Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.


Introduction
The present paper is a continuation of [16], where we have studied a class of constrained evolution problems, called sweeping processes, in the framework of a real Hilbert space X endowed with scalar product x, y and norm |x| = x, x . In order to describe the processes studied in [16], we assume that we are given right-continuous functions u : [0, T ] → X and w : [0, T ] → W , where W is a real Banach space and we suppose that u and w are regulated, i. e. they admit left limits at every point t ∈ (0, T ]. We also assume that a not necessarily convex moving constraint Z(w(t)) ⊂ X is given and that Z(w(t)) is r -prox-regular, i.e. Z(w(t)) is a closed sets having a neighborhood of radius r > 0 where the metric projection exists and is unique.
Assuming that the sets Z(w(t)) satisfy a suitable uniform non-empty interior condition, we have proved in [16] that for every initial condition x 0 ∈ Z(w(0)) there exists a right-continuous function ξ : [0, T ] → X of bounded variation (BV ) such that the variational inequality Since the normal cone N Z (x) of a closed Z ⊆ H at x ∈ Z is defined by the formula the variational inequality (0.1) can be formally interpreted as a BV integral formulation of the differential inclusioṅ ξ(t) ∈ −N C(t) (ξ(t)), ξ(0) = u(0) − x 0 (0. 3) with C(t) = u(t) − Z(w(t)), t ∈ [0, T ].
In [16,Section 5], we have shown under some technical assumptions, but dropping the uniform non-empty interior condition for Z(w(t)), that if the inputs u, w are absolutely continuous, then the output ξ is absolutely continuous and satisfies the pointwise variational inequality x(t) − z,ξ(t) + |ξ(t)| 2r |x(t) − z| 2 ≥ 0, x(t) + ξ(t) = u(t), x(0) = x 0 (0.4) for a. e. t ∈ (0, T ) and all z ∈ Z(w(t)). The existence and uniqueness result for (0.4) was stated and proved in [16,Corollary 5.3] and we recall the precise statement below in Proposition 2.3. A detailed survey of the literature related to non-convex sweeping processes was given in [16] and we do not repeat it here. Instead, we pursue further the study of (0.4) in the space of absolutely continuous functions. Let us mention only the publications that have particularly motivated our research, namely the pioneering paper [24] where the concept of sweeping process was elaborated, the detailed studies [7,25] of prox-regular sets, and a deep investigation of prox-regular sweeping processes carried out in [8,26].
It turns out that it is convenient in this context to represent the sets Z(w) = {x ∈ X : G(x, w) ≤ 1} as sublevel sets of a function G : X × W → [0, ∞) satisfying suitable technical assumptions. A detailed comparison of different continuity criteria has been done in the convex case in [5]. In the nonconvex case treated in the present paper we prove as our main result that the input-output mapping (u, w) → ξ is strongly continuous with respect to the W 1,1 -norms if G is continuously differentiable with respect to both x and w , and Lipschitz continuous if both gradients ∇ x G, ∇ w G are Lipschitz continuous. As a consequence of the Lipschitz input-output dependence, we apply the Banach contraction principle to prove the unique solvability of an implicit state dependent problem with w of the form w(t) = g(t, u(t), ξ(t)) with a given smooth function g : [0, T ]×X ×X → W . The authors are not aware of any result of this kind in the literature on prox-regular sweeping processes. Implicit problems in the convex case have been solved under suitable additional compactness assumptions in [18,19] and without compactness in [5]. The non-convex case has been considered for example in [1,2,12,13,21], but to our knowledge, in all existing publications, the sweeping process is regularized by some kind of compactification or viscous regularization. In our case, no compactification or other kind of regularization comes into play.
The paper is structured as follows. In Section 1, we identify sufficient conditions on the function G(·, w) which guarantee that the sublevel set Z(w) is r -prox-regular for every w ∈ W . Section 2 is devoted to finding additional hypotheses on the w -dependence of G which guarantee the validity of the existence and uniqueness result for Problem (0.4) in [16,Corollary 5.3]. The strong continuity of the (u, w) → ξ input-output mapping with respect to the W 1,1 -norm is proved in Section 3, and the local Lipschitz continuity of the mapping u → ξ in the implicit case w(t) = g(t, u(t), ξ(t)) is proved in Section 4. Definition 1.1. Let X be a real Hilbert space endowed with scalar product ·, · and norm |x| = x, x , let Z ⊂ X be a closed connected set, and let dist(x, Z) := inf{|x − z| : z ∈ Z} denote the distance of a point x ∈ X from the set Z . Let r > 0 be given. We say that Z is r -prox-regular if the following condition hold.

