Regularity properties of optimal controls for problems with time-varying state and control constraints

Abstract

Abstract. In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.

Introducing State Constraints in Optimal Control for Health Problems

“…Now we explore regularity properties of the multipliers. It is a simple matter to see that the main results in [12] apply and so we deduce that μ is absolutely continuous w.r.t. Lebesgue measure.…”

A unified numerical scheme for linear-quadratic optimal control problems with joint control and state constraints

“…Other improvements including allowing the control constraint to be a general fixed convex set and relaxing the differentiability assumption on the data are also presented in [32]. In a follow-up paper [69], the authors further refined the result to allow the control constraint to be time varying. In a series of recent papers [7,8,31] (see also [37]), the authors established several regularity results, such as Lipschitz continuity, Hölder continuity, and continuous differentiability, for general optimal control problems with nonlinear dynamics and convex cost integrand.…”

“…The literature is vast; recent monographs and selected papers include [7,8,10,11,13,14,16,24,31,32,37,38,42,46,[68][69][70]72]. As expected, the constrained optimal control problem is much more challenging than the unconstrained case, with the case of mixed algebraic state-control constraints being the most challenging to solve.…”

Convergence of discrete-time approximations of constrained linear-quadratic optimal control problems

“…In addition, our approach is implementable (in terms of relaxing the constraints to ensure the feasibility of the discretizations) under less restrictive conditions. Moreover, this approach leads to a new regularity result regarding the Lipschitz Continuity of the costate trajectory for the class of LQ optimal control problems with mixed control and state constraint, see [14], [25] . A different convergence problem in MPC that did receive considerable attention is the relation between finite horizon control problems and the corresponding infinite horizon problems, see e.g.…”