for every x ∈ Hd , the vector gd (x, 0) ∈ Hd is the Steiner point of the convex compact set Πd ◦ G(x) ⊂ Hd ,   d (v) the resulting function LIPco (H, H) −→ W 1,∞ Hd × B1 , Hd , G −→ gd is additive and positively   ˜ ˜ ∈ LIPco (H, H) and λ ≥ 0, the set-valued map Πd λ · G + G homogeneous, i.e., for any G, G : H d

d

Hd ; Hd is parametrized by λ · gd + g˜d : Hd × B1 −→ Hd . The proof will be given in a moment. For any indices d, k ∈ N, this lemma provides a function f˜d,k :   d [0, T ) −→ W 1,∞ Hd ×B1 , Hd related to fk : [0, T ) −→ LIPco (H, H) in a pointwise way. Due to condition (d) on fk , the function f˜d,k is piecewise constant “to the right” and so, it belongs to the Lebesgue–Bochner    d space L1 0, T ; C 0 Hd ×B1 , Hd . The relevant results of this subsection are the following statements: Lemma 4.20. Under the hypotheses of Theorem 3.3, let (Hd )d∈N , (Kk )k∈N and (fk )k∈N , (f˜d,k )d,k∈N be constructed as described recently.   For every dimension index d ∈ N, there exist a sequence k  ∞ of indices, a sequence fd, ∈N     d d of functions [0, T ) −→ W 1,∞ Hd × B1 , Hd and a function g˜d : [0, T ) −→ C 0 Hd × B1 , Hd with these properties:   d  d →∞ 1.) for all ρ > 0, f˜d,k Bd ×Bd −→ g˜d Bd ×Bd weakly in L1 0, T ; C 0 Bρ ×B1 , Hd , ρ 1 ρ 1   d  d →∞ 2.) for all ρ > 0, fd, Bd ×Bd −→ g˜d Bd ×Bd strongly in L1 0, T ; C 0 Bρ ×B1 , Hd , ρ

ρ

1

1

3.) for each ∈ N, fd, is a (finite) convex combination of f˜d,k , f˜d,k(+1) , f˜d,k(+2) , f˜d,k(+3) . . . and so, it is piecewise constant “to the right”, d 4.) for every ∈ N and t ∈ [0, T ), the function fd, (t) : Hd × B1 −→ Hd is bounded and Lipschitz continuous with   Lipfd, (t) ≤ const d, α  ,

  fd, (t)

∞

  ≤ const d, β .

Finally, Cantor’s diagonal method provides a joint (sub-) sequence of indices (again denoted by) k  ∞ for all dimension indices d ∈ N. This step will enable us to draw useful conclusions about the convergence of the underlying set-valued maps fk : [0, T ) −→ LIPco (H, H) we started with. Corollary 4.21. Under the hypotheses of Theorem 3.3, let (Hd )d∈N , (Kk )k∈N and (fk )k∈N , (f˜d,k )d,k∈N be constructed as described recently. Then there are a (joint) sequence k  ∞ of indices and a sequence (λ )∈N of finite scalar tuples   λ = λ,1 , . . . , λ,N ∈ [0, 1]N such that for every dimension index d ∈ N, there exists a function   d g˜d : [0, T ) −→ C 0 Hd ×B1 , Hd satisfying   d  d →∞ 1.) for all ρ > 0, f˜d,k Bd ×Bd −→ g˜d Bd ×Bd weakly in L1 0, T ; C 0 Bρ ×B1 , Hd , ρ

1

ρ

1

2.) for all ρ > 0 and d ∈ N, the linear combination fd, :=   d  d g˜d Bd ×Bd strongly in L1 0, T ; C 0 Bρ ×B1 , Hd , ρ

1

N  n=1

λ,n · f˜d,k(+n) fulfills fd, Bd ×Bd ρ

1

→∞

−→

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3.) for each ∈ N, the components of λ ∈ [0, 1]N induce a convex combination, i.e.,

N 

λ,n = 1, and

n=1

  d so, each fd, : [0, T ) −→ C 0 Hd × B1 , Hd , d ∈ N, is piecewise constant “to the right”, d 4.) for every d, ∈ N and t ∈ [0, T ), the function fd, (t) : Hd × B1 −→ Hd is bounded and Lipschitz continuous with   Lipfd, (t) ≤ const d, α  ,

  fd, (t)

∞

  ≤ const d, β .

Now we give the missing proofs for the lemmata in this subsection: Proof of Lemma 4.19. The projection Πd : H −→ Hd is continuous with Πd Lin(H,Hd ) ≤ 1 and so, the composition Πd ◦ G : H ; Hd is Lipschitz continuous with non-empty compact convex values and   Πd ◦ G ≤ G ∞ , ∞

  Lip Πd ◦ G ≤ LipG,

i.e., Πd ◦ G ∈ LIPco (H, Hd ). This implies for the restriction to the finite-dimensional subspace Hd ⊂ H: Πd ◦ G H ∈ LIPco (Hd , Hd ). d d The claimed function gd : Hd × B1 −→ Hd is a so-called Lipschitz parametrization of Πd ◦ G H : d Hd ; Hd . Its existence with properties (i)–(iii) results directly from [10, Theorem 9.7.2], for example. In regard to the additional properties (iv), (v), however, we suggest another explicit parametrization: Let Sd (C) ∈ Hd denote the Steiner point of a compact convex subset C ⊂ Hd as defined in, e.g., [10, § 9.4]

 Def. and, σ(C, p) ∈ R abbreviates the support function of C ⊂ Hd at p ∈ H, i.e., σ(C, p) = sup p, v v ∈ C . Both Sd :

  Kco (Hd ), dlPH −→ Hd

and σ :



 d Kco (Hd ), dlPH × B1 −→ R

are known to be Lipschitz continuous (see, e.g., [10, Theorem 9.4.1], [13, Corollary 3.2.8]). Hence, the d single-valued function gd : Hd × B1 −→ Hd ,          gd (x, p) := Sd Πd ◦ G(x) + σ Πd ◦ G(x), p − Sd Πd ◦ G(x) , p · p    d is Lipschitz continuous and fulfills gd x, B1 = Πd ◦ G(x) as well as gd (x, 0) = Sd Πd ◦ G(x) for every x ∈ Hd , i.e., conditions (i)–(iv) are satisfied. Furthermore, Sd : Kco (Hd ) −→ Hd is linear and for every p ∈ Hd , the function σ(·, p) : Kco (Hd ) −→ R is additive as well as positively homogeneous (see, e.g., [10, §§ 2.4, 9.4.1]). Together with the linearity of the ˜ ∈ LIPco (H, H) and λ > 0, the preceding formula projection Πd : H −→ H, we conclude that for any G, G d of the parametrization leads to the function Hd × B1 −→ Hd mapping (x, p) to          ˜ ˜ ˜ Sd Πd (λ G + G)(x) + σ Πd (λ G + G)(x), p − Sd Πd (λ G + G)(x) ,p p            = λ Sd Πd ◦ G(x) + σ Πd ◦ G(x), p − Sd Πd ◦ G(x) , p · p          ˜ ˜ ˜ + Sd Πd ◦ G(x) + σ Πd ◦ G(x), p − Sd Πd ◦ G(x) , p ·p = λ · gd (x, p) + g˜d (x, p) d ˜ respectively. This completes the proof of Lemma 4.19. 2 with gd , g˜d : Hd ×B1 −→ Hd being related to G, G

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Proof of Lemma 4.20. Due to assumption (ii) of Theorem 3.3, all the values of fk (t) ∈ LIPco (H, H) (k ∈ N, t ∈ [0, T )) are uniformly bounded in the sense that sup

sup

k∈N t ∈ [0,T )

Lipfk (t) ≤ α ,

sup

sup

k∈N t ∈ [0,T )

  fk (t)

∞

≤ β .

d ≥ 0 (depending on d, α Lemma 4.19 ensures a constant λ , β only) such that all single-valued functions d d -Lipschitz continuous and have their supremum norm f˜d,k (t) : Hd × B1 −→ Hd (t ∈ [0, T ), k ∈ N) are λ  bounded by β. d d Fix the radius ρ ∈ N arbitrarily. Then the set of restrictions to the compact subset Bρ × B1 ⊂ Hd × Hd ,

  d d i.e., f˜d,k (t) Bd ×Bd t ∈ [0, T ), k ∈ N , is relatively weakly compact in C 0 (Bρ × B1 , Hd ), · sup ρ 1

according to Proposition A.4.1. Next, we conclude from Proposition A.4.2 that f˜d,k Bd ×Bd k ∈ N is ρ 1   d d relatively weakly compact in L1 0, T ; C 0 (Bρ × B1 , Hd ) . Hence, there exist a sequence k  ∞ of indices   d d and some gd,ρ ∈ L1 0, T ; C 0 (Bρ × B1 , Hd ) with f˜d,k Bd ×Bd −→ gd,ρ ρ

  d d weakly in L1 0, T ; C 0 (Bρ × B1 , Hd )

1

( → ∞).

Now we take all radii ρ ∈ N into consideration appropriately. Indeed, Cantor’s diagonal method provides a joint (sub-) sequence of indices (again denoted by) k  ∞ such that for every ρ ∈ N, there is g˜d,ρ ∈   d d L1 0, T ; C 0 (Bρ × B1 , Hd ) satisfying f˜d,k Bd ×Bd −→ g˜d,ρ ρ

  d d weakly in L1 0, T ; C 0 (Bρ × B1 , Hd )

1

( → ∞).

In the next step, we compare the weak limits g˜d,ρ , g˜d,R for any R > ρ > 0:   d d g˜d,ρ = g˜d,R Bd×Bd ∈ L1 0, T ; C 0 (Bρ × B1 , Hd ) . ρ

1

Indeed, for every index ∈ N and R > r > 0, Mazur’s lemma (see, e.g., [1, Lemma 8.14], [102,   d Theorem V.1.2]) guarantees a finite convex combination fd,R, : [0, T ) −→ C 0 Hd × B1 , Hd of f˜d,k , f˜d,k(+1) , f˜d,k(+2) , . . . such that       g˜d,R − fd,R, Bd×Bd  R

1

Def.

L1 (0,T ;C 0 )

=

     gd,R (t) − fd,R, (t) Bd×Bd  ˜ R

1

sup

dL1 t

[0,T )

is bounded by 1 . The construction as convex combinations implies directly that the sequence of restrictions     d d fd,R, Bd×Bd ∈N converges to the same limit g˜d,ρ weakly in L1 0, T ; C 0 (Bρ × B1 , Hd ) for → ∞. The ρ

1

additional criterion about the L1 norm and R > ρ lead to the conclusion that       g˜d,R − fd,R, Bd×Bd  ρ

1

     gd,R (t) Bd×Bd − fd,R, (t) Bd×Bd  = ˜

Def. L1 (C 0 )

ρ

1

ρ

1

dL1 t

sup

[0,T )

  tends to 0 for → ∞ and so, the restriction g˜d,R Bd×Bd is also the weak limit of fd,R, Bd×Bd ∈N in ρ 1 ρ 1     d d d d L1 0, T ; C 0 (Bρ×B1 , Hd ) for → ∞. Hence, g˜d,ρ must coincide with g˜d,R Bd×Bd in L1 0, T ; C 0 (Bρ×B1 , Hd ) . ρ 1  d Now a function g˜d : [0, T ) −→ C 0 Hd × B1 , Hd ) can be defined for Lebesgue-almost every t ∈ [0, T ) uniquely by means of the condition

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 d  d g˜d (t) Bd×Bd := g˜d,R (t) Bd×Bd ∈ C 0 Bρ × B1 , Hd for every ρ, R ∈ N (ρ ≤ R). ρ

1

ρ

1

  d d In particular, the restriction g˜d Bd×Bd belongs to L1 0, T ; C 0 (Bρ × B1 , Hd ) for each radius ρ > 0. ρ 1 Finally, this function g˜d specifying “all” the weak limits so far gives us the opportunity to bridge the gap   between weak and norm convergence by means of a single sequence fd, ∈N not depending on the radius R > 0. Indeed, for each index ∈ N, it is again Mazur’s lemma which provides a finite convex combination   d  fd, : [0, T ) −→ C 0 Hd × B1 , Hd of f˜d,k , f˜d,k(+1) , f˜d,k(+2) , . . . satisfying       g˜d − fd, Bd×Bd  

1

Def. L1 (0,T ;C 0 )

=

       g˜d (t) − fd, (t) Bd×Bd  

1

sup

dL1 t <

1 

[0,T )

(in particular, we are now free to consider ρ = ρ( ) := respectively). Then we obtain for any ρ > 0 and every index ≥ ρ     gd Bd×Bd − fd, Bd×Bd  ˜ ρ

Def.

