Extreme solutions in control of moisture transport in concrete carbonation

Nonlinear Analysis: Real World Applications 47 (2019) 446–459

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Extreme solutions in control of moisture transport in concrete carbonation✩ Sergey A. Timoshin a,b ,∗, Toyohiko Aiki c a

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Lermontov str. 134, 664033 Irkutsk, Russia c Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681, Japan b

article

info

Article history: Received 31 July 2018 Received in revised form 9 December 2018 Accepted 10 December 2018 Available online 24 December 2018 Keywords: Control system Hysteresis Concrete carbonation Bang–bang controls

abstract This paper is concerned with a control system modeling concrete carbonation process when underlying hysteresis effects are taken into account. The system consists of a diffusion equation for moisture and an ordinary differential equation accounting for the hysteresis. We obtain the bang–bang principle for this system asserting the proximity of all attainable states to the so-called bang–bang states. The later are the states reached by controls valued in the extreme points of the control constraint. In our study we are able to dispense with the monotonicity of the right-hand side in the hysteresis equation. This assumption has repeatedly appeared in previous analysis of control systems with hysteresis. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In our fast growing industrial world it is difficult to underestimate the importance of building material such as concrete in the construction industry. From building individual houses to raising ever-taller skyscrapers concrete is used all around the globe. Carbonation of concrete is a critical issue in civil engineering as it poses a threat to durability and strength of concrete structures. Carbonation occurs when carbon dioxide in the air penetrates a concrete surface and dissolves in pore water forming carbonic acid in the concrete’s pores. The carbonic acid reacts then with calcium hydroxide – the material internally strengthening the concrete matrix – producing calcium carbonate as a result of the chemical reaction. This replacement increases concrete’s porosity and reduces its strength. The reaction of carbon dioxide and calcium dioxide only occurs in solution and so in very dry concrete carbonation will be slow. In saturated concrete the moisture presents a barrier to the penetration of carbon ✩ The research of the first author was partially supported by RFBR, Russia grant no. 18-01-00026. ∗ Corresponding author at: School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China. E-mail addresses: [email protected] (S.A. Timoshin), [email protected] (T. Aiki).

https://doi.org/10.1016/j.nonrwa.2018.12.003 1468-1218/© 2018 Elsevier Ltd. All rights reserved.

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dioxide and again carbonation will be slow. Measures to prevent ingress of water and carbon dioxide into the concrete structure and thus to control the carbonation rate might include, for instance, application of barrier coatings or using chemical liquids to deactivate or slow down the chemical reaction described above. A mathematical control problem modeling the corresponding process was considered in [1]. This model, in addition, takes account of hysteresis effects pertaining to the process and is described by the following system of differential equations: ρvt − (g(v)vx )x = h(v, w) · u

in Q(T ),

(1.1)

wt + ∂Ik(v) (w) ∋ F (v, w)

in Q(T ),

(1.2)

v(·, 0) = b0 ,

v(·, 1) = b1

on (0, T ),

(1.3)

v(0, ·) = v0 ,

w(0, ·) = w0

on (0, 1),

(1.4)

where the time T > 0 is fixed, Q(T ) := [0, T ] × (0, 1), ρ > 0 is a given physical constant, g, h, F are given functions, b0 , b1 , v0 , w0 are given boundary and initial conditions. Furthermore, K(v) = [f∗ (v), f ∗ (v)] is the interval of the real line for two prescribed curves w = f∗ (v) and w = f ∗ (v) describing the related hysteresis region. IK(v) is the indicator function of the set K(v), i.e. IK(v) (w) = 0 if w ∈ K(v) and IK(v) (w) = +∞ otherwise. The operator ∂IK(v) is the subdifferential in the sense of the convex analysis of IK(v) . Finally, the function u = u(t, x) plays the role of a control function. The unknown functions v = v(t, x) and w = w(t, x) in this model represent the relative humidity and the degree of saturation, respectively. The function g describes the moisture conductivity, ρ is the density of saturated vapor. Eq. (1.1) is the diffusion equation for moisture with the rate of water generation h controlled by the parameter u. The relation (1.2) models the so-called generalized play operator induced by the curves w = f∗ (u) and w = f ∗ (u), see [2–4] for details. The introduction of the latter operator to the model accounts for hysteretic relationship between u and w, playing in this case the roles of the input and output functions, respectively. The reader is referred to [1,5] for more physical background on the model. It is shown in [1] that control constraints for the function u arising in the modeling may naturally appear to be both nonconvex valued and dependent on the unknown states. Hence, system (1.1)–(1.4) is considered subject to the following control constraint: u ∈ U (t, x, v, w)

in Q(T ),

(1.5)

for a multivalued mapping U having closed, bounded, not necessarily convex values in R. For convenience, we denote system (1.1)–(1.5) by (P ) (original problem). The nonconvexity of control constraint in (1.5) has prompted the authors to consider a relaxation property for system (P ) in [1]. Namely, they proved that any solution of (1.1)–(1.4) subject to the constraint u ∈ co U (t, x, v, w)

in Q(T ),

(1.6)

can be approximated by solutions of (P ), where the symbol “co” stands for the convex hull of U , which is the intersection of all convex sets containing U . When developing numerical algorithms for solving nonconvex optimal control problems based on Pontryagin’s maximum principle, this property might prove to be of practical use in justifying the passage from nonconvex to convex optimal control problems. After the dynamics of the latter is further linearized, when possible, the optimality conditions and numerical schemes for them are obtained in a standard way. We denote system (1.1)–(1.4) supplied with (1.6) by (CP ) (convexified problem). As the next logical step in this research agenda, in the present paper we consider the so-called bang– bang principle for system (P ) which would allow to reduce computational complexity of the algorithms. In

