Impulsive control systems without commutativity assumptions

Abstract

This paper is concerned with optimal control problems for an impulsive system of the form $$\dot x(t) = f(t, x, u) + \sum\limits_{i = 1}^m {g_i } (t, x, u)\dot u_i ,u(t) \in U,$$ where the measurable controlu(·) is possibly discontinuous, so that the trajectories of the system must be interpreted in a generalized sense. We study in particular the case where the vector fieldsg i do not commute. By integrating the distribution generated by all the iterated Lie brackets of the vector fieldsg i , we first construct a local factorizationA 1 ×A 2 of the state space. If (x 1,x 2) are coordinates onA 1 ×A 2, we derive from (1) a quotient control system for the single state variablex 1, withu, x 2 both playing the role of controls. A density result is proved, which clarifies the relationship between the original system (1) and the quotient system. Since the quotient system turns out to be commutative, previous results valid for commutative systems can be applied, yielding existence and necessary conditions for optimal trajectories. In the final sections, two examples of impulsive systems and an application to a mechanical problem are given.