Perturbation analysis of a system of quasi-variational inequalities for optimal stochastic scheduling

Optimal scheduling of continuously evolving dynamical systems involves the analysis of certain partial differential inequalities on solution dependent domains. These are the "quasi-variatlonai inequalities" (QVI\´s) introduced for such problems by A. Bensoussan and J. L. Lions. They arise naturally in such applications as inventory control and unit commitment scheduling in electric energy systems. QVI\´s are, in these contexts, the analog of the dynamic programming partial differential equations of R. Bellman for continuous control problems. Analytical treatment of QVI\´s involves not only the solution of partial differential inequalities, but also the simultaneous analysis of the "free" boundaries of the domain of the solution. In effect, the optimal return function is the solution of the inequalities, and the optimal control/switching policy is defined by the free boundaries. This paper considers a system of two singularly perturbed, second-order, quasi-variational inequalities with a turning point in a bounded domain in R¹. Asymptotic approximations for the free boundaries are derived using standard perturbation analysis for boundary value problems. Also, local regularity properties of the limit solution are obtained. The results are illustrated by a parametric study of a simple model of commitment scheduling for a single unit in a power generation system.