Lemma 1.2.
A set Z ⊂ X is r -prox-regular if and only if for every y ∈ X such that d = dist(y, Z) < r there exists a unique x ∈ Z such that |y − x| = d and We represent the sets Z as the sublevel sets of a function G : X → [0, ∞) in the form We define the gradient ∇G(x) ∈ X of G at a point x ∈ X by the formula The following hypothesis is assumed to hold. Hypothesis 1.3. Let X be a real Hilbert space endowed with scalar product ·, · and norm |x| = x, x . We assume that (1.3) holds for a function G : X → [0, ∞), ∇G(x) exists for every x ∈ X , and there exist positive constants λ, c and a continuous increasing Throughout the paper, for a set S ⊂ X , the symbols ∂S , Int S , and S will denote respectively the boundary, the interior, and the closure of S . It is easy to check that under Hypothesis 1.
For n ∈ N put x n := x + 1 n ∇G(x). We have G(x n ) > 1 for n sufficiently large, hence x n / ∈ Z . Since x n converge to x as n → ∞, we conclude that x ∈ ∂Z .
In the convex case, we can choose G to be the Minkowski functional M Z (or gauge) associated with Z defined as M Z (x) = inf{s > 0 : 1 s x ∈ Z}. Then condition (iii) of Hypothesis 1.3 is automatically satisfied, since ∇M Z is monotone, and (ii) is just the uniform continuity condition of ∇M Z . For non-convex sets Z , condition (iii) excludes sharp concavities of ∂Z .
We now prove the following result.
Proposition 1.4. Let Hypothesis 1.3 hold and let r = c/λ. Then for every y ∈ X such that d = dist(y, Z) ∈ (0, r) there exists a unique x ∈ ∂Z such that In particular, Z is r -prox-regular.
The statement of Proposition 1.4 is not new. The finite-dimensional case was already solved in [28]. The fact that the conditions of Hypothesis 1.3 are sufficient for a set given by (1.3) to be prox-regular also in the infinite-dimensional case was shown in [4, Theorem 9.1] (see also [3]). The proof there refers to a number of deep concepts from non-smooth analysis along the lines, e. g., of [6, Chapter 2]. Here we present instead an elementary selfcontained proof using no other analytical tools but the properties of the scalar product, and the argument is split into several steps including two auxiliary Lemmas. Lemma 1.5. Let Hypothesis 1.3 hold and let r = c/λ. Then for all x ∈ ∂Z and z ∈ Z we have Proof of Lemma 1.5. For x ∈ ∂Z and z ∈ Z we have and the assertion follows.
Lemma 1.6. Let Hypothesis 1.3 hold, let V ⊂ X be the set of all y ∈ X for which there exists x ∈ Z such that |y − x| = dist(y, Z), and let U r := {y ∈ Z : dist(y, Z) < r}. Then the set V ∩ U r is dense in U r .
A highly involved proof of Lemma 1.6 can be found in a much more general setting, e. g., in [6, Theorem 3.1, p. 39]. For the reader's convenience, we show that in our special case, it can be proved in an elementary way. Proof of Lemma 1.6. We prove that for a given y ∈ U r and every ε > 0 there exists y * ∈ V ∩ U r such that |y − y * | < ε. (1.6) Let y ∈ X be arbitrarily chosen such that d := dist(y, Z) < r . For any α ∈ (0, r − d) we find x α ∈ ∂Z such that |y − x α | = d + α and put (see Figure 1) (1.7) Using Lemma 1.5 we check that y α ∈ V , |y α − x α | = d + α = dist(y α , Z). We have by (1.7) that y − y α = (d + α)(n α − n α ), hence, Figure 1: Illustration to Lemma 1.6.
