Some Applications of Hamilton–Jacobi Inequalities for Classical and Impulsive Optimal Control Problems

This article is devoted to some applications of Hamilton–Jacobi inequalities for control problems of ordinary and impulsive dynamical systems. We focus on the study of necessary and sufficient global optimality conditions. The optimality conditions are obtained by using certain families of Lyapunov type functions, where Lyapunov type functions can be strongly or weakly monotone solutions of the corresponding Hamilton–Jacobi inequalities. For classical problems of the optimal control theory, new variants of the Carathéodory and Krotov types optimality conditions and conditions of the so-called Hamilton–Jacobi canonical optimality theory are proposed and compared. For impulsive dynamical systems with trajectories of bounded variation, conditions for the strong and weak monotonicity of the Lyapunov type functions are obtained. These monotonicity conditions are formulated in terms of generalized Hamilton–Jacobi inequalities and they allow to investigate various questions of the position and optimal control theory for impulsive systems.