On a Non anticipative Filippov Theorem for Control Systems with state constraints

An important consequence of the famous Filippoy´s Theorem for a control system, states that given a fixed initial point y0, and a control u0, for any initial point y1, it is possible to find control u1such that the trajectory associated with (y1, u1) approximates the trajectory associated with (y0, u0) (with a suitable estimation of exponetial type). The goal of the present paper is to obtain such a result when trajectories of the controlsystem have to fullfill furthermore a given in advance state constraint. The main specificity of our results lies in the fact that u1is built from u0in a nonanticipative way i.e. at every time t, u1(t) depends only values of u0on the interval [0,t]. This nonanticipative property is crucial in particular for further applications to differential games.