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Necessary conditions for a nonclassical control problem with state constraints*
Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World The Federation of Automatic Control Toulouse, France, July 2017 The International International Federation of Congress Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 506–511 Necessary conditions for a Necessary conditions for a Necessary conditions for a control problem with state control problem with state control problem with∗ state
nonclassical nonclassical nonclassical constraints constraints constraints ∗∗
Chems Eddine Arroud ∗ Giovanni Colombo ∗∗ Chems Chems Eddine Eddine Arroud Arroud ∗ Giovanni Giovanni Colombo Colombo ∗∗ Chems Eddine Arroud ∗ Giovanni Colombo ∗∗ ∗ Department of Mathematics, Jijel University, Jijel, Algeria, and Mila ∗ ∗ Department of Mathematics, Jijel University, Jijel, Algeria, and Mila of Mathematics, Jijel (e-mail: University, Jijel, Algeria, and Mila University Center, Mila, Algeria [email protected]) ∗ Department Department of Mathematics, Jijel (e-mail: University, Jijel, Algeria, and Mila ∗∗ University Center, Mila, Algeria [email protected]) University Center, Mila, Algeria (e-mail: [email protected]) Dipartimento di Matematica “Tullio Levi-Civita”, University of ∗∗ University Center, Mila, Algeria (e-mail: [email protected]) ∗∗ Dipartimento di Matematica “Tullio Levi-Civita”, University Dipartimento di Matematica “Tullio Levi-Civita”, University of of Padova, Padova, Italy (e-mail: [email protected]). ∗∗ Dipartimento di Matematica “Tullio Levi-Civita”, University of Padova, Padova, Italy (e-mail: [email protected]). Padova, Padova, Italy (e-mail: [email protected]). Padova, Padova, Italy (e-mail: [email protected]). Abstract: We consider the problem of minimizing the cost h(x(T )) at the endpoint of a Abstract: consider problem of Abstract:x We We consider the problem of minimizing minimizing the cost cost h(x(T h(x(T )) )) at at the the endpoint endpoint of of aa trajectory subject to thethe finite dimensional dynamicsthe Abstract:x consider problem of minimizing trajectory subject to finite dimensional dynamics trajectory x We subject to the thethe finite dimensional dynamicsthe cost h(x(T )) at the endpoint of a x˙ ∈ dimensional −NC (x) + fdynamics (x, u), x(0) = x0 , trajectory x subject to the finite x (x) + f (x, u), x(0) x(0) = =x x0 ,, x˙˙ ∈ ∈ −N −NC C (x) + f (x, u), where NC denotes the normalx˙ cone to the convex set C. Such inclusion is termed, ∈ −NC (x) + f (x, u), x(0) = x00differential , where N denotes the normal cone to the convex set C. Such inclusion is whereMoreau, NC the normal cone the convex C. Such differential differential inclusion with is termed, termed, after sweeping process. Wetolabel it as a set “nonclassical” control problem state C denotes where N denotes the normal cone to the convex set C. Such differential inclusion is termed, C after Moreau, sweeping process. We label it as a “nonclassical” control problem with after Moreau, sweeping We side labelisitdiscontinuous as a “nonclassical” controltoproblem with state constraints, because the process. right hand with respect the state, andstate the after Moreau, sweeping We side labelis as a “nonclassical” controlto with constraints, because the right hand with the and the constraints, because theallprocess. right hand side isitdiscontinuous discontinuous with respect respect toproblem the state, state, andstate the constraint x(t) ∈ C for t is implicitly contained in the dynamics. constraints, because theall hand sidecontained is discontinuous with respect to the state, and the constraint x(t) ∈ tt is in constraint x(t) ∈C C for for allright is implicitly implicitly contained in the theofdynamics. dynamics. We prove necessary optimality conditions in the form Pontryagin Maximum Principle by constraint x(t) ∈ C for all t is implicitly contained in theof We prove necessary optimality conditions in Pontryagin Maximum by We prove essentially, necessary optimality conditions in the form ofdynamics. Pontryagin Maximum Principle by requiring, that C is independent of the time.form If the reference trajectory is inPrinciple the interior We prove necessary optimality conditions in the form of Pontryagin Maximum Principle by requiring, essentially, that C is independent of time. If the reference trajectory is in the interior requiring, essentially, that C is independent time.ones. If the trajectory is adjoint in the interior of C, necessary conditions coincide with theofusual Inreference the general case, the vector requiring, essentially, that C is independent ofusual time.ones. If the reference trajectory is adjoint in the interior of C, necessary conditions coincide with the In the general case, the vector of C, necessary conditions coincide with the usual ones. In the general case, the adjoint vector is a BV function and a signed vector measure appears in the adjoint equation. of a necessary coincide the usual ones.in the general case, the adjoint vector is function and vector measure appears the adjoint equation. is aC,BV BV functionconditions and a a signed signed vectorwith measure appears inIn the adjoint equation. is BV IFAC function and a signed vectorof measure the adjoint equation. © a2017, (International Federation Automaticappears Control)inHosting by Elsevier Ltd. All rights reserved. Keywords: Moreau’s sweeping process, Optimal control, Pontryagin Maximum Principle. Keywords: Keywords: Moreau’s Moreau’s sweeping sweeping process, process, Optimal Optimal control, control, Pontryagin Pontryagin Maximum Maximum Principle. Principle. Keywords: Moreau’s sweeping process, Optimal control, Pontryagin Maximum Principle. 1. INTRODUCTION the final cost h being smooth. There is a clear difference 1. the cost h smooth. There is clear 1. INTRODUCTION INTRODUCTION the final final cost control h being beingproblems smooth. with Therestate is aaconstraints clear difference difference with classical (see, 1. INTRODUCTION the final cost control h beingproblems smooth. with Therestate is aconstraints clear difference with classical (see, with classical control problems with state constraints (see, Vinter (2000)), where the constraint does not appear The sweeping process was introduced by Moreau in the e.g., with classical controlwhere problems with state constraints (see, e.g., Vinter (2000)), the constraint does not appear The sweeping process was introduced by Moreau in the e.g., Vinter (2000)), where the constraint does not appear explicitly in the dynamics: in this case the right hand side The sweeping process was introduced by Moreau in the Seventies as a model for dry friction and plasticity (see e.g., Vinter (2000)), where the constraint does not appear The sweeping process was introduced by Moreau in the explicitly in the dynamics: in this case the right hand side Seventies as for friction and explicitly in the dynamics: in thiswith caserespect the right side the dynamics is not Lipschitz to hand the state Seventies as aa model model for dry dry friction and plasticity plasticity (see Moreau (1974)) and later studied by several authors.(see In of explicitly in the dynamics: in this case the right hand side Seventies as a model for dry friction and plasticity (see of the dynamics is not Lipschitz with respect to the state Moreau (1974)) and later studied by several authors. In of the dynamics is not withgraph. respectThis to the but indeed hasLipschitz only closed factstate is a Moreau (1974)) and later studiedthebydifferential several authors. In variable, its perturbed version, it features inclusion of the dynamics is not Lipschitz with respect to the state Moreau (1974)) and later studiedthe several authors. In variable, but has closed This fact its version, it inclusion variable, but indeed indeed has only only closed graph. graph. This fact is is a a source of major difficulties in deriving necessary optimality its perturbed perturbed version, it features features thebydifferential differential inclusion but indeed has only closed graph. This fact is a + f (x(t)), t ∈ [0, T ] inclusion (1) variable, x(t) ˙ ∈ −N its perturbed version, it features the differential source of major difficulties in deriving necessary optimality C(t) (x(t)) source of major difficulties in deriving necessary optimality conditions for (3), (4). (1) x(t) ˙˙ ∈ x(t) ∈ −N −NC(t) (x(t)) (x(t)) + + ff (x(t)), (x(t)), tt ∈ ∈ [0, [0, T T ]] (1) source of major difficulties in deriving necessary optimality for (4). conditions for (3), (3), (4). t ∈ [0, T ] (1) conditions x(t) ˙ ∈ −NC(t) C(t) (x(t)) + f (x(t)), conditions for (3),(see, (4). e.g., Bagagiolo (2002), Gudovich In recent years coupled with the initial condition In recent years (see, Bagagiolo (2002), Gudovich coupled In al. recent years (see, ete.g., e.g., Bagagiolo (2002), Gudovich coupled with with the the initial initial condition condition (2011), Brokate al. (2013), Colombo et al. (2016), (2) et x(0) =condition x0 ∈ C(0). In recent years (see, et e.g., Bagagiolo (2002), Gudovich coupled with the initial et al. (2011), Brokate al. (2013), Colombo et al. (2016), et al. (2011), Brokate et al. (2013), Colombo et al. (2016), Colombo et al. (2016), Arroud et al. (2016), and Cao ∈ C(0). (2) x(0) = x (2) et x(0) = x00 ∈ C(0). al. (2011), Brokate et al. (2013), Colombo et al. (2016), Colombo et al. (2016), Arroud et al. (2016), and Cao Colombo et al. (2016), Arroud et al. (2016), and Cao (2) et x(0) = x0 ∈ C(0). al. (2017), and references therein) some papers dealing Here C(t) is a closed moving set, with normal cone Colombo et al. (2016), Arroud et al. (2016), and Cao et al. (2017), and references therein) some papers dealing et al. (2017), and references therein) some papers dealing with control problems involving the sweeping process were Here C(t) is a closed moving set, with normal cone Here C(t) is a closed moving set, with normal cone NC(t) (x) at x ∈ C(t). The space variable, in this paper, et al. (2017), and references therein) some papers dealing with control problems involving the sweeping process were with controlthe problems theinsweeping process were Here C(t)at isx a C(t). closedThe moving with in cone published, controlinvolving appearing the perturbation f (x) space variable, this N C(t) at R xn .