Feedback stabilization of multi-DOF nonlinear stochastic Markovian jump systems

A feedback control strategy is designed to asymptotically stabilize a multi-degree-of-freedom (DOF) nonlinear stochastic systems undergoing Markovian jumps. First, a class of hybrid nonlinear stochastic systems with Markovian jumps is reduced to a one-dimensional averaged Itô stochastic differential equation for controlled total energy. Second, the optimal control law is deduced by applying the dynamical programming principle to the ergodic control problem of the averaged systems with an undetermined cost function. Third, the cost function is determined by the demand of stabilizing the averaged systems. A Lyapunov exponent is introduced to analyze approximately the asymptotic stability with probability one of the originally controlled systems. To illustrate the application of the present method, an example of stochastically excited two coupled nonlinear oscillators with Markovian jumps is worked out in detail.