Almost mixed semi-continuous perturbation of Moreau's sweeping process

In this work, we introduce a new concept of semi-continuous set-valued mappings, called almost mixed semi-continuity, by taking maps that are upper semi-continuous with almost convex values in some points and lower semi-continuous in remaining points. We generalize earlier results obtained for both mixed semi-continuous maps and almost convex sets. We discuss the existence of solution for evolution problems driven by the so-called sweeping process subject to external forces, known as perturbation to the system, by this type of set-valued mappings. Finally, we give some topological properties of the attainable and solution sets in order to solve an optimal time problem.


1.
Introduction. It's well known that the existence of solution for the Cauchy problemẋ(t) ∈ F (t, x), t ∈ [0, T ]; x(0) = x 0 is obtained in both cases: when the right-hand side is upper semi-continuous with convex values or lower semicontinuous with nonconvex values. The concept of mixed semi-continuous mappings was used by Tolstonogov [15] and Fryskowski and Gorniewicz [10] for maps mixing both lower and upper semi-continuity regularity assumptions. The approach is based on the construction of a set-valued selection with convex values of the righthand side of the Cauchy problem. These results inspired many authors to study differential inclusions with mixed semi-continuous perturbation, particularly the so called perturbed sweeping process (P F )    −ẋ(t) ∈ N C(t) (x(t)) + F (t, x(t)) a.e. in [T 0 , T ], x(t) ∈ C(t) , ∀t ∈ [T 0 , T ], x(T 0 ) = u 0 ∈ C(T 0 ) , Thibault in [14]. Another way to obtain the existence of absolutely continuous solution is the use of the mapping K F defined on C R n ([T 0 , T ]) by x(t) ∈ F (t, s(t)) a.e. in [T 0 , T ]} which takes decomposable values.
On another side, Cellina and Ornelas [6] have introduced a new class of righthand side for which the Cauchy problem admits a solution, namely the almost convex sets (see the definition below). In our previous work [1], we have considered an evolution inclusion governed by the Moreau's sweeping process subject to almost convex perturbations, that is external forces applied on the system. Our first aim in this paper, is to combine between the almost convexity of sets with the mixed semi-continuity, to provide a new class of nonconvex problems admitting a solution, moreover, we establish the compactness of the attainable set for this class of maps.
The autonomous case of (P F ), when F does not depend separately on t and the sets C(t) := C are fixed, presents an interesting and useful application arising in the study of planning procedures in mathematical economy, see [3]. It was studied by [9] and [12] and takes the following form: For recent results, we refer to [1] and [2]. Our second aim is to give an existence result for (P G ) that will be used to obtain a solution of reaching any element of the attainable set in a minimum time which is known as the time optimality problem. Some notations, definitions and preliminary results are formulated in Section 2.
In Section 3, considering the linear growth condition satisfied by the element of minimal norm instead of F , we extend the existence result of (P F ) obtained in [11] and we study some topological properties for the solution set, then through an intermediate result we establish the existence of an absolutely continuous solution for a sweeping process perturbed by an almost mixed semi-continuous set-valued mapping. Finally in Section 4, we shall show how those results can be investigated to solve an optimal time problem.
2. Notations and preliminary results. Throughout the paper R n is the ndimensional Euclidean space,B its closed unit ball and B(x, r) the closed ball of center x and rayon r > 0. We denote by L([T 0 , T ]) the Lebesgue σ-field of [T 0 , T ], B(R n ) the Borel σ-field of R n and by C R n ([T 0 , T ]) the Banach space of all continuous mappings u : Let A ⊂ R n , we denote by co(A) the convex hull of A and co(A) its closed convex hull. Following [9], A is called almost convex if and only if for all b ∈ co(A) there exist λ 1 , λ 2 , 0 ≤ λ 1 ≤ 1 ≤ λ 2 such that An evident example of almost convex sets are convex sets, that is easy to see if we choose λ 1 = λ 2 = 1. Other examples are the boundary ∂P of a convex set P if 0 / ∈ P or the union ∂P ∪ {0} if 0 ∈ P. Lett ∈ [T 0 , T ], we denote by the attainable set of (P F ) at the timet, where Tt(u 0 ) is the set of the trajectories of the differential inclusion (P F ) on the interval [T 0 ,t].
