ON SMOOTHNESS OF SOLUTIONS TO PROJECTED DIFFERENTIAL EQUATIONS

. Projected diﬀerential equations are known as fundamental ma-thematical models in economics, for electric circuits, etc. The present paper studies the (higher order) derivability as well as a generalized type of derivability of solutions of such equations when the set involved for projections is prox-regular with smooth boundary.


1.
Introduction. Given a Hilbert space X, let us consider the constrained initial value problem: x(t) ∈ S for all t ∈ [0, T ], where T > 0, S is a closed subset of X, and f : [0, T ] × U → X is a single-valued mapping with U ⊂ X being an open set containing S. Concerning the existence of solutions, Nagumo's Theorem (see e.g. [4, Theorem 1.2.1]) states that, under some conditions of continuity of f and compactness of S, the problem (1) has a solution for any initial value x 0 ∈ S if and only if the following inclusion holds: where T B (S; x) stands for the Bouligand tangent cone of S at x. When this condition is not fulfilled and T B (S; x) is convex for all x ∈ S, we still can find generalized solutions of (1) by solving the following relaxed problem: ẋ(t) = P f (t, x(t)); T B (x(t); S) , a.e. t ∈ [0, T ], where P (·; K) denotes the metric projection to a closed convex set K. This type of relaxation was first introduced by C. Henry [23] for S convex, and then studied by B. Cornet [19] with certain nonconvex sets, namely tangentially regular sets. Nowadays, problems of this kind are known as Projected Differential Equations, and they form a particular case of Projected Differential Inclusions (see, e.g., [5]) and Perturbed Sweeping Processes (see, e.g., [18] and the references therein). Another relaxation of problem (1) is the implicitly constrained differential inclusion: ẋ(t) ∈ −N C (S; x(t)) + f (t, x(t)) a.e. t ∈ [0, T ], where N C (S; y) is the Clarke normal cone of S at y ∈ S and N C (S; y) = ∅ if y ∈ S. This differential inclusion is known to have diverse applications in electrical circuits [1], crowd motion [26], and hysteresis in elasto-plastic models [24], among others. Further, it is strongly related to the previous relaxed problem (3). Indeed, B. Cornet [19] proved the equivalence between these two problems. Its original proof was done in finite-dimensional spaces and for autonomous systems (that is, when the mapping f depends only on x), but the proof is still valid for non-autonomous systems in arbitrary Hilbert spaces. The result is the following: The existence of solutions for problem (4) as well as for its sweeping process extension ẋ(t) ∈ −N C (C(t); x(t)) + f (t, x(t)) a.e. t ∈ [0, T ], (with closed sets C(t) ⊂ X) is well studied in the literature (see, e.g., [7,14,33] and the references therein); the sweeping process itself given by (5) without the perturbation f (·, ·) has been introduced for an elasto-plastic mechanical system by J.J. Moreau in [29] who thoroughly developed its study in a series of subsequent fundamental papers. In contrast, little is known about the regularity of those solutions, beyond absolute continuity. In practice, when we deal with some of the aforementioned applications, properties like differentiability of the solutions are needed or at least desired. For example, in the crowd motion model [26], the derivative of the trajectory represents the velocity of people when exiting a congested building. Also, in [3], a certain behavior of the trajectory of a free endpoint Mayer problem for a controlled sweeping process is needed in relation of its interaction with the boundary. Indeed, to derive necessary optimality conditions for this type of control problem, the authors in [3] assume some outward / inward pointing conditions (see conditions M 1 and M 2 from [3]). These conditions impose a specific structure on the set I ∂ := {t ∈ [0, T ] : x * ∈ bd C(t)}, where (x * , u * ) is a solution of the controlled sweeping process, and bd C(t) denotes the boundary of the moving set C(t) (see Propositions 5.2 and 5.3 from [3]). We also refer to [3,15,16,17,9,10,11,27,34,21,8] for various other first results concerning controlled sweeping processes. Motivated by the foregoing discussion, in this work we start the study of regularity properties of solutions of problem (5), particularly (thanks to Theorem 1.1) focusing on the stationary case, that is, on problem (3). Here, we study two main properties: differentiability (in some sense) of trajectories of (3); and the structure of points where the trajectory touches the boundary. The study of the first property is motivated by known characterizations of differentiability of the metric projection onto convex sets and, by recent developments, on the differentiability of the metric projection onto nonconvex sets (see [32,20,2,12]). The study of the second property is related to the first one and motivated by the recent works cited above related to the optimal control of sweeping process.
When studying the differentiability of the solutions of problem (3), it is natural to think that this regularity property must be inherited from the regularity of the data of the problem, that is, of the set S and the mapping f . Based on [20,32], our study will consider the following two different cases: 1. when S is a C p+1 -submanifold itself (and therefore, S = bd S). 2. when S is a closed body, meaning that int S = ∅, S = cl(int S) and bd S is a C p+1 -submanifold of codimension 1.
