Optimal Control and Differential Games with Measures

We consider control problems with trajectories which involve ordinary measureable control functions and controls which are measures. The payoff involves a running cost in time and a running cost against the control measures. In the optimal control problem we are trying to minimize this payoff with both controls. In the differential game problem we are trying to minimize the cost with the ordinary controls assuming that the measure controls are chosen to maximize the cost. We will characterize the value functions in both cases using viscosity solution theory by deriving the Bellman and Isaacs equations. 0.Introduction and Summary The problems of this paper are motivated by models of physical controlled systems in which the trajectory is a function of bounded variation. The time of the jumps, if any, and the new spatial positions are under the control of the designer. In the optimal control problem the objective is to minimize a cost involving a running cost and a cumulative cost against the control measure. When the measure involves only jumps this will be a standard impulse control problem. But in this paper we are not restricting the measures merely to jumps but are allowing general Radon measures. In many problems of interest we can manipulate the system only with ordinary controls and we wish to do so to minimize a cost. But when the system is subject to disturbances one seeks to design the system so as to perform well under the worst possible circumstances. In this situation we assume that the disturbances are modelled by a measure term in the *Supported in part by AFOSR-86-0202, NSF DMS-9102967, and a grant from Loyola University **Supported in part by AFOSR-86-0202, an NSF grant, and a grant from Loyola University ***Supported in part by NSF DMS-9101360