Comparison Theorems for Viscosity Solutions of a System of Quasivariational Inequalities with Application to Optimal Control with Switching Costs

Abstract

DEDICATED TO PROFESSOR GEORGE LEITMANN WITH RESPECTIn this paper we prove a comparison principle between a viscosity sub-and supersolution for a system of quasivariational inequalities and apply it to show that a continuous lower value vector function of an optimal switching-cost control problem is characterized as the minimal, nonnegative, continuous, viscosity supersolution of the SQVI. ᮊ

Uniqueness of unbounded viscosity solutions for impulse control problem

“…Without loss of generality we assume that v 1 By definition, T v 1 is the value function for the optimal stopping time problem with stopping cost Mv 1 . Hence it satisfies the variational inequality:…”

“…We also deal with the finite horizon case in the same spirit as in [7], first proving a local comparison theorem in a cone and then a global comparison theorem, for quasivariational inequalities. Similarly for the switching problem Ball et al [1] have extended the result of [5] in to the case of unbounded solutions. Now we describe the control problem.…”

Robust Optimal Switching Control for Nonlinear Systems

“…An alternative derivation of this characterization relying on a general comparison principle for viscosity super-and subsolutions of SQVI is given in [2]. The usual formulation of the H ∞ -control problem also involves a stability constraint.…”