For a nonempty closed subset S of R n , we denote by d(·, S) the usual distance function associated and P roj S (u) the projection of u onto S defined by P roj S (u) = {y ∈ S : d(u, S) = u − y }. and δ * (x , S) = sup y∈S x , y the support function of S at x ∈ R n .
Let us recall some necessary definitions related to non smooth analysis. Let A be an open subset of a Hilbert space H and ϕ : A → (−∞, +∞] be a lower semicontinuous function, the proximal subdifferential ∂ P ϕ(x), of ϕ at x (see [8]) is the set of all proximal subgradients of ϕ at x, any ξ ∈ H is a proximal subgradient of ϕ at x if there exist positive numbers η and ς such that Let x be a point of S ⊂ H, we recall (see [8]) that the proximal normal cone to S at x is defined by N P S (x) = ∂ P δ S (x), where δ S denotes the indicator function of S, i.e. δ S (x) = 0 if x ∈ S and +∞ otherwise. Note that the proximal normal cone is also given by If ϕ is a real-valued locally-Lipschitz function defined on H, the Clarke subdifferential ∂ C ϕ(x) of ϕ at x is the nonempty convex compact subset of H given by t is the generalized directional derivative of ϕ at x in the direction v. The Clarke normal cone N C S (x) to S at x ∈ S is defined by polarity with T C S , that is, S denotes the clarke tangent cone and is given by . Recall now, that for a given ρ ∈]0, +∞] the subset S is uniformly ρ-prox-regular (see [13]) or equivalently ρ-proximally smooth if every nonzero proximal to S can be realized by a ρ-ball, this means that for all x ∈ S and all 0 = ξ ∈ N P S (x) one has for all x ∈ S. We make the convention 1 ρ = 0 for ρ = +∞. Recall that for ρ = +∞ the uniform ρ-prox-regularity of S is equivalent to the convexity of S.
The following proposition summarizes some important consequences of the uniform prox-regularity needed in the sequel. For the proof of these results we refer the reader to [4] and [13].
Proposition 1. Let S be a nonempty closed subset of H and x ∈ S. The following assertions hold: if the subset S is uniformly ρ-prox-regular with ρ ∈]0, +∞], then i) the proximal subdifferential of d(·, S) coincides with its Clarke subdifferential at all points ii) the proximal normal cone to S coincides with all the normal cone contained in the Clarke normal cone at all points x ∈ H, i.e., N S (x) = N P S (x) = N C S (x). Here and above ∂ C d(x, S) and N C S (x) denote respectively the Clarke subdifferential of d(·, S) and the Clarke normal cone to S (see [8] The following is an important closeness property of the subdifferential of the distance function associated with a set-valued mapping (see [4]). H be a Hausdorff-continuous set-valued mapping. Assume that K(z) is uniformly ρ-prox-regular for all z ∈ Ω. Then for a given 0 < σ < ρ, the following holds: for anyz ∈ Ω,x ∈ K(z) + (ρ − σ)B H , x n →x, z n →z with z n ∈ Ω (x n not necessarily in K(z n )) and ξ n ∈ ∂d(x n , K(z n )) with ξ n → wξ one hasξ ∈ d(x, K(z)), here → w means the weak convergence in H.

Remark 1.
This property means that for every ρ ∈]0, +∞], for a given 0 < σ < ρ, and for every set-valued mapping K : Ω H with uniformly ρ-prox-regular values, the set-valued mapping (z, The set-valued mapping F is said mixed semi-continuous (see [15]) if for every t ∈ [T 0 , T ], at each x ∈ R n such that F (t, x) is convex the set-valued mapping F (t, ·) is graphically closed, and whenever F (t, x) is not convex, F (t, ·) is lower semicontinuous on some neighborhood of x (see also example 4 in [10]). Let introduce the following definition. We say that F is almost mixed semi-continuous if for every t ∈ [T 0 , T ], at each x ∈ R n where F (t, x) is almost convex, F (t, ·) is graphically closed, and F (t, ·) is lower semi-continuous on some neighborhood of x whenever F (t, x) is not almost convex. Obviously, any mixed semi-continuous set-valued mapping is almost mixed semi-continuous. Let recall the following result which ensures the existence of a closed convex valued set-valued selection for a mixed semi-continuous set-valued mapping.
