OPTIMAL CONTROL OF A PERTURBED SWEEPING PROCESS VIA DISCRETE APPROXIMATIONS

. The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a con- trolled “play-and stop” operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained un- bounded diﬀerential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for diﬀerential inclu- sions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed ﬁnite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive eﬀective necessary optimality conditions by using second-order generalized diﬀerential tools of variational analysis that explicitly calculated in terms of the given problem data.

1. Introduction and problem formulation. In this paper we deal with a version of the sweeping process introduced by Jean-Jacques Moreau in the 1970s (see his comprehensive for that time lecture notes [23] with the references to the original publications) in the following form of the dissipative differential inclusion: where C(t) is a continuously moving convex set, and where the normal cone operator to a convex subset C ⊂ H of a Hilbert space is given by (2) The latter construction allows us to equivalently describe 1 as an evolution variational inequality [5,15], or as a differential variational inequality in the terminology of [24,29]. The original motivations for the introduction and study of the sweeping process come from applications to mechanical systems mostly related to friction and elastoplasticity, while further developments apply also to various problems of hysteresis, ferromagnetism, electric circuits, phase transitions, economics, etc.; see, e.g., [2,5,14,15,18,28,29] and the extensive bibliographies therein.
It has been realized in the sweeping process theory [10,16] that the Cauchy problem 1 admits a unique solution under mild assumptions on the given moving set C(t). This excludes considering any optimization problem for 1 and thus strictly distinguishes the sweeping process from the conventional optimal control theory for differential inclusionsẋ(t) ∈ F (x(t)). The latter theory supposes the existence of numerous solutions to the Cauchy problem and then studies minimization of some cost functionals over them; see, e.g., [22,31] with further references and discussions.
There are three approaches in the literature to introduce control actions in the sweeping process frameworks and then to conduct optimization with respect to these controls and the corresponding sweeping trajectories. The first approach considers controls in additive perturbations on the right-hand side of 1 without changing the moving set C(t). The results obtained in this direction mostly concern existence theorems and relaxation procedures while not optimality conditions; see [12] and the recent papers [2,6] with the references therein. We also place into this category the main results of [25,26] that establish the existence of optimal controls for rate-independent evolutions via some direct methods involving finite-difference approximations. The second approach developed in [4] for H = R n and then partly extended in [1] introduces controls in an ordinary differential equation associated with the sweeping process over a given set C(t) ⊂ R n . The obtained results provide necessary optimality conditions for the continuous-time problem in [4] and for the approximating finite-difference systems in [1].
The third approach to optimal control of the sweeping process, which is implemented in this paper, employs a control parametrization directly in the sweeping set C(t) making it dependent on control actions. It has been initiated in [8] for the case of a controlled hyperplane C(t) in R n and then strongly developed in the very recent paper [9] for the case of moving controlled polyhedra C(t) := x ∈ R n u i (t), x ≤ b i (t), i = 1, . . . , m , where the control functions u i (t) and b i (t) are absolutely continuous on the fixed time interval [0, T ]. Necessary optimality conditions for such optimal control problems of a new type are derived in [8,9] by using discrete approximations and appropriate generalized differential tools of variational analysis in the lines of [20,21,22]. Note that the discrete approximation approach to optimization of bounded differential inclusionsẋ ∈ F (x) developed in [20,22] essentially relies on the Lipschitz continuity of F , which fails in the case of 1 with controlled sweeping sets as in 3 and thus requires serious modifications. This paper concerns a new class of optimal control problems for the perturbed sweeping process −ẋ(t) ∈ N x(t); C(t) + f x(t), a(t) a.e. t ∈ [0, T ], x(0) := x 0 ∈ C(0), (4) with one part of controls a : [0, T ] → R d acting in the perturbation mapping f : R n × R d → R n and the other part of controls u : [0, T ] → R n acting in the moving set C(t) := C + u(t) with C := x ∈ R n x * i , x ≤ 0 for all i = 1, . . . , m , t, x(t), u(t), a(t),ẋ(t),u(t),ȧ(t) dt (6) with the proper terminal extended-real-valued cost function ϕ : R n → R := (−∞, ∞] and the running cost function : [0, T ] × R 4n+2d → R. Fixed r > 0, we impose the additional constraint on the u-controls: required by applications that occur to be significantly more difficult than the conventional ones in control theory: u(t) ≤ r, t ∈ [0, T ]. Note that the controlled moving set in 5 can be written in the polyhedral form . . , u m (t)), which shows that 5 reduces to 3 with controls acting only on the right-hand sides of the polyhedral inequalities. However, this reduction does not allow us to apply the results of [9] to our problem, even in the absence of perturbations, since there are no constraints on the controls b i (t) := −u i (t) in [9], while the imposed control constraint 7 cannot be ignored in our setting.
