A set evolution approach to the control of uncertain systems with discrete-time measurement

We investigate here a continuous time minimization problem in the presence of disturbances in the dynamics. The only information available to the controller is an incomplete observation of the state space at times given in advance. Also, the initial state is not supposed to be perfectly known. The corresponding control problem can be understood as a dynamic game of Min-Max type where the controller wants to minimize the cost - by choosing a strategy depending on a discrete-time incomplete measurement - against the worst case of disturbance and initial state. Our main goal is to pass from imperfect information in the measurement space to perfect information in the estimation space, hence we introduce a second problem based on estimation sets on the state. We prove that the value functions of both problems are equal. Finally, we provide a characterization of the value function through a system of Hamilton-Jacobi equations and inequalities in terms of Dini derivatives.