∈ ∈ space set, variable, innormal this paper, paper, N with controlthe problems involving thein sweeping process were C(t) (x)to IfC(t). C(t)The is convex, or mildly non-convex belongs published, control appearing the perturbation ff published, the control appearing in the perturbation n (x) at x ∈ C(t). The space variable, in this paper, N and/or in the moving set C. Several necessary conditions C(t) n . If C(t) C(t) is be convex, or mildlyhere), non-convex belongs to R Rthat convex, mildly non-convex belongs to themoving controlset appearing in necessary the perturbation f (in a sense notis made or precise and is published, n . Ifwill and/or in the C. Several conditions and/or in the moving set C. Several necessary conditions established, under different kinds of assumptions, or . Ifwill C(t) convex, or mildlyhere), non-convex belongs to Rthat (in a not be made precise and is (in a sense sense will notis map be made precise here), andthe is were and/or in the moving set C. Several necessary conditions Lipschitz as that a set-valued depending on t, and established, different of or were established, under under different kinds kinds of assumptions, assumptions, or Hamilton-Jacobi characterization of value function was (in a senseas will not map be made precise on here), andthe is awere Lipschitz a depending t, Lipschitz as that af set-valued set-valued depending t, and and the were established, under different kinds of assumptions, or perturbation is Lipschitzmap as well, then it on is well known aproved. Hamilton-Jacobi characterization of value function was a Hamilton-Jacobi characterization of value function was The present paper is devoted to prove a result Lipschitz as a set-valued map depending on t, and the perturbation ff is as then perturbation is Lipschitz Lipschitz as well, well, then it it is is well known Hamilton-Jacobi characterization of value function was that the Cauchy problem (1), (2) admits onewell andknown only aproved. The present paper is devoted to prove a result proved. The present is devoted to prove result by Arroud et paper al. (2016) and Brokate et al.a(2013). perturbation f is Lipschitz as well, then it is known that the problem (1), (2) one and only thatLipschitz the Cauchy Cauchy problem (1), (2) admits admits onewell and only inspired proved. The present is devoted to prove a(2013). result one solution (see, e.g., Thibault (2003)). Observe inspired by Arroud Arroud et paper al. (2016) (2016) andconditions Brokate etofal. al.Pontrya(2013). inspired by et al. and Brokate et More precisely, we prove necessary that the Cauchy problem (1), (2) admits one and only one Lipschitz solution (see, e.g., Thibault (2003)). Observe one Lipschitz (see,x(t) e.g.,∈Thibault by Arroud et al. (2016) and Brokate et al. (2013). that the statesolution constraint C(t) for(2003)). all t ∈ Observe [0, T ] is inspired More precisely, we necessary of More precisely,principle we prove provetype necessary conditions of PontryaPontryamaximum for (4) conditions subject to (3) and (2), one Lipschitz solution (see,x(t) e.g.,∈ Thibault that state constraint C(t) for all [0, T that the state constraint x(t) C(t) for(2003)). all ttif ∈ ∈x Observe [0, T ]] is is gin More precisely,principle we provetype necessary conditions of Pontryabuilt the in the dynamics, being N∈ (x) empty ∈ C(t): C(t) gin maximum for (4) subject to (3) and (2), gin maximum principle type for (4) subject to (3) andC(·) (2), the control appearing only within f , in the case where that the state constraint x(t) ∈ C(t) for all t ∈ [0, T ] is built in inathe the dynamics, beingthen NC(t) (x) empty empty if ifx(t) x ∈ ∈∈C(t): C(t): built dynamics, being N (x) x gin maximum principle type for (4) subject to (3) and (2), C(t) should solution x(·) exist, automatically C(t) the control appearing only within f , in the case where C(·) the control appearing only within f , in the case where C(·) is constant, smooth and convex (see Theorems 2 and 3). built in the dynamics, being N (x) empty if x ∈ C(t): C(t) should a x(·) exist, x(t) C(t) theconstant, control appearing onlyconvex within(see f , inTheorems the case where C(·) should a solution solution x(·) parameter exist, then then automatically automatically x(t)f∈ ∈, then C(t) is for all t. If a control u appears within smooth and 2 and 3). is constant, smooth and convex (see Theorems 2 and 3). The case where C satisfies milder convexity assumptions should a solution x(·) exist, then automatically x(t) ∈ C(t) for t. a parameter appears constant, smooth and convex (see Theorems 2 and 3). for all all t. If If to a control control parameter uthe appears within ff ,, then then is one is lead study problems of u type within The case where C satisfies milder convexity assumptions The case where C satisfies milder convexity assumptions and is not necessarily constant was treated in Arroud et al. for all t. If to a control parameter appears one is study of type case where C satisfies milder convexity assumptions one is lead lead to study problems problems of uthe the type within f , then The and is not necessarily constant was treated in Arroud et al. and is not necessarily constant was treated in Arroud et al. (2016) with an extra assumption, while Brokate et (x(t)) + f (x(t), u(t)), u(t) ∈ U (3) x(t) ˙ lead ∈ −N one is to study problems of the type C(t) and is not necessarily constant was treated in Arroud et (2016) with an extra assumption, while Brokate et al. (x(t)) + f (x(t), u(t)), u(t) ∈ U (3) ˙x(t) x(t) ∈ −N (2016) contains with an results extra assumption, whilecontrol Brokate et al. al. u(t) ∈ U (3) (2013) ˙ ∈ −NC(t) for a particular problem C(t) (x(t)) + f (x(t), u(t)), with an results extra assumption, whilecontrol Brokate et al. (3) (2016) x(t) ˙ ∈ −NC(t) (x(t)) + f (x(t), u(t)), u(t) ∈ U (2013) contains for a particular problem (2013) contains results for a particular control problem involving a fixed smooth and uniformly convex set C. subject to (2), aiming, for example, at (2013) contains for and a particular control problem involving aa fixed smooth uniformly C. subject involving fixedresults smooth and uniformly convex set C. More preccisely, differently from Colomboconvex et al. set (2016) subject to to (2), (2), aiming, aiming, for for example, example, at at involving a fixed smooth and uniformly convex set C. minimizing h(x(T )), (4) More subject to (2), aiming, for example, at preccisely, differently from Colombo et al. (2016) More preccisely, differently from Colombo et al. (2016) and Cao et al. (2017), where discrete approximations are minimizing h(x(T h(x(T )), )), (4) More preccisely, differently from Colombo et al. (2016) minimizing (4) and Cao et al. (2017), where discrete approximations are and Cao al. (2017), in et both Brokatewhere et al.discrete (2013) approximations and Arroud et are al. The second author minimizing )), by Padova University (4) used, is partially h(x(T supported and Cao et al. (2017), where approximations are used, Brokate al. (2013) and al. The second author is partially supported by Padova University used, in both Brokate et al.discrete (2013) technique. and Arroud Arroud etclasal. (2016)in theboth authors use a et penalization Theet project PRAT2015 “Control of dynamics with active constraints” The second author is partially supported by Padova University used, in both Brokate et al. (2013) technique. and Arroud etclasal. The second author is partially supported by Padova University (2016) the authors use a penalization technique. The clas(2016) the authors use a penalization The project PRAT2015 “Control of dynamics with active constraints” sical Moreau-Yosida regularization allows in Arroud et al. and by Istituto Nazionale di Alta Matematica. project PRAT2015 “Control of dynamics with active constraints” (2016) the authors use a penalization technique. The classical Moreau-Yosida regularization allows in Arroud et al. project PRAT2015 “Control of dynamics with active constraints” and sical Moreau-Yosida regularization allows in Arroud et al. and by by Istituto Istituto Nazionale Nazionale di di Alta Alta Matematica. Matematica. sical Moreau-Yosida regularization allows in Arroud et al. and by Istituto Nazionale di Alta Matematica.
Copyright © 2017, 2017 IFAC 508Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 508 Copyright ©under 2017 responsibility IFAC 508Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 508 10.1016/j.ifacol.2017.08.110
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511
507
(2016) to relax the uniform convexity assumption, at the price of requiring a strong outward pointing condition on f in order to treat the discontinuity of second derivatives of the squared distance function at the boundary of C(t). In Brokate et al. (2013), the authors adopt a suitable smoothing of the distance, which on one hand needs C(t) constant and uniformly convex and 0 ∈ C, while on the other avoids imposing further compatibility assumptions between f and C. In this paper we adapt to our situation the method developed in Brokate et al. (2013) and remove the assumption of strict convexity on C. The main technical part is Section 4.
where ε > 0 and u(t) ∈ U for all t. For each given u, this Cauchy problem admits a unique solution xε for each ε > 0 on a maximal interval of existence. It is not difficult to prove that this interval is [0, T ] (see the proof of Proposition 1).
2. PRELIMINARIES AND ASSUMPTIONS
over controls u, where x is a solution of (8). By standard results, Pε (u∗ ) admits a global minimizer uε , with the corresponding solution xε . Necessary conditions of the original problem will be obtained by passing to the limit along conditions for Pε (u∗ ).