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optimal linear control theory this principle states, roughly, that any attainable state of a control system can be reached by a bang–bang control, i.e. a control function valued in the set of extreme points of the constraint set. An extreme point of a convex set is a point that is not an interior point of any line segment lying entirely in this set. Thus, along with constraints (1.5), (1.6) we introduce the following alternative control constraint u ∈ ext co U (t, x, v, w) in Q(T ), (1.7) where “ext” is the collection of all extreme points of co U , and we show that solutions of (CP ) are close, in a prescribed sense, to solutions of (1.1)–(1.4), (1.7). We denote this last system by (EP ) (extremal problem). A mathematical model describing concrete carbonation process was proposed as a one-dimensional free boundary problem by Muntean–B¨ ohm in [6,7]. Large time behavior results for a variant of this model were obtained by Aiki–Muntean (see [8–11]). The moisture transport is an important component in the research of concrete carbonation. A mathematical model of the moisture transport in the form of the system (1.1)–(1.4) without control u was proposed in [5]. This model is a simplified version amenable to mathematical treatment of the model accounting for a hysteretic character for moisture transport in cementitious materials originated in [12] based on experimental observations. The existence and uniqueness of a solution for (uncontrolled) system (1.1)–(1.4) when F ≡ 0 and h = wf (t, x) with f ∈ L∞ (Q(T )), f ≥ 0 were established in [13]. In the case of more general nonlinear right-hand sides h and F , the corresponding existence-uniqueness results were obtained in [14]. The existence of a solution for the system (1.1)–(1.4) still without control in a three-dimensional domain was proved in [13,15] and the uniqueness was obtained by applying classical theory of parabolic equations in [16]. We note that the function F (v, w) in (1.2) is imposed to be nonincreasing in w in [1]. This somewhat demanding assumption on the right-hand side of the equation describing hysteresis relation has been repeatedly used through a number of works on control of PDE systems with hysteresis (cf., e.g., [17,18]). As one of the contributions of the present work, we dispense with this assumption and we do not impose any monotonicity requirements on the function F (v, w). At the end of the introduction, we mention the work [19] related to our study. In this reference, the bang–bang principle for a thermostat’s model featuring interplay of hysteresis and delay was considered. In contrast to our work, the control constraint in [19] is given by a fixed convex set and the model is described by a system of ODEs. 2. Preliminaries and hypotheses In this section, we introduce notation which we use throughout the paper and specify hypotheses on the data defining (P ). We also reformulate our problem in a functions spaces framework giving a precise meaning in which solutions of our control systems are understood. Then, we present the main theorem of the paper which guarantees the existence of solutions for our systems and states the bang–bang principle for them. Throughout, we denote by H the Hilbert space L2 (0, 1) with the standard inner product (·, ·)H and norm | · |H , and by V the Sobolev space H 1 (0, 1). A function φ : H → R ∪ {+∞} is called proper if the set {x ∈ H; φ(x) < +∞} is nonempty. By definition, the subdifferential ∂φ(x), x ∈ H, of a proper, convex, lower semicontinuous function φ is the set ∂φ(x) = {h ∈ H; (h, y − x)H ≤ φ(y) − φ(x), ∀y ∈ H}. The subdifferential ∂φ : H → 2H is a monotone operator: for any x, y such that ∂φ(x) ̸= ∅, ∂φ(y) ̸= ∅, and any h1 ∈ ∂φ(x), h2 ∈ ∂φ(y), the inequality (x − y, h1 − h2 )H ≥ 0 holds.

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For a Banach space X, denote by dX (x, A) the distance from a point x ∈ X to a set A ⊂ X. Then, the Hausdorff metric on the space of closed bounded subsets of X, denoted cb(X), is the function hausX (A, B) = max{sup dX (x, B), sup dX (y, A)}, x∈A

A, B ∈ cb(X).

y∈B

The notation ω-X means that the space X is equipped with the weak topology. The same notation is used for subsets of X with the topology induced by that of the space ω-X. Given a Banach space Y , a multivalued mapping Φ : X → Y is called lower semicontinuous if the set {x ∈ X; Φ(x) ∩ C ̸= ∅} is open in X for every open C ⊂ Y . A multivalued mapping Φ : [0, T ] → Y is called measurable if {t ∈ [0, T ]; Φ(t) ∩ C ̸= ∅} belongs to the σ-algebra of Lebesgue measurable subsets of [0, T ] for any open C ⊂ Y . A set F of measurable functions from [0, T ] to Y is called decomposable if together with any f1 , f2 it contains the function f1 · χE + f2 · χ[0,T ]\E , where χE stands for the characteristic function of the set E. On the space L2 (0, T ; H) along with the standard norm we will consider the so-called weak norm: ⏐∫ ′ ⏐ ⏐ t ⏐ ⏐ ⏐ |u|ω = sup ⏐ u(τ ) dτ ⏐ . ⏐ 0≤t≤t′ ≤T ⏐ t H

2

Denoting the space L (0, T ; H) equipped with this norm by

L2ω (0, T ; H)

we have the following easy result.