Besides the constraint issue, a major ingredient that distinguishes the novel framework in 4, 5 from the one in 1, 3 is the presence of controls in perturbations together with those in the moving sweeping set. This makes the new model challenging from the viewpoint of variational analysis and important for various applications. The primary application we have in mind is the crowd motion model (see, e.g., [17]), which corresponds to 4 with controls only in perturbations and whose simplified optimal control version is solved in our adjacent paper [7] based on the obtained optimality conditions. From a different viewpoint, the given description 5 of the moving set in 4 for each fixed function u(·) relates to the so-called play-and-stop operator; see [14,15,28] and the references therein. As discussed in [28], such operators constitute basic elements of the mathematical theory of rate independent hysteresis processes including, in particular, the celebrated Preisach model in ferromagnetism.
Yet another essential difference between the frameworks of [9] and of the current paper, even in the absence of controlled perturbations, is the choice of classes of feasible controls. The most natural class in the setting of [9] is the collection of controls u(t) absolutely continuous on [0, T ], which generate absolutely continuous trajectories x(t) of 3 under appropriate qualification conditions; cf. [9] for more details. In the setting 4, 5 of this paper by feasible controls u(·) and a(·) we understand functions that belong to the Sobolev spaces W 1,2 ([0, T ]; R n ) and W 1,2 ([0, T ]; R d ), respectively. It follows from the powerful well-posedness result of [12] that such a control pair generates a unique trajectory x(·) ∈ W 1,2 ([0, T ]; R n ) of the sweeping inclusion 4. Having this in mind, we formulate the sweeping optimal control problem (P ) as follows: minimize 6 over W 1,2 -controls (u(·), a(·)) on [0, T ] and the corresponding W 1,2 -trajectories x(·) of 4 with C(t) from 5 subject to the control equality constraint 7. Observe that, besides 7, there are implicit mixed (i.e., state-control) inequality constraints in (P ) given by which follow from 4 and 5 due to the second part of the normal cone definition 2.

TAN H. CAO AND BORIS S. MORDUKHOVICH
The main goal of this paper is to study the formulated optimal control problem (P ) and its slight parametric modification (P τ ) defined below by using the method of discrete approximations in the vein of [20,22] and its significant elaboration for the case of unperturbed non-Lipschitzian differential inclusions developed in [9]. The presence of controlled perturbations in 4 together with the control and mixed constraints in 7,8 essentially complicates the discrete approximation procedure. We aim to construct well-posed discrete approximations in such a way that every feasible (resp. locally optimal) solution to (P τ ) with τ ≥ 0 and (P 0 ) = (P ), can be strongly approximated in W 1,2 [0, T ] by feasible (resp. optimal) solutions to finitedifference control systems. Employing then appropriate first-order and second-order generalized differential constructions of variational analysis and explicitly calculating them via the problem data allow us to obtain effective necessary optimality conditions for discrete optimal solutions, which can be treated as suboptimality (almost optimality) conditions for the original sweeping control problem. Deriving exact optimality conditions for the continuous-time sweeping process in (P τ ) is a subject of [7]. Note that the finite-difference problems constructed in this paper provide more precise approximations of feasible and optimal solutions of (P τ ) in comparison with the corresponding settings of [9] and lead us in this way to new conditions for optimality in both discrete and continuous frameworks.
