A Maximum Principle for the Controlled Sweeping Process

Abstract

Abstract. We consider the free endpoint Mayer problem for a controlled Moreau process, the control acting as a perturbation of the dynamics driven by the normal cone, and derive necessary optimality conditions of Pontryagin's Maximum Principle type. The results are also discussed through an example. We combine techniques from [19] and from [6], which in particular deals with a different but related control problem. Our assumptions include the smoothness of the boundary of the moving set C(t), but, differently from [6], do not require strict convexity. Rather, a kind of inward/outward pointing condition is assumed on the reference optimal trajectory at the times where the boundary of C(t) is touched. The state space is finite dimensional.

Optimal control of a nonconvex perturbed sweeping process

“…t ∈ [0, T ], x(0) = x 0 ∈ C(0), (1. 3) with controls a : [0, T ] → R d acting in perturbations and controls u : [0, T ] → R n acting in the polyhedral moving set generated by the fixed vectors x * i as C(t) := C + u(t) with C := x ∈ R n x * i , x ≤ 0 for all i = 1, . .…”

“…Control functions appeared there either in additive perturbations [2,3,10,16,31], or in associated ordinary differential equations [1,5]. Necessary optimality conditions for optimal controls in such controlled sweeping models were derived in [3,5] by employing some other methods different from [7,8,12,13] under certain strong smoothness assumptions on the boundaries of compact uncontrolled sweeping sets. In the more recent paper [35], the author addressed relaxation issues for sweeping optimization problems with controls in additive perturbations and uncontrolled convex moving sets that were also included in optimization.…”

Extended Euler–Lagrange and Hamiltonian Conditions in Optimal Control of Sweeping Processes with Controlled Moving Sets

“…In this case conditions (6.3) and (5.5) yield q x (t) ≥ 0, and hence we have q x (2) ≤ 0 by (6). It shows that q x (t) = 0 on [1,2] and so ν(t) = −q x (t) = 0, which contradicts the condition ν(t) < 0. Thusū(t) remains constant on [1,2], and we find an optimal solution to this problem.…”

“…Then it follows from (4.21) and (4.23) that θ 2 = 0, which yields θ 1 = 0 and gives us a contradiction. △ We can directly check that the only optimal trajectory in this problem is given byx(t) = 3/2 on [0, 1/2],x(t) = 2 − t on [1/2, 1], and x(t) = 1 on [1,2], It is generated by the optimal controlū(t) = t − 2 on [0, 1] andū(t) = −1 on (1,2].…”

“…To the best of our knowledge, necessary optimality conditions involving the maximization of the corresponding Hamiltonian were first obtained for sweeping control systems in [7], where the authors considered a sweeping process with a strictly smooth, convex, and solid set C(t) ≡ C in (1.1) while with control functions entering linearly an adjacent ordinary differential equation. Further results with the maximum condition for global (as in [7]) minimizers were derived in [2] for the sweeping control systeṁ x(t) ∈ f x(t), u(t) − N x(t); C(t) a.e. t ∈ [0, T ], (1.8) where measurable controls u(t) enter the additive smooth term f while the uncontrolled moving set C(t) is compact, uniformly prox-regular regular (close enough to convexity), and possesses a C 3 -smooth boundary for each t ∈ [0, T ] under some other assumptions.…”