Lemma 2.1. If a bounded in L2 (0, T ; H) sequence un , n ≥ 1, converges to some u in L2ω (0, T ; H), then it converges to u in ω-L2 (0, T ; H) as well. Problem (P ) is considered under the following hypotheses: Hypotheses (A). (A1) ρ is a positive constant; (A2) g ∈ C 2 (0, ∞) and g(r) ≥ g0 for r > 0, where g0 > 0 is a constant. In addition, g(r) = G′ (r) for a continuous function G : (0, ∞) → R; (A3) h(v, w) is locally Lipschitz continuous, nonnegative and bounded on R2 ; (A4) F (v, w) is locally Lipschitz continuous in v and Lipschitz continuous in w with a Lipschitz constant LF < 1; (A5) f∗ , f ∗ ∈ C 2 (R) ∩ W 2,∞ (R) with 0 ≤ f∗ ≤ f ∗ ≤ w∗ on R, where w∗ is a positive constant; (A6) bi ∈ W 1,2 (0, T ) with bi ≥ κ0 on [0, T ] for some positive constant κ0 , i = 0,1; (A7) v0 ∈ V , w0 ∈ L∞ (0, 1) with v0 ≥ κ0 , w0 ≥ 0 a.e. on (0, 1), v0 (0) = b0 (0), v0 (1) = b1 (0) and f∗ (v0 ) ≤ w0 ≤ f ∗ (v0 ). In connection with constraint (1.5), we assume the following: Hypotheses (U). The multivalued mapping U : [0, T ]×(0, 1)×R×R → cb(R) has the following properties: (U1) the mapping (t, x) → U (t, x, v, w), v, w ∈ R, is measurable; (U2) the mapping (v, w) → U (t, x, v, w) is continuous in the Hausdorff metric on the space cb(R) a.e. on Q(T ); (U3) there exists m > 0 such that 0 ≤ U (t, x, v, w) ≤ m

a.e. on Q(T ), v, w ∈ R;

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(U4) there exists k ∈ L2 (0, T ; R+ ) such that hausR (U (t, x, v1 , w1 ), U (t, x, v2 , w2 )) ≤ k(t)(|v1 − v2 | + |w1 − w2 |) a.e. on Q(T ), vi , wi ∈ R, i = 1,2. To define solutions of our problems (P ), (CP ), and (EP ) consider the multivalued mapping U(t, v, w) = {u ∈ H; u(x) ∈ U (t, x, v(x), w(x)) a.e. on (0, 1)},

v, w ∈ H,

and the set K(v) = {w ∈ H; w(x) ∈ K(v(x)) a.e. on (0, 1)},

v ∈ H,

and let ∂IK (w) be the subdifferential of the indicator function of K. From [20, Theorem 1.5] and [21, Corollary 5.2] we infer that co U(t, v, w) = {u ∈ H; u(x) ∈ co U (t, x, v(x), w(x)) a.e. on (0, 1)} and ext co U(t, v, w) = {u ∈ H; u(x) ∈ ext co U (t, x, v(x), w(x)) a.e. on (0, 1)}, respectively. Definition 2.1. A triple {v, w, u} is called a solution of control system (P ) if v ∈ W 1,2 (0, T ; H) ∩ L∞ (0, T ; V ) ∩ L2 (0, T ; H 2 (0, 1)), v > 0, w ∈ W 1,2 (0, T ; H),

u ∈ L2 (0, T ; H),

and there exists a function ξ ∈ L2 (0, T ; H) such that wt + ξ = F (v, w)

in H a.e. on [0, T ],

ξ ∈ ∂IK(v) (w)

a.e. on [0, T ],

ρvt − G(v)xx = h(v, w) · u in H a.e. on [0, T ], v(·, 0) = b0 , v(0) = v0 ,

v(·, 1) = b1 w(0) = w0

u(t) ∈ U(t, v(t), w(t))

on (0, T ), in H,

in H for a.e. t ∈ [0, T ].

(2.1) (2.2) (2.3) (2.4) (2.5)

Solutions of control systems (CP ) and (EP ) are defined similarly, replacing the last inclusion with u(t) ∈ co U(t, v(t), w(t))

in H for a.e. t ∈ [0, T ]

(2.6)

and u(t) ∈ ext co U(t, v(t), w(t))

in H for a.e. t ∈ [0, T ],

(2.7)

respectively. Given Hypotheses (A) and (U), the main purpose of this work is to prove the following result. Theorem 2.1. Control systems (P ), (CP ) and (EP ) have solutions. Moreover, for any solution (v, w, u) of (CP ) there exists a sequence of solutions (vk , wk , uk ), k ≥ 1, of (EP ) such that (vk , wk ) → (v, w) in C([0, T ]; H × H) and uk → u weakly in L2 (0, T ; H). This last property is commonly referred to as bang–bang principle.

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3. State-independent control constraint’s system and its properties We note that the control constraints of our systems (P ), (CP ), and (EP ) all depend on the states v and w. On the other hand, from Hypothesis (U3) and the Krein–Milman theorem – the later implies, in particular, that ext co U (t, x, v, w) ⊂ U (t, x, v, w) – we see that all the controls in our problems are uniformly bounded. Accordingly, in this section we introduce the following state-independent “ambient” constraint u ∈ Sm := {u ∈ L2 (0, T ; H); 0 ≤ u(t, x) ≤ m

a.e. on Q(T )}

(3.1)

and explore some properties of the corresponding control system (1.1)–(1.4) coupled with (3.1). These properties will play a crucial role in establishing in the next section the main results of the present article. We denote the system (1.1)–(1.4), (3.1) by (AP ) (ambient problem). The notion of a solution to system (AP ) naturally extends from that given in Definition 2.1. Theorem 3.1. For any fixed u ∈ Sm system (1.1)–(1.4) has a unique solution. Moreover, for any solution (v, w, u) of (AP ) we have κ0 ≤ v ≤ M0 , 0 ≤ w ≤ M0 a.e. on Q(T ) (3.2) for a constant M0 > 0 independent of u. Proof . The existence of a unique solution to system (1.1)–(1.4) for a given fixed u ∈ Sm along with the uniform estimate (3.2) is proved in Sections 5, 6, and 3 of [14]. □ We note that cutting off outside the set [κ0 , M0 ] × [0, M0 ], if necessary, without loss of generality, we can now consider the functions h and F to be bounded and Lipschitz continuous on R2 . Proposition 3.1.