=

1

ρ

1

d

d

L1 (0,T ; C 0 (Bρ×B1 ))

     gd (t) Bd×Bd − fd, (t) Bd×Bd  ˜ ρ

1

ρ

1

dL1 t

sup

[0,T )

     ≤ gd (t) Bd×Bd − fd, (t) Bd×Bd  ˜ 



1

1

sup

dL1 t ≤

1

→∞

−→ 0.

2

[0,T )

Proof of Corollary 4.21. It is essentially based on the notion to combine the preceding proof of Lemma 4.20 with Cantor’s diagonal method (w.r.t. d ∈ N). First, we fix d ∈ N arbitrarily. Lemma 4.20 provides both a sequence k  ∞ of indices and a function   d g˜d : [0, T ) −→ C 0 Hd ×B1 , Hd such that for every radius ρ > 0, →∞ f˜d,k Bd ×Bd −→ g˜d Bd ×Bd ρ

1

ρ

1

  d d weakly in L1 0, T ; C 0 (Bρ ×B1 , Hd ) .

Next, we apply Cantor’s diagonal method with respect to d ∈ N and obtain a sequence of indices (again denoted by) k  ∞ such that for both every ρ > 0 and each d ∈ N, the following convergence for → ∞ holds f˜d,k Bd ×Bd −→ g˜d Bd ×Bd ρ

1

ρ

1

  d d weakly in L1 0, T ; C 0 (Bρ ×B1 , Hd ) .

Now (k )∈N and g˜d , d ∈ N, are specified as required in condition (1.) of Corollary 4.21.   It remains to construct the sequence (λ )∈N of tuples λ = λ,1 , . . . , λ,N ∈ [0, 1]N of various lengths N  N ∈ N, but with fixed total weight λ,n = 1. n=1

For each d ∈ N fixed arbitrarily, we have already discussed in the proof of Lemma 4.20 that for every ∈ N, Mazur’s lemma provides a (finite) convex combination fd, :=



Nd,

n=1

satisfying

  d λd,,n · f˜d,k(+n) : [0, T ) −→ C 0 Hd × B1 , Hd

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      g˜d − fd, Bd×Bd  

1

39

      =  g˜d (t) − fd, (t) Bd×Bd 

Def.

L1 (0,T ;C 0 )



1

sup

dL1 t <

1 .

[0,T )

In particular, this criterion implies for every ρ > 0 and ≥ ρ     gd Bd×Bd − fd, Bd×Bd  ˜ ρ

1

ρ

1

d

d

L1 (0,T ; C 0 (Bρ×B1 ))

≤

−→ 0

1 

( → ∞).

This can be interpreted as the basis of an inductive definition of (λd, )∈N w.r.t. d ∈ N.   Now we are going to use this notion for constructing a sequence λd+1, ∈N related to the subsequent   dimension d + 1 by means of λd, ∈N . Consider the auxiliary sequence of convex combinations 

Nd,

  d+1 λd,,n · f˜d+1,k(+n) : [0, T ) −→ C 0 Hd+1 × B1 , Hd+1

( ∈ N).

n=1

  d+1 d+1 For every positive radius ρ, it converges to g˜d+1 Bd+1 ×Bd+1 weakly in L1 0, T ; C 0 (Bρ × B1 , Hd+1 ) since ρ 1   so does f˜d+1,k d+1 d+1 . 

Bρ

×B1

∈N

d+1

As described before, we can always find a sequence of convex combinations whose restrictions to Bρ ×   d+1 d+1 d+1 B1 converge even strongly L1 0, T ; C 0 (Bρ × B1 , Hd+1 ) for every radius ρ > 0. Each of the latter convex combinations is now reformulated as a convex combination of f˜d+1,k(+1) , f˜d+1,k(+2) , f˜d+1,k(+3) , . . . .   The respective scalar coefficients lead to the next tuple λd+1, = λd+1,,1 , . . . , λd+1,,Nd+1, ∈ [0, 1]Nd+1, for each ∈ N. This inductive approach has an essential advantage: The strong convergence in L1 is “preserved” for the dimension index d in the sense that for every ρ > 0, 

Nd+1,

λd+1,,n · f˜d,k(+n)

n=1

−→ g˜d

→∞ d d Bρ×B1

d

d

Bρ×B1

  d d strongly in L1 0, T ; C 0 (Bρ ×B1 , Hd )



λd+1,,n · f˜d,k(+n) ∈ co fd, , fd,+1 , fd,+2 , fd,+3 . . . . n=1   This completes the inductive step for constructing the sequence λd+1, ∈N of tuples. Finally, Cantor’s diagonal method is again the key tool for providing a sequence (λ )∈N of tuples   λ = λ,1 , . . . , λ,N ∈ [0, 1]N not depending on the dimension index d ∈ N, but satisfying for every ρ > 0 and d ∈ N 

Nd+1,

due to

N 

λ,n · f˜d,k(+n)

n=1

  d d strongly in L1 0, T ; C 0 (Bρ ×B1 , Hd ) .

d

d

Bρ×B1

−→ g˜d

d

d

Bρ×B1

( → ∞)

2

4.3.3. The way to a “joint” limit function g : [0, T ) −→ LIPco (H, H) (i.e., independent of dimension d) We consider Hd ⊂ H (d ∈ N), (Kk )k∈N and (fk )k∈N , (f˜d,k )d,k∈N as constructed at the beginning of subsections 4.2 and 4.3.2 respectively. Corollary 4.21 provides the sequence k  ∞ of indices, the sequence   (λ )∈N of finite scalar tuples λ = λ,1 , . . . , λ,N ∈ [0, 1]N as well as a sequence (˜ gd )d∈N of functions   d 0 g˜d : [0, T ) −→ C Hd ×B1 , Hd with the properties (1.)–(4.) specified there. In a word, the initial sequence (fk )k∈N consists of functions fk : [0, T ) −→ LIPco (H, H) related to the approximate solutions Kk : [0, T ] −→ K(H) as formulated in Lemma 4.14. Hence, fk does not depend on

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40

the dimension index d. Our goal is to find a function g : [0, T ) −→ LIPco (H, H) which can be regarded as an accumulation point of (fk )k∈N in a sense appropriate for reachable sets. In particular, the “detour” via restrictions to the finite-dimensional subspaces Hd ⊂ H, d ∈ N, has led to limit functions g˜d : [0, T ) −→    d d C 0 Hd ×B1 , Hd which prove to induce functions g˜d (·) · , B1 : [0, T ) −→ LIPco (Hd , Hd ), but we want to avoid their dependence on d. The steps are first summarized in the following lemmata and, their proofs are postponed to the end of this subsection.  d Lemma 4.22. For every d ∈ N and Lebesgue-almost every t ∈ [0, T ), the set-valued map g˜d (t) · , B1 : Hd ; Hd belongs to LIPco (Hd , Hd ) (in the sense of Definition 2.6) with  d Lip˜ gd (t) · , B1 ≤ α ,

   g˜d (t) · , Bd1 

∞

 ≤ β.

Lemma 4.23. There exists a function g : [0, T ) −→ LIPco (H, H) with the following properties: (a) Lipg(t) ≤ α  and g(t) ∞ ≤ b holds for Lebesgue-almost every t ∈ [0, T ),   (b) for every K ∈ K(H), g(·) K : [0, T ) −→ LIPco (K, H), dlco,∞ is Lebesgue measurable, (c) for every d ∈ N and Lebesgue-almost every t ∈ [0, T ),  d Πd ◦ g(t) H = g˜d · , B1 : Hd ; Hd , d

(d) for every d ∈ N and ρ > 0,  lim

→∞

N     sup dlPH Πd λ,n · fk(+n) (t)(x) , Πd g(t)(x) dL1 t = 0, d

[0,T )

x ∈ Bρ

n=1

  (e) there exist a sequence j  ∞ of indices and a Lebesgue measurable set J ⊂ [0, T ) with L1 [0, T ) \J = 0 satisfying for any t ∈ J, every compact subset K ⊂ H and all d ∈ N, ⎧ (j )   N  ⎪  ⎪ ⎪ ⎪ λj ,n · fk(j +n) (t)(x) , Πd g(t)(x) = 0, lim sup dlPH Πd ⎪ ⎨ j→∞ x∈K ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim

n=1

sup

j→∞ x ∈ K

dlco

(j )  N

λj ,n · fk(j +n) (t)(x),



g(t)(x)

= 0.

n=1

Now we give the missing proofs of these lemmata.   d Proof of Lemma 4.22. For every d, k ∈ N, the function f˜d,k : [0, T ) −→ W 1,∞ Hd × B1 , Hd is defined as a pointwise parametrization of Πd fk H : [0, T ) −→ LIPco (Hd , Hd ) by means of Lemma 4.19. Hence, we d obtain for every t ∈ [0, T ) and x ∈ Hd  d f˜d,k (t) x, B1 = Πd fk (t)(x) ∈ Clbd,co (Hd ) = Kco (Hd ) since fk (t) ∈ LIPco (H, H) implies Πd fk (t) ∈ LIPco (H, Hd ) and due to Heine–Borel’s compactness theorem. Furthermore, assumption (ii) of Theorem 3.3 (about global bounds of the set-valued maps H ; H in the values of F) guarantees for any x, x1 , x2 ∈ Hd

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      d d dlPH f˜d,k (t) x1 , B1 , f˜d,k (t) x2 , B1 = dlPH Πd fk (t)(x1 ), Πd fk (t)(x2 )   ≤ dlPH fk (t)(x1 ), fk (t)(x2 )   ≤α  x1 − x2 ,      f˜d,k (t) x, Bd1  ≤ fk (t)(x) ≤ β . ∞ ∞ For each d ∈ N and ∈ N, Corollary 4.21 introduces the convex combination N 

fd, :=

  d λ,n · f˜d,k(+n) : [0, T ) −→ C 0 Hd × B1 , Hd

n=1

 d d whose restriction to Bρ × B1 (with arbitrary radius ρ > 0) converges to g˜d Bd ×Bd strongly in L1 0, T ; ρ 1  d  d C 0 Bρ ×B1 , Hd for → ∞.  d First, we conclude that for every ∈ N and t ∈ [0, T ), the set-valued map fd, (t) ·, B : Hd ; Hd is also 1

bounded and Lipschitz continuous with non-empty compact convex values and  d Lipfd, (t) · , B1 ≤ α ,

   fd, (t) · , Bd1 

∞

 ≤ β.

Next, ρ > 0 is fixed arbitrarily. As a consequence of the strong convergence in L1 , a subsequence of    d  d fd, (t) Bd ×Bd ∈N converges to g˜d (t) Bd ×Bd ∈ C 0 Bρ × B1 , Hd uniformly for Lebesgue-almost every t ∈ ρ

1

ρ

1

[0, T ). In regard to the set-valued maps, we obtain for those (Lebesgue-almost all) t ∈ [0, T ) lim inf →∞

    d d sup dlPH fd, (t) x, B1 , g˜d (t) x, B1 d

x ∈ Bρ

≤ lim inf →∞

  fd, (t)(x, p) − g˜d (t)(x, p) = 0 .

sup d

d

x ∈ Bρ , p ∈ B1

  d d As each of the sets fd, (t) x, B1 ⊂ Hd is compact and convex so is g˜d (t) x, B1 . Moreover, we conclude d

for any x1 , x2 , x ∈ Bρ

    d d dlPH g˜d (t) x1 , B1 , g˜d (t) x2 , B1

    d d ≤ lim sup dlPH fd, (t) x1 , B1 , fd, (t) x2 , B1 →∞

   g˜d (t) x, Bd1  ∞

  ≤α  x1 − x2 ,   d  ≤ lim sup fd, (t) x, B1 ∞ ≤ β . →∞

Obviously, these bounds do not depend on the radius ρ > 0 and thus, for Lebesgue-almost every t ∈ [0, T ),  d the set-valued map g˜d (t) · , B1 : Hd ; Hd belongs to LIPco (Hd , Hd ) with the global bounds as claimed. 2   The proof of Lemma 4.23 about g : [0, T ) −→ LIPco (H, H) is based on the results about Clbd,co (H), dlco presented in § 4.1. In particular, we use the close relation between the convergence w.r.t. dlco and the convergence of all projections on Hd w.r.t. dlPH . The preceding statements concern sequences of bounded closed and convex subsets of H and are now applied to the corresponding set-valued maps in a pointwise way. In this context, special aspects of uniform convergence come into play.