Related to the mapping f : [0, T ] × U → X, we will consider the following hypotheses: There exists a non-negative function β(·) ∈ L 1 ([0, T ], R) such that for all t ∈ [0, T ] and all x ∈ U , Under these hypotheses, at least in the case of a closed body, higher order derivability in the usual sense cannot be directly expected. Indeed, this is illustrated with the situation where, for a solution x, there is a nowhere dense set included in x −1 (bd S) with positive Lebesgue measure. Thus, we introduce a new notion of derivability of trajectories, called Ω-derivability (see Definition 4.5) in order to explore this situation. This notion will help us to derive the main results of the work.
The paper is organized as follows. In Section 2, we set some preliminary notations and definitions needed for the sequels. In Section 3 we study problem (3) when S is a C p+1 -submanifold, and we deduce that projected differential equations in this case have the same behavior as differential equations on manifolds (see Theorem 3.2). In Section 4 we study the problem (3) when S is a closed body with C p+1boundary. To overcome in this second case the natural difficulty of collisions we introduce the notion of Ω-derivability. Our main results are Theorem 4.7, where we deduce the Ω-derivability of the solutions, and Corollary 1, where Ω-derivability entails a piecewise differentiability in the analytic case. Finally, in Section 5 we derive some applications of our results. In Appendix A, we provide an example of an ill-behaved projected differential equation, where the usual notion of derivability is not well-adapted and the Ω-derivability is the appropriate concept.
2. Preliminaries. Throughout this paper, X will be a real Hilbert space endowed with the inner product ·, · and its associated norm · . By Riesz representation theorem, we identify the dual of X with X. We denote by B X (x, ε), the open ball of radius ε centered in x. We denote by B X and S X the closed unit ball and the unit sphere of X centered at zero, respectively. The letter p will always be an integer greater than 1.
Let S be a subset of a Banach space Y . We denote by int S, S and S • the interior, the closure and the (negative) polar set of S, respectively. Recall that the polar set of S is the subset of the dual space Y * given by where ·, · stands for the duality product between Y and Y * . We will not distinguish between duality products or inner products, using always the same notation. We will also write (in certain cases) clS instead of S to denote the closure. We say that S ⊂ X is a closed body if it is connected, closed, and For two Banach spaces Y and Z, we write L(Y ; Z) to denote the space of continuous linear operators from Y to Z, endowed with its usual norm. In the case when Z = Y , we simply write L(Y ). Also, for T ∈ L(Y ; Z), we write T * to denote the adjoint operator of T .
For a mapping f : U ⊂ Y → Z (where U is an open set) and u ∈ U , we write Df (u) to denote the (Frchet) derivative of f at u. If Y is a Hilbert space and Z = R, we write ∇f (u) to denote the gradient of f at u. Finally, for a curve γ : I ⊂ R → Y (where I is an interval of R) and t ∈ I, we will writeγ(t) or d dt γ(t) to denote the derivative of γ at t. Consequently, we denote by D k f (u) and by d k dt k γ(t) the k-th derivative of f at u and of γ at t, respectively.
Recall that, for a closed bounded interval I = [a, b] with a < b and an integer p ≥ 1, a mapping γ : I → Y is said to be p-continuously differentiable in I if it is of class C p in ]a, b[ and for each k ∈ {1, . . . , p} the unilateral derivatives and exist, and the mappings d k γ dt k extended from ]a, b[ to [a, b] in this way are also continuous. In a similar way, we define the C p property for γ : For a subset S of X and a point x ∈ X we denote by d( and by Proj(x; S) or Proj S (x) the set of all nearest points from S to x, that is, Whenever Proj(x; S) is a singleton, we call the only pointȳ ∈ Proj(x; S) the metric projection of x on S, and we denote it by P (x; S) or P S (x). We will use indistinctly these two notations. Moreover, in the particular case when S is a closed vector subspace of X, we will write Π S (x) instead of P (x; S), to emphasize that the metric projection of x coincides with the orthogonal projection onto the subspace.
In what follows, we will write T B (x; S), T C (x; S) and N C (x; S) to denote the Bouligand tangent cone, the Clarke tangent cone and the Clarke normal cone of x at S, respectively (see [13] for definitions and properties). Recall that for a point x ∈ S, the proximal normal cone of S at x is given by Let ρ : S → ]0, +∞] be a continuous function. The subset S is said to be ρ(·)prox-regular if for any x ∈ S and any ξ ∈ N P (S; x) ∩ B X one has that x ∈ Proj S (x + tξ), for any real t ≤ ρ(x).
We simply say that S is prox-regular if there exists a continuous function ρ : S → ]0, +∞] such that S is ρ(·)-prox-regular. In such a case, we define the enlargement U ρ(·) (S) := {y ∈ X : ∃x ∈ S such that x ∈ Proj S (y) and d S (y) < ρ(x)} .