Let (vi , wi ) ∈ W 1,2 (0, T ; H × H), i = 1,2, satisfy wit + ∂IK(vi ) (wi ) ∋ F (vi , wi )

a.e. on [0, T ],

wi0 (0) = w0i , where w0i ∈ L∞ (0, 1) with f∗ (vi (0)) ≤ wi0 ≤ f ∗ (vi (0)) a.e. on (0, 1), i = 1,2. Then, there exist positive constants 0 < T1 ≤ T and M (T1 ) such that ( ) |w1 (t) − w2 (t)|L∞ (0,1) ≤ M (T1 ) |v1 − v2 |L∞ (0,t1 ;L∞ (0,1)) + |w01 − w02 |L∞ (0,1) for 0 ≤ t ≤ t1 ≤ T1 . Proof . The inclusions in the statement can equivalently be rewritten as wit + ξi = F (vi , wi )(=: Fi ) ξi (t) ∈ ∂IK(vi ) (wi )

a.e. on Q(T ),

for a.e. t ∈ [0, T ], i = 1, 2.

We put v = v1 − v2 , w = w1 − w2 and fix t1 ∈ (0, T ]. For ℓ(t1 ) := |v|L∞ (0,t1 ;L∞ (0,1)) + |w01 − w02 |L∞ (0,1) define z1 (t) := w1 (t) − [w(t) − C(t + 1)ℓ(t1 )]+ ,

0 ≤ t ≤ t1 ,

(3.3)

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with C ≥ L, where L > 1 is a common Lipschitz constants of the functions f∗ , f ∗ , and F . Then, we have f∗ (v1 ) ≤ z1 ≤ f ∗ (v1 )

a.e. on Q(t1 ) := [0, t1 ] × (0, 1).

(3.4)

In fact, it is clear that z1 ≤ f ∗ (v1 ) a.e. on Q(t1 ). On the other hand, if w ≤ C(t + 1)ℓ(t1 ), then z1 ≥ f∗ (v1 ). Otherwise, we observe that z1 = w2 + C(t + 1)ℓ(t1 ) ≥ f∗ (v2 ) + C(t + 1)ℓ(t1 ) ≥ −Lℓ(t1 ) + f∗ (v1 ) + C(t + 1)ℓ(t1 ) ≥ f∗ (v1 ). and (3.4) follows. Similarly, we define z2 (t) = w2 (t) + [w(t) − C(t + 1)ℓ(t1 )]+

for 0 ≤ t ≤ t1 ,

and obtain f∗ (v2 ) ≤ z2 ≤ f ∗ (v2 )

a.e. on Q(t1 ).

(3.5)

Using the definition and monotonicity of the operator ∂IK(v) we rewrite (3.3) as the variational inequality ξi (z − wi ) ≤ 0

for all z ∈ K(vi ), i = 1, 2.

Observing that in view of (3.4), (3.5)zi ∈ K(vi ), i = 1, 2, we deduce then that ξi (wi − zi ) ≥ 0

a.e. on Q(t1 ), i = 1, 2.

Consequently, we see that w1t [w − C(t + 1)ℓ(t1 )]+ ≤ F1 [w − C(t + 1)ℓ(t1 )]+ ,

}

−w2t [w − C(t + 1)ℓ(t1 )]+ ≤ −F2 [w − C(t + 1)ℓ(t1 )]+

a.e. on Q(t1 ),

Summing up these two inequalities we further obtain 1 d 2 |[w − C(t + 1)ℓ(t1 )]+ | ≤ (F1 − F2 − Cℓ(t1 ))[w − C(t + 1)ℓ(t1 )]+ 2 dt ≤(L|v| + F (v1 , w1 ) − F (v1 , w2 ) − Cℓ(t1 ))[w − C(t + 1)ℓ(t1 )]+ ≤ ((L − C)ℓ(t1 ) + LF |w|) [w − C(t + 1)ℓ(t1 )]+ ( ) C −L ≤ − (t + 1)ℓ(t1 ) + LF w [w − C(t + 1)ℓ(t1 )]+ T +1 ( ) C −L C(T + 1)LF ≤ w − C(t + 1)ℓ(t1 ) [w − C(t + 1)ℓ(t1 )]+ C(T + 1) C −L a.e. on Q(t1 ). The assumption LF < 1 (cf. Hypothesis (A4)) allows us to choose C > L and T1 such that C(T1 + 1)LF ≤ 1. C −L Indeed, setting C = Then, we infer that

N δ L

for some N > 1, where δ > 0 is such that LF = 1 − δ, we can take T1 ≤

δ(N −1) N (1−δ) .