The rest of the paper is organized as follows. Section 2 lists basic assumptions and preliminaries from the sweeping process theory used below. In Section 3 we justify the existence of optimal solutions to (P τ ) as τ ≥ 0 and discuss its relaxation stability. This section also contains the definition of intermediate local minimizers and their relaxed modification studied in the paper.
In Section 4 we develop a constructive finite-difference procedure to strongly in W 1,2 [0, T ] approximate any feasible control (u(t), a(t)) and the corresponding trajectory x(t) of 4 by feasible solutions to discrete approximation systems that are piecewise linearly extended to [0, T ]. The next Section 5 establishes the strong W 1,2approximation of the local optimal solution to (P τ ) by extended optimal solutions to its discrete counterparts constructed therein. This makes a bridge between optimization of the continuous-time and discrete-time sweeping control systems while justifying in this way an effective usage of discrete approximations to solve the original sweeping control problem.
After reviewing Section 6 the generalized differential tools of variational analysis used in the paper and their calculations via the given data of (P τ ), we derive in Section 7 necessary optimality conditions for the constructed discrete approximations. Finally, in Section 8 we discuss the possibility to employ the obtained results in the limiting procedure to derive nondegenerate necessary optimality conditions for a given local minimizer in (P τ ) with their subsequent applications to the crowd motion model. The notation of this paper is standard in variational analysis and optimal control; see, e.g., [21,27,31]. Recall that the symbol B(x, ε) denotes the closed ball of the space in question centered at x with radius ε > 0 while IN signifies the collection of all natural numbers {1, 2, . . .}.

2.
Standing assumptions and preliminaries. Throughout the paper we impose the following standing assumptions on the initial data of the optimal control problem (P ) in 4-8: (H1) The mapping f : R n × R d → R n is continuous on R n × R d and locally Lipschitz continuous in the first argument, i.e., for every ε > 0 there is a constant Furthermore, there is a constant M > 0 ensuring the growth condition (H2) The terminal cost function ϕ : R n → R and the running cost function : [0, T ] × R 4n+2d → R in 6 are lower semicontinuous (l.s.c.) while is bounded from below on bounded sets. Now we are ready to formulate the powerful well-posedness result for the sweeping process under consideration that reduces to [12,Theorem 1]. Proposition 1. (well-posedness of the controlled sweeping process). Under the assumptions in (H1), let u(·) ∈ W 1,2 ([0, T ]; R n ) and a(·) ∈ W 1,2 ([0, T ]; R d ), and let M > 0 be taken from 10. Then the perturbed sweeping inclusion 4 with C(t) from 5 admits the unique solution x(·) ∈ W 1,2 ([0, T ]; R n ) generated by (u(·), a(·)) and satisfying the estimates Proof of Proposition 1. To deduce this result from [12, Theorem 1], with taking into account the solution estimates therein, it remains to verify that C(t) in 5 generated by the chosen W 1,2 -control u(·) varies in an absolutely continuous way [12], i.e., there is an absolutely continuous function v : with dist(x; Ω) standing for the distance from x ∈ R n to the closed set Ω ⊂ R n and with the function in our case. To verify 11, pick any y ∈ R n and c ∈ C and then easily get the estimates dist y; which imply in turn by the definition of the distance function that Using this and then changing the positions of t and s give us the resulting inequality   3. Discrete approximations of feasible solutions. In this section we construct a sequence of discrete approximations of the sweeping differential inclusion in 4, 5 with the constraints in 7 and 8, but without appealing to the minimizing functional 6. The main result of this section is justifying the strong W 1,2 -approximation of any feasible control and the corresponding sweeping trajectory by their finite-difference counterparts, which are piecewise linearly extended to the continuous-time interval [0, T ]. First we reduce 4 to a more conventional form of differential inclusions. Introduce the new variable z := (x, u, a) ∈ R n × R n × R d and define the set-valued mapping Consider the collection of active constraint indices of polyhedron 5 atx ∈ C given by it is not difficult to observe (see, e.g., [13, Proposition 3.1]) the explicit representation of 12 via the active index set 13 at x−u ∈ C. Then we can rewrite 4 in the following equivalent form with respect to the variable z = (x, u, a): with the initial condition z(0) = (x 0 , u(0), a(0)) satisfying x 0 − u(0) ∈ C, i.e., such that x * i , x 0 −u(0) ≤ 0 for all i = 1, . . . , m. Proposition 1 allows us to have solutions of the differential inclusion 15 in the class of Note that, although the resulting system 15 is written in the conventional form of the theory of differential inclusions, it does not satisfy usual assumptions therein. Indeed, the right-hand side of 15 is intrinsically unbounded in all its components, including the first (perturbed normal cone) one in which is highly non-Lipschitzian. Furthermore, the constrained system under consideration contains the intrinsic inequality state constraints 8 together with the equality one 7 on the whole time interval [0, T ].