It is known (see, e.g., [18]) that when S is ρ(·)-prox-regular, U ρ(·) (S) is an open neighborhood of S, the metric projection P S = P (·; S) is well-defined and locally Lipschitz-continuous on U ρ(·) (S), and the function 1 We say that a set S is tangentially regular at x ∈ S if T B (x; S) = T C (x; S). Also, we say that S is normally regular at x if N P (x; S) = N C (x; S), so in this case S is also tangentially regular at x. Whenever S is a prox-regular set, then it is normally regular (and hence tangentially regular) at each of its points. For further information on prox-regular sets, we refer the reader to [18,31].
A subset M of X is said to be a C p -submanifold if there exists a closed vector subspace Z of X such that for any point m ∈ M there exist an open neighborhood U of m and a C p -mapping ϕ : U → X with ϕ(m) = 0 such that In the above definition, the pairs (U, ϕ) are called local charts and Z is called the model space. This local representation of M is not unique, in the sense that they may exist several local charts and model spaces fitting this definition.
For a C p -submanifold and a point m ∈ M we define the tangent space of M at m as which is a closed vector subspace of X. Consequently, we define the normal space It is not hard to see that a C p -submanifold is tangentially regular (even normally regular when p ≥ 2) and that Let M 1 and M 2 be two C p -submanifolds of X, with model spaces Z 1 and Z 2 , respectively. A mapping f : M 1 → M 2 is said to be of class C k (with k ∈ {1, . . . , p}) if for any m ∈ M 1 and any local charts (U 1 , ϕ 1 ) and (U 2 , ϕ 2 ) of M 1 and M 2 with m ∈ U 1 and f (U 1 ) ⊂ U 2 , we have that A closed body S is said to have a C p -smooth boundary if bd S is a C p -submanifold. In such a case, it is known that bd S is a submanifold of X of codimension 1, that is, any model space Z used to represent bd S is a hyperplane. Thus, we can define for S the mapping n : bd S → S X where for any x ∈ bd S, n(x) is the only element such that We call the vector n(x) the exterior normal vector of S at x. It is not hard to prove that the mapping n is of class C p−1 (see, e.g., [20]). Now, let us recall in the form of a theorem the following properties (a) and (b) from [20] and (c) from [32]. They will serve as key tools in the development in the next sections.
(c) Further, for the C p+1 -submanifold M , one also has (see [32]) that ∀m ∈ M, DP M (m) = Π TmM . Remark 1. Note that problems (3) and (4) could be defined using other normal cones, like the Proximal normal cone defined above, or the Limiting normal cone (see, e.g., [28] for the definition). However, under the hypothesis of C p+1 -smooth boundary, the set S becomes prox-regular. This yields that both these normal cones of S coincide with Clarke normal cone (see, e.g., [18]), and so all these alternative formulations would be equivalent to the ones presented in this work. Figure 1 shows two examples of prox-regular sets: one with smooth boundary (left set) and the other with non-smooth boundary (right set). The nonsmoothness of the boundary in the right set comes from the corners of it. Note that this set can be recovered as the intersection of two closed bodies with smooth boundary (one closed ball intersected with the complement of another open ball), which yields that nonsmoothness can appear quite easily. However, as we already mention before, in this first study we will focus our attention only to the first example.
Let us also recall what we understand for a solution of problem (3), problem (4) or problem (5). The following definition considers only the perturbed sweeping processes, but we can easily adapt it to projected differential equations.
is said to be a local solution of (5) if it is a locally absolutely continuous mapping (hence absolutely continuous if I = [0, t * ]) satisfying (a) for all t ∈ I, x(t) ∈ S(t) and If p ≥ 1 and x : I → X is a solution of (3), (4) or (5) which is of class C p , we consider its derivativeẋ : I → X as the only representative of class C p−1 such that 3. The manifold case. In this section we study problem (3) in the case when S is a C p+1 -submanifold. Before showing the main theorem of this section, we will need the following lemma: is also of class C p , where ρ(·) is the prox-regularity function associated with M .
Proof. Let (U, ϕ) be a local chart of M and let Z be the model space of M . To abbreviate notation, let us denote φ := ϕ −1 . For every m ∈ M ∩ U , let us define the continuous linear operator L m : Z → Y given by L m := Dφ(m) Z . It is easy to see that L * m = Π Z • Dφ(m) * and also, since m L m is an automorphism of Z, and therefore we can define the continuous linear operator Since DP M (m) = Π TmM by Theorem 2.1(c), we conclude that A m = A(m), finishing the first part of the proof.
The second part of the proof follows as a simple application of the chain rule and Theorem 2.1(a).
Theorem 3.2. Consider the initial value problem (3) and assume that for some integer p ≥ 1: where ρ : S → ]0, +∞] stands for the prox-regularity function of S. Then, there exists a local solution x : I → X of (3), and it is unique in I. Furthermore, the solution x(·) is of class C p+1 in I.
If in addition the mapping f also satisfies the linear growth condition (F 2 ), that is, Proof. Since X is a metric space, there exists an open set V ⊂ X such that S ⊂ V and V ⊂ U . Since Hilbert spaces admit C ∞ -partitions of unity (see [35]), there exists a C ∞ -mapping ϕ from X to [0, 1] such that Let us define then the function Applying Lemma 3.1 and recalling that P S is of class C p in U ρ(·) (S) by Theorem 2.1(a), we have that g is of class C p on [0, T ] × X. In particular, it is also locally Lipschitz. Consider the auxiliary differential equation This problem has a local solution x : I → X which is unique in I, where I is an interval closed at its left endpoint 0, and x is (p + 1)-continuously differentiable.