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Lemma 4.24. In addition to the assumptions of Lemma 4.1, let (Fm )m∈N be any sequence in LIPco (H, H)  Suppose that for every dimension index d ∈ N, there exists with supm LipFm ≤ α  and supm Fm ∞ ≤ β. a set-valued map Gd ∈ LIPco (Hd , Hd ) fulfilling for every radius ρ > 0 lim

m→∞

  sup dlPH Πd Fm (x), Gd (x) = 0 . d

x ∈ Bρ

Then there exists a set-valued map G ∈ LIPco (H, H) satisfying for each d ∈ N, ρ > 0 and x ˜∈H ⎧ ⎨

  x), G(˜ x) −→ 0 dlco Fm (˜   dl F (x), G(x) −→ 0 sup co m ⎩ d

(m → ∞).

x ∈ Bρ

 In particular, it has the bounds LipG ≤ α  and G ∞ ≤ β.  ˜ ∈ Proof of Lemma 4.24. For each vector x ∈ d∈N Hd , preceding Lemma 4.7 provides a unique set G(x)   ˜ Clbd,co (H) satisfying lim dlco Fm (x), G(x) = 0 since the inclusion Hd ⊂ Hd+1 holds for any d ∈ N. In m→∞ ˜ particular, we obtain Πd G(x) = Gd (x) whenever x ∈ Hd . ˜ :  This pointwise construction leads to a set-valued map G d∈N Hd ; H. It belongs to   ˜ LIPco d∈N Hd , H . Indeed, each value of G is a non-empty bounded closed and convex subset of H  ˜ Fm (x) ≤ β for each vector x in its domain. by construction. Lemma 4.2 implies G(x) ∞ ≤ lim inf m→∞

∞

˜ assume that there are x, x In regard to the claimed α -Lipschitz continuity of G, ˜ ∈ Hd (for some   ˜ ˜ x) > α G(˜  x − x d ∈ N) with dlPH G(x), ˜ . Representing the Pompeiu–Hausdorff distance between convex sets in terms of support functions (e.g., [13, Corollary 3.2.8]) leads to some p ∈ B ⊂ H with ˜ ˜ x), p) > α ˜ being defined as the pointwise limit of σ(G(x),  x − x p) − σ(G(˜ ˜ . As a consequence of G (Fm )m∈N w.r.t. dlco , there is a sufficiently large index m ∈ N with σ(Fm (x), p) − σ(Fm (˜ x), p) > α  x − x ˜ . On the other hand, however, we conclude from p ≤ 1   σ(Fm (x), p) − σ(Fm (˜ x), p) ≤ dlPH Fm (x), Fm (˜ x) ≤ Lip Fm · x − x ˜ ≤ α  x − x ˜   ˜ :  ˜ ∈ – a contradiction. Hence, G Hd −→ Clbd,co (H), dlPH is α -Lipschitz continuous and so, G d∈N   LIPco d∈N Hd , H . ˜: The set-valued map G d∈N Hd ; H has a unique continuous extension G : H ; H with non-empty ˜ as an α bounded convex values. Indeed, we interpret G -Lipschitz continuous (single-valued) function    Clbd,co (H), dlPH and apply the standard result in [102, Theorem 10.9.1] (or, e.g., [31, d∈N Hd −→  § 65]) to H = d∈N Hd since [13, Theorem 3.2.4] and the separation theorem for convex sets imply the   completeness of Clbd,co (H), dlPH .  As immediate consequences, we have G ∈ LIPco (H, H), LipG ≤ α , G ∞ ≤ β.    Next, we show dlco Fm (˜ x), G(˜ x) −→ 0 for m → ∞ and each vector x ˜ ∈ H \ d∈N Hd . Fix ε > 0 arbitrarily. There exists a vector x ∈ Hd with d ∈ N sufficiently large and x − x ˜ < 4 (1ε+ α)  . For each vector p ∈ B ⊂ H, we conclude from the α -Lipschitz continuity of both Fm (m ∈ N) and G w.r.t. dlPH σ(Fm (˜ x), p) − σ(G(˜ x), p)   ≤ σ(Fm (x), p) − σ(G(x), p) + p · dlPH Fm (˜ x), Fm (x)   + p · dlPH G(˜ x), G(x)   ˜ − x  x ≤ σ(Fm (x), p) − σ(G(x), p) + 2 p · α

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 ≤ σ(Fm (x), p) − σ(G(x), p) + 2 α and so, lim sup σ(Fm (˜ x), p) − σ(G(˜ x), p) ≤ m→∞

43

ε 4 (1 + α )

lim σ(Fm (x), p) − σ(G(x), p) + ε = ε since x ∈ Hd .

m→∞

In other words, σ(Fm (˜ x), p) −→ σ(G(˜ x), p) for m → ∞ and every p ∈ H. This is equivalent to   dlco Fm (˜ x), G(˜ x) −→ 0 according to Lemma 4.1. Finally, it remains to prove for every d ∈ N and ρ > 0   sup dlco Fm (x), G(x) −→ 0. d

x ∈ Bρ

Choose ε > 0 arbitrarily and select an index n0 ∈ N with

∞ n=n0

2−n < 2ε . Definition 2.16 of dlco leads to

n0    |σ(M1 , pn ) − σ(M2 , pn )| dlco M1 , M2 − 2−n 1 + |σ(M1 , pn ) − σ(M2 , pn )| n=1

< ε 2

 for any sets M1 , M2 ∈ Clbd,co (H). Due to H = d∈N Hd , there is a sufficiently large dimension index d˜ ≥ d 

 d Def. Def. d˜ ε satisfying dist p1 , p2 , . . . , pn0 , Hd˜ ≤ 4 (1+  . We obtain for all x ∈ Bρ = Bρ ∩ Hd ⊂ Bρ ∩ Hd˜ = Bρ β) 

and n ∈ 1, . . . , n0     σ Fm (x), pn − σ G(x), pn       ≤ σ Fm (x), Πd˜ pn − σ G(x), Πd˜ pn + 2 β pn − Πd˜ pn      ε ≤ σ Fm (x), Π∗d˜ pn − σ G(x), Π∗d˜ pn + 2 β  4 (1 + β)     ε ≤ σ Πd˜ Fm (x), pn − σ Πd˜ G(x), pn + 2

   ε ≤ max 1, pn · dlPH Πd˜ Fm (x), Πd˜ G(x) + . 2 Πd˜ G(x) = Gd˜(x) and the assumption about the convergence of the respective projections on Hd˜ (uniform on any closed ball in Hd˜) lead to     ε σ Fm (x), pn − σ G(x), pn ≤ 0 + 2 m→∞ x∈Bd 1 ≤ n ≤ n0 ρ σ(Fm (x), pn ) − σ(G(x), pn ) ε ≤ sup lim sup sup 2 m→∞ x∈Bd 1 ≤ n ≤ n0 1 + σ(Fm (x), pn ) − σ(G(x), pn ) lim sup sup

sup

ρ

  ε ε lim sup sup dlco Fm (x), G(x) < + = ε 2 2 m→∞ x∈Bd ρ

   with ε > 0 having been fixed arbitrarily, i.e., dlco Fm (x), G(x) m∈N converges to 0 uniformly w.r.t. d

x ∈ Bρ . 2 Corollary 4.25. Under the assumptions of Lemma 4.24, the convergence   sup dlco Fm (x), G(x) −→ 0

x∈K

(m → ∞)

holds for every compact subset K ⊂ H, i.e., it is locally uniform w.r.t. x ∈ H.

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Proof of Corollary 4.25. Select K ∈ K(H) and ε > 0 arbitrarily. As K is compact in H, we can always find sufficiently large indices n0 , d ∈ N and a radius ρ > 0 successively such that ∞ 

2−n <

n=n0

ε min{ε, 1} d , K ⊂ Bρ + Br with r := . 4 8 (1 + α ) (1 + max{ p1 , . . . , pn0 }) d

In particular, for each vector x ∈ K, there is some π(x) ∈ Bρ with π(x) − x < r. The auxiliary func  σ is 1-Lipschitz continuous and so, the composition Clbd,co (H), dlPH × tion [0, ∞) −→ [0, 1), σ −→ 1+σ   Clbd,co (H), dlPH −→ R defined as σ(C1 , p) − σ(C2 , p) (C1 , C2 ) −  → 1 + σ(C1 , p) − σ(C2 , p) also satisfies the Lipschitz condition with the constant ≤ 2 p . Now we conclude from the α -Lipschitz continuity of all Fm (m ∈ N) and G   sup dlco Fm (x), G(x)

x∈K

n0 

< sup

x∈K n=1 n0 

≤ sup

x∈K n=1

≤ sup

x∈K

ε |σ(Fm (x), pn ) − σ(G(x), pn )| + 1 + |σ(Fm (x), pn ) − σ(G(x), pn )| 4

2−n 2−n

 ε  |σ(F (π(x)), p ) − σ(G(π(x)), p )| m n n + 2 pn α r + 1 + |σ(Fm (π(x)), pn ) − σ(G(π(x)), pn )| 4

  dlco Fm (π(x)), G(π(x)) + 2 ·

max

1 ≤ n ≤ n0

pn · α r +

ε 4

  ε ε + . x), G(˜ x) + ≤ sup dlco Fm (˜ 4 4 d x ˜∈B ρ

Finally, the second convergence property formulated in Lemma 4.24 reveals     ε lim sup sup dlco Fm (x), G(x) ≤ lim sup sup dlco Fm (˜ x), G(˜ x) + 2 d m→∞ x∈K m→∞ x ˜∈B ρ

< ε,    i.e., dlco Fm (x), G(x) m∈N tends to 0 uniformly w.r.t. x ∈ K. 2 We complete this subsection 4.3.3 by proving its main statement: Proof of Lemma 4.23. As a consequence of the construction and Lemma 4.22, we have a function g˜d :   d [0, T ) −→ C 0 Hd ×B1 , Hd for each dimension index d ∈ N satisfying • for Lebesgue-almost every t ∈ [0, T ),     g˜d (t) · , Bd1  ≤ β, ∞

• for every radius ρ > 0, for → ∞.

N  n=1

λ,n · f˜d,k(+n)

 d g˜d (t) · , B1 ∈ LIPco (Hd , Hd ),

d d Bρ×B1

−→ g˜d

d d Bρ×B1

 d Lip˜ gd (t) · , B1

≤

α ,

  d d strongly in L1 0, T ; C 0 (Bρ ×B1 , Hd )

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We have already remarked in the proof of Lemma 4.22 that for every d, k ∈ N, the function f˜d,k : [0, T ) −→   d W 1,∞ Hd × B1 , Hd is defined as a pointwise parametrization of Πd fk H : [0, T ) −→ LIPco (Hd , Hd ) by d means of Lemma 4.19. Hence, we have for every t ∈ [0, T ) and x ∈ Hd  d f˜d,k (t) x, B1 = Πd fk (t)(x) ∈ Clbd,co (Hd ) = Kco (Hd ) . In particular, the last convergence mentioned above implies 

N      d dL1 t = 0 sup dlPH Πd λ,n · fk(+n) (t)(x) , g˜d (t) x, B1

lim

→∞

d

[0,T )

x ∈ Bρ

n=1

for every dimension index d ∈ N and any radius ρ > 0. After considering all radii ρ ∈ N and dimension indices d ∈ N successively, Cantor’s diagonal method provides both a sequence j  ∞ of indices and a Lebesgue measurable set J ⊂ [0, T ) with   L1 [0, T ) \ J = 0 such that for all t ∈ J, ρ > 0 and d ∈ N,  lim

j→∞

sup

dlPH Πd

d

x ∈ Bρ

(j )  N

   d λj ,n · fk(j +n) (t)(x) , g˜d (t) x, B1 = 0.

n=1

For each time instant t ∈ J ⊂ [0, T ), Lemma 4.24 and Corollary 4.25 guarantee the unique set-valued map g(t) ∈ LIPco (H, H) fulfilling   • Lipg(t) ≤ α , g(t)∞ ≤ β and (j )   N λj ,n · fk(j +n) (t)(x) converges to g(t)(x) ∈ Clbd,co (H) w.r.t. dlco locally uniformly w.r.t. • j∈N

n=1

x ∈ H.