R n be a closed valued set-valued mapping global measurable, mixed semi-continuous, and there exists f : [T 0 , T ] × R n → R + a Carathéodory function which is integrably bounded on bounded subsets of R n and satisfying Then for any > 0 and any compact set K ⊂ C([T 0 , T ], R n ) there is a non empty closed convex valued set-valued mapping Φ : which has sequentially closed graph with respect to the norm of uniform convergence in K and the weak topology σ(L 1 and such that for any u ∈ K and φ ∈ Φ(u) one has for a.e. t ∈ [T 0 , T ] for almost every t ∈ [T 0 , T ].
3. Existence results. Let begin by the following weaker version of Theorem 3.1 in [11], obtained by taking an unbounded perturbation: we replace the linear growth intersection condition by the linear growth condition of only the element with minimal norm. We provide the existence of solution and the compactness of the attainable set.
R n be a set-valued mapping with nonempty closed valued satisfying: (H 1 ) there exists some constant ρ ∈]0, +∞] such that for each t ∈ [T 0 , T ] the sets C(t) are uniformly ρ-prox regular; (H 2 ) C(t) varies in an absolutely continuous way, that is, there exists an absolutely continuous non negative function η : Let F : [T 0 , T ] × R n R n be a closed valued set-valued mapping satisfying the following assumptions: is lower semi-continuous on some neighborhood of x; (iii) there are two nonnegative constants p, q and for all (t, Then, for each u 0 ∈ C(T 0 ), there is an absolutely continuous solution of (P F ), moreover, for any fixed timet ∈ [T 0 , T ], the attainable set of (P F ) att is compact. Proof.
Step 1. Since the assumption (H 3 ) in [11] is satisfied by our condition (iii), taking f (t, x) = p + q x and applying Theorem 3.1 in [11], (P F ) admits an absolutely continuous solution u : where δ : [T 0 , T ] → R + be the absolutely continuous solution of the ordinary differential equation with σ(t) = + p + q u 0 + t T0 |η(s)|ds and > 0 fixed.
Step 2. Proving that the set of the trajectories of the differential inclusion (P F ) on the interval [T 0 ,t], : u is an absolutely continuous solution of (P F )} is compact. Let (u n ) be a sequence in Tt(u 0 ), for every n ∈ N and t ∈ [T 0 ,t], by then (u n (t)) is relatively compact, and for all t, t ∈ [T 0 ,t] such that t ≤ t we have so, we get the equicontinuity of the sequence (u n ). We conclude that (u n ) is relatively compact in C R n ([T 0 ,t]). By Ascoli's theorem, (u n ) admits a subsequence (again denoted by) (u n ) that converges uniformly to u such that (u n ) converges Consider the set where γ(t) =η(t) + 2δ(t), K is a compact set, by Theorem 2.1 there exists a nonempty closed convex set-valued mapping Φ : , with the properties cited there, such that for every φ n ∈ Φ(u n ) and all n ∈ N φ n (t) ∈ F (t, u n (t)) and φ n (t) by Proposition 1 we geṫ u n (t) + φ n (t) ∈ −µ(t)∂d(u n (t), C(t)) a.e. t ∈ [T 0 ,t].
It is clear that the set-valued mapping J has closed, uniformly ρ-prox-regular values, and satisfies (H 2 ) with an absolutely continuous function V (·) such that V (t) = t T0 µ(s)ds. Hence, we obtain the inclusion −ż(t) ∈ µ(t)∂d(z(t), J(t)) a.e. t ∈ [T 0 ,t], This shows that Tt(u 0 ) is compact. From the compactness of Tt(u 0 ) we deduce that of A u0 (t). Now, we address the particular case of a fixed set C(t) := C, and an autonomous perturbation F (t, x) := G(x). Such problems arises in the analysis of resource allocation mechanisms in economics and crowd motion modeling variational inequalities. The following result is crucial in the statement of our next result.
Proposition 3. Let C be a closed subset of R n uniformly ρ-prox-regular, and G : R n R n be a measurable set-valued mapping satisfying the following assumptions: (i) at each x ∈ R n such that G(x) is compact and almost convex the set-valued mapping G(·) is upper semi-continuous, and whenever G(x) is not almost convex G(·) is lower semi-continuous on some neighborhood of x; (ii) there are two nonnegative constants p, q and for all x ∈ R n P roj G(x) (0) ≤ p + q x .