Notation. We define the distance from a set C ⊂ Rn as d(x) = inf{ y − x : y ∈ C} and signed distance from C as dS (x) = d(x) if x ∈ C and dS (x) = − inf{ y − x : y ∈ C} if c ∈ C. The normal cone to a convex set C is defined as NC (x) = ∅ if x ∈ C and NC (x) = {v ∈ Rn : v, y − x ≤ 0 ∀y ∈ C} if x ∈ C. Assumptions on the set C. Let n
C = {x ∈ R : g(x) ≤ 0}, where g : Rn → R is of class C 2 with gradient ∇g = 0 on the boundary ∂C of C, and with the Hessian matrix ∇2 g(x) positive semidefinite for all x ∈ Rn . Assume furthermore that g(·) is coercive, so that C is compact (and convex) and that g(0) < 0, so that 0 ∈ C and C has nonempty interior. Observe that under our assumption the signed distance dS (x) from C is of class C 2 in a neighborhood of ∂C. Assumptions on the dynamics and the cost. The control set U ⊂ Rn is compact and f is continuous and bounded, say by a constant β, and is of class C 1 with respect to x, with ∇x f (x, u) ≤ L for all x, u. The cost h is smooth. 2
Let now ψ(x) be a C smoothing of dS in the interior of C (which is < 0 in int C and is such that ∇ψ(x) is the unit external normal to C at x for every x ∈ ∂C). Set also 1 Ψ(x) = ψ 3 (x) 1(0,+∞) (ψ(x)). 3 Observe that Ψ(·) is of class C 2 and convex in the whole of Rn and that both ∇Ψ(·) and ∇2 Ψ(·) vanish on C. Moreover one has d(x)∇ψ(x) = d(x)∇d(x),
(5)
∇Ψ(x) = d2 (x)∇d(x),
(6)
∇2 Ψ(x) = 2d(x)∇d(x) ⊗ ∇d(x) + d2 (x)∇2 d(x),
(7)
because in C, and in particular at the points where ∇d(x) does not exist (namely, in ∂C), both sides of the above expressions vanish, and outside C they coincide.
1 minimize h(x(T )) + 2
T 0
u(t) − u∗ (t) 2 dt,
(8) 509
(9)
3.1 A priori estimates for the regularized problem Proposition 1. Let εn → 0 and let (un , xn ) be a solution of the problem Pεn . Then, up to a subsequence, un converges strongly in L2 (0, T ) to u∗ and xn converges weakly in W 1,2 (0, T ) to x∗ . Proof. Since 0 ∈ C and so ∇Ψ(0) = 0, by the convexity of Ψ we obtain that ∇Ψ(x), x ≥ 0 for all (x). Thus
=
t 0
xn (t) − x0 =
x (t) −1 n , ∇Ψ(xn (t)) + f (xn (t), un (t)) dt ≤ β, xn (t) εn
which, in particular, implies that xn is defined in the whole of [0, T ]. Moreover, x˙ n 2L2 =
≤
T
x˙ n (t),
0
T 0
−1 ∇Ψ(xn (t)) + f (xn (t), un (t)) dt εn
−1 d Ψ(xn (t)) + f (xn (t), un (t)), x˙ n (t) dt εn dt −1 1 = Ψ(xn (T )) + Ψ(0) + β εn εn
T 0
x˙ n (t) dt
√ ≤ β T x˙ n L2 ,
where we have used the fact that 0 ∈ C and that ψ(xn (T )) ≥ 0. The above estimate implies that the sequence x˙ n is uniformly bounded in L2 (0, T ). Thus, up to a subsequence, xn converges weakly in W 1,2 (0, T ) to x ¯. Observe now that the uniform boundedness of x˙ n L2 (0,T ) together with (6), recalling (5) imply that d(xn (·))2 L2 (0,T ) ≤ Kεn
3. THE REGULARIZED PROBLEM Consider the regularized dynamics −1 x(t) ˙ = ∇Ψ(x(t)) + f (x(t), u(t)), x(0) = x0 , ε
For every ε > 0 and every global minimizer x∗ , u∗ of (4) subject to (3) and (2), we consider the approximate problem Pε (u∗ )
(10)
for a suitable constant K. Thus x(t) ∈ C for all t. Again up to a subsequence, un converges weakly in L2 (0, T ) to some u ¯. By using the very same argument of Proposition 4.3 in Arroud et al. (2016), one can prove that x ¯ is the
Proceedings of the 20th IFAC World Congress 508 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511 Toulouse, France, July 9-14, 2017
solution of (3), (2) corresponding to u ¯, that u ¯ = u∗ , and so that x ¯ = x∗ , and that the convergence is indeed strong. Remark. Eq. (10) implies that, up to a subsequence, √ √ (11) d(xn (·)) L2 (0,T ) ≤ T K εn . From the uniform convergence of xn (again up to a subsequence) we also get d(xn (·)) L∞ → 0 for n → ∞. This is an important difference between this approach and the use of Moreau-Yosida approximation, which instead yields the stronger estimate d(xn (·)) L∞ ∼ εn (see Sene et al. (2014) or (Arroud et al., 2016, Proposition 4.1)). Observe that the assumption that C is constant appears essential in order to obtain uniform a priori estimates for x˙ n L2 within the present approach, which essentially uses the weaker penalization d3 instead of the Moreau-Yosida one, namely d2 . 3.2 Necessary conditions for the regularized problem The approximate problem Pε (u∗ ) satisfies the assumptions for necessary conditions of classical unconstrained optimal control problems. The same computations of Section 6 in Arroud et al. (2016) yield that for every ε and every minimizer (uε , xε ) there exists an absolutely continuous adjoint vector pn : [0, T ] → Rn such that −1 ∇2 Ψ(xε (t)) + ∇x f (xε (t), uε (t)) pε (t) (12) −p˙ε (t) = ε a.e. on [0, T ], together with the final condition −pε (T ) = ∇h(xε (T )),
(13)
and the maximality condition
Recalling (13), we obtain from the above inequality and Gronwall’s lemma that there exists a constant K1 independent of n such that (14) pn ∞ ≤ K1 . Now we address ourselves to prove an a priori estimate on p˙ n L1 . To this aim, we define ξn (t) = pn (t), ∇ψ(xn (t)), so that, ξ˙n (t) = p˙ n (t), ∇ψ(xn (t)) + pn (t), ∇2 ψ(xn (t))x˙ n (t). Set now δn (t) = d(xn (t)), δn (t) = ∇d(xn (t)), δn (t) = ∇2 d(xn (t)). With this notation, thanks to (5), (6), and (7), eq. (12) can be rewritten as δn2 (t) δn (t)ξn (t) δn (t)pn (t) − 2 δn (t) + εn εn +∇x f (xn (t), un (t))pn (t). (15) Inserting p˙n and x˙ n in the expression for ξ˙n we obtain (omitting the t-dependence and using the fact that ∇2 d(x) ∇d(x) = 0 for all x where it makes sense, and that δn ∇ψ(xn ) 2 = δn ∇d(x)n) = δn ) −p˙ n (t) = −
δn ξn δ2 = − n δn pn , δn + εn εn +∇x f (xn , un )pn , ∇ψ(xn ) − pn , ∇2 ψ(xn )f (xn , un ). Observe now that the first summand in the right hand side of the above expression is bounded in L1 (0, T ) uni−ξ˙n + 2
δ 2
formly with respect to n, because, recalling (11), nεnL1 is uniformly bounded. In turn, the second and the third summands are seen to be bounded in L∞ (0, T ), uniformly with respect to n, by invoking (14). By multiplying both sides by sign(ξn ), we thus obtain the second estimate T 1 δn (s)|ξn (s)| ds ≤ K2 , (16) εn
pε (t), ∇u f (xε (t), uε (t))uε (t) − uε (t) − u∗ (t), uε (t) =
max{pε (t), ∇u f (xε (t), uε (t))u − uε (t) − u∗ (t), u} u∈U
for a.e. t ∈ [0, T ]. 4. PASSING TO THE LIMIT
t
From now on we consider a sequence εn → 0 such that the minimum (un , xn ) of the approximate problem converges as in the statement of Proposition 1. We set pn := pεn . 4.1 A priori estimates for the adjoint vectors of the approximate problem
for a suitable constant K3 independent of n.
We obtain from (12) and (7) that
4.2 Passing to the limit along the adjoint equation
T
∇2 Ψ(xn (s))pn (s), pn (s) pn (s) t pn (t) + ∇x f (xn (t), un (t))pn (t), dt ≤ pn (t)
−1 pn (t) − pn (T ) = εn
for a suitable constant K2 , independent of n. As a consequence, all three summands in the right hand side of the adjoint equation (15) are bounded in L1 (0, T ), uniformly with respect to n, and so we reach our final estimate (17) p˙ n L1 (0,T ) ≤ K3
By possibly extracting a further subsequence, we can assume that the sequence of measures p˙ n dt converges weakly∗ in the sense of Radon measures to a signed vector measure µ, which is the distributional derivative of the BV function p(t) =: lim pn (t), i.e., µ = dp, where the limit of the pn is pointwise in [0, T ]. By arguing as in (Arroud et al., 2016, Proposition 7.3), we obtain first that the sequence of measures δn (t)ξn (t) δn (t) dt εn
(since ∇2 Ψ is positive semidefinite and ∇x f is bounded) T ≤ L pn (t) dt. t
510
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511
converges weakly∗ to a finite signed vector Radon measure, which can be written as ξ(t)n∗ (t) dν, where ξ ∈ L1ν (0, T ), ξ ≥ 0 ν-a.e., n∗ (t) denotes the unit outward normal vector to C at x∗ (t) if x∗ (t) ∈ ∂C and 0 if x∗ (t) ∈ int C, and ν is a finite vector measure. Moreover δn2 (t) δ (t)pn (t) η(t)∇2 d(x∗ (t))p(t) εn n
in L2 (0, T ), where η ∈ L∞ ν (0, T ) and η ≥ 0 a.e., with η ≡ 0 when x∗ is in int C. 4.3 Passing to the limit along the maximality condition By taking into account Proposition 1, we obtain that the limit adjoint vector p is such that p(t), ∇u f (x∗ (t), u∗ (t))u∗ (t) =
= max{p(t), ∇u f (x∗ (t), u∗ (t))u} for a.e. t ∈ [0, T ]. (18) u∈U
5. THE MAIN RESULT We deduce from Sections 3 and 4 the following necessary conditions: Theorem 2. Under the assumptions stated in Section 2, let (x∗ , u∗ ) be a global minimizer for (4) subject to (3) and to (2). Then there exist a BV adjoint vector p : [0, T ] → Rn , together with a finite signed Radon measure ν on [0, T ], and measurable vectors ξ, η : [0, T ] → Rn (with ξ ∈ L1ν (0, T ), ξ(t) ≥ 0 for ν-a.e. t, η ∈ L∞ (0, T ), and η(t) ≥ 0 for a.e. t) such that ξ(t) = η(t) = 0 for all t with x∗ (t) ∈ int C, satisfying the following properties: • (adjoint equation)
− +
ϕ(t), ∇2x d(x∗ (t))p(t)η(t) dt
[0,T ]
ϕ(t), ∇x f (x∗ (t), u∗ (t))p(t) dt,
[0,T ]
The following is the second part of our necessary conditions. It is only a partial result, with respect to the rich set of conditions proved in Brokate et al. (2013). Theorem 3. Under the assumptions stated in Section 2, let (x∗ , u∗ ) be a global minimizer for (4) subject to (3) and to (2), and let p the adjoint vector given by Theorem 2. Define pN (t) = p(t), n∗ (t), t ∈ [0, T ]. The following properties hold: (1) pN (t) = 0 for all t ∈ E0 , and p is absolutely continuous on E0 , where it satisfies the classical adjoint equation −p(t) ˙ = ∇x f (x∗ (t), u∗ (t)) p(t). (19) (2) At every interior (or such that a left or right neighborhood is contained in E∂ ) point t of E∂ , jumps of p may occur only in the normal direction n∗ (t), namely p(t−) − p(t+) = pN (t−) − pN (t+) n∗ (t). (3) The adjoint vector p is absolutely continuous on every open interval contained in E∂ , and for a.e. t in such interval we have −p(t) ˙ = n˙ ∗ (t), p(t)n∗ (t) + Γ(t)p(t) −
−Γ(t)p(t), n∗ (t)n∗ (t), (20) where Γ(t) = ∇x f (x∗ (t), u∗ (t)) − η(t)∇2x d(x∗ (t)).
