Applications of Lyapunov Type Functions for Optimization Problems in Impulsive Control Systems

Abstract

This paper deals with an application of Lyapunov type functions for optimality conditions of impulsive processes. A impulsive optimal control problem with trajectories of bounded variation and impulsive controls (regular vector measures) is considered. The problem under consideration is characterized by two main features. First, the dynamical control system is linear with respect to the impulsive control and may have not the so-called well-posedness property of Frobenius type. Second, there are intermediate state constraints on the one-sided limits of the trajectory at fixed instants of time. Sufficient optimality conditions corresponding to the Hamilton--Jacobi canonical optimality theory are presented. These optimality conditions involve some sets of Lyapunov type functions. These functions are strongly monotone solutions of the corresponding proximal Hamilton-Jacobi inequalities. Moreover, we introduce compound (defined piecewise in the variable $t$) Lyapunov type functions, which are more applicable for dynamical systems with discontinuous trajectories and intermediate state constraints. Examples illustrating the optimality conditions are discussed.

Keywords

• measure-driven impulsive control system
• trajectories of bounded variation
• optimal control conditions
• intermediate state constraints
• monotone of Lyapunov type functions