Having in mind further applications including those in [7], it makes sense to consider a parametric version of the equality constraint in 7 with a small parameter τ ≥ 0 while replacing 7 by which reduces to 7 when τ = 0. Fix any τ ∈ [0, min{r, T }], k ∈ IN and denote by j τ (k) := [kτ /T ] the smallest index j such that t k j ≥ τ and by j τ (k) : The next theorem on the strong discrete approximation of feasible sweeping solutions is a counterpart of [9, Theorem 3.1] for the perturbed sweeping process in 4, 5 constrained by 8, 16 with additional quantitative estimates expressed via the system data. The reader can see that both the formulation and proof in the new setting are significantly more involved in comparison with [9]. Observe also the novel approximation conclusion 20, which holds also in the setting of [9] while being missed therein. This conclusion will allow us to construct a more precise discrete approximation of a local minimizer in Theorem 5.1, which is crucial to derive a new transversality condition for the original continuous-time sweeping control problem (P τ ) in [7].
Theorem 3.1. (W 1,2 -strong discrete approximation of feasible sweeping solutions). Under the validity of (H1), let the triplez(·) = (x(·),ū(·),ā(·)) be a given feasible solution to the constrained sweeping system from 4, 5, and 16 with a fixed parameter τ ∈ [0, min{r, T }], and let the constant K be taken from 9. Define the discrete partitions of [0, T ] by and suppose thatz(·) has the following properties at the mesh points (while observing that all these properties hold ifz and there is a constant µ > 0 independent of k such that Then there exist a sequence of piecewise linear functions z k (t) : with µ := max 3µ(1 + 4KT )e K , 4µ(e K + 1) , where the symbol "var " stands for the total variation of the function in question.
Proof of Theorem 3.1. Let y k (·) := (y k 1 (·), y k 2 (·), y k 3 (·)) be piecewise linear on [0, T ] and such that Define next w k (t) = (w k 1 (t), w k 2 (t), w k 3 (t)) :=ẏ k (t) as a piecewise constant and right continuous function on [0, T ] via the derivatives at non-mesh points and deduce from 19 that var (w k 2 ; [0, T ]) ≤ µ for every k ∈ IN . It follows from the definition of w k (·) that due to the existence of the right derivative ofẋ(0) by the imposed assumption on the validity of 15 at the mesh points. Furthermore, we get from 8 that and use for simplicity the notation t j := t k j as j = 1, . . . , k. To construct the claimed trajectories x k (t) of 22, we proceed by induction and suppose that the value of x k (t j ) is known. Define now the vectors and assume without loss of generality that u k (t j ) = r for j = j τ (k), . . . , j τ (k), which clearly yields Since the sets F (z) in 12 are closed and convex, we select the unique projection and deduce from 26 and j = 0, . . . , k shows that the inclusions in 22 are fulfilled and condition 20 holds. Furthermore, we deduce from 14 and 27 that at the mesh points. To verify that the triples (x k (t), u k (t), a k (t)), k ∈ IN , constructed above satisfy all the conclusions of the theorem, let us first show that for all j = 0, . . . , k. Indeed, picking any t ∈ [t j , t j+1 ] for j = 0, . . . , k − 1, we have the representation It then follows from 29 that Using the Lipschitz continuity of f with respect to x imposed in 9 gives us and thus, by taking the first condition in 19 into account, we arrive at the inequalities (32) Now we proceed by induction to verify that for j = 3, . . . , k. Starting with j = 3, observe from 19, 31, and 32 that which justifies the validity of 33 at j = 3. Suppose next that 33 holds for t j as j ≥ 3 and show that it is also satisfied for t j+1 . Indeed, employing 19 and 