On the Generalized Solutions for Uncertain Systems with Applications to Optimal Control and Estimation Problems∗

Abstract:- The paper deals with the control and estimation problems for impulsive control system under un-certainty described by differential systems with measures. Uncertainty conditions are of a set–membership type, i.e. uncertain parameters and functional disturbances are taken to be unknown but bounded with given bounds (e.g., the model may contain unpredictable errors without their statistical description). Mod-els of this kind arise in many applications ranging from space navigation to investment problems as well as ecological management. In this paper we consider solution concepts for such measure driven uncertain dynamical systems. The approaches are based on the techniques of approximation of the discontinuous generalized trajectory tubes by the solutions of usual differential systems without measure terms and pro-vide convenient frameworks to derive optimality conditions in the form of either a Maximum Principle or Hamilton-Jacobi-Bellman equations. Another class of problems related to the addressed systems concerns the state estimation. The redesign of the approach for impulsive control systems in the framework of the new solution concept is presented through the characterization of the reachable set as a level set of the value function regarded as a solution of Hamilton-Jacobi-Bellman equations