Near optimal control for resonant oscillatory systems

The paper is concerned with optimal control problems for near-resonant motions of two-frequency quasi-conservative nonlinear weakly controlled systems. It is assumed that the conservative subsystem exhibits a single primary resonance with a known near resonant domain. Escape from this near resonant domain can be identified with the failure of resonance, and the main goal of the implication of control forces is to hold the system in this restricted domain. Away from the resonant domain the optimal control problems can be treated by the averaging method that is known to fail near the resonant surface. In this paper we develop a special version of the hierarchical averaging approach. The hierarchical averaging procedure is extended to the optimal control problems. As the result, the averaged system of the maximum principle is obtained, and the convergence of the approximate solution to the solution of the initial optimal control problem is proved.