Abstract
A weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints is provided. In the formulation of the minimax control problem we allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Then a new constraint qualification of Mangassarian-Fromovitz type is introduced which allowed us to prove the necessary conditions of optimality. We also derived the optimality conditions under a full rank conditions type and showed that it is, as usual, a particular case of the Mangassarian-Fromovitz type condition case. Illustrative examples are presented.
Highlights
IntroductionConsider the following minimax optimal control problem with mixed constraints (P R). such that, for each α∈A x (t; α) = f (t, x(t, α), u(t), v(t), α), a.e. t ∈ [S, T ]
We introduce a new type of constraint qualifications generalizing the Mangasarian-Fromovitz conditions (MFC) tailored to derive the necessary conditions of optimality for the general minimax problem (PR)
As Proposition 4.2 below, that when A is finite and any of the constraint qualifications A1), A2) or (MFC) is valid, the necessary optimality conditions hold true
Summary
Consider the following minimax optimal control problem with mixed constraints (P R). such that, for each α∈A x (t; α) = f (t, x(t, α), u(t), v(t), α), a.e. t ∈ [S, T ]. (A, ρ) is an arbitrary compact metric space, g : Rn × A → R, (f, b, l) : [S, T ] × Rn × Rku × Rkv × A → Rn × Rmb × Rml , are given functions, V (t) ⊂ Rkv for all t ∈ [S, T ] is a time-dependent set, C(α) ⊂ Rn is a Keywords and phrases: Minimax optimal control problems; mixed constrained; maximum principle; nonsmooth analysis. We set out to obtain necessary optimality conditions for the minimax problem (PR) with mixed constraints, in which the set of parameters A is an arbitrary metric space. In the same section we state the main result of the work, the necessary conditions for constrained minimax control problems, when A is an arbitrary metric space, as Theorem 4.4.
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