To conclude, it is enough to prove that x(t) ∈ S for all t ∈ I. Let Ψ : I → R be the function given by for all u therein (see (7)), for every t ∈ I with x(t) ∈ U , we can write d dt where the last equality to 0 follows from the inclusion DP S (P S (y))(v) ∈ T y S for all y ∈ S, v ∈ X and from the inclusion Thus,ẋ(s) = 0 for all s ∈ J, which yields that x(s) is constant in J. In particular, d dt Ψ(t) = 0. In any case, we get that d dt Ψ ≡ 0, which yields that Ψ is constant in I. So, since Ψ(0) = 0, we conclude that Ψ ≡ 0. The conclusion follows. Now, assume in addition that f satisfies the growth condition (F 2 ). Then, the mapping g also satisfies on the whole set [0, T ] × X the same growth condition.
Indeed, recalling that for every x ∈ S and DP S (x) = Π TxS (see Theorem 2.1(c)), we can write By a classical application of Grönwall's lemma (see, e.g., [5,Chapter 2.4]), the linear growth condition of g yields that the local solution x is Lipschitz-continuous on I, and so, if we consider x : I → X as the maximal solution of problem (9), then it has to be global (namely, I = [0, T ]). Otherwise, again by classic arguments, we could extend the solution x which would contradict its maximality. This finishes the proof.
Remark 2. Note that Theorem 3.2 remains true if we replace problem (3) by where t 0 ∈ [0, T ]. Indeed, we may consider any initial condition x(t 0 ) = x 0 maintaining the same proof. In such a case (when t 0 = 0), we understand a local solution x : I → X as in Definition 2.2, but with This result should be compared with [14, Theorem 3.1], where local existence results for projected differential equations are obtained. A small modification also allows us to derive local results without considering global prox-regularity. The proof of our theorem is rather simple in comparison to the one of the theorem cited above, thanks to the advantage of smoothness of the manifold.
This result should also be compared with classic ordinary differential equations on manifolds (see, e.g., [25,Ch. 9]): We could approach the proof of Theorem 3.2 by showing that (t, m) → DP M (m) • f (t, m) is a time-dependent flow in the manifold M . This is possible based on the recent results in [32,20] and Lemma 3.1. However, the present proof has the interest of using only the tools of variational analysis and classic differential equations. 4. The nonconvex body case . Now, we study the case when S is a nonconvex body. The main difference between this case and the previous one (when S is a manifold itself) is that, for a solution x : I → X of problem (3), the instants when the trajectory meets the boundary of S furnish a discontinuity of the derivativė x(·). In what follows, we will refer to these instants in I as the collisions of the trajectory. To better understand this difficulty, we can divide the interval of the solution as follows: • C in (x): the set of instants t ∈ I for which there exists δ > 0 such that To simplify notation, we will write also B(  instants t ∈ I when the trajectory x(·) lies in the interior of S. For each t ∈ A(x) whereẋ(t) exists, we see from the inclusion x(t) ∈ int S that T B (S; x(t)) = X, and henceẋ (t) = f (t, x(t)).
The set B int (x) are the instants t ∈ I when the trajectory lies locally in the boundary of S, that is, when there exists a neighborhood V around t such that x V is a curve in bd S. The set B iso (x) are the instants when the trajectory touches the boundary coming from int S, and then goes immediately back to int S again. The set B iso (x) is by definition isolated. The set C in (x) stands for the instants of the incoming collisions: the trajectory is coming from the interior int S, meets bd S and then remains a while in bd S. Similarly, the set C out (x) stands for the instants of the outgoing collisions: the trajectory is coming from the bd S and then it enters in int S.
The sets C in (x) and C out (x) are easy to picture, and so we will call their union the set of instants of regular collisions, denoted by C reg (x). Finally, the set C irr (x) stands for the instants of irregular collisions, in the sense that they represent the moments when the trajectory makes infinitely many exchanges between int S and bd S, before and/or after each one of them. Namely, an instantt ∈ I belongs to C irr (x) if and only if there exist a monotone (i.e. either increasing or decreasing) sequence (t n ) in I \ {t} converging tot such that for all n ∈ N x(t n ) ∈ int S if n is even, if n is odd.
We will study differentiability properties of x for each one of these sets. In the rest of this section we will adopt the following notation concerning the boundary of S: ρ : bd S → ]0, +∞] will be the prox-regularity function of bd S and we will write n : bd S → S X to denote the mapping assigning to each x ∈ bd S the unit exterior normal vector of S at x (see Section 2). Also, we will simply write P instead of P bd S .