31 tells us that which shows that estimate 33 holds for t j+1 , and thus it is justified for all j = 3, . . . , k. Now picking any j ∈ {3, . . . , k} and using the first inequality in 19, we get Combining it with 32, we arrive at 30. This readily implies that , the relationships in 21 are satisfied with ε k defined in 30. Furthermore, it follows from 30 and 31 that for t ∈ [t j , t j+1 ] and j = 0, . . . , k − 1. Next we consider relationships for the ucomponent of z k (·). The first and third conditions in 19 yield which justifies the first estimate in 24. To verify the second estimate therein, we deduce from 27, 30, and the second inequality in 19 that which readily gives us the claimed result in 24. It remains to justify the W 1,2 -convergence of z k (t) toz(t) in 23. Using 27 for j = 0 with x k (t 0 ) = x 0 , the construction of a k (t), and the Newton-Leibniz formula, it suffices to show that the sequence of (ẋ k (t),u k (t)) converges to (ẋ(t),u(t)) strongly in L 2 [0, T ]. To this end we have as k → ∞ due to 18, 25, and the definition of w k (t). It follows furthermore that due to the above convergence of {ẋ k (·)}. This verifies 23 and completes the proof of the theorem.

4.
Existence of optimal sweeping solutions and relaxation. In this section we start studying optimal solutions to the original sweeping control problem (P ). By taking into account the discussion above and further applications, our main attention is paid to the parametric family of problems (P τ ) as τ ≥ 0 with (P 0 ) = (P ), which are different from (P ) only in that the control constraint 7 is replaced by those in 16. First we establish the following existence theorem of optimal solutions for (P τ ) in the the class of W 1,2 [0, T ] functions.
We can see that the underlying assumption of Theorem 4.1 is the convexity of the integrand with respect to velocities. This assumption, which is not needed for deriving necessary optimality conditions, can be generally relaxed (and even fully dismissed in rather broad nonconvex settings from the viewpoint of actual solving optimization problems for differential inclusions) due the so-called Bogoluybov-Young relaxation procedure. To describe it in the setting of (P τ ), denote by F (t, x, u, a,ẋ,u, a) the convexification of the integrand in 6 on the set F (x, u, a) from 12 with respect to the velocity variables (ẋ,u,ȧ) for all t, x, u, a, i.e., the largest convex and l.s.c. function majorized by (t, x, u, a, ·, ·, ·) on this set; we put := ∞ at points out of F (x, u, a). Define now the relaxed sweeping problem (R τ ) by over all the triples z(·) = (x(·), u(·), a(·)) ∈ W 1,2 [0, T ] satisfying the constraints in 16. Of course, there is no difference between problems (P τ ) and (R τ ) if the integrand is convex with respect to (ẋ,u,ȧ). Furthermore, Theorem 4.1 ensures the existence of optimal solutions to (R τ ). The strong relationship between the original and relax/convexified problems, known as relaxation stability, is that in many situations the optimal values of the cost functionals therein agree. This phenomenon has been well recognized for differential inclusions with Lipschitzian right-hand sides in state variables (see [30]), which is never the case for the sweeping process. A more subtle result of this type is obtained in [11, Theorem 4.2] for differential inclusions satisfying the modified one-sided Lipschitz property, which however is also restrictive in applications to sweeping control. The relaxation stability result that directly concerns sweeping control problems is given in [12,Theorem 2] while it deals only with the case of controlled perturbations. In general, relaxation stability in sweeping optimal control is an open question.