Having defined g(t) ∈ LIPco (H, H) for t ∈ [0, T ) \ J arbitrarily, we obtain a function g : [0, T ) −→ LIPco (H, H) satisfying the claimed conditions (a), (b). Lemma 4.7, in particular, implies Πd g(t)(x) =  d g˜d (t) x, B1 for every t ∈ J and x ∈ H. Hence, conditions (c), (d) on g are also fulfilled. Finally, we conclude the remaining condition (e) from the last limit mentioned above and the α -Lipschitz continuity of all fk (t), g(t) ∈ LIPco (H, H) – by essentially the same arguments for approximating K ∈ K(H) as in the proof of Corollary 4.25. 2 4.3.4. Reachable sets of the limit function g : [0, T ) −→ LIPco (H, H) Now we compare the reachable sets of the non-autonomous evolution inclusions 

x ∈ A x + fk (t)(x) in [0, T ) x(0) ∈ M0



x ∈ A x + g(t)(x) in [0, T ) x(0) ∈ M0

at time instant t ∈ [0, T ) abbreviated as ϑfk (t, M0 ) and ϑg (t, M0 ) ⊂ H respectively. In a word, we adapt the approach to Proposition 2.14 about converging reachable sets presented in § 4.2 because the pointwise convergence of the set-valued coefficient maps w.r.t. dlco is not really obvious here. Throughout this subsection 4.3.4, we still suppose the hypotheses and notations formulated in Theorem 3.3 and §§ 4.2, 4.3.2. For each index ∈ N, the function fk : [0, T ) −→ LIPco (H, H) is piecewise constant “to the right” by construction and so, the corresponding set-valued map F˜ : [0, T ) × H ; H satisfies the conditions (i)–(iv)

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 In particular, its reachable sets prove to be compact due to of Proposition 2.9 with λ(·) := α , k(·) := β. Corollary 2.10. In Lemma 4.23 (b), the limit function g : [0, T ) −→ LIPco (H, H) has proved to be Lebesgue measurable locally w.r.t. dlco for the closed convex value sets. According to the next lemma, g is sufficiently regular w.r.t. time t for ensuring the compactness of all related reachable sets. In particular, g induces the Carathéodory ˜ : [0, T ) × H ; H, (t, x) → g(t)(x). set-valued map G From our point of view, however, it is not immediately obvious how the weak convergence of (fk )∈N ˜ (i.e., as to g in a Lebesgue–Bochner space implies the appropriate form of convergence for (F˜ )∈N to G required in Proposition 2.14). Alternatively, Lemma 4.28 below verifies the counterpart of Lemma 4.9 for ˜ directly and, our special example. Then all further arguments of proof in § 4.2 can be applied to (F˜ )∈N , G we obtain the main result of this subsection: Corollary 4.26. The assumptions of Theorem 3.3 and § 4.3.2 guarantee for every initial set M0 ∈ K(H) and t ∈ [0, T )   dlPH ϑF˜ (t, M0 ), ϑG˜ (t, M0 ) −→ 0

( → ∞).

The auxiliary steps are formulated and then proved: Lemma 4.27. The function g : [0, T ) −→ LIPco (H, H) induces the set-valued map ˜ : [0, T ) × H ; H, G

(t, x) → g(t)(x)

with the following properties: (i) (ii) (iii) (iv)

˜ is a non-empty closed and convex subset of H, each value of G ˜ x) is contained in B  ⊂ H, for every t ∈ [0, T ) and x ∈ H, the set G(t, β ˜ x) : [0, T ) ; H is (strongly) Lebesgue measurable, for every x ∈ H, the set-valued map G(·, ˜ ·) : H ; H is α for all t ∈ [0, T ), G(t, -Lipschitz continuous w.r.t. dlPH .

Hence, the reachable set ϑG˜ (t, M0 ) is compact in H at each time instant t ∈ [0, T ) and for every initial set M0 ∈ K(H). ˜ : [0, T ) × H ; H, (t, x) → Lemma 4.28. Consider F˜ : [0, T ) × H ; H, (t, x) → fk (t)(x) ( ∈ N) and G 0 g(t)(x) as constructed in § 4.3 so far. Then for any d ∈ N, u ∈ C ([0, T ], Hd ) and Lebesgue measurable set J ⊂ [0, T ] given,  dlPH

  Πd F˜ s, u(s) ds,

J



   ˜ s, u(s) ds −→ 0 Πd G

( → ∞) .

J

Proof of Lemma 4.27. The properties (i), (ii) and (iv) result from the construction of g (in Lemma 4.23) in a rather obvious way. ˜ x) : [0, T ) ; H is Feature (iii) remains to be verified, i.e., for every x ∈ H, the set-valued map G(·, strongly Lebesgue measurable. The locally uniform convergence w.r.t. dlco formulated in Lemma 4.23 (e) implies for each vector x ∈ H and Lebesgue-almost every t ∈ [0, T )

lim

j→∞

dlco

(j )  N

n=1

 λj ,n · fk(j +n) (t)(x), g(t)(x)

= 0,

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or, equivalently due to the uniform bounds of fk (k ∈ N), g and Lemma 4.1,

lim

j→∞

σ

(j )  N

 λj ,n · fk(j +n) (t)(x), p

  = σ g(t)(x), p

n=1

for every p ∈ H. Each of the convex combinations on the left-hand side is piecewise constant “to the right” by construction and so, it is scalarly measurable – in the sense that its support function w.r.t. any fixed p ∈ H is Lebesgue measurable in [0, T ) (see, e.g., [47, Definition II.2.31] or [23, § V.1]). Hence, their pointwise limit Lebesgue-almost everywhere, i.e., [0, T ) ; H, t → g(t)(x), is also scalarly measurable. Its values are non-empty bounded (norm-) closed and convex subsets of the Hilbert space H and hence, they are weakly compact due to Mazur’s lemma. Finally, [47, Proposition II.2.39] implies the Lebesgue measurability of g(·)(x) : H ; H (in the sense of general Lemma 4.29 below) since H is assumed to be separable. (Alternatively, this conclusion can be drawn by means of Pettis’ measurability theorem [102, § V.4 Theorem] applied to all the single-valued functions of the so-called Castaing representation in Lemma 4.29 (e).) ˜ : [0, T ) × H ; H satisfies all assumptions of Proposition 2.9 and thus, Corollary 2.10 In a word, G guarantees the compactness of every reachable set ϑG˜ (t, M0 ) ⊂ H for t ∈ [0, T ) and M0 ∈ K(H). 2 Lemma 4.29 (Measurability: equivalent criteria [23, Th.III.30] [10, Th.8.1.4]). Let (Ω, A, μ) be a measurable space with μ ≥ 0 σ-finite and A complete. Suppose X to be a complete separable metric space and F : Ω ; X to have non-empty closed values. Then the following properties are equivalent:

Def.  F −1 (M ) = ω ∈ Ω F (ω) ∩ M = ∅ ∈ A for every open set M ⊂ X, F −1 (M ) ∈ A for every closed set M ⊂ X, F −1 (M ) ∈ A for every Borel set M ⊂ X,   dist x, F (·) : Ω −→ R is μ measurable for every x ∈ X, there exists a sequence (fk )k∈N of μ measurable selections fk : Ω −→ X of F with F (ω) =

 fk (ω) k ∈ N for every ω ∈ Ω, (f) the graph of F belongs A ⊗ B(X).

(a) (b) (c) (d) (e)

  Proof of Lemma 4.28. Choose the dimension indices d˜ ≤ d and the function u ∈ C 0 [0, T ], Hd˜ arbitrarily. Corollary 4.21 (1.) states for every radius ρ > 0 →∞ f˜d,k Bd ×Bd −→ g˜d Bd ×Bd ρ

1

ρ

1

  d  d weakly in L1 0, T ; C 0 Bρ ×B1 , Hd

  d with f˜d,k : [0, T ) −→ W 1,∞ Hd × B1 , Hd denoting the (pointwise) parametrization of Πd ◦ fk (t) H : d   d 0 Hd ; Hd by means of Lemma 4.19. Furthermore, Lemma 4.23 (c) confirms that g ˜ (t) ∈ C H × B , H d d d 1 is a parametrization of Πd ◦ g(t) H : Hd ; Hd for Lebesgue-almost every t ∈ [0, T ). d Choose ρ > 0 sufficiently large with u(t) ≤ ρ − 1 for all t ∈ [0, T ]. For each “control function”  d v ∈ L1 0, T ; B1 and Lebesgue measurable subset J ⊂ [0, T ), the functional   d  d L1 0, T ; C 0 Bρ ×B1 , Hd −→ R,

 h −→

  h(s) u(s), v(s) ds

J

 d  d is linear and continuous whenever C 0 Bρ ×B1 , Hd is supplied with the supremum norm. Hence, we conclude the two convergences for → ∞

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  ⎧    d d ⎪ ˜ ⎪ g˜d (s) u(s), B1 ds −→ 0, fd,k (s) u(s), B1 ds, dist ⎪ ⎪ ⎨ J J      d d ⎪ ˜ ⎪ g˜d (s) u(s), B1 ds, fd,k (s) u(s), B1 ds −→ 0, ⎪ dist ⎪ ⎩ J

J

and this is equivalent to the claimed convergence for → ∞  dlPH J

  Πd F˜ s, u(s) ds,



   ˜ s, u(s) ds −→ 0 . 2 Πd G

J

˜ = g(·)(·) : [0, T ) × H ; H and its 4.3.5. The convergence of (Kk (·))∈N to the reachable sets of G consequence for state constraints Now we focus on the sequence (Kk )k∈N of curves [0, T ] −→ K(H) constructed as part of approximate solutions in § 4.3.1. At the beginning of § 4.3.2, the key properties of Kk (·) and fk : [0, T ) −→ LIPco (H, H) were summarized as follows: (a) Kk (0) = M0 ,   (b) dist Kk (t), V ≤ k1 for all t ∈ [0, T ],   (c) for all t ∈ [0, T ), the map fk (t) ∈ LIPco (H, H) belongs to F B1/k (Kk (t)) and satisfies lim h1 · h↓0    dlPH Kk (t + h), ϑfk (t) h, Kk (t) = 0. (d) fk is piecewise constant “to the right”, i.e.: For each t ∈ [0, T ), there exists some δ > 0 such that fk [t, t+δ) is constant. Earlier versions of the viability theorem are mostly restricted to vector-valued states (see, e.g., [3,16,45,89]) or to set-valued states in K(Rn ) (see, e.g., [64],[67, § 5.2]). At this step of the proof, it usually lays the basis for extracting a uniformly converging subsequence of (Kk )k∈N by means of the Arzelà–Ascoli theorem (see, e.g., [42]). This well-known characterization of compactness, however, requires that the pointwise union of all values are relatively compact in the respective state space, i.e., in our context, we need for each t ∈ [0, T ] that

   Kk (t) k ∈ N is relatively compact in K(H), dlPH . Now this property cannot be concluded just from uniform boundedness (as usual) since H is a real Hilbert space of (possibly) infinite dimension. Hence, we use the results about converging reachable sets in § 4.3.4 instead. They are all based on the appropriate form of convergence for (fk )∈N to g and the general conclusions in § 4.2, but not on the   Arzelà–Ascoli theorem for K(H), dlPH . As a consequence of Proposition 2.3 and Corollary 2.10, the infinitesimal condition (c) implies directly Kk (t) = ϑfk (·) (t, M0 ) = ϑfk (·) (t, M0 ) ∈ K(H) for every t ∈ [0, T ] and k ∈ N. We conclude immediately from Corollary 4.26     dlPH Kk (t), ϑG˜ (t, M0 ) = dlPH ϑfk (·) (t, M0 ), ϑG˜ (t, M0 )   →∞ = dlPH ϑF˜ (t, M0 ), ϑG˜ (t, M0 ) −→ 0. In Theorem 3.3, the set of constraints V ⊂ K(H) is assumed to be closed w.r.t. dlPH . Hence, condition (b) ensures for every t ∈ [0, T ] K(t) := ϑG˜ (t, M0 ) ∈ V.