We are now ready to prove Proposition 1.4. Proof of Proposition 1.4. To prove that Z is r -prox-regular, consider any d ∈ (0, r) and put Let f : Γ → X be the mapping defined by the formula by virtue of Lemma 1.5. Furthermore, by Lemma 1.6, for y from a dense subset of Γ d there exists x ∈ Γ and a unit vector n(x) such that y = x + dn(x) and |y − z| ≥ d for all z ∈ Z . We consequently have |x We find a sequence of elements y j ∈ f (Γ), j ∈ N, which converges to y , y j = f (x j ). For j, k ∈ N we have in particular By Lemma 1.5 we have We conclude that {x j } is a Cauchy sequence in X , hence it converges to some x ∈ Γ and the continuity of f yields y = f (x). We have thus proved that for each y ∈ Γ d there exists a unique x ∈ Γ such that y = f (x), and the assertion follows from Lemma 1.6.

Absolutely continuous inputs
We now consider a family of sets {Z(w) : w ∈ W } parameterized by elements w of a Banach space W with norm | · | W and defined as the sublevel sets where W ′ is the dual of W , and ·, · is the duality W → W ′ . We assume the following hypothesis to hold.
Hypothesis 2.1. Let X be a real Hilbert space endowed with scalar product ·, · and norm |x| = x, x and let W be a real Banach space with norm | · | W . We assume that (2.1) holds for a locally Lipschitz continuous function G : The property (v) in Hypothesis 2.1 is a kind of uniform coercivity of the function G which will play a role in the next Lemma. Let us observe that Hypothesis 2.1 and Proposition 1.4 imply that the set Z(w) given by (2.1) is r -prox-regular for every w ∈ W . Lemma 2.2. Let Hypothesis 2.1 hold. Then for every K > 0 there exists a constant C K > 0 such that Proof. Let K > 0 be given and let max{|w 1 | W , |w 2 | W } ≤ K . We first check that Indeed, if this was not true, we can assume that there exists x ∈ Z(w 1 ) such that dist(x, Z(w 2 )) ≥ D K + α for some α > 0. By Hypotheses 2.1 (iv)-(v) we then have which is a contradiction. We now consider the cases d H (Z(w 1 ), Z(w 2 )) ≥ r or d H (Z(w 1 ), Z(w 2 )) < r separately. Let us start with the case Then for x ∈ Z(w 1 ) we have by Hypotheses 2.1 (iv)-(v) that In the case we proceed as follows. For every ε > 0 there exists We have G(x ′ ε , w 2 ) = 1, G(x ε , w 2 ) > 1, and by (2.8), Hypotheses 2.1 (i) and 2.1 (iii), On the other hand, we have G(x ε , w 1 ) ≤ 1 = G(x ′ ε , w 2 ), hence, by Hypothesis 2.1 (iv), and combining (2.7) with (2.9) and with the arbitrariness of ε we complete the proof.
We cite without proof the following result.
Proposition 2.3. Let {Z(w); w ∈ W } be a family of r -prox-regular sets and let (2.5) hold for every K > 0 and every w 1 , w 2 ∈ W . Then for every u ∈ W 1,1 (0, T ; X), w ∈ W 1,1 (0, T ; W ), and every initial condition x 0 ∈ Z(w(0)) there exists a unique solution ξ ∈ W 1,1 (0, T ; X) such that 10) The statement was proved in [16,Corollary 5.3] under the assumption that the constant C K in (2.5) can be chosen independently of K . This is indeed not a real restriction, since the input values w(t) belong to an a priori bounded set. We obtain global Lipschitz continuity under the hypotheses of The above developments have shown that the assumptions of Proposition 2.3 are fulfilled if Hypothesis 2.1 holds. The existence and uniqueness of solutions to (0.4) is therefore guaranteed for all u ∈ W 1,1 (0, T ; X), w ∈ W 1,1 (0, T ; W ), and every initial condition x 0 ∈ Z(w(0)). We now prove the following identity which plays a substantial role in our arguments.
