A Maximum Principle for Optimal Control Problems Involving Sweeping Processes with a Nonsmooth Set

We generalize a maximum principle for optimal control problems involving sweeping systems previously derived in de Pinho et al. (Optimization 71(11):3363–3381, 2022, https://doi.org/10.1080/02331934.2022.2101111) to cover the case where the moving set may be nonsmooth. Noteworthy, we consider problems with constrained end point. A remarkable feature of our work is that we rely upon an ingenious smooth approximating family of standard differential equations in the vein of that used in de Pinho et al. (Set Valued Var Anal 27:523–548, 2019, https://doi.org/10.1007/s11228-018-0501-8).


Introduction
In recent years, there has been a surge of interest in optimal control problems involving the controlled sweeping process of the form ẋ(t) ∈ f (t, x(t), u(t)) − N C(t) (x(t)), u(t) ∈ U, x(0) ∈ C 0 . ( In this respect, we refer to, for example, [3], [4], [5], [8], [9], [16], [23], [10] (see also accompanying correction [11]), [6], [15] and [14].Sweeping processes first appeared in the seminal paper [18] by J.J. Moreau as a mathematical framework for problems in plasticity and friction theory.They have proved of interest to tackle problems in mechanics, engineering, economics and crowd motion problems; to name but a few, see [1], [5], [16], [17] and [21].In the last decades, systems in the form (1.1) have caught the attention and interest of the optimal control community.Such interest resides not only in the range of applications but also in the remarkable challenge they rise concerning the derivation of necessary conditions.This is due to the presence of the normal cone N C(t) (x(t)) in the dynamics.Indeed, the presence of the normal cone renders the discontinuity of the right hand of the differential inclusion in (1.1) destroying a regularity property central to many known optimal control results.
Lately, there has been several successful attempts to derive necessary conditions for optimal control problems involving (1.1).Assuming that the set C is time independent, necessary conditions for optimal control problems with free end point have been derived under different assumptions and using different techniques.In [10], the set C has the form C = {x : ψ(x) ≤ 0} and an approximating sequence of optimal control problems, where (1.1) is approximated by the differential equation for some positive sequence γ k → +∞, is used.Similar techniques are also applied to somehow more general problems in [23].A useful feature of those approximations is explored in [12] to define numerial schemes to solve such problems.
More recently, an adaptation of the family of approximating systems (1.2) is used in [14] to generalize the results in [10] to cover problems with additional end point constraints and with a moving set of the form C(t) = {x : ψ(t, x) ≤ 0}.
The case where I = 1 in (1.3) and ψ 1 is C 2 is covered in [14].Here, we assume I > 1 and that the functions ψ i are also C 2 .Although going from I = 1 in (1.3) to I > 1 may be seen as a small generalization, it demands a significant revision of the technical approach and, plus, the introduction of a constraint qualification.This is because the set (1.3) may be nonsmooth.We focus on sets (1.3), satisfying a certain constraint qualification, introduced in assumption (A1) in section 2 below.This is, indeed, a restriction on the nonsmoothness of (1.3).A similar problem with nonsmooth moving set is considered in [15].Our results cannot be obtained from the results of [15] and do not generalize them.
This paper is organized in the following way.In section 2, we introduce the main notation and we state and discuss the assumptions under which we work.In this same section, we also introduce the family of approximating systems to ẋ(t) ∈ f (t, x(t), u(t)) − N C(t) (x(t)) and establish a crucial convergence result, Theorem 2.2.In section 3, we dwell on the approximating family of optimal control problems to (P ) and we state the associated necessary conditions.The Maximum Principle for (P ) is then deduced and stated in Theorem 4.1, covering additionally, problems in the form of (P ) where the end point constraint x(T ) ∈ C T is absent.Before finishing, we present an illustrative example of our main result, Theorem 4.1.

Preliminaries
In this section, we introduce a summary of the notation and state the assumptions on the data of (P ) enforced throughout.Furthermore, we extract information from the assumptions establishing relations crucial for the forthcoming analysis.