t+σ t+σ T − ϕ (s), dp(s) + ϕT (s), n∗ (s)ξ(s) dν
• (transversality condition)
−p(T ) = ∇h(x∗ (T )),
t−σ
t−σ
t+σ ϕT (s), ∇x f (x∗ (s), u∗ (s))p(s) ds =
• (maximality condition) p(t), ∇u f (x∗ (t), u∗ (t))u∗ (t) =
u∈U
E∂ := {t ∈ [0, T ] : x∗ (t) ∈ ∂C}. Of course, E0 is open and E∂ is closed, but one has to take into account the possibility that E∂ be irregular (e.g., totally disconnected). Such phenomenon, in stratified state constrained control theory, is sometimes referred to as Zeno phenomenon, namely the switching from a stratum (the boundary of C, in this case) to other strata (the interior of C in this case) occurs at a complicated set (see, e.g., Barnard et al. (2013)).
(2). Since n∗ (t) is continuous on E∂ , there exist n − 1 continuous unit vectors v1 (t),. . . , vn−1 (t) such that Rn = Rn∗ (t) ⊕ span v1 (t), . . . , vn−1 (t) for all t ∈ E∂ . Let t ∈ E∂ and σ > 0 be such that [t − σ, t + σ] ⊂ E∂ . Let ϕ : [0, T ] → Rn be continuous, with support contained in [t − σ, t + σ]. Set ϕT (t) = ϕ(t) − ϕ(t), n∗ (t)n∗ (t). By putting ϕT (t) in place of ϕ in the adjoint equation we obtain
[0,T ]
= maxp(t), ∇u f (x∗ (t), u∗ (t))u
E0 := {t ∈ [0, T ] : x∗ (t) ∈ int C}
Proof. (1). The first assertion is obvious, since on E0 we have n∗ ≡ 0 and the absolute continuity together with (19) follow from the properties of the functions ξ and η proved in Theorem 2.
for all continuous functions ϕ : [0, T ] → Rn − ϕ(t), dp(t) = − ϕ(t), n∗ (t)ξ(t) dν(t) [0,T ]
509
t−σ
for a.e. t ∈ [0, T ].
Some more precise statements on the measure ν require some assumptions on the reference trajectory x∗ . In fact, consider the sets 511
t+σ ϕT (s), η(s)∇2 d(x∗ (s))p(s) ds. (21) − t−σ
Observe now that ϕT , n∗ (t) ≡ 0, so that, by letting σ → 0 in the above equation and using the continuity
Proceedings of the 20th IFAC World Congress 510 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511 Toulouse, France, July 9-14, 2017
of ϕT , we obtain ϕT (t), p(t+) − p(t−) = 0, namely p(t+) − p(t−), ϕ(t) = p(t+) − p(t−), ϕ(t), n∗ (t) n∗ (t) . By taking subsequently ϕ such that ϕ(t) = n∗ (t), and ϕ(t) = v i (t), i = 1, . . . , n − 1, we obtain that p(t−) − p(t+) = pN (t−) − pN (t+) n∗ (t), namely jumps of p may occur only in the direction n∗ (t), for all t in the interior of E∂ . If t is a left or a right endpoint of E∂ , one can extend n∗ as a constant to the left or to the right of t and repeat the same argument. (3). Fix now an interval [s, t] ⊆ E∂ . The regularity condition on ∂C allows us to integrate by parts on (s, t), so that t s
n∗ (τ ), dp(τ ) +
t s
n˙ ∗ (τ ), p(τ ) dτ =
= n∗ (t+), p(t+) − n∗ (s−), p(s−) = 0. (22) Observe that both summands in the right hand side of (22) vanish, as a consequence of (1) and of the fact that p has bounded variation, since s and t belong to the interior of I∂ . In other words, the two measures n∗ , dp and n˙ ∗ , p dt coincide in any open interval contained in E∂ . Therefore, for all continuous ϕ with support contained in (s, t), we obtain from (21) and (22) that − =−
t s
t s
ϕ(τ ), n∗ (τ ) n∗ (τ ), dp(τ ) − =
t s
s
for all t ∈ [0, 1] p (t) is constant on [0, t¯) ∪ (t¯, 1) y
py (1) = −1. The maximum condition reads as (−1, −1), (ux∗ , uy∗ ) =
t s
max
|u1 |≤1,|u2 |≤1
(−1, −1), (u1 , u2 )
for t = 1
y
(t)), (ux∗ , uy∗ ) =
max
|u1 |≤1,|u2 |≤1
(−1, py (t)), (u1 , u2 )
for 0 ≤ t < 1, t = t¯, which gives ux∗ = −1, while no information is available for uy∗ (t). If we assume that uy∗ is constant, then an expected optimal control uy∗ = −1 is found. Of course all other optimal controls uy∗ , namely uy∗ (t) = −1 for 0 ≤ t < t¯ and uy∗ (t) ≤ 0 for t¯ < t < 1 satisfy our necessary conditions, and in this case t¯ = y0 .
ϕ(τ ), dp(τ ) = ϕT (τ ), dp(τ )
ϕ(τ ), n∗ (τ ) p(τ ), n˙ ∗ (τ ) dτ
REFERENCES
s
−
px (t) = −1
(−1, p
t + ϕ(τ ) − ϕ(τ ), n∗ (τ ) n∗ (τ ), ∇x f (x∗ (τ ), u∗ (τ ))p(τ ) dτ t
is straightforward. Moreover, since the state constraint is not active, necessary optimality conditions are well known. If instead 0 ≤ y0 < 1, then our analysis becomes relevant. An inspection to the level sets of the cost h shows that as long as the trajectory does not hit ∂C the only optimal control is (−1, −1). Therefore, there exists at most one t¯ such that the optimal solution hits ∂C and after t¯ it remains on ∂C. On ∂C, namely for x = 0, we have ∇2x dC (0, y) ≡ 0. Thanks to Theorems 2 and 3 we obtain, for the optimal trajectory (x∗ , y∗ ) corresponding to the optimal control (ux∗ , uy∗ ) an adjoint vector (px , py ), such that (px , py ) is absolutely continuous on (0, t¯) ∪ (t¯, 1), and such that p˙ x = 0, p˙y = 0 a.e. on [0, T ], px (1) = py (1) = −1, px is continuous at t = 1 and py (1−) + 1 = 1, namely py (1−) = 0. Thus the adjoint vector (px , py ) satisfies:
ϕ(τ ) − ϕ(τ ), n∗ (τ ) n∗ (τ ), η(τ )∇2x d(x∗ (τ ))p(τ ) dτ
Since ϕ and the interval (s, t) are arbitrary, we obtain (20). 6. AN EXAMPLE The state space is R2 (x, y), the constraint is C := {(x, y) : y ≥ 0}, the upper half plane. We wish to minimize h(x(1), y(1)) = x(1) + y(1) subject to
x(t), ˙ y(t) ˙ ∈ −NC x(t), y(t) + ux (t), uy (t) x(0), y(0) = 0, y0 , y0 ≥ 0, x y where the controls u (t), u (t) belong to [−1, 1] × [−1, 1] =: U .
This problem satisfies all our assumptions.
Observe first that if y0 ≥ 1, the constraint C does not play any role, and the optimality of the control (−1, −1) 512
Ch. Arroud and G. Colombo. A Maximum Principle for the controlled Sweeping Process. SVVA, in print. DOI: 10.1007/s11228-017-0400-4. F. Bagagiolo. Dynamic programming for some optimal control problems with hysteresis. NoDEA, 9:149–174, 2002. R.C. Barnard and P.R. Wolenski. Flow invariance on stratified domains. Set-Valued Var. Anal., 21:377–403, 2013. M. Brokate and P. Krejˇc´ı. Optimal control of ODE systems involving a rate independent variational inequality. Discrete and continous dynamical systems series B, 18: 331–348, 2013. Tan H. Cao and B.Sh. Mordukhovich. Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discrete and continous dynamical systems series B, 22:267–306, 2017. G. Colombo, R. Henrion, N.D. Hoang, and B.Sh. Mordukhovich. Optimal control of the sweeping process over polyhedral controlled sets. J. Differential Equations, 260:3397–3447, 2016. G. Colombo and M. Palladino. The minimum time function for the controlled Moreau’s sweeping process. SIAM J. Control, 54:2036–2062, 2016. A. Gudovich and M. Quincampoix. Optimal control with hysteresis nonlinearity and multidimensional play operator. SIAM J. Control Optim. 49:788–807, 2011.
