Unmaximized Inclusion Type Conditions for Nonconvex Constrained Optimal Control Problems

Necessary conditions of optimality in the form of an unmaximized Inclusion (UI) are derived for optimal control problems with state constraints. The conditions presented here generalize earlier optimality conditions to problems that may be nonconvex. The derivation of UI type conditions in the absence of the convexity assumption is of particular importance when deriving necessary conditions for constrained problems. We illustrate this feature by establishing, as an application, optimality conditions for problems that in addition to state constraints incorporate mixed state-control constraints. 1991 Mathematics Subject Classification. 49K15. The dates will be set by the publisher. Introduction In this paper we derive necessary conditions of optimality involving unmaximized Inclusion-type conditions for optimal control problems with pure state constraints and we report on some main applications. These subsume and substantially extend the results in [6], [8], [5] and [3]. The problem of interest is: (P)  Minimize g(x(0), x(1)) + ∫ 1 0 L(t, x(t), u(t))dt subject to ẋ(t) = f(t, x(t), u(t)) a.e. t ∈ [0, 1] h(t, x(t)) ≤ 0 a.e. t ∈ [0, 1] u(t) ∈ U(t) a.e. t ∈ [0, 1] (x(0), x(1)) ∈ C. Here g : R × R → R, L : [0, 1]× R × R → R, f : [0, 1]× R × R → R, h : [0, 1]× R → R are given functions, C ⊂ R × R a given set and U : [0, 1]⇒ R is a given multifunction. First order necessary conditions for the optimal control problem (P) when the data may be nonsmooth have undergone continuous development. Such conditions, written in the form of maximum principles, are based on