$\Bbb L^2$ Sufficient Conditions for End-Constrained Optimal Control Problems With Inputs in a Polyhedron

Abstract

An $\hbox{{\bbb L}}^2$-local optimality sufficiency theorem is proved for a class of structured infinite-dimensional nonconvex programs with constraints of the form $u\in\Omega$ and h(u)=0, where $\Omega$ is a set of Lebesgue measurable essentially bounded vector-valued functions $u(\cdot ): [0,1]\rightarrow \hbox{{\bbb R}}^m$ with range in a polyhedron U, and h is a smooth map of the space of essentially bounded functions $u(\cdot )$ into ${\bbb R}^k$. The sufficiency theorem is based on formal counterparts of the finite-dimensional Karush--Kuhn--Tucker sufficient conditions in a Cartesian product of polyhedra, a strengthened variant of Pontryagin's necessary condition, and structure and continuity conditions on the first and second differentials of the objective function and equality constraint functions. The new sufficient conditions are directly applicable to nonconvex continuous-time Bolza optimal control problems with control-quadratic Hamiltonians, unqualified affine inequality constraints on vector-valued control inputs, and equality constraints on the terminal state vector or equivalent isoperimetric constraints on integrals of functions depending on the state and control variables.