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J.-J. Moreau. On unilateral constraints, friction and plasticity. In New variational techniques in mathematical physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973):171–322. Edizioni Cremonese, Rome, 1974. M. Sene and L. Thibault. Regularization of dynamical systems associated with prox-regular moving sets. J. Nonlin. and Convex Anal., 15:647–663, 2014. L. Thibault. Sweeping process with regular and nonregular sets. J. Differential Equations, 193:1–26, 2003. R.B. Vinter. Optimal control. Birkh¨ auser, Boston, 2000.
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ScienceDirect
IFAC PapersOnLine 50-1 (2017) 506–511 Necessary conditions for a Necessary conditions for a Necessary conditions for a control problem with state control problem with state control problem with∗ state
nonclassical nonclassical nonclassical constraints constraints constraints ∗∗
Chems Eddine Arroud ∗ Giovanni Colombo ∗∗ Chems Chems Eddine Eddine Arroud Arroud ∗ Giovanni Giovanni Colombo Colombo ∗∗ Chems Eddine Arroud ∗ Giovanni Colombo ∗∗ ∗ Department of Mathematics, Jijel University, Jijel, Algeria, and Mila ∗ ∗ Department of Mathematics, Jijel University, Jijel, Algeria, and Mila of Mathematics, Jijel (e-mail: University, Jijel, Algeria, and Mila University Center, Mila, Algeria [email protected]) ∗ Department Department of Mathematics, Jijel (e-mail: University, Jijel, Algeria, and Mila ∗∗ University Center, Mila, Algeria [email protected]) University Center, Mila, Algeria (e-mail: [email protected]) Dipartimento di Matematica “Tullio Levi-Civita”, University of ∗∗ University Center, Mila, Algeria (e-mail: [email protected]) ∗∗ Dipartimento di Matematica “Tullio Levi-Civita”, University Dipartimento di Matematica “Tullio Levi-Civita”, University of of Padova, Padova, Italy (e-mail: [email protected]). ∗∗ Dipartimento di Matematica “Tullio Levi-Civita”, University of Padova, Padova, Italy (e-mail: [email protected]). Padova, Padova, Italy (e-mail: [email protected]). Padova, Padova, Italy (e-mail: [email protected]). Abstract: We consider the problem of minimizing the cost h(x(T )) at the endpoint of a Abstract: consider problem of Abstract:x We We consider the problem of minimizing minimizing the cost cost h(x(T h(x(T )) )) at at the the endpoint endpoint of of aa trajectory subject to thethe finite dimensional dynamicsthe Abstract:x consider problem of minimizing trajectory subject to finite dimensional dynamics trajectory x We subject to the thethe finite dimensional dynamicsthe cost h(x(T )) at the endpoint of a x˙ ∈ dimensional −NC (x) + fdynamics (x, u), x(0) = x0 , trajectory x subject to the finite x (x) + f (x, u), x(0) x(0) = =x x0 ,, x˙˙ ∈ ∈ −N −NC C (x) + f (x, u), where NC denotes the normalx˙ cone to the convex set C. Such inclusion is termed, ∈ −NC (x) + f (x, u), x(0) = x00differential , where N denotes the normal cone to the convex set C. Such inclusion is whereMoreau, NC the normal cone the convex C. Such differential differential inclusion with is termed, termed, after sweeping process. Wetolabel it as a set “nonclassical” control problem state C denotes where N denotes the normal cone to the convex set C. Such differential inclusion is termed, C after Moreau, sweeping process. We label it as a “nonclassical” control problem with after Moreau, sweeping We side labelisitdiscontinuous as a “nonclassical” controltoproblem with state constraints, because the process. right hand with respect the state, andstate the after Moreau, sweeping We side labelis as a “nonclassical” controlto with constraints, because the right hand with the and the constraints, because theallprocess. right hand side isitdiscontinuous discontinuous with respect respect toproblem the state, state, andstate the constraint x(t) ∈ C for t is implicitly contained in the dynamics. constraints, because theall hand sidecontained is discontinuous with respect to the state, and the constraint x(t) ∈ tt is in constraint x(t) ∈C C for for allright is implicitly implicitly contained in the theofdynamics. dynamics. We prove necessary optimality conditions in the form Pontryagin Maximum Principle by constraint x(t) ∈ C for all t is implicitly contained in theof We prove necessary optimality conditions in Pontryagin Maximum by We prove essentially, necessary optimality conditions in the form ofdynamics. Pontryagin Maximum Principle by requiring, that C is independent of the time.form If the reference trajectory is inPrinciple the interior We prove necessary optimality conditions in the form of Pontryagin Maximum Principle by requiring, essentially, that C is independent of time. If the reference trajectory is in the interior requiring, essentially, that C is independent time.ones. If the trajectory is adjoint in the interior of C, necessary conditions coincide with theofusual Inreference the general case, the vector requiring, essentially, that C is independent ofusual time.ones. If the reference trajectory is adjoint in the interior of C, necessary conditions coincide with the In the general case, the vector of C, necessary conditions coincide with the usual ones. In the general case, the adjoint vector is a BV function and a signed vector measure appears in the adjoint equation. of a necessary coincide the usual ones.in the general case, the adjoint vector is function and vector measure appears the adjoint equation. is aC,BV BV functionconditions and a a signed signed vectorwith measure appears inIn the adjoint equation. is BV IFAC function and a signed vectorof measure the adjoint equation. © a2017, (International Federation Automaticappears Control)inHosting by Elsevier Ltd. All rights reserved. Keywords: Moreau’s sweeping process, Optimal control, Pontryagin Maximum Principle. Keywords: Keywords: Moreau’s Moreau’s sweeping sweeping process, process, Optimal Optimal control, control, Pontryagin Pontryagin Maximum Maximum Principle. Principle. Keywords: Moreau’s sweeping process, Optimal control, Pontryagin Maximum Principle. 1. INTRODUCTION the final cost h being smooth. There is a clear difference 1. the cost h smooth. There is clear 1. INTRODUCTION INTRODUCTION the final final cost control h being beingproblems smooth. with Therestate is aaconstraints clear difference difference with classical (see, 1. INTRODUCTION the final cost control h beingproblems smooth. with Therestate is aconstraints clear difference with classical (see, with classical control problems with state constraints (see, Vinter (2000)), where the constraint does not appear The sweeping process was introduced by Moreau in the e.g., with classical controlwhere problems with state constraints (see, e.g., Vinter (2000)), the constraint does not appear The sweeping process was introduced by Moreau in the e.g., Vinter (2000)), where the constraint does not appear explicitly in the dynamics: in this case the right hand side The sweeping process was introduced by Moreau in the Seventies as a model for dry friction and plasticity (see e.g., Vinter (2000)), where the constraint does not appear The sweeping process was introduced by Moreau in the explicitly in the dynamics: in this case the right hand side Seventies as for friction and explicitly in the dynamics: in thiswith caserespect the right side the dynamics is not Lipschitz to hand the state Seventies as aa model model for dry dry friction and plasticity plasticity (see Moreau (1974)) and later studied by several authors.(see In of explicitly in the dynamics: in this case the right hand side Seventies as a model for dry friction and plasticity (see of the dynamics is not Lipschitz with respect to the state Moreau (1974)) and later studied by several authors. In of the dynamics is not withgraph. respectThis to the but indeed hasLipschitz only closed factstate is a Moreau (1974)) and later studiedthebydifferential several authors. In variable, its perturbed version, it features inclusion of the dynamics is not Lipschitz with respect to the state Moreau (1974)) and later studiedthe several authors. In variable, but has closed This fact its version, it inclusion variable, but indeed indeed has only only closed graph. graph. This fact is is a a source of major difficulties in deriving necessary optimality its perturbed perturbed version, it features features thebydifferential differential inclusion but indeed has only closed graph. This fact is a + f (x(t)), t ∈ [0, T ] inclusion (1) variable, x(t) ˙ ∈ −N its perturbed version, it features the differential source of major difficulties in deriving necessary optimality C(t) (x(t)) source of major difficulties in deriving necessary optimality conditions for (3), (4). (1) x(t) ˙˙ ∈ x(t) ∈ −N −NC(t) (x(t)) (x(t)) + + ff (x(t)), (x(t)), tt ∈ ∈ [0, [0, T T ]] (1) source of major difficulties in deriving necessary optimality for (4). conditions for (3), (3), (4). t ∈ [0, T ] (1) conditions x(t) ˙ ∈ −NC(t) C(t) (x(t)) + f (x(t)), conditions for (3),(see, (4). e.g., Bagagiolo (2002), Gudovich In recent years coupled with the initial condition In recent years (see, Bagagiolo (2002), Gudovich coupled In al. recent years (see, ete.g., e.g., Bagagiolo (2002), Gudovich coupled with with the the initial initial condition condition (2011), Brokate al. (2013), Colombo et al. (2016), (2) et x(0) =condition x0 ∈ C(0). In recent years (see, et e.g., Bagagiolo (2002), Gudovich coupled with the initial et al. (2011), Brokate al. (2013), Colombo et al. (2016), et al. (2011), Brokate et al. (2013), Colombo et al. (2016), Colombo et al. (2016), Arroud et al. (2016), and Cao ∈ C(0). (2) x(0) = x (2) et x(0) = x00 ∈ C(0). al. (2011), Brokate et al. (2013), Colombo et al. (2016), Colombo et al. (2016), Arroud et al. (2016), and Cao Colombo et al. (2016), Arroud et al. (2016), and Cao (2) et x(0) = x0 ∈ C(0). al. (2017), and references therein) some papers dealing Here C(t) is a closed moving set, with normal cone Colombo et al. (2016), Arroud et al. (2016), and Cao et al. (2017), and references therein) some papers dealing et al. (2017), and references therein) some papers dealing with control problems involving the sweeping process were Here C(t) is a closed moving set, with normal cone Here C(t) is a closed moving set, with normal cone NC(t) (x) at x ∈ C(t). The space variable, in this paper, et al. (2017), and references therein) some papers dealing with control problems involving the sweeping process were with controlthe problems theinsweeping process were Here C(t)at isx a C(t). closedThe moving with in cone published, controlinvolving appearing the perturbation f (x) space variable, this N C(t) at R xn .