Necessary conditions for optimal control problems with discontinuous trajectories

Optimal control problems arise in a number of applications, for example midcourse guidance of space vehicles and resource economics, in which the state trajectories are permitted to be discontinuous. These trajectories can be interpreted as responses to impulsive controls. Necessary conditions of optimality have previously been derived for nonlinear problems of this nature through a change of independent variable technique which reduces the impulsive control problem to a conventional one. This earlier approach is limited to problems in which the data is regular in its time dependence, the control constraint set is independent of time and the velocity set has a convexity-type property. We announce new necessary conditions which dispose with all these hypotheses; they are proved by methods of approximation and a limiting argument.