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˚ 4.3.6. K := ϑG˜ ( · , M0 ) “solves” the set differential inclusion K(t) ∈ F(K(t))   ˜ ·) ∈ F K(t) for Lebesgue-almost In regard to main Theorem 3.3, the essential aspect still missing is G(t, every t ∈ [0, T ] since then K(·) satisfies the “integral” notion of a solution mentioned in Proposition 2.3 (1.). Lemma 4.30. Under the assumptions of Theorem 3.3 and § 4.3.2, the curve K : [0, T ] −→ K(H),

t −→ ϑG˜ (t, M0 )

˜ : [0, T ) × H ; H satisfy: and the convex-valued Carathéodory map G (a) K is sequentially continuous w.r.t. ΔCl (in the sense of Definition 2.2), ˜ ·) = g(t) ∈ LIPco (H, H) belongs to (b) for Lebesgue-almost every t ∈ [0, T ), the set-valued map G(t,   F K(t) .   ˚ In regard to the “full” Definition 2.15 of a solution to the set differential inclusion K(t) ∈ F K(t) , we still need slightly stronger assumptions about the range of F for applying the equivalence in Proposition 2.3:   Corollary 4.31. In addition to the assumptions of Theorem 3.3 and § 4.3.2, suppose F K(H) ⊂ LIPco,cp (H, H), i.e., all set-valued maps in the range of F have compact and convex values in H. ˚ Then, K: [0, T ] ; H, t → ϑG˜ (t, M0 ) is a compact-valued solution to the set differential inclusion K(t) ∈   F K(t) in the sense of Definition 2.15, i.e., it also fulfills the morphological criterion for Lebesgue-almost every t ∈ [0, T ) lim

h↓0

   1 · dlPH K(t + h), ϑG(t,·) h, K(t) = 0. ˜ h

Proof of Lemma 4.30. Property (a) results directly from Proposition A.1.4. It remains to show g(t) ∈   F K(t) for Lebesgue-almost every t ∈ [0, T ). By assumption, the graph of F is closed with respect to locally uniform scalar convergence – in the following sense: Choose any M ∈ K(H), G ∈ LIPco (H, H). Let (Mk )k∈N and (Gk )k∈N be arbitrary sequences in K(H) and LIPco (H, H) respectively satisfying for every compact set K0 ⊂ H ⎧ ⎨ Gk ∈ F(Mk ) for each k ∈ N,    = 0. dlPH (Mk , M ) + sup dlco Gk (x), G(x) ⎩ lim k→∞

x ∈ K0

Then, the tuple (M, G) belongs to the graph of F or, equivalently, G ∈ F(M ). The preceding condition (c) about the construction of (Kk )k∈N and (fk )k∈N states for every t ∈ [0, T ) and k ∈ N   fk (t) ∈ F B1/k (Kk (t)) . Lemma 4.23 (e) guarantees a sequence j  ∞ of indices and a Lebesgue measurable set J ⊂ [0, T ] with   L1 [0, T ] \ J = 0 satisfying for any t ∈ J and every compact subset K0 ⊂ H,

lim

sup

j→∞ x ∈ K0

dlco

(j )  N

 λj ,n · fk(j +n) (t)(x), g(t)(x)

= 0.

n=1

In particular, the coefficients λj ,n ∈ [0, 1] of the convex combinations do not depend on t, x and K0 . Hence, we obtain for every t ∈ J

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g(t) K ∈ 0

co j∈N

⊂

co j∈N

fk(j +n) (t) K n ∈ N 0  n∈N

 F Bk

1 (j +n)

 (Kk(j +n) (t)) K

0

with the closure of the pointwise convex hull in LIPco (K0 , H) referring to the metric dlco,∞ (introduced in   Definition 2.16). The convergence of Kk (t) k∈N to K(t) w.r.t. dlPH implies g(t) K

0

  co F Bε (K(t)) K

∈

0

⊂ LIPco (K0 , H)

ε>0

for every t ∈ J and K0 ∈ K(H).     ! Finally we show ε > 0 co F Bε (K(t)) ⊂ F K(t) similarly to the arguments proving [67, Lemma 5.20]: For arbitrary ρ > 0 and K0 ∈ K(H), we introduce the abbreviation         Bρ F K(t) ; K0 := G ∈ LIPco (H, H) ρ > dist G K , F K(t) K 0 0   Def. = inf sup dlco G(x), Z(x) . Z ∈ F(K(t)) x∈K0

    It is convex in the sense that for any G1 , G2 ∈ Bρ F K(t) ; K0 and λ ∈ [0, 1], the pointwise convex combination λ · G1 + (1 − λ) · G2 : H ; H,

x → λ · G1 (x) + (1 − λ) · G2 (x)

      also belongs to Bρ F K(t) ; K0 since the value F K(t) ⊂ LIPco (H, H) is assumed to be convex in Hypothesis 3.3 (i). Then for every ρ > 0 and K0 ∈ K(H), there exists ε > 0 sufficiently small with       F Bε (K(t)) ⊂ Bρ F K(t) ; K0 . Indeed, otherwise there would be two sequences (Mk )k∈N , (Gk )k∈N in K(H) and LIPco (H, H) with       dlPH Mk , K(t) ≤ k1 , Gk ∈ F(M ) \ B F K(t) ; K for every k ∈ N. For an arbitrary set W ∈ K(H), k ρ 0     we consider the sequence Gk W k∈N of functions (W, · ) −→ Clbd,co (H), dlco . Each member is α -Lipschitz continuous and, all its values are contained in Bβ ⊂ H. Hence, the sequence even belongs     to C 0 W, (Clbd,co (Bβ ), dlco ) . Clbd,co (Bβ ), dlco is compact according to Lemma 4.3 and so, we conclude   from the Arzelà–Ascoli theorem in metric spaces (see, e.g., [42]) that Gk W k∈N has a subsequence con    verging in C 0 (W, Clbd,co (H)), dlco,∞ . K(H), dlPH is known to be separable (see, e.g., [52, Exercises 2.22, 2.23] or [91, page 66]). Thus, Cantor’s diagonal method provides a subsequence of (Gk )k∈N converging to some G ∈ LIPco (H, H) uniformly on every compact subset W of H. Assuming the graph of F to be closed       in the sense specified initially, G must belong to F K(t) – but this contradicts G ∈ / Bρ F K(t) ; K0 . Hence, the last inclusion is proved indirectly for some ε > 0 sufficiently small.     Due to the above-mentioned (pointwise) convexity of Bρ F K(t) ; K0 , we obtain for every t ∈ J and K0 ∈ K(H) g(t) K

0

  co F Bε (K(t)) K

∈

0

ε>0

    Bρ F K(t) ; K0 K

⊂

0

ρ>0

For every compact set K0 ⊂ H, each time instant t ∈ J and index j ∈ N, the definition of       Bρ F K(t) ; K0 K ensures a set-valued map Zj ∈ F K(t) with 0

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sup

x ∈ K0

  dlco g(t)(x), Zj (x) ≤

1 j

51

.

  Due to the separability of K(H), dlPH , Cantor’s diagonal method leads to a sequence (again denoted by)     (Zj )j∈N in F K(t) ⊂ LIPco (H, H) satisfying lim sup dlco g(t)(x), Zj (x) = 0 for every K0 ∈ K(H). j→∞ x ∈ K0   Assuming the graph of F to be closed in the above-mentioned sense, we conclude g(t) ∈ F K(t) for every t ∈ J, i.e., for Lebesgue-almost every t ∈ [0, T ). 2 From now on, we consider the additional assumption that all values of F are compact-valued, i.e.,     F K(H) ⊂ LIPco,cp (H, H). K(H), dlPH is known to be separable (see, e.g., [52, Exercises 2.22, 2.23]

 or [91, page 66]). As a consequence, G ∈ LIPco,cp (H, H) LipG ≤ α  proves to be separable w.r.t. dl∞ . This is the relevant advantage of our additional hypothesis. In comparison with Lemma 4.23 (b), we obtain the Lebesgue measurability of g(·) even w.r.t. the supremum Pompeiu–Hausdorff metric dl∞ – in the following sense:   Lemma 4.32. Assume in addition that F K(H) ⊂ LIPco,cp (H, H), i.e., all set-valued maps in the range of F have compact and convex values in H.   Then, g: [0, T ) −→ LIPco,cp (H, H), dl∞ , t −→ g(t) is Lebesgue measurable.    for Proof of Lemma 4.32. Lemma 4.30 (b) ensures g(t) ∈ F K(t) ⊂ LIPco,cp (H, H) and Lipg(t) ≤ α Lebesgue-almost every t ∈ [0, T ). For every auxiliary map Φ ∈ LIPco,cp (H, H) and t ∈ [0, T ), we express the Pompeiu–Hausdorff distance between convex sets by means of support functions [13, Corollary 3.2.8]     ˜ x) dl∞ Φ, g(t) = sup dlPH Φ(x), G(t, x∈H

= sup

˜ x), p) . sup σ(Φ(x), p) − σ(G(t,

x ∈ H p ∈ B1

˜ x), p) In particular, for arbitrary t ∈ [0, T ), the function H × B1 −→ R, (x, p) −→ σ(Φ(x), p) − σ(G(t,   is Lipschitz continuous and, H is supposed to be separable. Hence, [0, T ) −→ R, t −→ dl∞ Φ, g(t) can be represented as the supremum of (at most) countably many Lebesgue measurable functions (due to Lemma 4.27 (iii)). Thus, it is also Lebesgue measurable for every Φ ∈ LIPco,cp (H, H). Finally, Lemma 4.29 guarantees the claimed Lebesgue measurability of g. 2 Proof of Corollary 4.31. It is based on the equivalence “(1.) ⇐⇒ (3.)” in Proposition 2.3. The missing hypothesis (iii) there can be concluded from Lusin’s theorem about the so-called almost continuity   of Lebesgue measurable functions applied to g: [0, T ) −→ LIPco,cp (H, H), dl∞ (see, e.g., [14, Theorem 7.1.13] or, supplementarily, [22, Theorem 4.1, 4.2] for set-valued maps) since continuous functions on compact domains are known to be even uniformly continuous. 2 4.4. The characterization of solutions to set differential inclusions in § 2.5 Finally, Proposition 2.17 in § 2.5 remains to be proved. We are going to conclude the more sophisticated part of its claimed equivalence from main Theorem 3.3. In more detail, the focus in this subsection is on the equivalence: Lemma 4.33. Let the set-valued map F : [0, T ] × K(H) ; LIPco (H, H) satisfy the assumptions of Proposition 2.17 and, suppose K : [0, T ] ; H to have non-empty compact values. Consider the following two statements:

