Game representations for state constrained continuous time linear regulator problems

TL;DR

This paper introduces a supremum-of-quadratics representation for convex barrier constraints, transforming certain constrained linear regulator problems into unconstrained two-player linear quadratic games, and characterizes optimal policies through game equivalence.

Contribution

It develops a novel supremum-of-quadratics representation for convex barrier constraints and applies it to reformulate constrained regulator problems as unconstrained games with explicit feedback solutions.

Findings

Convex barrier constraints can be represented as a supremum of quadratic functions.

Constrained problems are equivalent to unconstrained two-player linear quadratic games.

Explicit state feedback characterizations for optimal policies are derived.

Abstract

A supremum-of-quadratics representation for convex barrier-type constraints is developed and applied within the context of a class of continuous time state constrained linear regulator problems. Using this representation, it is shown that a linear regulator problem subjected to such a convex barrier-type constraint can be equivalently formulated as an unconstrained two-player linear quadratic game. By demonstrating equivalence of the upper and lower values of this game, state feedback characterizations for the optimal policies of both players are developed. These characterizations are subsequently illustrated by example.