Our current study here and its continuation in [7] concern local optimal solutions to (P τ ) involving a local version of relaxation stability. Following [20], we say that z(·) is a relaxed intermediate local minimizer This notion distinguishes local minimizers that lie between classical weak and strong minima in continuous-time variational problems and can be strictly different from both of them even in fully convex settings; see [22] for discussions, examples, and references. It is clear that from the viewpoint of deriving necessary optimality conditions we can confine ourselves to the case of α = 1.

5.
Discrete approximations of local optimal solutions. In this section we construct a sequence of well-posed discrete approximations of each problem (P τ ) as 0 ≤ τ ≤ τ with τ = min{r, T } and then employ this method to the study of relaxed intermediate local minimizers for this problem. Given any r.i.l.m.z = (x(·),ū(·),ā(·)) for (P τ ) and the discrete mesh ∆ k from 17, for every k ∈ IN define the discrete sweeping control problem (P τ k ) as follows: minimize x where > 0 is taken from definition 38 with α = 1 while ε k and µ are taken from Theorem 3.1. Let us first show that each problem (P τ k ) admits an optimal solution for all large k ∈ IN ; this issue is unavoidable in employing the method of discrete approximations to study local minimizers for (P τ ). Proposition 2. (existence of optimal solutions to discrete approximations). Suppose that (H1) holds and that (H2) is also satisfied around the given local minimizerz(·) for (P τ ). Then each problem (P τ k ) admits an optimal solution provided that k ∈ IN is sufficiently large.
Proof of Proposition 2. Theorem 3.1 tells us that the set of feasible solutions to (P τ k ) is nonempty for all large k ∈ IN . Moreover, the constraints in 42-44 ensure that this set is bounded. To justify the claimed existence of optimal solutions to (P τ k ) by the Weierstrass existence theorem, it remains to verify that this set is closed. To proceed, take a sequence z ν (·) = z ν := (x ν 0 , . . . , x ν k , u ν 0 , . . . , u ν k−1 , a ν 0 , . . . , a ν k−1 ) of feasible solutions for (P τ k ) converging to some z(·) = z := (x 0 , . . . , x k , u 0 , . . . , u k−1 , a 0 , . . . , a k−1 ) as ν → ∞ and show that z is feasible to (P τ k ) as well. Observe that and so x j − u j ∈ C for all j = 0, . . . , k − 1. Picking now i ∈ {1, . . . , m}\I(x j − u j ), we have x * i , x j − u j < 0, which yields x * i , x ν j − u ν j < 0 for ν sufficiently large. Then it follows that i ∈ {1, . . . , m}\I(x ν j − u ν j ) and hence I(x ν j − u ν j ) ⊂ I(x j − u j ) for ν ∈ IN sufficiently large. By taking 12 and 14 into account, we get the equalities . This shows therefore that Passing there to the limit as ν → ∞ and using the closedness of N (x j − u j ; C) give us which ensures that x j+1 ∈ x j − h k F (x j , u j , a j ) and thus completes the proof of the proposition.
The next theorem is a key result of the method of discrete approximations in sweeping optimal control. It shows that optimal solutions to (P τ ) and (P τ k ) are so closely related that solving the continuous-time control problem (P τ ) for small τ ≥ 0 can be practically replaced by solving its finite-dimensional discrete counterparts (P τ k ) when k is sufficiently large. Moreover, it justifies the possibility to derive necessary optimality conditions for local minimizers of (P τ ) by passing to the limit from those in (P τ k ) as k → ∞.