d 2 2 |[w − C(t + 1)ℓ(t1 )]+ | ≤ C∗ |[w − C(t + 1)ℓ(t1 )]+ | a.e. on Q(t1 ) dt for 0 ≤ t1 ≤ T1 and a positive constant C∗ . The application of Gronwall’s inequality leads to 2

|[w − C(t + 1)ℓ(t1 )]+ | ≤ 0

a.e. on Q(t1 )

for 0 ≤ t1 ≤ T1 . We remark here that w(0) ≤ Cℓ(t1 ). In particular, we have w ≤ C(t + 1)ℓ(t1 )

a.e. on Q(t1 )

for 0 ≤ t1 ≤ T1 . The same estimate for −w is similarly obtained. We complete the proof of the proposition by setting M (T1 ) = C(T1 + 1). □

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For any solution (v, w) of (1.1)–(1.4) with a fixed control u ∈ Sm the estimate

Proposition 3.2.

|vt |L2 (0,T ;H) + |vx |L∞ (0,T ;H) + |vxx |L2 (0,T ;H) + |vx |L4 (Q(T )) + |vx |L2 (0,T ;L∞ (0,1)) ≤ Rm holds for a positive constant Rm which does not dependent on u. Proof . First, we show that vx ∈ L4 (Q(T )) ∩ L2 (0, T ; L∞ (0, 1)).

(3.6)

To this aim, we note that by the Gagliardo–Nirenberg inequality there exists a positive constant M∗ such that 1/4 3/4 |vx (t)|L4 (0,1) ≤ M∗ (|vxx (t)|H |vx (t)|H + |vx (t)|H ) for a.e. t ∈ [0, T ]. Hence, ∫ 0

T

4

|vx |L4 (0,1) dt ≤M∗4

∫

T

3

4

(|vxx |H |vx |H + |vx |H )dt ∫ T 3 4 ≤M∗4 (|vx |L∞ (0,T ;H) |vxx |H dt + |vx |L∞ (0,T ;H) · T ), 0

0

Since v ∈ L∞ (0, T ; V ) ∩ L2 (0, T ; H 2 (0, 1)), we see from this inequality that vx ∈ L4 (Q(T )). Then, the Sobolev embedding theorem from H 1 (0, 1) to L∞ (0, 1) implies that vx ∈ L2 (0, T ; L∞ (0, 1)). Next, setting z = G(v) we obtain ρvt − zxx = h(u, w)u

in Q(T ).

Multiplying both sides of this equality by (z−zb )t , where zb (t, x) = G(b1 (t))x+G(b0 (t))(1−x), (t, x) ∈ Q(T ), we have ∫ 1 ∫ 1 1 d 2 2 ρg0 |vt |H + |(z − zb )x |H ≤ h(v, w)u(z − zb )t dx + ρvt zbt dx 2 dt 0 0 2

2

a.e. on [0, T ]. Here, we used that vt zt = g(v)|vt | ≥ g0 |vt | and z − zb = 0 at x = 0,1. We note that |h(v, w)u(z − zb )t | ≤ |h|L∞ (R2 ) m (|g(v)vt | + |g(b0 )b0t | + |g(b1 )b1t |) ≤ |h|L∞ (R2 ) m Cg (|vt | + |b0t | + |b1t |)

a.e. on Q(T ),

where Cg = sup{|g(r)|; κ0 ≤ r ≤ M0 } with κ0 and M0 from Theorem 3.1. Furthermore, |ρvt zbt | ≤ |vt |(|g(b0 )b0t | + |g(b1 )b1t |) ≤ |vt |Cg (|b0t | + |b1t |)

a.e. on Q(T ).

Invoking Young’s inequality, from the last three inequalities we infer that ρg0 1 d 2 2 2 2 |vt |H + |(z − zb )x |H ≤ R1 (1 + m2 )(|b0t | + |b1t | + 1) 2 2 dt a.e. on [0, T ] for some positive constant R1 . Consequently, there exists a positive constant R2 depending on m only such that ∫ T 2 2 |vt |H dt + |zx (t)|H ≤ R2 for 0 ≤ t ≤ T. (3.7) 0

Also, because zxx = ρvt − h(v, w)u, we can find a positive number R3 such that ∫ T 2 |zxx |H dt ≤ R3 . 0

(3.8)

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In addition, similarly to the proof of (3.6), we get T

∫

1

∫

4

|zx | dxdt ≤ R4 ,

(3.9)

0

0

where R4 is a positive constant. Now, letting q = G−1 it is clear that 2

2

vxx = q ′′ (z)|zx | + q ′ (z)zxx ≤ Cq (|zx | + |zxx |), where Cq =

1 g0

+

1 g03

(3.10)

sup{|g ′ (r)|; κ0 ≤ r ≤ M0 }. Then (3.7)–(3.10) imply that |vx |L∞ (0,T ;H) + |vxx |L2 (0,T ;H) ≤ R5 ,

for a positive constant R5 and the proposition follows.

□

Theorem 3.2. There exists a positive constant Cm depending on m only such that for any two solutions (vi , wi , ui ), i = 1,2, of (AP ) we have 2

∫

2

|v1 (t) − v2 (t)|H + |w1 (t) − w2 (t)|H ≤ Cm

0

t

2

|u1 (τ ) − u2 (τ )|H dτ

(3.11)

for all t ∈ [0, T ]. Proof . Setting v = v1 − v2 , w = w1 − w2 , u = u1 − u2 , fi = h(vi , wi )ui , i = 1,2, from (1.1) we obtain ρvt − (g(v1 )v1x − g(v2 )v2x )x = f1 − f2

on Q(T ).