Proof. Assume that x(τ ) ∈ int S and x(θ) ∈ bd S, so G(τ ) > 0 and G(θ) = 0. Since d(·; bd S) is Lipschitz, the function G is locally absolutely continuous on I. Further, at each t ∈ I where x is derivable, G is derivable too. Writing it results that the set of t ∈]τ, θ[ whereẋ(t) exists andĠ(t) < 0 has positive Lebesgue measure. The other case is similar.
Recall that we are studying the differentiability of the solutions x : I → X in the sense of Definition 2.2. In particular, I is always a subinterval of [0, T ] closed at its left endpoint 0. Proof. On the one hand, ift ∈ A(x), then there exists δ > 0 such that J = ]t − δ,t + δ[ ∩I ⊂ A(x). Then, x J is a local solution of the classic problem ẏ(t) = f (t, y(t)), t ∈ I, y(t) = x(t).
This yields that x(·) is of class C p+1 neart. On the other hand, ift ∈ B int (x), there exists δ > 0 such that J = ]t − δ,t + δ[ ∩I ⊂ B int (x). Then, x J is a local solution of ẏ(t) = P f (t, y(t)); T B (bd S; y(t)) , a.e. t ∈ I According to Theorem 3.2 and Remark 2, x J is of class C p+1 , which yields that x(·) is of class C p+1 neart. Finally, assume thatt ∈ B iso (x). Note that in such a case,t cannot be an extreme point of I. Thus, by continuity of x(·) and the definition of B iso (x), we may choose an interval J = ]t − δ,t + δ[ ⊂ I such that x(J) ⊂ S ∩ U ρ(·) (bd S) and J ∩ x −1 (bd S) = {t}. Then, for all t ∈ J we can write f (t, x(t)) = DP (P (x(t)))f (t, x(t)) + f (t, x(t)), n(P (x(t))) =:g(t) n(P (x(t))), where as said above n : bd S → S X is the mapping assigning at each point in bd S the unit exterior normal vector of bd S at this point. Since x(t) ∈ S for all t ∈ J, it results that x(t) ∈ int S for all t ∈ J \ {t}. We also note for each t ∈ J \ {t} that d(x(t),bd S) , so at each t ∈ J \ {t} whereẋ(t) exists we see by (11) and (7) that where G(·) := d(x(·); bd S). By Lemma 4.2 we can find two sequences (t − n ) ⊂ ]t−δ,t[ and (t + n ) ⊂ ]t,t + δ[ converging tot such that g(t − n ) > 0 and g(t + n ) < 0, for all n ∈ N.
Since g is continuous, we deduce that g(t) = 0, and so, x J is a local solution of Then, as for the caset ∈ A(x), we deduce that x(·) is of class C p+1 neart. This finishes the proof.
Clearly, whenevert ∈ C reg (x), there is no hope to have continuous derivability of x att, since the derivativeẋ is changing discontinuously due to the effect of the boundary. Nevertheless, we still can have piecewise differentiability, if the set C reg (x) is small. The following lemma ensures precisely this feature. Proof. We will show that C in (x) must be at most countable. The proof for C out (x) is similar. Fix t 1 , t 2 ∈ C in (x) and let δ 1 , δ 2 > 0 be such that ]t i , t i + δ i [ ⊂ B int (x) (with i = 1, 2). Without losing generality, assume that t 1 < t 2 . It is not hard to see that t 2 / ∈ ]t 1 , t 1 + δ 1 [, which yields that Now, assume that C in (x) = {t α } α∈Λ with t α = t α for α = α , and let {δ α } α∈Λ be a family of positive values such that J α := ]t α , t α + δ α [ ⊂ B int (x). From the reasoning above, we have that {J α : α ∈ Λ} is a pairwise disjoint family of subsets of the separable metric space I, thus the set Λ is at most countable. Now, let us study the set C irr (x), which does not seem to be a set of regularity of the solution x(·). It would be tempting to try to prove that C irr (x) is "small", like the set C reg (x). Unfortunately, in Appendix A we present an example for which C irr (x) is in fact large (with positive Lebesgue measure). Furthermore, in this example, the trajectory x(·) fails to be continuously differentiable at each point of C irr (x).
Nevertheless, the main problem when looking for smoothness of x(·) is not the set C irr (x) itself, but the density of the set of instants of regular collisions. Thus, to continue our study of smoothness in C irr (x), we would like to avoid the set C reg (x). To do so, we need to introduce a weaker notion of differentiability. Definition 4.5 ((Ω, k)-continuous derivability). Let Y be a Banach space, I be an interval of R, Ω be a subset of I and k be a nonnegative integer. Let us consider a curve y : Ω → Y . We say that the curve y is 1. Ω-continuous at t if y is continuous at t with respect to the induced topology on Ω. Similarly, for an open set U (relative to I) we say that y is Ω-continuous in U if it is so for each t ∈ U ∩ Ω. 2. Ω-derivable at t ∈ Ω if there exists a unique element d ∈ Y such that lim Ω t →t In such a case, we call d the Ω-derivative of y at t, and we denote it by D Ω y(t); we also notice in this case that y is Ω-continuous at t. 3. (Ω, k)-derivable at t ∈ Ω, with k ≥ 2, if there exist a sequence of curves y j : Ω → Y with j = 0, 1, . . . , k − 1 and a neighborhood U of t (relative to I) such that • y 0 = y; • For every j, y j is Ω-continuous in U ; • For every j ∈ {1, . . . , k − 1} and every t ∈ U ∩ Ω, y j (t ) = D Ω y j−1 (t ).