∈ ∈ space set, variable, innormal this paper, paper, N with controlthe problems involving thein sweeping process were C(t) (x)to IfC(t). C(t)The is convex, or mildly non-convex belongs published, control appearing the perturbation ff published, the control appearing in the perturbation n (x) at x ∈ C(t). The space variable, in this paper, N and/or in the moving set C. Several necessary conditions C(t) n . If C(t) C(t) is be convex, or mildlyhere), non-convex belongs to R Rthat convex, mildly non-convex belongs to themoving controlset appearing in necessary the perturbation f (in a sense notis made or precise and is published, n . Ifwill and/or in the C. Several conditions and/or in the moving set C. Several necessary conditions established, under different kinds of assumptions, or . Ifwill C(t) convex, or mildlyhere), non-convex belongs to Rthat (in a not be made precise and is (in a sense sense will notis map be made precise here), andthe is were and/or in the moving set C. Several necessary conditions Lipschitz as that a set-valued depending on t, and established, different of or were established, under under different kinds kinds of assumptions, assumptions, or Hamilton-Jacobi characterization of value function was (in a senseas will not map be made precise on here), andthe is awere Lipschitz a depending t, Lipschitz as that af set-valued set-valued depending t, and and the were established, under different kinds of assumptions, or perturbation is Lipschitzmap as well, then it on is well known aproved. Hamilton-Jacobi characterization of value function was a Hamilton-Jacobi characterization of value function was The present paper is devoted to prove a result Lipschitz as a set-valued map depending on t, and the perturbation ff is as then perturbation is Lipschitz Lipschitz as well, well, then it it is is well known Hamilton-Jacobi characterization of value function was that the Cauchy problem (1), (2) admits onewell andknown only aproved. The present paper is devoted to prove a result proved. The present is devoted to prove result by Arroud et paper al. (2016) and Brokate et al.a(2013). perturbation f is Lipschitz as well, then it is known that the problem (1), (2) one and only thatLipschitz the Cauchy Cauchy problem (1), (2) admits admits onewell and only inspired proved. The present is devoted to prove a(2013). result one solution (see, e.g., Thibault (2003)). Observe inspired by Arroud Arroud et paper al. (2016) (2016) andconditions Brokate etofal. al.Pontrya(2013). inspired by et al. and Brokate et More precisely, we prove necessary that the Cauchy problem (1), (2) admits one and only one Lipschitz solution (see, e.g., Thibault (2003)). Observe one Lipschitz (see,x(t) e.g.,∈Thibault by Arroud et al. (2016) and Brokate et al. (2013). that the statesolution constraint C(t) for(2003)). all t ∈ Observe [0, T ] is inspired More precisely, we necessary of More precisely,principle we prove provetype necessary conditions of PontryaPontryamaximum for (4) conditions subject to (3) and (2), one Lipschitz solution (see,x(t) e.g.,∈ Thibault that state constraint C(t) for all [0, T that the state constraint x(t) C(t) for(2003)). all ttif ∈ ∈x Observe [0, T ]] is is gin More precisely,principle we provetype necessary conditions of Pontryabuilt the in the dynamics, being N∈ (x) empty ∈ C(t): C(t) gin maximum for (4) subject to (3) and (2), gin maximum principle type for (4) subject to (3) andC(·) (2), the control appearing only within f , in the case where that the state constraint x(t) ∈ C(t) for all t ∈ [0, T ] is built in inathe the dynamics, beingthen NC(t) (x) empty empty if ifx(t) x ∈ ∈∈C(t): C(t): built dynamics, being N (x) x gin maximum principle type for (4) subject to (3) and (2), C(t) should solution x(·) exist, automatically C(t) the control appearing only within f , in the case where C(·) the control appearing only within f , in the case where C(·) is constant, smooth and convex (see Theorems 2 and 3). built in the dynamics, being N (x) empty if x ∈ C(t): C(t) should a x(·) exist, x(t) C(t) theconstant, control appearing onlyconvex within(see f , inTheorems the case where C(·) should a solution solution x(·) parameter exist, then then automatically automatically x(t)f∈ ∈, then C(t) is for all t. If a control u appears within smooth and 2 and 3). is constant, smooth and convex (see Theorems 2 and 3). The case where C satisfies milder convexity assumptions should a solution x(·) exist, then automatically x(t) ∈ C(t) for t. a parameter appears constant, smooth and convex (see Theorems 2 and 3). for all all t. If If to a control control parameter uthe appears within ff ,, then then is one is lead study problems of u type within The case where C satisfies milder convexity assumptions The case where C satisfies milder convexity assumptions and is not necessarily constant was treated in Arroud et al. for all t. If to a control parameter appears one is study of type case where C satisfies milder convexity assumptions one is lead lead to study problems problems of uthe the type within f , then The and is not necessarily constant was treated in Arroud et al. and is not necessarily constant was treated in Arroud et al. (2016) with an extra assumption, while Brokate et (x(t)) + f (x(t), u(t)), u(t) ∈ U (3) x(t) ˙ lead ∈ −N one is to study problems of the type C(t) and is not necessarily constant was treated in Arroud et (2016) with an extra assumption, while Brokate et al. (x(t)) + f (x(t), u(t)), u(t) ∈ U (3) ˙x(t) x(t) ∈ −N (2016) contains with an results extra assumption, whilecontrol Brokate et al. al. u(t) ∈ U (3) (2013) ˙ ∈ −NC(t) for a particular problem C(t) (x(t)) + f (x(t), u(t)), with an results extra assumption, whilecontrol Brokate et al. (3) (2016) x(t) ˙ ∈ −NC(t) (x(t)) + f (x(t), u(t)), u(t) ∈ U (2013) contains for a particular problem (2013) contains results for a particular control problem involving a fixed smooth and uniformly convex set C. subject to (2), aiming, for example, at (2013) contains for and a particular control problem involving aa fixed smooth uniformly C. subject involving fixedresults smooth and uniformly convex set C. More preccisely, differently from Colomboconvex et al. set (2016) subject to to (2), (2), aiming, aiming, for for example, example, at at involving a fixed smooth and uniformly convex set C. minimizing h(x(T )), (4) More subject to (2), aiming, for example, at preccisely, differently from Colombo et al. (2016) More preccisely, differently from Colombo et al. (2016) and Cao et al. (2017), where discrete approximations are minimizing h(x(T h(x(T )), )), (4) More preccisely, differently from Colombo et al. (2016) minimizing (4) and Cao et al. (2017), where discrete approximations are and Cao al. (2017), in et both Brokatewhere et al.discrete (2013) approximations and Arroud et are al. The second author minimizing )), by Padova University (4) used, is partially h(x(T supported and Cao et al. (2017), where approximations are used, Brokate al. (2013) and al. The second author is partially supported by Padova University used, in both Brokate et al.discrete (2013) technique. and Arroud Arroud etclasal. (2016)in theboth authors use a et penalization Theet project PRAT2015 “Control of dynamics with active constraints” The second author is partially supported by Padova University used, in both Brokate et al. (2013) technique. and Arroud etclasal. The second author is partially supported by Padova University (2016) the authors use a penalization technique. The clas(2016) the authors use a penalization The project PRAT2015 “Control of dynamics with active constraints” sical Moreau-Yosida regularization allows in Arroud et al. and by Istituto Nazionale di Alta Matematica. project PRAT2015 “Control of dynamics with active constraints” (2016) the authors use a penalization technique. The classical Moreau-Yosida regularization allows in Arroud et al. project PRAT2015 “Control of dynamics with active constraints” and sical Moreau-Yosida regularization allows in Arroud et al. and by by Istituto Istituto Nazionale Nazionale di di Alta Alta Matematica. Matematica. sical Moreau-Yosida regularization allows in Arroud et al. and by Istituto Nazionale di Alta Matematica.
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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511
507
(2016) to relax the uniform convexity assumption, at the price of requiring a strong outward pointing condition on f in order to treat the discontinuity of second derivatives of the squared distance function at the boundary of C(t). In Brokate et al. (2013), the authors adopt a suitable smoothing of the distance, which on one hand needs C(t) constant and uniformly convex and 0 ∈ C, while on the other avoids imposing further compatibility assumptions between f and C. In this paper we adapt to our situation the method developed in Brokate et al. (2013) and remove the assumption of strict convexity on C. The main technical part is Section 4.
where ε > 0 and u(t) ∈ U for all t. For each given u, this Cauchy problem admits a unique solution xε for each ε > 0 on a maximal interval of existence. It is not difficult to prove that this interval is [0, T ] (see the proof of Proposition 1).
2. PRELIMINARIES AND ASSUMPTIONS
over controls u, where x is a solution of (8). By standard results, Pε (u∗ ) admits a global minimizer uε , with the corresponding solution xε . Necessary conditions of the original problem will be obtained by passing to the limit along conditions for Pε (u∗ ).