Multiplying this equation by −vxx and integrating the result we have ρ d 2 dt

∫

1

2

1

∫

|vx | dx+ 0

(g(v1 )v1x − g(v2 )v2x )x vxx dx ∫ 1 =− (f1 − f2 )vxx dx a.e. on [0, T ]. 0

(3.12)

0

To evaluate the second term of the left-hand side of (3.12) we observe that ∫

1

1

∫ (g(v1 )v1x − g(v2 )v2x )x vxx dx =

2

g(v1 )|vxx | dx

0

0

∫

1

(g(v1 ) − g(v2 ))v2xx vxx dx

+ 0

∫

1

2

2

g ′ (v1 )(|v1x | − |v2x | )vxx dx

+ 0

∫ +

1

2

(g ′ (v1 ) − g ′ (v2 ))|v2x | vxx dx

0

=: I1 + I2 + I3 + I4

a.e. on [0, T ].

(3.13)

Then, from (A2) we deduce that ∫ I1 ≥ g0

1

2

|vxx | dx 0

a.e. on [0, T ].

(3.14)

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455

Next, we note that since κ0 ≤ vi ≤ M0 on Q(T ), i = 1,2, there exists a positive constant Cg such that |g ′ (vi )| ≤ Cg , |g(v1 ) − g(v2 )| ≤ Cg |v|, |g ′ (v1 ) − g ′ (v2 )| ≤ Cg |v| a.e. on [0, T ] for i = 1,2. Furthermore, employing the Sobolev embedding theorem, H¨ older’s and Young’s inequalities we infer that ∫ 1 2Cg2 g0 2 2 2 |vx |H |v2xx |H , (3.15) |I2 | ≤ Cg |v||v2xx ||vxx |dx ≤ |vxx |H + 8 g0 0 ∫ 1 |I3 | ≤ Cg (|v1x | + |v2x |)|vx ||vxx |dx 0

4Cg2 g0 2 2 2 2 ≤ |vxx |H + (|v1x |L∞ (0,1) + |v2x |L∞ (0,1) )|vx |H , 8 g0 ∫ 1 2Cg2 g0 2 2 4 2 |I4 | ≤ Cg |v||v2x | |vxx |dx ≤ |vxx |H + |v2x |L4 (0,1) |vx |H 8 g0 0 a.e. on [0, T ]. Denoting by Lh the Lipschitz constant of h we also have ∫ 1 ∫ 1 ∫ − (f1 − f2 )vxx dx ≤ Lh m (|v| + |w|)|vxx |dx + |h|L∞ (R2 ) 0

0

(3.16) (3.17)

1

|u||vxx |dx

0 2

≤

2|h|L∞ (R2 ) 2 2L2h m2 g0 2 2 2 |vxx |H + (|vx |H + |w|H ) + |u|H 2 g0 g0

a.e. on [0, T ]. Now, taking account of (3.13)–(3.18) from (3.12) we obtain ( ) g0 ρ d 2 2 2 2 2 |vx (t)|H + |vxx (t)|H ≤ E(t)|vx (t)|H + C1 |w(t)|H + |u(t)|H 2 dt 2

(3.18)

(3.19)

for a.e. t ∈ [0, T ], where C1 is a positive constant and E(t) =

) 2C 2 4Cg2 ( g 2 2 4 |v1x (t)|L∞ (0,1) + |v2x (t)|L∞ (0,1) + |v2x (t)|L4 (0,1) g0 g0 2Cg2 2Ch2 2 + |u2xx (t)|H + , t ∈ [0, T ]. g0 g0

From Proposition 3.2 it follows that E ∈ L1 (0, T ). Therefore, applying Gronwall’s inequality to (3.19) we conclude that ( ∫ t )∫ t 2 2C1 2 2 2 exp E(τ )dτ |vx (t)|H ≤ (|w(τ )|H + |u(τ )|H )dτ, t ∈ [0, T ]. ρ ρ 0 0 Proposition 3.1 implies then that 2

t

∫

|vx (t)|H ≤ C2

0

∫

2

|u(τ )|H dτ + C2

t

0

2

|v|L∞ (0,τ ;L∞ (0,1)) dτ

2 |v|L∞ (0,t;L∞ (0,1)) ,

for a constant C2 > 0. Letting H(t) = t ∈ [0, T ] and invoking the Sobolev embedding theorem we further derive ∫ t ∫ t 2 H(t) ≤ C2 |u(τ )|H dτ + C2 H(τ )dτ, t ∈ [0, T ]. 0

0

Applying Gronwall’s inequality and Proposition 3.1 again we infer that ∫ t 2 2 2 |v(t)|L∞ (0,1) + |w(t)|L∞ (0,1) ≤ C3 |u(τ )|H dτ, 0

where C3 is a positive constant and the theorem thus follows.

