Near optimal control for resonant systems

We consider optimal control problems for resonance motions. As a model, we study resonances in a two-frequency quasi conservative nonlinear system. It is supposed that the conservative subsystem exhibits a single primary resonance with a known near resonance domain [1]. Escape from this near resonance domain can be identified with the failure of resonance, and the main goal of the implication of control forces is to keep the system within this restricted domain. Away from the near-resonance domain the optimal control problems can be studied by the averaging method [2] As known, this method fails in a small vicinity of the resonance surface. In this paper, we develop a special version of the hierarchical averaging method. This method has been developed [1], [3], [4] as a tool for the analysis of near-resonance motions. In this paper we extend the hierarchical averaging procedure to control problems for oscillatory systems with resonances. As a result, the averaged system of the maximum principle for resonance motions is obtained. The convergence of the approximate solution to the solution of the initial optimal control problem is proved.