On ${\text{L}}^2 $ Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems

Abstract

${\text{L}}^2 $-local convergence and active constraint identification theorems are proved for gradient projection iterates in sets of ${\text{L}}^\infty $ functions on $[0,1]$ with range in a polyhedron $U \subset \mathbb{R}^m $. These theorems extend earlier results for $U = [0,\infty ) \subset \mathbb{R}^1 $ and are based on an infinite-dimensional variant of the Karush–Kuhn–Tucker second-order sufficient conditions in polyhedral subsets of $\mathbb{R}^n $. The new sufficient conditions and convergence results proved here are directly applicable to continuous-time optimal control problems with smooth nonconvex nonquadratic objective functions and Hamiltonians that are quadratic in the control input vector u. In particular, these theorems apply to nonconvex nonquadratic regulator problems with control-linear state equations and vector-valued inputs $u(t)$ satisfying unqualified affine inequality constraints at almost all t in $[0,1]$.