□

t ∈ [0, T ],

456

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4. Existence and bang–bang principle for the control problem In this section, we prove Theorem 2.1. First, we note that with the help of the Krein–Milman theorem one can easily obtain the following inclusions ext co U(t, v, w) ⊂ U(t, v, w) ⊂ co U(t, v, w) for a.e. t ∈ [0, T ], v, w ∈ H. Hence, to prove the existence of solutions for control systems (P ), (CP ), and (EP ), in view of Definition 2.1, it is enough to show that there exists a solution to (EP ) only. To this end, let T be the operator which with each u ∈ Sm (cf. (3.1)) associates the unique solution of system (1.1)–(1.4) provided by Theorem 3.1: T (u) := (v(u), w(u)), u ∈ Sm . (4.1) Define R := {(v, w) ∈ C([0, T ]; H × H); (v, w) = T (u), u ∈ Sm } to be the set of all solutions to (1.1)–(1.4) when u ranges over Sm . It is proved in [1, Theorem 3.3] that the operator T : Sm → C([0, T ]; H × H) is weak–strong continuous. This implies, in particular, that the set R is compact in C([0, T ]; H × H). On the other hand, given Hypothesis (U) it is a routine matter to verify that the mapping co U(t, v, w) is measurable in t, for v, w ∈ H and hausH -continuous in (v, w) for a.e. t ∈ [0, T ]. Moreover, hausH (co U(t, v1 , w1 ), co U(t, v2 , w2 )) ≤ k(t)(|v1 − v2 |H + |w1 − w2 |H ) (4.2) a.e. on [0, T ], vi , wi ∈ H, i = 1,2, where k ∈ L2 (0, T ; R+ ) is from Hypothesis (U 4). Consequently, the application of [21, Proposition 8.2] guarantees the existence of a continuous mapping α : R → L2 (0, T ; H) such that α(v, w) ∈ ext co U(t, v(t), w(t)) (4.3) a.e. on [0, T ], (v, w) ∈ R. From Hypothesis (U3) we see that α(v, w) ∈ Sm , (v, w) ∈ R. Furthermore, we conclude that the superposition α ◦ T : Sm → Sm is weak–weak continuous. Since Sm is obviously convex and compact in the weak topology of the space L2 (0, T ; H), from the Schauder fixed point theorem it follows that there exists a fixed point u∗ ∈ Sm of the operator α ◦ T : u∗ = α(T (u∗ )).

(4.4)

Finally, setting (v∗ , w∗ ) := T (u∗ ), from (4.1), (4.3), and (4.4) we infer that (v∗ , w∗ , u∗ ) is a solution to system (EP ). Now, to prove the bang–bang principle for system (P ) we fix an arbitrary solution (v∗ , w∗ , u∗ ) to problem (CP ) and a number n ≥ 1. From (4.2) we deduce that for any (v, w) ∈ R there exists u ∈ co U(t, v, w) such that ( ) 1 2 2 2 |u∗ (t) − u|H < + M k 2 (t) |v∗ (t) − v|H + |w∗ (t) − w|H (4.5) n a.e. on [0, T ] for a positive constant M . Let Vn : [0, T ] × H × H → H be the multivalued mapping defined as follows Vn (t, v, w) = {u ∈ H : u satisfies inequality (4.5)}. (4.6) Furthermore, define the multivalued mapping Un : [0, T ] × H × H → H by the rule Un (t, v, w) = co U(t, v, w) ∩ Vn (t, v, w).

(4.7)

From (4.6) we see that the sets Vn (t, v, w) and Un (t, v, w) are non-empty for a.e. t ∈ [0, T ] and any (v, w) ∈ R. Moreover, the mapping Vn has convex open values. Since the mapping co U(t, v, w) is measurable in t and

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457

continuous in (v, w), from a Scorza–Dragoni type theorem for multivalued mappings from [22] and Luzin’s theorem we infer that for any ε > 0, there exists a compact set Tε ⊂ [0, T ], µ([0, T ] \ Tε ) ≤ ε, such that the restriction of co U(t, v, w) to Tε ×H ×H is lower semicontinuous and the restriction of the function k(t) to Tε is continuous, respectively. Then, from (4.5), (4.6) it follows that the restriction of the mapping Vn (t, v, w) to Tε × H × H has open graph. Hence, the restriction of Un (t, v, w) to Tε × H × H is lower semicontinuous. The same property also holds for the mapping U n (t, v, w) = Un (t, v, w), where the closure is taken in the space H. Clearly, we have that U n (t, v, w) ⊂ co U(t, v, w). Therefore, Corollary 4.1 of [23] implies that the multivalued mapping Γn : R → L2 (0, T ; H): Γn (v, w) := {u ∈ L2 (0, T ; H); u(t) ∈ U n (t, v(t), w(t)) a.e. on [0, T ]} is lower semicontinuous. As it obviously has closed decomposable values, from [21, Proposition 2.2] we further infer that there exists a continuous mapping αn : R → L2 (0, T ; H) such that αn (v, w) ∈ Γn (v, w), (v, w) ∈ R. From the definition of Γn we also see that αn (v, w) ∈ Sm , (v, w) ∈ R and ( ) 1 2 2 2 |u∗ (t) − αn (v, w)(t)|H < + M k 2 (t) |v∗ (t) − v(t)|H + |w∗ (t) − w(t)|H n

(4.8)

a.e. on [0, T ]. By [24, Theorem 0.2], there exists a continuous mapping βn : R → L2 (0, T ; H) such that βn (v, w)(t) ∈ ext co U(t, v(t), w(t)) a.e. on [0, T ],

(4.9)

1 , (4.10) n where recall that | · |ω is the weak norm on the space L2 (0, T ; H) introduced in Section 2. As above, consider now the superposition βn ◦ T which is continuous from ω-Sm to ω-Sm . The Schauder fixed point theorem implies the existence of a fixed point un of the operator βn ◦T . Setting (v(un ), w(un )) := T (un ) we have un = βn (v(un ), w(un )), n ≥ 1, and in view of (4.9) the triple {v(un ), w(un ), un } yields a solution to Problem (EP ). The weak–strong continuity of the operator T : Sm → C([0, T ]; H × H) implies the existence of a subsequence (without relabeling) {v(un ), w(un ), un }, n ≥ 1, such that |αn (v, w) − βn (v, w)|ω <

un = βn (v(un ), w(un )) → u in ω-L2 (0, T ; H), (v(un ), w(un )) → (v(u), w(u))

in C([0, T ]; H × H).