Notation
For a set S ⊂ R n , ∂S, cl S and int S denote the boundary, closure and interior of S.
If g : R p → R q , ∇g represents the derivative and ∇ 2 g the second derivative.If g : R × R p → R q , then ∇ x g represents the derivative w.r.t.x ∈ R p and ∇ 2 x g the second derivative, while ∂ t g(t, x) represents the derivative w.r.t.t ∈ R.
The Euclidean norm or the induced matrix norm on R p×q is denoted by |•|.We denote by B n the closed unit ball in R n centered at the origin.The inner product of x and y is denoted by x, y .For some A ⊂ R n , d(x, A) denotes the distance between x and A. We denote the support function of Standard concepts from nonsmooth analysis will also be used.Those can be found in [7], [19] or [22], to name but a few.The Mordukhovich normal cone to a set S at s ∈ S is denoted by N S (s) and ∂f (s) is the Mordukhovich subdifferential of f at s (also known as limiting subdifferential ).
For any set A ⊂ R n , cone A is the cone generated by the set A.
We now turn to problem (P ).We first state the definition of admissible processes for (P ) and then we describe the assumptions under which we will derive our main results.Definition 2.1 A pair (x, u) is called an admissible process for (P ) when x is an absolutely continuous function and u is a measurable function satisfying the constraints of (P ).
Assumptions on the data of (P) A1: The function ψ i , i = 1, . . ., I, are C 2 .The graph of C(•) is compact and it is contained in the interior of a ball rB n+1 , for some r > 0. There exist constants β > 0, η > 0 and ρ ∈]0, 1[ such that and, for and Assumption (A1) concerns the functions ψ i defining the set C and it plays a crucial role in the analysis.All ψ i are assumed to be smooth with gradients bounded away from the origin when ψ i takes values in a neighorhood of zero.Moreover, the boundary of C may be nonsmooth at the intersection points of the level sets x : ψ i (t, x) = 0 .However, nonsmoothness at those corner points is restricted to (2.2) which excludes the cases where the angle between the two gradients of the functions defining the boundary of C is obtuse; see figure 1.
On the other hand, (2.3) guarantees that the Gramian matrix of the gradients of the functions taking values near the boundary of C(t) is diagonally dominant and, hence, the gradients are linearly independent.
In many situations, as in the example we present in the last section, we can guarantee the fulfillment of (A1), in particular (2.4), replacing the function Not allowed Allowed Figure 1: Examples of two diferent sets C. On the left size, a set that does not satisfies (2.2).On the right side, the set C is nonsmooth and it fulfils (2.2). where Here, h is an C 2 function, with h s an increasing function defined on [−2β, −β].For example, h may be a cubic polinomial with positive derivative on the interval ] − 2β, −β[.For all t ∈ [0, T ], set It is then a simple matter to see that and that the functions ψi (•) satisfy the assumption (A1).
The assumption that the graph of C(•) is compact and contained in the interior of a ball is introduced to avoid technicalities in our forthcoming analysis.In applied problems, this may be easily side tracked by considering the intersection of the graph of C(•) with a tube around the optimal trajectory.We now proceed introducing an approximation family of controlled systems to (1.1).Let x(•) be a solution to the differential inclusion Under our assumptions, measurable selection theorems assert the existence of measurable functions u and Considering the trajectory x, some observations are called for.Let µ be such that The properties of the graph of C(•) in (A1) guarantee the existence of such maximum.

Define the function
consider a sequence {σ k } such that σ k ↓ 0 and choose another sequence {γ k } with γ k ↑ +∞ and where Let x k be a solution to the differential equation for some u k (t) ∈ U a.e.t ∈ [0, T ].Take any t ∈ [0, T ] such that ẋk (t) exists and ψ j (t, Assume k is such that j ∈ I(t, x k (t)).Then, whenever γ k is sufficiently large, we have Above, we have used the definition of µ and the inequality i ∈I(t,x k (t)) which holds for γ k sufficiently large.Now, if x k (0) ∈ C k (0), we assure that x k (t) ∈ C k (t), for all t ∈ [0, T ], and It follows that, for k sufficienttly large, we have We are now a in position to state and prove our first result, Theorem 2.2 below.This is in the vein of Theorem 4.1 in [23] (see also Lemma 1 in [10] when ψ is independent of t and convex) deviating from it in so far as the approximating sequence of control systems (2.6) differs from the one introduced in [10]1 .The proof of Theorem 2.2 relies on (2.7).

be a sequence of solutions of Cauchy problems
then there exists a subsequence {x k } (we do not relabel) converging uniformly to x, a unique solution to the Cauchy problem where u is a measurable function such that u(t) ∈ U a.e.t ∈ [0, T ].
If, moreover, all the controls u k are equal, i.e., u k = u, then the subsequence converges to a unique solution of (2.9), i.e., any solution of Then there exist subsequences (we do not relabel) weakly- * converging in L ∞ to some v and ξ i .Hence for an absolutely continuous function x.Obviously, (2.12) Integrating this inequality, we get Passing to the limit as k → ∞, we obtain Let t ∈ [0, T ] be a Lebesgue point of x and ξ.Passing in the last inequality to the limit as τ ↓ 0, it leads to Since z ∈ R n is an arbitrary vector and the set f (t, x(t), U ) is convex, we conclude that By the Filippov lemma there exists a measurable control u(t) ∈ U such that Furthermore, observe that ξ i is zero if ψ i (t, x(t)) < 0. If for some u such that u(t) ∈ U a.e., u k = u for all k, then the sequence x k converges to the solution of Indeed, to see this, it suffices to pass to the limit as k → ∞ and then as τ ↓ 0, in the equality We now prove the uniqueness of the solution.We follow the proof of Theorem 4.1 in [23].Notice, however, that we now consider a special case and not the general case treated in [23].Suppose that there exist two different solutions of (2.9): x 1 and x 2 .We have 1 2 If, for all i, ψ i (t, x 1 (t)) < 0 and ψ i (t, x 2 (t)) < 0, then ξ i 1 (t) = ξ i 2 (t) = 0 and we obtain 1 2 Suppose that ψ j (t, x 1 (t)) = 0. Then by the Taylor formula we get where θ ∈ [0, 1].Since ψ j (t, x 2 (t)) ≤ 0, we have Now, if ψ j (t, x 2 (t)) = 0, we deduce in the same way that Thus we have 1 2 Hence |x 1 (t) − x 2 (t)|= 0. 2