• The curve y k−1 is Ω-derivable at t. In such a case, the Ω-derivative of y k−1 at t is uniquely determined by the curve y, and we call it the (Ω, k)-derivative of y at t, denoted by D k Ω y(t). By convention, we will say that a curve y is (Ω, 0)-derivable (resp. (Ω, 1)derivable) at t if it is Ω-continuous (resp. Ω-derivable) at t. 4. (Ω, k)-continuously derivable at t if there exist a curveỹ : Ω → Y and a neighborhood U of t (relative to I) such that • For every t ∈ U , y is (Ω, k)-derivable at t , andỹ(t ) = D k Ω y(t ); and • the curveỹ is Ω-continuous at t.
The definition above has no interest when the set Ω is "too small". In fact, if Ω has an isolated point, no curve can be Ω-derivable at that point. Nevertheless, if Ω is large enough, we may obtain interesting properties. The following proposition shows that the Ω-derivability has reasonable stability properties when Ω is a dense set of I. Proposition 1. Let Y be a Banach space and let Ω be a subset of an interval I. If Ω has no isolated points, then the following properties hold: 1. If t ∈ Ω and y : I → Y is an Ω-continuous curve such that exists, then y is Ω-derivable at t with D Ω y(t)s = sd. 2. If y 1 and y 2 are two (Ω, k)-continuously derivable curves in Y , and α : Ω → R is also (Ω, k)-continuously derivable, then the mapping t → α(t)y 1 (t) + y 2 (t) is (Ω, k)-continuously derivable. 3. If y is an (Ω, k)-continuously derivable curve in Y and G : Y → Z is a mapping of class C k , then G • y is an (Ω, k)-continuously differentiable curve in Z. 4. If y : I → Y is derivable at each t ∈ Ω, then it is Ω-derivable and D Ω y(t) = y(t) for each t ∈ Ω.
5. If y : I → Y is a curve of class C k , then it is (Ω, k)-continuously derivable and D j Ω y(t) = d j dt j y(t) for each j ∈ {1, . . . , k} and each t ∈ Ω. Proof. We will prove the first statement. All other properties follow directly from it just repeating the classic proofs for the usual definition of derivatives.
t − t satisfies equation (14). Thus, we only need to prove the uniqueness. Assume that there exists another vector d ∈ Y satisfying equation (14). Since Ω is not isolated, then there exists a sequence (t n ) ⊂ Ω \ {t} converging to t, and so we can write Then, d = d, finishing the proof.
With the notion of Ω-derivability, we now continue the study of the smoothness of x in C irr (x). To do so, we will need the following lemma describing the behavior of the normal component of the velocity of x in such moments.
Lemma 4.6. Let U be an open set of X containing bd S on which P (·) is well defined and p-continuously differentiable. Let x : I → X be a solution of (3) with x(I) ⊂ U . Let Ω = I \ C reg (x) and let the function g : I → R be defined by Then, for eacht ∈ C irr (x), we have that g(t) = 0. Furthermore, if x(·) is (Ω, k)continuously derivable with 1 ≤ k ≤ p, then g is (Ω, k)-continuously derivable with Proof. Fixt ∈ C irr (x). Without loss of generality, there exists an increasing sequence (t n ) converging tot such that x(t n ) ∈ int S when n is even and x(t n ) ∈ bd S if n is odd, for which we may assume that (t n ) ⊂ Ω. Now, as seen in (13) for each t whereẋ(t) exists and x(t) ∈ int S one has hence by Lemma 4.2 we can construct a sequence (s n ) ⊂ Ω such thatt n ≤ s n ≤t n+1 and • g(s n ) > 0 if n is even, and • g(s n ) < 0 if n is odd.
Since g is continuous, we deduce that g(t) = 0. Let us assume now that x(·) is (Ω, k)-continuously derivable. Then, since Ω has no isolated points, we can apply chain rule of Proposition 1 to deduce that g is (Ω, k)-continuously derivable.
We claim that the following statement holds: for each j ∈ {0, . . . , k} there exists a mapping h j : I × U → X of class C k−j such that and such that there exists a sequence (t n ) ⊂ Ω with t n t and satisfying • h j (t n , x(t n )) > 0 if n is even, and • h j (t n , x(t n )) < 0 if n is odd.