Notation. We define the distance from a set C ⊂ Rn as d(x) = inf{ y − x : y ∈ C} and signed distance from C as dS (x) = d(x) if x ∈ C and dS (x) = − inf{ y − x : y ∈ C} if c ∈ C. The normal cone to a convex set C is defined as NC (x) = ∅ if x ∈ C and NC (x) = {v ∈ Rn : v, y − x ≤ 0 ∀y ∈ C} if x ∈ C. Assumptions on the set C. Let n
C = {x ∈ R : g(x) ≤ 0}, where g : Rn → R is of class C 2 with gradient ∇g = 0 on the boundary ∂C of C, and with the Hessian matrix ∇2 g(x) positive semidefinite for all x ∈ Rn . Assume furthermore that g(·) is coercive, so that C is compact (and convex) and that g(0) < 0, so that 0 ∈ C and C has nonempty interior. Observe that under our assumption the signed distance dS (x) from C is of class C 2 in a neighborhood of ∂C. Assumptions on the dynamics and the cost. The control set U ⊂ Rn is compact and f is continuous and bounded, say by a constant β, and is of class C 1 with respect to x, with ∇x f (x, u) ≤ L for all x, u. The cost h is smooth. 2
Let now ψ(x) be a C smoothing of dS in the interior of C (which is < 0 in int C and is such that ∇ψ(x) is the unit external normal to C at x for every x ∈ ∂C). Set also 1 Ψ(x) = ψ 3 (x) 1(0,+∞) (ψ(x)). 3 Observe that Ψ(·) is of class C 2 and convex in the whole of Rn and that both ∇Ψ(·) and ∇2 Ψ(·) vanish on C. Moreover one has d(x)∇ψ(x) = d(x)∇d(x),
(5)
∇Ψ(x) = d2 (x)∇d(x),
(6)
∇2 Ψ(x) = 2d(x)∇d(x) ⊗ ∇d(x) + d2 (x)∇2 d(x),
(7)
because in C, and in particular at the points where ∇d(x) does not exist (namely, in ∂C), both sides of the above expressions vanish, and outside C they coincide.
1 minimize h(x(T )) + 2
T 0
u(t) − u∗ (t) 2 dt,
(8) 509
(9)
3.1 A priori estimates for the regularized problem Proposition 1. Let εn → 0 and let (un , xn ) be a solution of the problem Pεn . Then, up to a subsequence, un converges strongly in L2 (0, T ) to u∗ and xn converges weakly in W 1,2 (0, T ) to x∗ . Proof. Since 0 ∈ C and so ∇Ψ(0) = 0, by the convexity of Ψ we obtain that ∇Ψ(x), x ≥ 0 for all (x). Thus
=
t 0
xn (t) − x0 =
x (t) −1 n , ∇Ψ(xn (t)) + f (xn (t), un (t)) dt ≤ β, xn (t) εn
which, in particular, implies that xn is defined in the whole of [0, T ]. Moreover, x˙ n 2L2 =
≤
T
x˙ n (t),
0
T 0
−1 ∇Ψ(xn (t)) + f (xn (t), un (t)) dt εn
−1 d Ψ(xn (t)) + f (xn (t), un (t)), x˙ n (t) dt εn dt −1 1 = Ψ(xn (T )) + Ψ(0) + β εn εn
T 0
x˙ n (t) dt
√ ≤ β T x˙ n L2 ,
where we have used the fact that 0 ∈ C and that ψ(xn (T )) ≥ 0. The above estimate implies that the sequence x˙ n is uniformly bounded in L2 (0, T ). Thus, up to a subsequence, xn converges weakly in W 1,2 (0, T ) to x ¯. Observe now that the uniform boundedness of x˙ n L2 (0,T ) together with (6), recalling (5) imply that d(xn (·))2 L2 (0,T ) ≤ Kεn
3. THE REGULARIZED PROBLEM Consider the regularized dynamics −1 x(t) ˙ = ∇Ψ(x(t)) + f (x(t), u(t)), x(0) = x0 , ε
For every ε > 0 and every global minimizer x∗ , u∗ of (4) subject to (3) and (2), we consider the approximate problem Pε (u∗ )
(10)
for a suitable constant K. Thus x(t) ∈ C for all t. Again up to a subsequence, un converges weakly in L2 (0, T ) to some u ¯. By using the very same argument of Proposition 4.3 in Arroud et al. (2016), one can prove that x ¯ is the
Proceedings of the 20th IFAC World Congress 508 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511 Toulouse, France, July 9-14, 2017
solution of (3), (2) corresponding to u ¯, that u ¯ = u∗ , and so that x ¯ = x∗ , and that the convergence is indeed strong. Remark. Eq. (10) implies that, up to a subsequence, √ √ (11) d(xn (·)) L2 (0,T ) ≤ T K εn . From the uniform convergence of xn (again up to a subsequence) we also get d(xn (·)) L∞ → 0 for n → ∞. This is an important difference between this approach and the use of Moreau-Yosida approximation, which instead yields the stronger estimate d(xn (·)) L∞ ∼ εn (see Sene et al. (2014) or (Arroud et al., 2016, Proposition 4.1)). Observe that the assumption that C is constant appears essential in order to obtain uniform a priori estimates for x˙ n L2 within the present approach, which essentially uses the weaker penalization d3 instead of the Moreau-Yosida one, namely d2 . 3.2 Necessary conditions for the regularized problem The approximate problem Pε (u∗ ) satisfies the assumptions for necessary conditions of classical unconstrained optimal control problems. The same computations of Section 6 in Arroud et al. (2016) yield that for every ε and every minimizer (uε , xε ) there exists an absolutely continuous adjoint vector pn : [0, T ] → Rn such that −1 ∇2 Ψ(xε (t)) + ∇x f (xε (t), uε (t)) pε (t) (12) −p˙ε (t) = ε a.e. on [0, T ], together with the final condition −pε (T ) = ∇h(xε (T )),
(13)
and the maximality condition
Recalling (13), we obtain from the above inequality and Gronwall’s lemma that there exists a constant K1 independent of n such that (14) pn ∞ ≤ K1 . Now we address ourselves to prove an a priori estimate on p˙ n L1 . To this aim, we define ξn (t) = pn (t), ∇ψ(xn (t)), so that, ξ˙n (t) = p˙ n (t), ∇ψ(xn (t)) + pn (t), ∇2 ψ(xn (t))x˙ n (t). Set now δn (t) = d(xn (t)), δn (t) = ∇d(xn (t)), δn (t) = ∇2 d(xn (t)). With this notation, thanks to (5), (6), and (7), eq. (12) can be rewritten as δn2 (t) δn (t)ξn (t) δn (t)pn (t) − 2 δn (t) + εn εn +∇x f (xn (t), un (t))pn (t). (15) Inserting p˙n and x˙ n in the expression for ξ˙n we obtain (omitting the t-dependence and using the fact that ∇2 d(x) ∇d(x) = 0 for all x where it makes sense, and that δn ∇ψ(xn ) 2 = δn ∇d(x)n) = δn ) −p˙ n (t) = −
δn ξn δ2 = − n δn pn , δn + εn εn +∇x f (xn , un )pn , ∇ψ(xn ) − pn , ∇2 ψ(xn )f (xn , un ). Observe now that the first summand in the right hand side of the above expression is bounded in L1 (0, T ) uni−ξ˙n + 2
δ 2
formly with respect to n, because, recalling (11), nεnL1 is uniformly bounded. In turn, the second and the third summands are seen to be bounded in L∞ (0, T ), uniformly with respect to n, by invoking (14). By multiplying both sides by sign(ξn ), we thus obtain the second estimate T 1 δn (s)|ξn (s)| ds ≤ K2 , (16) εn
pε (t), ∇u f (xε (t), uε (t))uε (t) − uε (t) − u∗ (t), uε (t) =
max{pε (t), ∇u f (xε (t), uε (t))u − uε (t) − u∗ (t), u} u∈U
for a.e. t ∈ [0, T ]. 4. PASSING TO THE LIMIT
t
From now on we consider a sequence εn → 0 such that the minimum (un , xn ) of the approximate problem converges as in the statement of Proposition 1. We set pn := pεn . 4.1 A priori estimates for the adjoint vectors of the approximate problem
for a suitable constant K3 independent of n.