(4.11) (4.12)

Invoking Lemma 2.1, from (4.10), (4.11) we infer that zn := αn (v(un ), w(un )) → u in ω-L2 (0, T ; H).

(4.13)

Hence, (v(zn ), w(zn )) := T (zn ) → (v(u), w(u))

in C([0, T ]; H × H).

From (3.11), (4.8) we now deduce that 2

2

|v∗ (t) − v(un )(t)|H + |w∗ (t) − w(un )(t)|H ∫ t ( ) M1 C m 2 2 ≤ k 2 (τ ) |v∗ (τ ) − v(zn )(τ )|H + |w∗ (τ ) − w(zn )(τ )|H dτ + M M1 Cm n 0 2 2 + |v(zn )(t) − v(un )(t)|H + |w(zn )(t) − w(un )(t)|H

(4.14)

458

S.A. Timoshin and T. Aiki / Nonlinear Analysis: Real World Applications 47 (2019) 446–459

for a constant M1 > 0. Passing in this inequality to the limit as n → ∞ and taking account of (4.12), (4.14) we have that 2

2

|v∗ (t) − v(u)(t)|H + |w∗ (t) − w(u)(t)|H ∫ t ( ) 2 2 ≤ M M1 Cm k 2 (τ ) |v∗ (τ ) − v(u)(τ )|H + |w∗ (τ ) − w(u)(τ )|H dτ. 0

Then, the application of Gronwall’s inequality yields v∗ = v(u),

w∗ = w(u).

(4.15)

Hence, (4.12) implies that (v(un ), w(un )) → (v∗ , w∗ )

in C([0, T ]; H × H).

Finally, with the help of Lebesgue’s dominated convergence theorem from (4.8), (4.11), (4.13)–(4.15) we conclude that un → u∗ in ω-L2 (0, T ; H). The last two convergences complete the proof of Theorem 2.1. Acknowledgments The authors want to thank the anonymous referees for their valuable suggestions and remarks which helped to improve the manuscript. References [1] T. Aiki, S.A. Timoshin, Relaxation for a control problem in concrete carbonation modeling, SIAM J. Control Optim. 55 (6) (2017) 3489–3502. [2] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, in: Appl. Math. Sci, vol. 121, Springer-Verlag, New York, 1996. [3] P. Krejˇ c´ı, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, in: Gakuto Int. Ser. Math. Sci. Appl, vol. 8, Tokyo, 1996. [4] A. Visintin, Differential Models of Hysteresis, in: Appl. Math. Sci, vol. 111, Springer-Verlag, Berlin, 1994. [5] T. Aiki, K. Kumazaki, Mathematical modelling of concrete carbonation process with hysteresis effect, RIMS, Kyoto Univ. s¯ urikaisekikenky¯ usho, k¯ oky¯ uuroku 1792 (2012) 99–107. [6] A. Muntean, A Moving-Boundary Problem: Modeling, Analysis and Simulation of Concrete Carbonation (Ph.D. thesis), Cuvillier Verlag, G¨ otingen, 2006, Faculty of Mathematics, University of Bremen, Germany. [7] A. Muntean, M. B¨ ohm, A moving-boundary problem for concrete carbonation: Global existence and uniqueness of solutions, J. Math. Anal. Appl. 350 (1) (2009) 234–251. [8] T. Aiki, A. Muntean, Existence and uniqueness of solutions to a mathematical model predicting service life of concrete structures, Adv. Math. Sci. Appl. 19 (2009) 109–129. [9] T. Aiki, A. Muntean, Large time behavior of solutions to concrete carbonation problem, Commun. Pure Appl. Anal. 9 (2010) 1117–1129. √ [10] T. Aiki, A. Muntean, A free-boundary problem for concrete carbonation: Rigorous justification of t-law of propagation, Interfaces Free Bound. 15 (2012) 167–180. [11] T. Aiki, A. Muntean, Large-time asymptotics of moving-reaction interfaces involving nonlinear Henry’s law and timedependent Dirichlet data, Nonlinear Anal. 93 (2013) 3–14. [12] K. Maekawa, T. Ishida, T. Kishi, Multi-scale modeling of concrete performance, J. Adv. Concr. Technol. 1 (2003) 91–126. [13] T. Aiki, K. Kumazaki, Mathematical model for hysteresis phenomenon in moisture transport of concrete carbonation process, Physica B 407 (2012) 1424–1426. [14] T. Aiki, S.A. Timoshin, Existence and uniqueness for a concrete carbonation process with hysteresis, J. Math. Anal. Appl. 449 (2) (2017) 1502–1519. [15] T. Aiki, K. Kumazaki, Well-posedness of a mathematical model for moisture transport appearing in concrete carbonation process, Adv. Math. Sci. Appl. 21 (2011) 361–381. [16] K. Kumazaki, T. Aiki, Uniqueness of a solution for some parabolic type equation with hysteresis in three dimensions, Netw. Heterog. Media 9 (2014) 683–707. [17] A.A. Tolstonogov, Properties of solutions of a control system with hysteresis, J. Math. Sci. (N. Y.) 196 (3) (2014) 405–433.

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