Approximating Family of Optimal Control Problems
In this section we define an approximating family of optimal control problems to (P ) and we state the corresponding necessary conditions.Let (x, û) be a global solution to (P ) and consider sequences {γ k } and {σ k } as defined above.Let xk (•) be the solution to It follows from Theorem 2.2 that k ↓ 0. Take α > 0 and define the problem Clearly, the problem (P α k ) has admissible solutions.Consider the space and the distance Endowed with d W , W is a complete metric space.Take any (c, u) ∈ W and a solution y to the Cauchy problem Under our assumptions, the function is continuous on (W, d W ) and bounded below.Appealing to Ekeland's Theorem we deduce the existence of a pair (x k , u k ) solving the following problem Lemma 3.1 Take γ k → ∞, σ k → 0 and k → 0 as defined above.For each k, let (x k , u k ) be the solution to (AP k ).Then there exists a subsequence (we do not relabel) such that Proof We deduce from Theorem 2.2 that {x k } uniformly converges to an admissible solution x to (P ).Since U and C 0 are compact, we have U ⊂ KB m and C 0 ⊂ KB n .Without loss of generality, u k weakly- * converges to a function ũ ∈ L ∞ ([0, T ], U ). Hence it weakly converges to ũ in L 1 .From optimality of the processes (x k , u k ) we have Since (x, û) is a global solution of the problem, passing to the limit, we get Hence x(0) = x(0), ũ = û a.e., and u k converges to û in L 1 , and some subsequence converges to û almost everywhere (we do not relabel).2 We now finish this section with the statement of the optimality necessary conditions for the family of problems (AP k ).These can be seen as a direct consequence of Theorem 6.2.1 in [22].Proposition 3.2 For each k, let (x k , u k ) be a solution to (AP k ).Then there exist absolutely continous functions p k and scalars λ k ≥ 0 such that (a) (nontriviality condition) where the superscript * stands for transpose, (c) (maximization condition) is attained at u k (t), for almost every t ∈ [0, T ], (d) (transversality condition) To simplify the notation above, we drop the t dependance in p k , ṗk , x k , u k , x and û.Moreover, in (b), we write ψ k instead of ψ(t, x k (t)), f k instead of f (t, x k (t), u k (t)).The same holds for the derivatives of ψ and f .

