On Optimality of Stochastic Non-Linear Tracking Control Systems

The tracking problem for differential stochastic equations in the present of stochastic uncertainty of white noise, and control input have been considered. In this work, our consideration have been focused on the case where both original dynamic state stochastic system and the desired stochastic dynamic system, are driven  by white noise stochastic process. The main aim of this work is to make the behavior of the original dynamic system following the behavior of the desired one for arbitrary controller, using tracking control system approach. The tracking and stabilizing controller that guarantee the optimum tracking error system between the original system and the desired one have been derived and developed.The necessary theorems for optimum tracking have been stated and proved supported with some concluding remarks. The controller can also been divided into robust one and optimal one. The optimum controller can be obtained as a solution of some linear deterministic differential Riccati equation, while the robust one can be obtained so that some controllability properties are ensured. The Riccati equation associated with linear stochastic optimal controller and tracking one, have also been desired and discussed.Finally some illustration ranking for time varying system and for law order differential system to larger one, have been illustrated, with details and corresponding Riccati equation for justification of the present work.