First- and second-order necessary conditions for control problems with constraints

Abstract:

Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form $G(x,u) \in \Gamma$, where $\Gamma$ is a closed convex set of a Banach space with nonempty interior. The inequality constraints $g(s,x,u) \leqslant 0$ depend on a parameter $s$ belonging to a compact metric space $S$. The equality constraints are split into two sets of equations $K(x,u) = 0$ and $H(x,u) = 0$, where the first equation is an abstract control equation, and $H$ is assumed to have a full rank property in $u$. The objective function is ${\max _{t \in T}}f(t,x,u)$ where $T$ is a compact metric space, $f$ is upper semicontinuous in $t$ and Lipschitz in $(x,u)$. The results are in terms of a function $\sigma$ that disappears when the parameter spaces $T$ and $S$ are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.