We obtain from (12) and (7) that
4.2 Passing to the limit along the adjoint equation
T
∇2 Ψ(xn (s))pn (s), pn (s) pn (s) t pn (t) + ∇x f (xn (t), un (t))pn (t), dt ≤ pn (t)
−1 pn (t) − pn (T ) = εn
for a suitable constant K2 , independent of n. As a consequence, all three summands in the right hand side of the adjoint equation (15) are bounded in L1 (0, T ), uniformly with respect to n, and so we reach our final estimate (17) p˙ n L1 (0,T ) ≤ K3
By possibly extracting a further subsequence, we can assume that the sequence of measures p˙ n dt converges weakly∗ in the sense of Radon measures to a signed vector measure µ, which is the distributional derivative of the BV function p(t) =: lim pn (t), i.e., µ = dp, where the limit of the pn is pointwise in [0, T ]. By arguing as in (Arroud et al., 2016, Proposition 7.3), we obtain first that the sequence of measures δn (t)ξn (t) δn (t) dt εn
(since ∇2 Ψ is positive semidefinite and ∇x f is bounded) T ≤ L pn (t) dt. t
510
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511
converges weakly∗ to a finite signed vector Radon measure, which can be written as ξ(t)n∗ (t) dν, where ξ ∈ L1ν (0, T ), ξ ≥ 0 ν-a.e., n∗ (t) denotes the unit outward normal vector to C at x∗ (t) if x∗ (t) ∈ ∂C and 0 if x∗ (t) ∈ int C, and ν is a finite vector measure. Moreover δn2 (t) δ (t)pn (t) η(t)∇2 d(x∗ (t))p(t) εn n
in L2 (0, T ), where η ∈ L∞ ν (0, T ) and η ≥ 0 a.e., with η ≡ 0 when x∗ is in int C. 4.3 Passing to the limit along the maximality condition By taking into account Proposition 1, we obtain that the limit adjoint vector p is such that p(t), ∇u f (x∗ (t), u∗ (t))u∗ (t) =
= max{p(t), ∇u f (x∗ (t), u∗ (t))u} for a.e. t ∈ [0, T ]. (18) u∈U
5. THE MAIN RESULT We deduce from Sections 3 and 4 the following necessary conditions: Theorem 2. Under the assumptions stated in Section 2, let (x∗ , u∗ ) be a global minimizer for (4) subject to (3) and to (2). Then there exist a BV adjoint vector p : [0, T ] → Rn , together with a finite signed Radon measure ν on [0, T ], and measurable vectors ξ, η : [0, T ] → Rn (with ξ ∈ L1ν (0, T ), ξ(t) ≥ 0 for ν-a.e. t, η ∈ L∞ (0, T ), and η(t) ≥ 0 for a.e. t) such that ξ(t) = η(t) = 0 for all t with x∗ (t) ∈ int C, satisfying the following properties: • (adjoint equation)
− +
ϕ(t), ∇2x d(x∗ (t))p(t)η(t) dt
[0,T ]
ϕ(t), ∇x f (x∗ (t), u∗ (t))p(t) dt,
[0,T ]
The following is the second part of our necessary conditions. It is only a partial result, with respect to the rich set of conditions proved in Brokate et al. (2013). Theorem 3. Under the assumptions stated in Section 2, let (x∗ , u∗ ) be a global minimizer for (4) subject to (3) and to (2), and let p the adjoint vector given by Theorem 2. Define pN (t) = p(t), n∗ (t), t ∈ [0, T ]. The following properties hold: (1) pN (t) = 0 for all t ∈ E0 , and p is absolutely continuous on E0 , where it satisfies the classical adjoint equation −p(t) ˙ = ∇x f (x∗ (t), u∗ (t)) p(t). (19) (2) At every interior (or such that a left or right neighborhood is contained in E∂ ) point t of E∂ , jumps of p may occur only in the normal direction n∗ (t), namely p(t−) − p(t+) = pN (t−) − pN (t+) n∗ (t). (3) The adjoint vector p is absolutely continuous on every open interval contained in E∂ , and for a.e. t in such interval we have −p(t) ˙ = n˙ ∗ (t), p(t)n∗ (t) + Γ(t)p(t) −
−Γ(t)p(t), n∗ (t)n∗ (t), (20) where Γ(t) = ∇x f (x∗ (t), u∗ (t)) − η(t)∇2x d(x∗ (t)).
t+σ t+σ T − ϕ (s), dp(s) + ϕT (s), n∗ (s)ξ(s) dν
• (transversality condition)
−p(T ) = ∇h(x∗ (T )),
t−σ
t−σ
t+σ ϕT (s), ∇x f (x∗ (s), u∗ (s))p(s) ds =
• (maximality condition) p(t), ∇u f (x∗ (t), u∗ (t))u∗ (t) =
u∈U
E∂ := {t ∈ [0, T ] : x∗ (t) ∈ ∂C}. Of course, E0 is open and E∂ is closed, but one has to take into account the possibility that E∂ be irregular (e.g., totally disconnected). Such phenomenon, in stratified state constrained control theory, is sometimes referred to as Zeno phenomenon, namely the switching from a stratum (the boundary of C, in this case) to other strata (the interior of C in this case) occurs at a complicated set (see, e.g., Barnard et al. (2013)).
(2). Since n∗ (t) is continuous on E∂ , there exist n − 1 continuous unit vectors v1 (t),. . . , vn−1 (t) such that Rn = Rn∗ (t) ⊕ span v1 (t), . . . , vn−1 (t) for all t ∈ E∂ . Let t ∈ E∂ and σ > 0 be such that [t − σ, t + σ] ⊂ E∂ . Let ϕ : [0, T ] → Rn be continuous, with support contained in [t − σ, t + σ]. Set ϕT (t) = ϕ(t) − ϕ(t), n∗ (t)n∗ (t). By putting ϕT (t) in place of ϕ in the adjoint equation we obtain
[0,T ]
= maxp(t), ∇u f (x∗ (t), u∗ (t))u
E0 := {t ∈ [0, T ] : x∗ (t) ∈ int C}
Proof. (1). The first assertion is obvious, since on E0 we have n∗ ≡ 0 and the absolute continuity together with (19) follow from the properties of the functions ξ and η proved in Theorem 2.
for all continuous functions ϕ : [0, T ] → Rn − ϕ(t), dp(t) = − ϕ(t), n∗ (t)ξ(t) dν(t) [0,T ]
509
t−σ
for a.e. t ∈ [0, T ].
Some more precise statements on the measure ν require some assumptions on the reference trajectory x∗ . In fact, consider the sets 511
t+σ ϕT (s), η(s)∇2 d(x∗ (s))p(s) ds. (21) − t−σ
Observe now that ϕT , n∗ (t) ≡ 0, so that, by letting σ → 0 in the above equation and using the continuity
Proceedings of the 20th IFAC World Congress 510 Chems Eddine Arroud et al. / IFAC PapersOnLine 50-1 (2017) 506–511 Toulouse, France, July 9-14, 2017
of ϕT , we obtain ϕT (t), p(t+) − p(t−) = 0, namely p(t+) − p(t−), ϕ(t) = p(t+) − p(t−), ϕ(t), n∗ (t) n∗ (t) . By taking subsequently ϕ such that ϕ(t) = n∗ (t), and ϕ(t) = v i (t), i = 1, . . . , n − 1, we obtain that p(t−) − p(t+) = pN (t−) − pN (t+) n∗ (t), namely jumps of p may occur only in the direction n∗ (t), for all t in the interior of E∂ . If t is a left or a right endpoint of E∂ , one can extend n∗ as a constant to the left or to the right of t and repeat the same argument. (3). Fix now an interval [s, t] ⊆ E∂ . The regularity condition on ∂C allows us to integrate by parts on (s, t), so that t s
n∗ (τ ), dp(τ ) +
t s
n˙ ∗ (τ ), p(τ ) dτ =
= n∗ (t+), p(t+) − n∗ (s−), p(s−) = 0. (22) Observe that both summands in the right hand side of (22) vanish, as a consequence of (1) and of the fact that p has bounded variation, since s and t belong to the interior of I∂ . In other words, the two measures n∗ , dp and n˙ ∗ , p dt coincide in any open interval contained in E∂ . Therefore, for all continuous ϕ with support contained in (s, t), we obtain from (21) and (22) that − =−
t s
t s
ϕ(τ ), n∗ (τ ) n∗ (τ ), dp(τ ) − =
t s
s
for all t ∈ [0, 1] p (t) is constant on [0, t¯) ∪ (t¯, 1) y
py (1) = −1. The maximum condition reads as (−1, −1), (ux∗ , uy∗ ) =
t s
max
|u1 |≤1,|u2 |≤1
(−1, −1), (u1 , u2 )
for t = 1
y
(t)), (ux∗ , uy∗ ) =
max
|u1 |≤1,|u2 |≤1
(−1, py (t)), (u1 , u2 )
for 0 ≤ t < 1, t = t¯, which gives ux∗ = −1, while no information is available for uy∗ (t). If we assume that uy∗ is constant, then an expected optimal control uy∗ = −1 is found. Of course all other optimal controls uy∗ , namely uy∗ (t) = −1 for 0 ≤ t < t¯ and uy∗ (t) ≤ 0 for t¯ < t < 1 satisfy our necessary conditions, and in this case t¯ = y0 .
ϕ(τ ), dp(τ ) = ϕT (τ ), dp(τ )
ϕ(τ ), n∗ (τ ) p(τ ), n˙ ∗ (τ ) dτ
REFERENCES
s
−
px (t) = −1
(−1, p
t + ϕ(τ ) − ϕ(τ ), n∗ (τ ) n∗ (τ ), ∇x f (x∗ (τ ), u∗ (τ ))p(τ ) dτ t
is straightforward. Moreover, since the state constraint is not active, necessary optimality conditions are well known. If instead 0 ≤ y0 < 1, then our analysis becomes relevant. An inspection to the level sets of the cost h shows that as long as the trajectory does not hit ∂C the only optimal control is (−1, −1). Therefore, there exists at most one t¯ such that the optimal solution hits ∂C and after t¯ it remains on ∂C. On ∂C, namely for x = 0, we have ∇2x dC (0, y) ≡ 0. Thanks to Theorems 2 and 3 we obtain, for the optimal trajectory (x∗ , y∗ ) corresponding to the optimal control (ux∗ , uy∗ ) an adjoint vector (px , py ), such that (px , py ) is absolutely continuous on (0, t¯) ∪ (t¯, 1), and such that p˙ x = 0, p˙y = 0 a.e. on [0, T ], px (1) = py (1) = −1, px is continuous at t = 1 and py (1−) + 1 = 1, namely py (1−) = 0. Thus the adjoint vector (px , py ) satisfies:
ϕ(τ ) − ϕ(τ ), n∗ (τ ) n∗ (τ ), η(τ )∇2x d(x∗ (τ ))p(τ ) dτ
Since ϕ and the interval (s, t) are arbitrary, we obtain (20). 6. AN EXAMPLE The state space is R2 (x, y), the constraint is C := {(x, y) : y ≥ 0}, the upper half plane. We wish to minimize h(x(1), y(1)) = x(1) + y(1) subject to
x(t), ˙ y(t) ˙ ∈ −NC x(t), y(t) + ux (t), uy (t) x(0), y(0) = 0, y0 , y0 ≥ 0, x y where the controls u (t), u (t) belong to [−1, 1] × [−1, 1] =: U .
This problem satisfies all our assumptions.
Observe first that if y0 ≥ 1, the constraint C does not play any role, and the optimality of the control (−1, −1) 512
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