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(a) K is sequentially continuous w.r.t. ΔCl (in the sense of Definition 2.2) and, for Lebesgue-almost   every t ∈ [0, T ), there exists a set-valued map G ∈ F t, K(t) ⊂ LIPco (H, H) with lim h1 · h↓0    dlPH K(t + h), ϑG h, K(t) = 0. (b) K is continuous w.r.t. the Pompeiu–Hausdorff metric dlPH and, there exists a convex-valued ˜ : [0, T ] × H ; H such that Carathéodory map G   ˜ • G(t, ·) ∈ F t, K(t) ∈ LIPco (H, H) for Lebesgue-almost every t ∈ [0, T ] and   • for every t ∈ [0, T ], the set K(t) ⊂ H coincides with the closed reachable set ϑG˜ t, K(0) ⊂ H of the ˜ x). non-autonomous evolution inclusion x ∈ A x + G(·,  Then, statement (a) always implies statement (b). Moreover, under the additional hypothesis F [0, T ] ×  K(H) ⊂ LIPco,cp (H, H), the implication “(b) =⇒ (a)” also holds. Proof of Lemma 4.33 (Part 1). We first focus on the claimed implication “(b) =⇒ (a)” under the additional   hypothesis F [0, T ] × K(H) ⊂ LIPco,cp (H, H) because it is easier to conclude from preceding arguments. Indeed, G(·, x) : [0, T ] ; H is supposed to be Lebesgue measurable for every x ∈ H. According to the   ˜ ·) is Lebesgue measurable. Hence, we can proof of Lemma 4.32, [0, T ) −→ LIPco,cp (H, H), dl∞ , t −→ G(t, ˜ : [0, T ] ×H ; H satisfies follow the arguments for Corollary 4.31 and conclude from Lusin’s theorem that G hypothesis (iii) of Proposition 2.3 as well. Finally, the claim results from the implication “(1.) =⇒ (3.)” there (i.e., in Proposition 2.3). 2 A key tool for proving the implication “(a) =⇒ (b)” is the following adaption of the Scorza-Dragoni theorem to upper semicontinuous set-valued maps (cited in [40, Theorem 2.1], see also, e.g., [46,50,96] supplementarily): Lemma 4.34 ([84]). Let X and Y be separable metric spaces. Suppose for a set-valued map F : [0, T ] ×X ; Y that for Lebesgue-almost all t ∈ [0, T ], the graph of F(t, ·) : X ; Y is a closed subset of X × Y .  [0, T ] × X ; Y with (possibly empty) closed values and the Then there exists a set-valued map F: subsequent properties:  x) ⊂ F(t, x). (i) For Lebesgue-almost all t ∈ [0, T ] and every x ∈ X, F(t, (ii) For every Lebesgue measurable set J ⊂ [0, T ] and all measurable maps u : J −→ X, v : J −→ Y with      t, u(t) for Lebesgue-almost v(t) ∈ F t, u(t) for Lebesgue-almost every t ∈ J, it holds v(t) ∈ F every t ∈ J.  (iii) For any ε > 0, there is a closed subset Jε ⊂ [0, T ] with Lebesgue measure L1 [0, T ] \ Jε ) < ε such  that the set-valued restriction F : Jε × X ; Y has closed graph. J ×X ε

Preparing the proof of Lemma 4.33 “(a) =⇒ (b)” by means of Lemma 4.34, the metric dlco,∞ on   LIPco (H, H) reveals an advantage over dl∞ , i.e., separability: Corollary 4.4 states that Clbd,co (BR ), dlco is both complete and separable. Each set-valued map G ∈ LIPco (H, H) can be interpreted as a single-valued     continuous function H −→ Clbd,co (H), dlPH whose graph is a closed subset of H × Clbd,co BG∞ w.r.t. the metric · + dlco . Thus, standard (partly indirect) arguments lead to the following consequence: 

Lemma 4.35. For any parameters Λ, R > 0, the set G ∈ LIPco (H, H) LipG ≤ Λ, G ∞ ≤ R supplied with dlco,∞ is a metric space that is both complete and separable.   Furthermore, Clbd,co (BR ), dlco is even compact for every radius R > 0 due to Lemma 4.3. Thus, we conclude from the Arzelà–Ascoli theorem in metric spaces (see, e.g., [42]):

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Lemma 4.36. For any parameters Λ, R > 0 and non-empty compact set K0 ⊂ H, the set

LIPco (K0 , H) LipG ≤ Λ, G ∞ ≤ R is compact w.r.t. dlco,∞ .

53



G ∈

This detail provides the well-known equivalence between closed graph and upper semicontinuity of setvalued maps (as formulated in, e.g., [10, Proposition 1.4.8] or [47, Proposition I.2.22, I.2.23]) and so, we can “extend” Lemma 4.34 in regard to convex values – in the following sense: Corollary 4.37. Let X be a separable metric space, H a separable Hilbert space, J ⊂ R closed and Λ, R > 0, K0 ∈ K(H). Suppose the set-valued map  : J ×X ; F



G ∈ LIPco (K0 , H) LipG ≤ Λ, G ∞ ≤ R =: BΛ,R



to have closed graph (but with possibly empty values).  is upper semicontinuous (in the sense of Bouligand and Kuratowski, e.g., [10, DefiniThen F tion 1.4.1]) and so is the pointwise closed convex hull   : J × X ; BΛ,R ⊂ LIPco (K0 , H), co F  (t, x) → cl λ · G1 + (1 − λ) · G2

 dl co,∞

 x), λ ∈ [0, 1] . G1 , G2 ∈ F(t,

   : J × X ; LIPco (K0 , H), dlco,∞ also has closed graph. In particular, co F Proof of Lemma 4.33 (Part 2). On our way to conclude the implication “(a) =⇒ (b)” from the arguments for Theorem 3.3, fix ε > 0 arbitrarily. K is continuous w.r.t. dlPH due to Lemma A.2.3. Assumption (a)    [0, T ] × K(H), dlPH ; and Lemma 4.34 lead to a compact subset Jε ⊂ [0, T ] and a set-valued map F:   LIPco (H, H), dlco,∞ satisfying   • the Lebesgue measure L1 [0, T ] \ Jε < 2ε ,   • for every t ∈ Jε , there is at least one G ∈ F t, K(t) ⊂ LIPco (H, H) with lim

h↓0

   1 · dlPH K(t + h), ϑG h, K(t) = 0, h

 • the set-valued restriction F has closed graph, Jε ×K(H)  • F has closed (possibly empty) values and properties (i), (ii) in Lemma 4.34. Since the boundary of the open set R \ Jε ⊂ R is (at most) countable, there always exists an open subset   Jε ⊂ R with Jε ⊂ Jε ⊂ [0, T ] and L1 [0, T ] \ Jε < ε.    := R × H with (r, u), (s, v) " := r s + u, v and obtain a separable Now we supply the product space H H  generates  := (1, A) : R × D(A) −→ R × H Def. real Hilbert space. Furthermore, the closed operator A = H    a strongly continuous semigroup S(t) t≥0 sharing the relevant properties with the original C0 semigroup    is normal with S(t) t≥0 on H (like quasi-contraction according to assumption (H4)). In particular, A compact resolvent due to hypothesis (H5). ε ⊂ K(H)  of constraints Next, we define an appropriate set V ε := V

  t ∈ Jε

 {t} × K(t) ∪





 {t} × M .

t ∈ [0,T ]\Jε M ∈ K(H)

 because Jε is open in R and K is It is closed w.r.t. the Pompeiu–Hausdorff metric dlPH on K(H) continuous w.r.t. dlPH .

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 ε : K(H)  ; LIPco (H,  H)  are constructed by means of a Last, but not least, the set-valued maps Gε , G Lipschitz continuous restriction map      dlPH −→ R × K(H), | · | + dlPH , K(H),

" −→ M



τM " , ΠM "



with

 " = inf π1 (M "), τM " := inf t ∈ R ∃ x ∈ H : (t, x) ∈ M 

  " " = π2 M ∩ ({τM ΠM " := x ∈ H (τM " , x) ∈ M " } × H)  −→ R, π2 : H  −→ H denoting the respective orthogonal projections. In a word, τ " ∈ R specifies and π1 : H M "⊂H  and, Π " ⊂ H contains all vectors x ∈ H of these tuples. the smallest real component of tuples in M M  ε : K(H)    ; LIPco (H,  H)  are constructed as extensions of F  Gε , G and F respectively (in a Jε ×K(H)

broad sense):

    {0} × F τM " , ΠM " " 

Gε M := {0} × G ∈ LIPco (H, H) LipG ≤ Λ, G ∞ ≤ R     τ ", Π "   {0} × F M M  " 

Gε M := {0} × G ∈ LIPco (H, H) LipG ≤ Λ, G ∞ ≤ R 

Jε ×K(H)

 if τM " ∈ Jε else  if τM " ∈ Jε else

with the assumed global bounds of F : [0, T ] × K(H) ; LIPco (H, H) Λ := R :=

sup

sup

t ∈ [0,T ] M ∈ K(H)

G ∈ F(t,M )

sup

sup

t ∈ [0,T ] M ∈ K(H)

G ∈ F(t,M )

Lip G, G ∞ .

 ε are closed and have possibly empty Gε has non-empty closed and convex values whereas the values of G      ε : K(H),  dlPH ; LIPco (H,  H),  dlco,∞ components (as a consequence of Lemma 4.34, 4.35). The graph of G      is closed since so is the graph of F : Jε × K(H), dlPH ; LIPco (H, H), dlco,∞ and because the Jε ×K(H) subset Jε of Jε is open in R.  ε might be empty prevents us from applying The detail that some set components of the values of F   of constraints. We are  ε ⊂ K(H) Theorem 3.3 directly to evolution inclusions related to A, Gε and the set V going to avoid this obstacle by following the tracks of its proof in § 4.3 in combination with property (ii)  in Lemma 4.34. of F  related to A,  Gε and the The arguments of § 4.3.1 can still be applied to evolution inclusions in H     dlPH ,   set Vε . In more details, for every index k ∈ N, there exist four functions Kk , Pk : [0, T ] −→ K(H),  H)  and δk : [0, T ) −→ [0, T ] satisfying fk : [0, T ) −→ LIPco (H,  k (0) = {0} × K(0) ∈ R × H = H,  (a) K   1    ε and   (b’) dist Kk (t), Vε ≤ k , dlPH Kk (t), Pk (t) < k1 , Pk (t) ∈ V   1   dlPH Pk (t), Pk (t − δk (t)) < k for all t ∈ [0, T ],    H)  belongs to Gε Pk (t − δk (t)) and satisfies lim 1 · (c’) for all t ∈ [0, T ), the map fk (t) ∈ LIPco (H, h↓0 h      k (t + h), ϑ  = 0. dlPH K fk (t) h, Kk (t)  (d) fk is piecewise constant “to the right”, i.e., in the following sense: For each t ∈ [0, T ), there exists some δ > 0 such that fk is constant. (Hence, fk has at most countably many points of discontinuity [t, t+δ)

in [0, T ).)

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 k is Lipschitz continuous w.r.t. ΔCl . K  Pk is piecewise constant “to the left”, i.e., in the following sense: For each t ∈ (0, T ), there exists some δ > 0 such that Pk (t−δ,t] is constant. (g) For each t ∈ [0, T ), δk (t) ∈ [0, t] can be chosen as δ ≥ 0 such that fk [t − δ,t] is constant with a value in   Gε Pk (t − δ) . In particular, δk (·) increases linearly on each open subinterval between two of the (at most countably many) points of discontinuity of fk , Pk and so, it is Lebesgue measurable. (e) (f)

 Gε , V ε implies directly the special forms K  k (t) = {t} × Kk (t), Pk (t) = {t} × Pk (t) and The choice of A, fk (t) ∈ {0} × fk (t) with some Kk (t), Pk (t) ∈ K(H) and fk (t) ∈ LIPco (H, H) for each t ∈ [0, T ]. This extension of § 4.3.1 holds for essentially two reasons: First, we do not need the graph of Gε to be closed here. Second, we can apply the generalizing arguments in Remark 4.18 because the relevant ε ⊂ R × K(H) with time component contingent intersection condition is satisfied for every tuple (t, M ) ∈ V   t ∈ [0, T ] \ ∂ Jε and, the exceptional set ∂ Jε ⊂ R is (at most) countable.      dlco,∞ and the composition Pk ( · − δk (·)): [0, T ) −→ K(H),  H),  dlPH Both fk : [0, T ) −→ LIPco (H,  that are Lebesgue measurable. Hence, we conclude from property (ii) in Lemma 4.34 (for F)          ε Pk (t − δk (t)) ⊂ G  ε B 1 Pk (t) ε B2 K  k (t) ⊂ G fk (t) ∈ G k k is satisfied for Lebesgue-almost every t ∈ [0, T ). The conclusions in §§ 4.3.2–4.3.5 stay the same in this situation and so, we obtain a sequence of indices  H)  ;H   such that the set-valued maps [0, T ) × H (k )∈N and a limit function gε : [0, T ) −→ LIPco (H,  ˜) → fk (t)(˜ x) and (t, x ˜) → gε (t)(˜ x) respectively satisfy defined as (t, x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

  ϑfk t, {0} × K(0)   ϑ t, {0} × K(0) g  ε     k (t), ϑg t, {0} × K(0) lim dlPH K  ε →∞      k (t), V ε  k (t), Pk (t) = lim dist K lim dlPH K   