Let us prove it inductively. Suppose first that such a condition holds for j ∈ {0, . . . , k−1} and fix n ∈ N. If n is even and noting that the mapping t → h j (t, x(t)) is at least locally absolutely continuous, we have that Thus, there exists s n ∈ Ω ∩ [t n , t n+1 ] such that d dt h j (s n , x(s n )) is well defined and d dt h j (s n , x(s n )) < 0. If n is odd, we can use the same reasoning to find s n ∈ Ω ∩ [t n , t n+1 ] such that d dt h j (s n , x(s n )) is well defined and d dt h j (s n , x(s n )) > 0. Thus, we can construct a sequence (s n ) in Ω such that t n < s n < t n+1 and such that • d dt h j (s n , x(s n )) < 0 if n is even, and • d dt h j (s n , x(s n )) > 0 if n is odd. Let us consider also the set S := {s n : n ∈ N} ∩ B int (x). Since (s n ) is an increasing sequence, for each s ∈ S we can choose δ > 0 such that the neighborhood Further, for each s ∈ S we can construct a C ∞ -function ϕ s : I → [0, 1] such that ϕ s (s) = 1 and ϕ s (t) = 0 for each t ∈ I \ V s .
With this construction, and denoting V = s∈S V s and ϕ := s∈S ϕ s , we define the function h j+1 : I × U → R given by Note that h j+1 is of class C k−(j+1) (in particular, it is continuous) and that for each t ∈ A(x) whereẋ(t) exists we have t ∈ I \ V along with x(t) ∈ int S. Hencė x(t)) by (11), which yields . Furthermore, for each n ∈ N, we have that otherwise.
In both cases, we get that d dt h j (s n , x(s n )) = h j+1 (s n , x(s n )). Then, we deduce that h j+1 (s n , x(s n )) < 0 if n is even and h j+1 (s n , x(s n )) > 0 if n is odd. By replacing t n = s n+1 for each n ∈ N, we conclude that the statement holds true for j + 1.
For j = 0, just consider h 0 : I × U → R given by h 0 (t, u) = f (t, u), n(P (u)) , and take for (t n ) the sequence (s n ) constructed in the first part of the proof. The claim is then proved. Now, to conclude, fix j ∈ {0, . . . , k − 1} and consider h j and a sequence (t n ) as in the claim. Since h j is at least continuous, h j (t, x(t)) = lim n h j (t n , x(t n )) = 0. Then, consider the sequence (t n ) defined at the beginning of the proof. Recalling thatt 2n ∈ A(x) for every n ∈ N, we can write which proves the second part of the lemma. Finally, the case where the sequence (t n ) decreases tot is similar, and so the proof is finished. Now, we present the main theorem of this section, where we prove the (Ω, p + 1)continuous derivability of the trajectory x, with Ω = I \ C reg (x).
Theorem 4.7. Let us consider problem (3) and suppose that the set S is a nonconvex body with C p+1 -smooth boundary. If the mapping f satisfies (F 1 ), that is, f is of class C p on I × U , then problem (3) admits at least a local solution and any such solution x : I → X is (Ω, p + 1)-continuously derivable with Ω = I \ C reg (x).
An interesting corollary of the above theorem, is the case of analytic equations in finite dimensions, that is when the boundary of S is analytic and when the perturbation function f is also analytic. In such a case, we can assure that there are merely finitely many collisions, entailing that the solution x must be piecewise analytic.

Corollary 1.
[Analytic case] Let X = R n and suppose that the set S is a nonconvex body with analytic boundary. If the vector field f is also analytic, then for any Thus, x is piecewise analytic, that is, there exist a finite sequence 0 = t 0 < t 1 < . . . < t n = t * such that Proof. Since I = [0, t * ] is compact, if C reg (x) is infinite, any cluster point of C reg (x) would be an element of C irr (x). So, we only need to prove that C irr (x) = ∅. By contradiction, let us suppose that there existst ∈ C irr (x).
Let U = U ρ (bd S), where ρ(·) is the prox-regularity function of bd S. Let J 0 := [t − δ 0 ,t + δ 0 ] ∩ I small enough such that x(J 0 ) ⊂ U . By Lemma 4.6, the function g(t) = f (t, x(t)), n(P (x(t))) is (Ω, k)-continuously derivable for every k ∈ N with Ω = I \ C reg (x), and D k Ω g(t) = 0, ∀k ∈ N. Now, consider the auxiliary problem and let y : J 1 → X be a local solution, which is analytic by the Cauchy-Kowalevski theorem (see, e.g. [22, Ch. 1.D]). Without lose of generality, we may assume that J 1 is small enough such that y(J 1 ) ⊂ U . Let us define the function h : J 1 → R by h(t) = f (t, y(t)), n(P (y(t))) for all t ∈ J 1 . Note that h is analytic since n is analytic by hypothesis, and P = P bd S is also analytic (see [30,Lemma 1]).
We claim that for each k ∈ N, there exists a C ∞ -function F k : The case k = 0 is direct: Consider F 0 (t, u) = f (t, u), n(P (u)) . Now, fix k ∈ N and assume that such a function F k exists. As in Lemma 4.6, we note that and we define the . Then, by chain rule and so, our claim is proved for k + 1. This finishes the proof of the claim. Now, sincet ∈ C irr (x), there exists a sequence (t n ) ⊂ A(x) ∩ J 1 converging tot. Thus, for each k ∈ N we can write d k dt k h(t) = F k (t, y(t)) = F k (t, x(t)) = lim n F k (t n , x(t n )) = lim n D k Ω g(t n ) = D k Ω g(t) = 0.