Proof. 1) Since, for all t ∈ D(u), G(u(t)) is almost convex, then there exist two nonempty set-valued mappings ∆ 1 : D(u) [0, 1] and ∆ 2 : There is no loss of generality in assuming that, for

Consider its graph
with σ : (t, λ 1 ) → d(λ 1 g(t), G(u(t))), then Gph∆ 1 is measurable as the intersection of two measurable subsets. In addition, its values are closed subsets of [0, 1] because the values of G are closed. Then, we conclude that ∆ 1 is measurable on D(u).
by the properties of the normal cone and the assumption on g, we get, for all ∈ N C (u(θ(t))) + G(u(θ(t))) ∈ N C (ũ(t)) + G(ũ(t)). b) Suppose that Ω = ∅, setting τ = sup Ω, since Ω is closed relative to D(u) then τ ∈ Ω, here the complement of Ω is open relative to D(u), then it consists of at most a countably many non-overlapping open intervals ]a i , b i [, with the possible exception of one of the form ]τ, b ii [. For each i, apply step 2 to the interval ]a i , b i [ to infer the existence of two measurable subsets of ]a i , b i [ with characteristic functions χ i 1 (·) and χ i 2 (·) such that χ i 1 (·) + χ i 2 (·) = χ ]ai,bi[ (·). Settinġ Since D(u) is closed, then m = inf D(u) and M = sup D(u) are in D(u) and we get the following cases: where the sum is over all intervals contained in the [m, τ ], since λ 2 (t) ≥ 1 and .
Theorem 3.2. Let C be a closed subset of R n uniformly ρ-prox-regular and G : R n R n be a measurable set-valued mapping, satisfying the assumptions (i) and (ii). Then for all u 0 ∈ C, 1) the problem (P G ) has at least an absolutely continuous solution; 2) for all t ∈ [T 0 , T ] the attainable set at t, A u0 (t) coincides with A co u0 (t), the attainable set at t of the convexified problem (P co ).
Proof. 1) Observing that with D(·) defined as above, the set-valued mapping F : R n R n given by is mixed semi-continuous. Since G(x) ⊂ F (x) for all x ∈ R n , then Consequently and by Theorem 3.1, there exists an absolutely continuous function u : [T 0 , T ] → R n solution of the problem (P co ). And by Proposition 3, the problem (P G ) admits at least an absolutely continuous solutionũ : [T 0 , T ] → R n such that u(T ) = u(T ).
2) For every t ∈ [T 0 , T ], the attainable set at t, A u0 (t), is contained in the attainable set at t of the convexified problem, A co u0 (t), it is enough to show that A co u0 (t) ⊂ A u0 (t).
Let u(t) ∈ A co u0 (t), so u(·) is an absolutely solution of the problem (P co ). Proposition 3 is applied on [T 0 , t] to find a solutionũ(·) of the problem (P G ) such that u(t) =ũ(t) ∈ A u0 (t). Then, A co u0 (t) ⊂ A u0 (t).

Corollary 1.
Let C be a closed subset of R n , uniformly ρ-prox-regular. Let D be a subset of R n and Z : R n R n be a nonempty measurable set-valued mapping such that at each x ∈ D, Z(·) is upper semi-continuous at x with compact values and whenever x / ∈ D, Z(·) is lower semi-continuous on some neighborhood of x, and let a continuous single-valued map h : Gph(Z) → R n satisfying the following assumption: H h ) there are nonnegative constants p and q such that h(x, z) ≤ p + q x , ∀(x, z) ∈ Gph(Z).
We associate with these data the set-valued mapping G : R n R n defined by ∀x ∈ R n , G(x) = {h(x, z)} z∈Z(x) .
Assume that G(·) is compact and almost convex for every x ∈ D. Let u 0 , u 1 be given in R n , and assume that for some T 0 ≤ t ≤ T, u 1 ∈ A u0 (t). Then, the problem of reaching u 1 from u 0 in a minimum time admits a solution.
Proof. Under the hypotheses on h and Z, the set-valued mapping G is almost mixed semi-continuous and P roj G(x) (0) ≤ p + q x , ∀(x, z) ∈ Gph(Z).