Regularity properties of solutions to linear quadratic optimal control problems with state constraints

Abstract

We establish regularity properties of solutions of linear quadratic optimal control problems involving state inequality constraints. Under simply stated and directly verifiable hypotheses on the data, it is shown that if the state constraint has index k > 0 then the optimal control u is k times differentiable; the kth derivative may be discontinuous but it is a function of bounded variation (and consequently it has left and right limits at each point in its domain). If on the other hand the state constraint has index k = 0 then the optimal control is continuous. The latter property is perhaps surprising because it implies that for the class of problems considered optimal state trajectories cannot abruptly change direction when they strike the boundary of the state constraint region. These findings are significant because they justify assumptions which underlie analysis of junction conditions (i.e. properties of state trajectories at contact points with the boundary of the state constraint set) provided elsewhere in the literature.