Thus, d k dt k h(t) = 0 for each k ∈ N. Since h is analytic, there exists an open interval J 2 =]t − δ 2 ,t + δ 2 [⊂ J 1 on which h is identically zero. This yields that y J2 is a solution of the projected differential equation ẏ(t) = DP (P (y(t)))f (t, y(t)), ∀t ∈ R y(t) = x(t), which yields that y J2 is a curve in S. Since f (t, y(t)) =ẏ(t) = DP (P (y(t)))f (t, y(t)) = P f (t, y(t)); T C (y(t)); bd S) , ∀t ∈ J 2 , then y J2∩I is a local solution of problem (10) with t 0 =t (see Remark 2), which yields that y J2∩I = x J2∩I . Since J 2 ∩ I is relatively open in I, we conclude that t ∈ B int (x), which is a contradiction since B(x) ∩ C irr (x) = ∅. We deduce that C irr (x) is empty, which finishes the proof.

5.
Applications to sweeping processes. In this section, we illustrate how Theorem 4.7 and Corollary 1 can be applied to a particular case of sweeping processes, namely those where the moving set S : [0, T ] ⇒ X can be described as a translation and homothetic transformation of the starting set S(0) = S 0 .
Real applications can be found in electrical circuits. Indeed, several nonsmooth electrical circuits can be written as perturbed sweeping processes (see, e.g., [1,7]). For example, let us consider a circuit with an ideal diode, an inductor and a current source (see Figure 3), where x is the current through the inductance and a current source i(t).
L u The dynamic is given by The third relation in (16) is a complementarity relation which can be written equivalently as u(t) ∈ −N (R + ; y(t)). Therefore, the system (16) is equivalent to the following sweeping process: Assume S 0 is a closed body with C p+1 -smooth boundary and f : [0, T ] × X → Xsatisfies the hypothesis (F 1 ). Then, problem (5) has a maximal solution x : I → X which is (Ω, p + 1)-continuously derivable, Ω = I \ N and N (= C reg (x)) is at most a countable subset of I.
Furthermore, if X is finite dimensional, α, ξ and f are analytic and S 0 has analytic boundary, then the solution x is piecewise analytic.
Then, sinceẏ (t) = DP S0 (y(t))g(t, y(t)) if y(t) ∈ bd S 0 g(t, y(t)) if y(t) ∈ int S 0 , Grönwall's lemma allows us to ensure that the solution y of Problem (17) has to be global (namely, I = [0, T ]). The same applies then for the solution x of the original problem. This finishes the proof.
Observe that for the example of electrical circuits given by the system (16), Proposition 2 can be directly applied by setting f ≡ 0, α ≡ 1, ξ ≡ i(·) and S 0 = R + .
Furthermore, in this particular example, it is not hard to see that the set N , given by the regular collisions of x(·), is related to the sign changes of the function d dt i. Indeed, rewriting (17) for the change of variables y(t) = x(t) − i(t), we get ẏ(t) ∈ − d dt i(t) − N C (R + , y(t)) for almost all t ∈ [0, T ], y(0) = 0.
This dynamical system is easier to understand in terms of the collisions: An incoming collisiont ∈ C in (y) requires d dt i(t) to be strictly positive in the interval ]t − δ,t[ for some δ > 0. Similarly, an outgoing collisiont ∈ C out (y) requires d dt i(t) to be strictly negative in the interval ]t,t + δ[ for some δ > 0. Intuitively, since incoming and outgoing collisions alternate (it is not possible to have two consecutive incoming collisions nor outgoing ones), we can see that |N | ≤ 2 · # Sign changes of d dt i .
Formally, defining the sign function sgn : R → R by we can write the following proposition: In such a case, the solution x is piecewise (p + 1)-continuously differentiable.
6. Final comments. In this work, we investigate the regularity of the solutions, when we deal with a smooth projected differential equation, in the sense that the perturbation f and the set S are smooth. The notion of Ω-derivability allowed us to deduce Theorem 4.7 and Corollary 1, which are the main results of this work. Furthermore, we presented applications of our results to an important class of perturbed sweeping processes, studied in the literature. We hope that these fundamental results can bring to the community some new perspectives on the different situations that require smoothness of the solutions or regularity of the collisions of the trajectories. The natural continuation of this work is to address the general perturbed sweeping process. To replicate the results that we already have for the stationary case, we need to introduce a suitable notion of "smooth movement" for the set-valued map S : [0, T ] ⇒ X.
Another important problem is to study the case when the set S 0 can be described as an intersection of finite closed bodies with smooth boundary, which is the situation of most of the applications. A particular case of such sets are the semi-algebraic sets, which have shown to be very pertinent objects when dealing with applications outside the manifold setting. We will treat these problems in a future work. Applications to optimal control will also be explored.