→∞

→∞

 k (t) = K   ∈ K(H) = 0 = 0

ε ⊂ K(H)  implies for every t ∈ Jε for every t ∈ [0, T ]. The special choice of V   {t} × K(t) = ϑgε t, {0} × K(0)        dlPH . Moreover, the construction of gε ε = 0 and V ε is closed in K(H), because dist ϑgε t, {0} × K(0) , V  0 ⊂ H,  preserves that for Lebesgue-almost every t ∈ [0, T ] and every compact subset K gε (t) K "

0

∈

co ∈N

   0 , H),  dlco,∞ . ∈ LIPco (K fk(+n) (t) K "0 n ∈ N

This inclusion implies the form gε (t) = {0} × gε (t) for Lebesgue-almost every t ∈ [0, T ) with a Lebesgue measurable function gε : [0, T ) −→ LIPco (H, H). In addition, the same arguments as in the proof of Lemma 4.30 (b) (in § 4.3.6) confirm for 0 ⊂ H  Lebesgue-almost every t ∈ [0, T ] and every compact set K gε (t) K "

0

     ε " ϑg t, {0} × K(0)  0 , H)  ∈ co G ∈ LIPco (K ε K 0

   ε " : K(H)  ; LIPco (K  0 , H),  with the pointwise closed convex hull co G K 0

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 ε (M " → cldl "), λ ∈ [0, 1] " G 1 , G2 ∈ G M λ G1 K "0 + (1 − λ) G2 K co,∞ 0  ε " due to Corollary 4.37. We obtain because the latter has the closedness of its graph in common with G K0  0 ∈ K(H)  for Lebesgue-almost every t ∈ Jε and all K gε (t) K "

  Gε {t} × K(t) K "0   Def. gε (t) = π2 gε (t) ∈ F t, K(t) ⊂ LIPco (H, H) . 0

=⇒

∈

  Finally, standard arguments complete this proof of Lemma 4.33 as a consequence of L1 [0, T ] \ Jε < ε with ε > 0 having been fixed arbitrarily. 2 Acknowledgments Parts of the recent extensions to a separable Hilbert space are closely related with the 66th workshop “Advances in Convex Analysis and Optimization” in Erice, Italy in July 2016. I thank the organizers Prof. Annamaria Barbagallo, Prof. Hélène Frankowska and Prof. Gioconda Moscariello for inviting me to this outstanding event and giving me the opportunity of presenting my results. Some of these results had been developed in connection with a research stay at University Paris 1 Panthéon-Sorbonne. I thank Prof. Joël Blot and Prof. Georges Haddad for the invitation and the hospitality. Last, but not least I would also like to express my gratitude to Prof. Jean-Pierre Aubin especially for the interesting discussions opening me new perspectives how the notions of viability theory and mutational analysis can be applied in and beyond maths. This article is dedicated to the 25th anniversary of his “mutational calculus” in metric spaces initiated in [4]. Appendix A A.1. Tools for differential inclusions in Banach spaces A.1.1. Filippov’s theorem for evolution inclusions in a separable Banach space In this subsection, we formulate a key result about the existence of solutions to evolution inclusions. Its counterpart for values in Rn (rather than a Banach space) is known as Filippov’s theorem and can be found in various references like [9,89,101]. For states in a separable Banach space, however, we prefer the extensions of Frankowska [39] (see [33,48,81,97–99] complementarily): Theorem A.1.1 (Immediate consequence of [39, Theorem 1.2]). Consider a separable Banach space (X, · )   and a strongly continuous semigroup S(t) t≥0 of bounded linear operators X −→ X with the infinitesimal generator A. Suppose 0 ≤ t0 < T and set M := sup S(t) Lin(X,X) < ∞. For a Lebesgue–Bochner 0 ≤ t ≤ T −t0

integrable function h : [t0 , T ] −→ X, let the continuous curve y : [t0 , T ] −→ X be a mild solution of y  (t) = A y(t) + h(t)

in [t0 , T ].

˜ : [t0 , T ] × X ; X: Assume for the set-valued map G ˜ is a non-empty closed subset of X, (i) each value of G ˜ x) : [t0 , T ] ; X is (strongly) Lebesgue measurable, (ii) for every x ∈ X, the set-valued map G(·, 1 ˜ ·) : X ; X (iii) there exists λ ∈ L ([t0 , T ]) such that for Lebesgue-almost every t ∈ [t0 , T ], the map G(t, is λ(t)-Lipschitz continuous w.r.t. the Pompeiu–Hausdorff distance dlPH ,

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(iv) there is a Lebesgue integrable function γ : [t0 , T ] −→ R with   ˜ y(t)) ≤ γ(t) dist h(t), G(t,

for Lebesgue-almost every t ∈ [t0 , T ].

Then for every x0 ∈ X and each ε > 0, there exist a mild solution x : [t0 , T ] −→ X of the evolution inclusion   ˜ t, x(t) x (t) ∈ A x(t) + G

(A.1)

and its associated Lebesgue–Bochner integrable selection g : [t0 , T ] −→ X with the additional features (1.) x(t0 ) = x0 , (2.) for all t ∈ [t0 , T ], x(t) − y(t) ≤



t x0 − y(0) +

 # M · t λ(s) γ(s) ds + ε (t − t0 ) · M · e t0

ds

,

t0

(3.) for Lebesgue-almost every t ∈ [t0 , T ], t



g(t) − h(t) ≤ λ(t) · x0 − y(0) +

 γ ds + ε (t − t0 ) ·

t0

M ·e

#t

M·

t0

λ(s) ds

+ γ(t) + ε.

In the special case X = Rn , the corresponding statements hold with ε = 0 and so, we (re-) find the wellestablished form of Filippov’s existence theorem for differential inclusions (see, e.g., [9, § 2.4 Theorem 1], [33, § 8.5] or [101, Theorem 2.4.3]). A.1.2. Continuity properties of closed reachable sets The preceding Theorem A.1.1 lays the basis for comparing closed reachable sets of evolution inclusions for various initial sets and set-valued maps. Proposition A.1.2 ([68, § 4.2, Proposition 2]). In addition to the assumptions of Theorem A.1.1 about   (X, · ) and S(t) t≥0 , let CS ≥ 1 and ωS ∈ R denote constants such that S(t) Lin(X,X) ≤ CS · eωS ·t holds for every t ≥ 0. ˜ 1, G ˜ 2 : [0, T ] × X ; X Suppose for the set-valued maps G ˜ j is a non-empty closed and convex subset of X, (i) each value of G ˜ j is integrably bounded, i.e., there exists k ∈ L1 ([0, T ]), k ≥ 0, with G ˜ j (t, ξ) ⊂ Bk(t) ⊂ X for any (ii) G t ∈ [0, T ], ξ ∈ X, j ∈ {1, 2}, ˜ j (·, ξ) : [0, T ] ; X is (strongly) Lebesgue measurable, (iii) for every ξ ∈ X, the set-valued map G 1 (iv) there is λ ∈ L ([0, T ]) such that for Lebesgue-almost every t ∈ [0, T ], the map Gj (t, ·) : X ; X is λ(t)-Lipschitz continuous w.r.t. dlPH .   ˜ 1 (t, ξ), G ˜ 2 (t, ξ) ≤ δ(t) for a.e. t ∈ [0, T ]. Let δ ∈ L1 ([0, T ]) satisfy sup dlPH G ξ∈X

Then the closed reachable sets of any sets M1 , M2 ∈ Cl(X) fulfill the following estimate at every time instant t ∈ [0, T ]:

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  dlPH ϑG˜ 1 (t, M1 ), ϑG˜ 2 (t, M2 ) 

t

≤ dlPH (M1 , M2 ) +

 δ(s) ds · econst(CS ,ωS ,T )·(1+λL1 ([0,t]) ) .

0

Corollary A.1.3 ([68, § 4.2, Corollary 3]). In addition to the hypotheses of Proposition A.1.2 about (X, · ),     ˜1, G ˜ 2 and δ(·), suppose S(t) S(t) t≥0 , G to be quasi-contractive (as in (H4)). t≥0 Then the closed reachable sets of any M1 , M2 ∈ Cl(X) satisfy for all t ∈ [0, T ]   dlPH ϑG˜ 1 (t, M1 ), ϑG˜ 2 (t, M2 ) 

t

≤ dlPH (M1 , M2 ) +

 δ(s) ds · econst(ωS ,T ) · (t + λL1 ([0,t]) ) .

0

The second aspect concerns the continuity of reachable sets with respect to time. Since reachable sets are constructed by means of mild solutions, we regard it as suitable to consider the “time-oriented” distance function ΔCl instead of the Pompeiu–Hausdorff distance dlPH . Proposition A.1.4 ([68, § 2.2, Lemma 3]). Under the assumptions of Proposition A.1.2 about the Banach   space (X, · ), the strongly continuous semigroup S(t) t≥0 , its related constants CS ≥ 1, ωS ∈ R and the ˜ : [0, T ] × X ; X, the closed reachable sets of every M ∈ Cl(X) satisfy set-valued map G ΔCl



   ≤ CS e|ωS | T · s, ϑG˜ (s, M ) , t, ϑG˜ (t, M )

 k(τ ) dτ [s,t]

for every s, t ∈ [0, T ] (s ≤ t). In particular, the set-valued map [0, T ] ; X,

t → ϑG˜ (t, M )

with all values in Cl(X) is sequentially continuous w.r.t. ΔCl . A.2. Implementing C0 semigroups in sequential convergence in a real Banach space   Let (X, · ) be a real Banach space. S(t) t≥0 is supposed to denote a strongly continuous semigroup of bounded linear operators X −→ X with the infinitesimal generator A. CS ≥ 1 and ωS ∈ R abbreviate constants such that S(t) Lin(X,X) ≤ CS · eωS ·t holds for every t ≥ 0. Proposition A.2.2 below specifies the equivalence of various approaches to sequential convergence in  X := R × X. It is presented for the special case of quasi-contractive semigroups (i.e., CS = 1) in [67, Proposition 3.117]. For the sake of a self-contained and convenient presentation, we give a detailed proof for the general situation here.  ×X  −→ [0, ∞) are defined  := R × X and ρ ≥ 0, the functions ΔX , ΔX,ρ , ηX,ρ : X Definition A.2.1. For X as    |s − t| + S(t − s) x − y  if s ≤ t   ΔX (s, x), (t, y) =   |s − t| + x − S(s − t) y  if s > t    |s − t| + S(ρ + t − s) x − S(ρ) y  if s ≤ t   ΔX,ρ (s, x), (t, y) =   |s − t| + S(ρ) x − S(ρ + s − t) y  if s > t

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    ηX,ρ (s, x), (t, y) = |s − t| + S(ρ) x − S(ρ) y  In particular, ηX,0  i.e., of a norm on X,

 ×X  :X  −→ R coincides with the metric induced by the probably most popular choice (t, y) " := |t| + y . X

     and any bounded sequence x  · " , the  = (t, x) ∈ X n n∈N in X, Proposition A.2.2. For every element x X following features are equivalent:   −x n X lim x " = 0,   , x n = 0 for every ρ > 0, (ii) lim ηX,ρ x n→∞   , x n = 0 for every ρ > 0, (iii) lim ΔX,ρ x n→∞   n = 0. , x (iv) lim ΔX x (i)

n→∞

n→∞

Proof. “(i) =⇒ (ii)” and “(iv) =⇒ (iii)” are obvious consequences of Definition A.2.1 since each linear operator S(ρ) : X −→ X (ρ ≥ 0) is bounded.   Def.  that ηX,ρ ( Assume for x  = (t, x) and a bounded sequence = (t , x ) in X x, x n ) = “(ii) =⇒ (i)” x  n n n n∈N   |t − tn | + S(ρ) x − S(ρ) xn  −→ 0 (n → ∞) holds for every ρ > 0. The resolvent R(λ, A) of the generator A of (S(t))t≥0 is known to have the representation as a limit of Bochner integrals τ R(λ, A) y =

lim

τ →∞

e− λ t S(t) y dt

0

for every y ∈ X and λ ∈ C with Re λ > ωS (see, e.g., [37, Theorem II.1.10]). Hence, Lebesgue’s theorem   about dominated convergence leads to R(ωS + 2 CS , A) x − xn  −→ 0 for n → ∞. It implies the norm convergence x − xn −→ 0 since R(ωS + 2 CS , A) : X −→ X is a bijective contraction with R(ωS + 2 CS , A) Lin(X,X) ≤ Re(ωS +2CSCS ) − ωS = 12 . “(iii) =⇒ (iv)” It also results from the integral representation of the resolvent operator R(ωS + 2 CS , A).   Indeed, assuming for a norm bounded sequence x n = (tn , xn ) n ∈ N   Def. ΔX,ρ ( x, x n ) = |t − tn | + S(ρ + (tn −t)+ ) x − S(ρ + (t−tn )+ ) xn 

n→∞

−→ 0

for every ρ > 0 (with the abbreviation r+ := max{r, 0} for r ∈ R), we obtain    R(ωS + 2 CS , A) S((tn −t)+ ) x − S((t−tn )+ ) xn  n→∞ −→ 0 and thus, ΔX ( x, x n )

Def.

