IMPULSIVE CONTROL-SYSTEMS WITHOUT COMMUTATIVITY ASSUMPTIONS

This paper is concerned with optimal control problems for an impulsive system of the form x(t) = f(t, x, u) + SIGMA(i=1)m g(i)(t, x, u)u(i), u(t) is-an-element-of U, where the measurable control u(.) is possibl y discontinuous, so that the trajectories of the system must be interp reted in a generalized sense. We study in particular the case where th e vector fields g(i) do not commute. By integrating the distribution g enerated by all the iterated Lie brackets of the vector fields g(i), w e first construct a local factorization A1 x A2 of the state space. If (x1, x2) are coordinates on A1 x A2, we derive from (1) a quotient co ntrol system for the single state variable x1, with u, x2 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 resul ts valid for commutative systems can be applied, yielding existence an d necessary conditions for optimal trajectories. In the final sections , two examples of impulsive systems and an application to a mechanical problem are given.