Maximum Principle for (P )
In this section, we establish our main result, a Maximum Principle for (P ).This is done by taking limits of the conclusions of Proposition 3.2, following closely the analysis done in the proof of [10,Theorem 2].
Observe that 1 2 where M is the constant of (A2).Taking into account hypothesis (A1) and (2.7) we deduce the existence of a constant K 0 > 0 such that 1 2 This last inequality leads to Since, by (a) of Proposition 3.2, |p k (T )|≤ 1, we deduce from the above that there exists Now, we claim that the sequence { ṗk } is uniformly bounded in L 1 .To prove our claim, we need to establish bounds for the three terms in (3.3).Following [10] and [14], we start by deducing some inequalities that will be of help.
Denote I k = I(t, x k (t)) and S j k = sign ∇ x ψ j k , p k .We have Observe that (see (2.3) and (2.4)) Using this and integrating the previous equality, we deduce the existence of M 1 > 0 such that: We are now in a position to show that Using (A1) and (4.2), we deduce that for k large enough.Summarizing, there exists a M 2 > 0 such that Mimicking the analysis conducted in Step 1, b) and c) of the proof of Theorem 2 in [10] and taking into account (b) of Proposition 3.2 we conclude that there exist constants N 1 > 0 such that for k sufficiently large, proving our claim.Before proceeding, observe that it is a simple matter to assert the existence of a constant N 2 such that This inequality will be of help in what follows.
Let us now recall that ξ i k (t) = γ k e γ k (ψ i (t,x k (t))−σ k ) and that the second inequality in (2.11) holds.We turn to the analysis of Step 2 in the proof of Theorem 2 in [10] (see also [14]).Adapting those arguments, we can conclude the existence of some function p ∈ BV ([0, T ], R n ) and, for i = 1, . . ., I, functions where ∇ ψi (t) = ∇ψ i (t, x(t)).The finite signed Radon measures η i are weak- * limits of Observe that the measures ∇ψ i (x(t), p(t) dη i (t) (4.6) are nonnegative.
For each i = 1, . . ., I, the sequence ξ i k is weakly- * convergent in L ∞ to ξ i ≥ 0. Following [14], we deduce from (4.5) that, for each i = 1, . . ., I, It turns out that ξ i ∇ x ψi , p = 0 a.e.. (4.7) Consider now the sequence of scalars {λ k }.It is an easy matter to show that there exists a subsequence of {λ k } converging to some λ ≥ 0. This, together with the convergence of p k to p, allows us to take limits in (a) and (c) of Proposition 3.2 to deduce that It remains to take limits of the transversality conditions (d) in Proposition 3.2.First, observe that From the basic properties of the Mordukhovich normal cone and subdifferential (see [19], section 1.Finally, and mimicking Step 3 in the proof of Theorem 2 in [10], we remove the dependence of the conditions on the parameter α.This is done by taking further limits, this time considering a sequence of α j ↓ 0.
We then summarize our conclusions in the following Theorem.
Noteworthy, condition e) is not considered in any of our previous works.
Problem (P f ) differs from (P ) because x(T ) is not constrained to take values in C T .We apply Theorem 4.1 to (P f ).Since x(T ) is free, we deduce from (f) in the above Theorem that −p(T ) = λ∂φ(x(T )).Suppose that λ = 0. Then p(T ) = 0 contradicting the nontriviality condition (a) of Theorem 4.1.Without loss of generality, we then conclude that the conditions of Theorem 4.1 hold with λ = 1.We summarize our findings in the following Corollary.
Corollary 4.2 Let (x, û) be the optimal solution to (P f ).Suppose that assumption A1-A6 are satisfied.

Example
Let us consider the following problem • (x 0 , y 0 , z 0 ) ∈ intC, with x 0 < −δ, y 0 = 0 and z 0 > 0, We choose T > 0 small and, nonetheless, sufficiently large to guarantee that, when σ = 0, the system can reach the interior of C T but not the segment {(x, 0, 0) | x ∈ [−δ, 0]}.Since σ and T are small, it follows that the optimal trajectory should reach C T at the face δy − y 2 x = δy 2 of C T .
To significantly increase the value of the x(T ), the optimal trajectory needs to live on the boundary of C for some interval of time.Then, before reaching and after leaving the boundary of C, the optimal trajectory lives in the interior of C. Since δ is small, the trajectory cannot reach C T from any point of the sphere x 2 + y 2 + (z + h) 2 = 1 with z > 0. This means that, while on the boundary of C the trajectory should move on the sphere x 2 + y 2 + (z + h) 2 = 1 untill reaching the plane z = 0 and then it moves on the intersection of the two spheres.
While in the interior of C, the control can change sign from −1 to 1 or from 1 to −1.Certainly, the control should be 1 right before reaching the boundary and −1 right before arriving at C T .Changes of the control from 1 to −1 or −1 to 1 before reaching the boundary translate into time waste and leads to smaller values of x(T ).It then follows that the optimal control should be of the form for some value t ∈]0, T [.
After the modification (2.5), the data of the problem satisfy the conditions under which Theorem 4.1 holds.We now show that the conclusions of Theorem 4.1 completly identify the structure (5.1) of the optimal control.
From Theorem 4.1 we deduce the existence of λ ≥ 0, p, q, r ∈ BV ([0, T ], R), finite signed Randon measures η 1 and η 2 , null respectively in where û is the optimal control.Let t 1 be the instant of time when the trajectory reaches the shere x 2 + y 2 + (z + h) 2 = 1, t 2 the instant of time when the trajectory reaches the intersection of the two spheres and t 3 be the instant of time the trajectory leaves the boundary of C. We have 0 < t 1 < t 2 < t 3 < T .
Next we show that the multiplier q changes sign only once and so identifing the structure (5.1) of the optimal control in a unique way.We start by looking at the case when t = T .We have p q (T ) = λ 0 + µ y 2 −δ .
From the previous analysis we deduce that q should be positive almost everywhere on the boundary.It then follows that to find the optimal solution we have to analyze admissible trajectories with the controls with the structure (5.1) and choose the optimal value of t.
or simply L ∞ when the domains are clearly understood) is the Lebesgue space of essentially bounded functions h : [a, b] → R p .We say that h ∈ BV ([a, b]; R p ) if h is a function of bounded variation.The space of continuous functions is denoted by C([a, b]; R p ).