To proceed, observe first that x(t) − u(t) = lim k→∞ (x k (t) −ū k (t)) ∈ C by the closedness of the polyhedron C. It follows from the above that there are a function ν : IN → IN and a sequence of real numbers {α(k) j | j = k, . . . , ν(k)} such that as k → ∞. Then by the closedness and convexity of the normal cone we have the relationships where I(·) is taken from 13, and where λ j i = 0 if i ∈ I( x(t) − u(t))\I(x j (t) − u j (t)) for j = k, . . . , ν(k) and all large k ∈ IN . It shows by 14 that z(·) satisfies 15 and hence the constraints in 8.

Consider further the integral functional
which is l.s.c. in the weak topology of L 2 ([0, T ]; R 2n+d ) due to the convexity of the integrand in y. Hence by the construction of z(·). Passing to the limit in 43 and 44 as k → ∞ and using 49, we get This means that z(·) belongs to the given neighborhood ofz(·) in W 1,2 ([0, T ]; R 2n+d ). Furthermore, the definition of F in 37 and its convexity in the velocity variables yield Thus the passage to the limit in the cost functional of (P τ k ) and the assumption on c > 0 in the negation of 48 together with (H2) bring us to the relationships (50) Applying now Theorem 3.1 to the r.i.l.m.z(·) gives us a sequence {z k (·)} of feasible solutions to (P τ k ) whose extensions to the whole interval [0, T ] strongly approximatē z(·) in the W 1,2 topology with the additional convergence in 20. Sincez k (·) is an optimal solution to (P τ k ), we have It follows from the structure of the cost functionals in (P τ k ), the strong W 1,2convergence of z k (·) →z(·) together with 20 in Theorem 3.1, and the continuity assumptions on ϕ and imposed in this theorem that J k [z k ] → J[z] as k → ∞. which clearly contradicts the fact thatz(·) is a r.i.l.m. for problem (P τ ). This justifies the validity of 48 and thus completes the proof of the theorem. 6. Generalized differentiation and calculations. After establishing close connections between optimal solutions to the original and discretized sweeping control problems, our further goal is to derive effective necessary optimality conditions to each problem (P τ k ) defined in 39-45. Looking at this problem, we can see that it is intrinsically nonsmooth, even if both terminal and running costs are assumed to be differentiable, which is not the case here. The main reason for this is the unavoidable presence of the geometric constraints 40 with F given by 12 whose increasing number comes from the discretization of the sweeping differential inclusion 4. We can deal with such problems by using the robust generalized differential constructions, which are basic in variational analysis and its applications; see, e.g., the books [3,21,27] and the references therein. Here we first recall their definitions with a brief overview of the needed properties and then deduce from [13] major coderivative calculations for the mapping F in 12 via the initial data of the sweeping process. This together with available calculus rules of generalized differentiation plays a crucial role in deriving verifiable necessary optimality conditions for the sweeping control problems under consideration.
Given a set-valued mapping/multifunction G : R n ⇒ R m , denote by When Ω is convex, 54 reduces to the normal cone of convex analysis, but it is often nonconvex in nonconvex settings. The crucial feature of 54 and the associated subdifferential and coderivative constructions for functions and multifunctions (see below) is full calculus based on variational and extremal principles. For a set-valued mapping F : R n ⇒ R m with its graph locally closed around (x,ȳ), the coderivative of F at (x,ȳ) generated by 54 is defined by When F : R n → R m is single-valued and continuously differentiable (C 1 ) aroundx, we have via the adjoint/transposed Jacobian matrix ∇F (x) * , whereȳ = F (x) is omitted.
Our main emphases here is on evaluating the coderivative of the set-valued mapping F from 12 entirely via the given data of the perturbed sweeping process. Note that the partial normal cone structure of the mapping F reveals the second-order subdifferential nature of the aforementioned construction in the sense of [19]. For simplicity in further applications, suppose below that the perturbation function f is smooth while observing that the available calculus rules allow us to consider Lipschitzian perturbations.