=

  n→∞ |t − tn | + S((tn −t)+ ) x − S((t−tn )+ ) xn  −→ 0 .

   be arbitrary with ηX,ρ ( “(ii) =⇒ (iii)” Let the sequence x n = (tn , xn ) n ∈ N and x  = (t, x) ∈ X x, x n ) −→ 0 for every ρ > 0. First, we assume tn ≥ t for all n ∈ N in addition. Then, it holds for every ρ > 0 ΔX,ρ ( x, x n )    = |t − tn | +  S(ρ + tn − t) x − S(ρ) xn      S(ρ + t − S(ρ) xn  + S(ρ) x − ≤ |t − tn | + S(ρ) x − t) x n   ωS ρ  = ηX,ρ ( x − S(tn − t) x x, x n ) + CS e −→ 0 (n → ∞).

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Similarly we obtain under the additional assumption tn ≤ t for all n ∈ N ΔX,ρ ( x, x n )   − S(ρ+t−tn ) xn  = |t − tn | +  S(ρ) x    S(ρ) x − S(ρ + t − tn ) x + ≤ |t − tn | + S(ρ+t−tn ) x − S(ρ+t−tn ) xn     ≤ |t − tn | + CS eωS (t−tn ) S(ρ) x − S(ρ) xn  + CS eωS ρ  x − S(t − tn ) x   ≤ CS e|ωS (t−tn )| ηX,ρ ( x, x n ) + CS eωS ρ x − S(t − tn ) x −→ 0 (n → ∞).  Applying these cases to subsequences, we conclude ΔX,ρ x , x n ) −→ 0 (n → ∞) for each ρ > 0 – and without additional assumptions.    be arbitrary with ΔX,ρ ( “(iii) =⇒ (ii)” Let the sequence x n = (tn , xn ) n ∈ N and x  = (t, x) ∈ X x, x n ) −→ 0 for each ρ > 0. First, we suppose tn ≥ t for all n ∈ N in addition and obtain for every ρ > 0 ηX,ρ ( x, x n )    = |t − tn | +    S(ρ) x − S(ρ) xn   S(ρ + tn − t) x ≤ |t − tn | + S(ρ + tn − t) x − S(ρ) xn  + S(ρ) x −   x, x n ) + CS eωS ρ x − S(tn − t) x = ΔX,ρ ( −→ 0 (n → ∞). Complementarily we conclude under the additional assumption tn ≤ t (n ∈ N) ηX,ρ ( x, x n )   = |t − tn | + S(ρ) xn − S(ρ) x   ≤ |t − tn | + S( ρ − t + tn ) 2

≤ |t − tn | + CS eωS ρ

≤ CS e|ωS | ( 2 +|t−tn |) −→ 0

 ρ  S( +t−tn ) xn − S( ρ +t−tn ) x 2 2   S( ρ +t−tn ) xn − S( ρ ) x + 2 2  ρ  S( +t−tn ) x − S( ρ ) x 2 2   ρ   ρ x, x n ) + S( 2 +t−tn ) x − S( ρ2 ) x ΔX, 2 (

Lin(X,X) (ρ 2 −t+tn )

(n → ∞).

   , x n ) n∈N tends to 0 for n → ∞ and every parameter ρ > 0 – without any additional Finally, ηX,ρ x assumptions about (tn )n∈N . 2 This equivalence between convergence of vectors can be extended to non-empty compact subsets of a real Banach space X and the distance function ΔCl (as specified in Definition 2.2) in comparison with the Pompeiu–Hausdorff metric dlPH . Lemma A.2.3. Let K : [0, T ] ; X have non-empty compact values. Then the following features are equivalent: (a) K is sequentially continuous w.r.t. ΔCl (in the sense of Definition 2.2). (b) K is continuous w.r.t. the Pompeiu–Hausdorff metric dlPH . Proof. “(a) =⇒ (b)” Let (tk )k∈N be any sequence in [0, T ] with limit t. In regard to an indirect proof, we     first assume ΔCl (t, K(t)), (tk , K(tk )) −→ 0 for k → ∞ and dist K(t), K(tk ) ≥ ε > 0 for all k ∈ N. For   each index k ∈ N, the compactness of K(t) ⊂ X provides an element xk ∈ K(t) with dist xk , K(tk ) =   dist K(t), K(tk ) . Moreover, we can always find some yk ∈ K(tk ) satisfying

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      ΔX (t, xk ), (tk , yk ) ≤ ΔCl t, K(t) , tk , K(tk ) −→ 0 (k → ∞).   The sequence (xk )k∈N in K(t) has a converging subsequence xk ∈N with limit x ∈ K(t). In particular, the triangle inequality of the norm · on X implies for any k ∈ N with t ≤ tk       ΔX (t, x), (tk , yk ) ≤ S(tk − t)Lin(X,X) x − xk + ΔX (t, xk ), (tk , yk ) and for any k ∈ N with t > tk     ΔX (t, x), (tk , yk ) ≤ x − xk + ΔX (t, xk ), (tk , yk ) .   Hence, we obtain ΔX (t, x), (tk , yk ) −→ 0 for → ∞. Proposition A.2.2 guarantees for → ∞ successively

=⇒

  x − yk  −→ 0    xk − yk  −→ 0  

  contradicting dist(xk , K(tk )) = dist K(t), K(tk ) ≥ ε > 0 for all k ∈ N.     In the second part, we also suppose ΔCl (t, K(t)), (tk , K(tk )) −→ 0 for k → ∞, but dist K(tk ), K(t) ≥ ε > 0 for all k ∈ N. The compactness of K(tk ) ⊂ X provides a vector yk ∈ K(tk ) for each index k ∈ N such     that dist yk , K(t) = dist K(tk ), K(t) holds. Now choose xk ∈ K(t) with ΔX





  |t − tk | + dist S(t − tk ) yk , K(t) if t ≥ tk   (t, xk ), (tk , yk ) = |t − tk | + dist yk , S(tk − t) K(t) if t < tk   ≤ ΔCl (t, K(t)), (tk , K(tk )) . 

  Due to compactness of K(t) ⊂ X, there exists a converging subsequence xk ∈N with a limit x ∈ K(t). The same arguments about ΔX as before lead to   ΔX (t, x), (tk , yk ) −→ 0

( → ∞)

    and so, we conclude x − yk  −→ 0 ( → ∞) from Proposition A.2.2. This contradicts dist yk , K(t) =   dist K(tk ), K(t) ≥ ε for every k ∈ N however. “(b) =⇒ (a)” Alternatively to indirect proofs using equivalent concepts of convergence in R × X, we now use another general feature of strongly continuous semigroups of bounded linear operators restricted to compact sets (see, e.g., [37, Lemma I.5.2]): For any compact sets CR ⊂ R, CX ⊂ X, the single-valued map CR × CX −→ X, (t, x) −→ S(t) x is uniformly continuous. The standard proof of this property reveals that for every ε > 0, there exist both δ > 0 and a radius   ρ > 0 such that all (s, x), (t, y) ∈ CR × Bρ CX fulfill |t − s| < δ and x − y < δ

=⇒

  S(s) x − S(t) y  <

ε 2

.

Indeed, this supplementary radius ρ = ρ(CR , CX , ε) > 0 can be selected as the Lebesgue number related to the finite cover of CR × CX which is induced pointwise by the condition on the perturbations to be bounded by ε from above (see, e.g., [13, Theorem 2.3.1]). Fix t ∈ [0, T ], ε > 0 arbitrarily and let ρ, δ > 0 denote the radii related to CR := [0, T ], CX := K(t), ε.   First, we consider any sequence (tk )k∈N in [t, T ] converging to t. Assuming dlPH K(tk ), K(t) −→ 0 for   



k → ∞, there always exists an index k0 ∈ N with dlPH K(tk ), K(t) < min ρ, δ and |tk − t| < min δ, 2ε for all k ≥ k0 . Together with the definition of the Pompeiu–Hausdorff distance, it implies for all k ≥ k0

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  Def.   ΔCl (t, K(t)), (tk , K(tk )) = |tk − t| + dlPH S(tk − t) K(t), K(tk ) 

< min δ, 2ε + 2ε ≤ ε. Next, we make similar conclusions for any sequence (tk )k∈N in [0, t] converging to t. The corresponding index k0 ∈ N now leads to the estimate    Def.  ΔCl (t, K(t)), (tk , K(tk )) = |tk − t| + dlPH K(t), S(t − tk ) K(tk ) 

< min δ, 2ε + 2ε ≤ ε for all k ≥ k0 . Both cases guarantee the sequential continuity of K (w.r.t. ΔCl ) at every time instant t ∈ [0, T ] in the sense of Definition 2.2. 2 A.3. Invariant subspaces of a C0 semigroup in a Banach space Proposition A.3.1 ([82, § 4.5, Theorem 5.1]). Let X be a real Banach space, Y a closed subspace of X and   S(t) t≥0 a strongly continuous semigroup of bounded linear operators X −→ X with generator A.   Y is an invariant subspace of the semigroup S(t) t≥0 in the sense of S(t) Y ⊂ Y for every t ≥ 0 if and only if there is a real number α such that for every λ > α, Y is an invariant subspace of the resolvent −1 Def.  R(λ; A) = λ IdX − A of A. Proposition A.3.2 ([72, Theorem 4.6.1]). In addition to the assumptions of Proposition A.3.1, let CS ≥ 1, ωS ∈ R denote constants such that S(t) Lin(X,X) ≤ CS · eωS ·t holds for every t ≥ 0. If Y is a closed and invariant subspace of the resolvent R(λ; A) for some λ > ωS , then Y is an invariant subspace of A and S(t) for all t ≥ 0, i.e.,   A D(A) ∩ Y ⊂ Y,

S(t) Y ⊂ Y

for every t ≥ 0.

  Moreover, S(t) Y t≥0 is a strongly continuous semigroup on Y whose generator is A D(A)∩Y . (It is called a subspace semigroup [37, I.1.11, II.2.3].) A.4. Results about weak compactness of Banach-valued functions Proposition A.4.1 ([53, Theorem 4], special case of [26, Theorem 4.3]). Let S be a compact Hausdorff space and X a Banach space.   A subset W ⊂ C 0 (S, X) is weakly compact in C 0 (S, X), · sup if it is bounded, equi-continuous and if for every s ∈ S, the set {f (s) | f ∈ W } is relatively weakly compact in X. Proposition A.4.2 ([34, Theorem 2], [100, Proposition 7]). Let (Ω, Σ, μ) be a finite measure space and X an arbitrary real Banach space. For any weakly compact and convex subset W ⊂ X, the subset 



h ∈ L1 (μ, X) h(ω) ∈ W for μ-almost every ω ∈ Ω

is relatively weakly compact in L1 (μ, X). Proposition A.4.3 ([100, Corollary 5]). Let (Ω, Σ, μ) be a probabilistic space and X an arbitrary real Banach space as in preceding Proposition A.4.2. 

1 Set W := g ∈ L (μ, X) |g(ω)| ≤ 1 for μ-almost every ω ∈ Ω .

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  A sequence gn (·) n∈N in W ⊂ L1 (μ, X) converges weakly to g ∈ L1 (μ, X) if and only if for any sub    

sequence gnk (·) k∈N given, there exists a sequence hk (·) k∈N with hk ∈ co gnk , gnk+1 . . . such that for μ-almost every ω ∈ Ω, hk (ω) −→ g(ω)

(k → ∞)

weakly in X.

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