Optimal control for control actions with mixed impulse functions and bounded piecewise continuous functions

A number of optimization problems of the mechanics of space flight and the motion of walking robots and manipulators have an irregular structure: classical maximum principle (or classical dynamical programming) do not formally make it possible to find optimal controls. As we explain, many of these tasks need control which have an impulse character. This impulse character means the control will permit the system response to have some instantaneous jumps and leads us to consider generalized solutions of differential equation models. And more general case is optimal control where the control actions are mixed impulse functions and bounded piecewise continuous function to be considered here.;Here we developed a new approach to deal with these generalized optimal control problems having some instantaneous jumps. We established the generalized maximum principle based on standard variational theory. We provide a synthesis procedure to find both the impulse controllers and usual controller in feedback form. We also obtain the generalized dynamical programming based on the standard dynamical programming. Furthermore, we also developed a generalized Hinfinity control theory to deal with linear control systems when including impulse disturbances. As an inverse problem of the generalized optimal control task, we formulate one using the nastiest disturbance (including impulse disturbances) to test a stabilized control system's robustness. We have also developed the necessary conditions and synthesis procedure to solve this generalized (nastiest) disturbance task. Several examples were given to illustrate the theories.