Lemma 2.4. Let Hypothesis 2.1 hold and let u, w, ξ, x be as in Proposition 2.3. Let ∇ w G : X × W → W ′ be continuous. Then for almost all t ∈ (0, T ) with the choice (2.14) Proof. We first check that the denominator in (2.14) is bounded away from zero. Indeed, thanks to Hypothesis 2.1(ii) we find δ c > 0 such that the implication holds for all x 1 , x 2 ∈ Z(w(t)). Then we have With this choice ofx, we have for all t ∈ [0, T ]. For a. e. t ∈ (0, T ) one of the following two cases occurs: (1)ξ(t) = 0, Moreover, for t ∈ B , the vectorξ(t) points in the direction of the unit outward normal vector n(x(t), w(t)) to Z(w(t)) at the point x(t), that is, which holds for a. e. t ∈ B by the above argument. Since dist(x(t), ∂Z(w(t))) = 0 for t ∈ B , we obtain (2.13) directly from (2.19). For t ∈ (0, T ) \ B , identity (2.13) is trivial sinceξ(t) = 0 a. e. on t ∈ (0, T ) \ B .
Under Hypothesis 2.1, the mapping (x, w) → dist(x, Z(w)) is locally Lipschitz continuous. This can be easily proved as follows. Let x, x ′ ∈ X , w, w ′ ∈ W be given. Put d = dist(x, Z(w)), d ′ = dist(x ′ , Z(w ′ )), and assume for instance that d ≥ d ′ . For an arbitrary ε > 0 we find (2.21) and the assertion follows from (2.5). Moreover the following statement holds true.
Lemma 2.5. Let Hypothesis 2.1 hold, and let K > 0 be given. Then there exists m K > 0 such that for all w, w ′ ∈ W satisfying the inequalities We argue as in (2.21) and obtain The proof will be complete if we prove that for a suitable value of m K and for w, w ′ satisfying (2.22) we have We claim that the right choice of m K is with C K from Lemma 2.2 and any d * < r with r as in Proposition 1.4. Indeed, from Lemma 2.2 it follows that ρ := d H (Z(w), Z(w ′ )) ≤ d * . Consider any z ∈ ∂Z(w) and assume that z / ∈ ∂Z(w ′ ). We distinguish two cases: z ∈ Int Z(w ′ ) and z / ∈ Z(w ′ ). For z ∈ Int Z(w ′ ) and t ≥ 0 we put Then for t < r we have dist(z(t), Z(w)) = |z(t) − z| = t. Since d H (Z(w), Z(w ′ )) = ρ ≤ d * < r , there exists necessarily t ≤ ρ such that z(t) ∈ ∂Z(w ′ ), and we conclude that dist(z, ∂Z(w ′ )) ≤ ρ. In the case z / ∈ Z(w ′ ) we use Proposition 1.4 and find z ′ ∈ ∂Z(w ′ ) such that dist(z, Z(w ′ )) = dist(z, ∂Z(w ′ )) = |z − z ′ | ≤ ρ and (2.23) follows.
It is easy to see that a counterpart of inequality (2.23) does not hold for general sets. It suffices to consider R 1 > R 2 > 0 and Z 1 = B R 1 (0), Z 2 = Z 1 \ B R 2 (0), where for x ∈ X and R > 0 we denote by B R (x) the open ball {y ∈ X : |x − y| < R}. Then The solution mapping of (2.10)-(2.12) is continuous in the following sense.
Lemma 2.7. Let {v n ; n ∈ N ∪ {0}} ⊂ L 1 (0, T ; X), {g n ; n ∈ N ∪ {0}} ⊂ L 1 (0, T ; R) be given sequences such that Then lim n→∞ T 0 |v n (t) − v 0 (t)| dt = 0. Notice that Lemma 2.7 does not follow from the Lebesgue Dominated Convergence Theorem, since we do not assume the pointwise convergence. The proof is elementary and we repeat it here for the reader's convenience. Proof of Lemma 2.7. We first prove that property (i) holds for every ϕ ∈ L ∞ (0, T ; X). For a fixed ϕ ∈ L ∞ (0, T ; X) and δ > 0 we use Lusin's Theorem to find a function ψ ∈ C([0, T ]; X) and a set M δ ⊂ [0, T ] such that meas(M δ ) < δ and ψ(t) = ϕ(t) for all t ∈ [0, T ] \ M δ , ψ ≤ ϕ . We then have Since δ can be chosen arbitrarily small and g 0 ∈ L 1 (0, T ), the integral of g 0 over M δ can be made arbitrarily small and we obtain (2.26) Let us note that the transition from (i) to (2.26) is related to the Dunford-Pettis Theorem, see [10]. To prove Lemma 2.7 we put for t ∈ [0, T ] if v 0 (t) = 0.
