Monotonicity of Lyapunov Type Functions for Impulsive Control Systems

The paper is devoted to the study of impulsive dynamical systems with trajectories of bounded variation and impulsive controls (regular vector measures). A new concept of solutions for these systems is introduced. According to this concept, the solution is an upper semicontinuous set-valued mapping. The relationship between the new solution concept and conventional one is established. We prove that the set of solutions is a closure of the set of the absolutely continuous solutions. Here, the closure is understood in the sense of the convergence in Hausdorff metric for graphs of the supplemented absolutely continuous trajectories. In this paper, we focus mainly on the study of some monotonicity properties of a continuous function with respect to a nonlinear impulsive control system with trajectories of bounded variation. Definitions of strong and weak monotonicity and $V$-monotonicity are proposed and discussed. The set of conventional variables $t$ and $x$ of Lyapunov type functions is now supplemented with the variable $V,$ which, on the one hand, is responsible for the impulsive dynamics of the system and has the property of the time variable and, on the other hand, characterizes some resource of the impulsive control. We show that such double interpretation of variable $V$ leads to different definitions of monotonicity, which are called monotonicity and $V$-monotonicity. For smooth Lyapunov type functions, infinitesimal conditions of monotonicity in the form of Hamilton–Jacobi differential inequalities are presented.