Feedback Necessary Optimality Conditions for Nonlinear Measure-Driven Processes

Abstract

We consider a non-convex optimal impulsive control problem for nonlinear differential equations, driven by vector-valued Borel measures, under no commutativity assumptions of the Frobenius type. For this problem, we derive nonlocal necessary optimality conditions operating with a specific class of impulsive feedback controls, generated by certain functions of the Lyapunov type. These feedback controls are constructed in a way similar to the dynamical programming, but with the use of weakly monotone solutions to the corresponding Hamilton-Jacobi equation, instead of the Bellman’s function. We offer the notion of weakly monotone function with respect to a measure-driven differential equation, and give constructive criteria for this type of monotonicity. Based on a space-time representation of impulsive processes, we propose the concept of impulsive feedback control and present nonlocal necessary optimality conditions, which are shown to filter out non-optimal extrema of the impulsive Maximum Principle.