Stochastic Optimal Time-Delay Control and Stabilization of Quasi-Integrable Hamiltonian Systems

Innovative procedures for the stochastic optimal time-delay control and stabilization are proposed for a quasi-integrable Hamiltonian system subject to Gaussian white noises. First, the problem of stochastic optimal control with time delay is formulated. Then, the problem is converted into a stochastic optimal control without time delay, and the converted control problems are then solved by applying the stochastic averaging method and the stochastic dynamical programming principle. The optimal time-delay stabilization of quasi-integrable Hamiltonian systems is formulated as an ergodic control with a cost function determined by minimizing the largest Lyapunov exponent of the controlled system. As an example, a two-degree-of-freedom quasi-integrable Hamiltonian system with time delay in feedback control forces is investigated in detail to illustrate the procedures and their effectiveness.