Having in mind representation 15 of the mapping F in terms of the generating vectors x * i of the convex polyhedron 5 with the active constraint indices I(x) in 14, consider the following subsets: where I 0 (y) and I > (y) are defined in 56 withx = x − u, and where γ i ∈ R for i ∈ I 0 (y) while γ i ≥ 0 for i ∈ I > (y). Furthermore, 57 holds as an equality and the domain dom D * G x − u, w − f (x, a) can be computed by Proof of Theorem 6.1. Picking any y ∈ dom D * G(x − u, w − f (x, a)) and z * ∈ D * F (x, u, a, X)(y) and then denoting G Employing now in 59 the coderivative estimate for the normal cone mapping G obtained in [13,Theorem 4.5] with the exact coderivative calculation given in [13,Theorem 4.6] under the linear independence of the generating vectors x * i and also taking into account the structure of the mapping f in 59, we arrive at 57 and the equality therein under the aforementioned assumption.
7. Necessary optimality conditions. In this section we derive necessary conditions for optimal solutions to each discrete approximation problems (P τ k ) with k ∈ IN and 0 ≤ τ ≤ τ = min{r, T }. As shown in Theorem 5.1, for large k ∈ IN and any fixed τ ∈ [0, τ ] the constructed optimal solutionsz k (·) to (P τ k ) are practically undistinguished (in the W 1,2 norm) from the optimal solutionz(·) to the continuous-time sweeping control problem (P τ ), and so the necessary optimality conditions forz k (·) obtained below can be well treated as "almost optimality" necessary conditions for the solutionz(·) to (P τ ) playing virtually the same role in applications.
(85) Now we are ready to justify all the conditions claimed in the theorem. Observe first that 71 clearly yields 61. Next we extend the vector p by a zero component by putting p 0 := (x * 0k , u * 0k , a * 0k ). Then the conditions in 63 follow from 76, 79, and 82. Furthermore, the conditions in 64 follow from 84 and 85. Using the relationships which hold due to 75, 78, 81, and 83, and then substituting them into the lefthand side of 72, we arrive at 65 for all j = 0, . . . , k − 1. To verify the nontriviality condition 60, suppose by contradiction that λ = 0, α = 0, ξ = 0, p uk 0 = 0, p ak 0 = 0, and p x j = 0 for j = 0, . . . , k − 1. Then 76 yields that p x k = 0 and thus p x j = 0 for all j = 0, . . . , k. Observe further that x * 0k = p x 0 = 0, and so the conditions in 74, 75, and 83 imply that x * jj = 0 and X * jj = 0 for j = 0, . . . , k − 1. The validity of 84 and 85 ensures that p u j = 0 and p a j = 0 for j = 1, . . . , k, which in turn show by 77, 78, 80, and 81 that u * jj = 0 and a * jj = 0 for j = 0, . . . , k − 1. As mentioned above, all the components of y * j different from (x * jj , u * jj , a * jj , X * jj ) are zero for j = 0, . . . , k − 1. Hence we have y * j = 0 for j = 0, . . . , k − 1 and similarly y * k = 0 since the only potential nonzero component of this vector is x * 0k = p x 0 = 0. We get therefore that y * j = 0 for all j = 0, . . . , k, which violates the nontriviality condition for (M P ) and thus completes the proof of the theorem.
The final result of this section employs the effective coderivative calculations for the sweeping control mapping taken from Theorem 6.1 that allows us to obtain necessary optimality conditions in (P τ k ) expressed entirely via the given problem data and the minimizer under consideration under the additional assumption on the smoothness of f , which is imposed for simplicity. Furthermore, we derive an enhanced nontriviality relation in the case of linear independence of the generating vectors x * i for the underlying convex polyhedron C from 5. Theorem 7.2. (optimality conditions for discretized sweeping inclusions via their initial data). Letz k = (x k ,ū k ,ā k ) be an optimal solution to problem (P τ k ) in the general framework of Theorem 7.1 with F given by 14 via the active constraint indices I(·) in 13 and locally smooth perturbation function f , let the active index subsets I 0 (·) and I > (·) be taken from 56, and let the triples (θ xk j , θ uk j , θ ak j ) be defined in 68. Then there exist dual elements (λ k , ξ k , p k ) as in Theorem 7.1 together