We are now ready to prove one of our main results, namely Theorem 2.6. Proof of Theorem 2.6. By Lemma 2.4 we check that s n given by the formula s n (t) = ∇ x G(x n (t), w n (t)) dist(x n (t), ∂Z(w n (t))) + |∇ x G(x n (t), w n (t))| 2 ẇ n (t), ∇ w G(x n (t), w n (t)) (2.27) satisfy a. e. the identity ξ n (t),ẋ n (t) + s n (t) = 0.
We have proved in the previous section that the solution mapping (u, w) → ξ of Problem (2.10)-(2.12) is strongly continuous with respect to the W 1,1 -norm provided Hypothesis 2.1 holds and ∇ w G is a continuous function. Here we show that if ∇ x G, ∇ w G are Lipschitz continuous, then the solution mapping of Problem (2.10)-(2.12) is locally Lipschitz continuous with respect to the W 1,1 -norm.
Here are the precise assumptions.
Hypothesis 3.1. Let Hypothesis 2.1 hold. Assume that the partial derivatives ∇ x G(x, w) ∈ X , ∇ w G(x, w) ∈ W ′ exist for every (x, w) ∈ X × W and there exist positive constants K 0 , K 1 , C 0 , C 1 such that In the following two lemmas we derive some useful formulas.

Lemma 3.2. Let Hypothesis 3.1 (i) hold, and let
Then for a. e. t ∈ (0, T ) we have either Moreover, for a. e. t ∈ (0, T ) we have that
where A and B are defined as in Lemma 3.2.
We are now ready to prove the following main result.
Theorem 3.4. Let Hypothesis 3.1 hold, let (u i , w i ) ∈ W 1,1 (0, T ; X) × W 1,1 (0, T ; W ) and x 0 i ∈ Z(w i (0)) be given for i = 1, 2, let ξ i ∈ W 1,1 (0, T ; X) be the respective solutions to (2.10)-(2.12) with x i = u i − ξ i for i = 1, 2. Then for a. e. t ∈ (0, T ) we have Proof. By Lemma 3.3, we have where B[u, w] is defined as in Lemma 3.2. Hence (3.9) follows since from Hypothesis 3.1 and the triangle inequality applied to B[u, w] we infer that Corollary 3.5. For every R > 0 there exists a constant C(R) > 0 such that for every (u i , w i ) ∈ W 1,1 (0, T ; X) × W 1,1 (0, T ; W ) and every x 0 i ∈ Z(w i (0)) for i = 1, 2 such that max{ Proof. In the situation of Theorem 3.4 put K 2 = max{K 0 , K 1 }/c, and Then from (3.9) it follows that We now use Gronwall's argument and put M(t) = t 0 m(τ ) dτ . It follows from (3.11) that d dt Integrating from 0 to T we obtain the assertion.
Remark 3.6. The local Lipschitz continuity of the input-output mapping cannot be expected if ∇ x G is not Lipschitz even if G is convex. A counterexample is constructed in [15,Theorem 2.2]. On the other hand, the global W 1,1 -Lipschitz continuity of the sweeping process holds if Z(w) is a convex polyhedron. This is shown for instance in [9] for Z independent of w , and it is generalized in [17] to the case of non-orthogonal projections to the convex polyhedron, i.e. when the time derivative of ξ lies in a prescribed cone of admissible directions.
Hypothesis 4.1. A continuous function g : [0, T ] × X × X → W is given such that its partial derivatives ∂ t g, ∂ u g, ∂ ξ g exist and satisfy the inequalities for every u, v, ξ, η ∈ X and a. e. t ∈ (0, T ) with given functions a, b ∈ L 1 (0, T ) and given constants γ, ω, C ξ , C u > 0 such that where c, K 1 are as in Hypothesis 2.1 (i) and Hypothesis 3.1.
We now prove the main result of this section.