Stochastic stabilization of quasi-generalized Hamiltonian systems

Abstract

A procedure for designing a feedback control to asymptotically stabilize, with probability one, quasi-generalized Hamiltonian systems subject to stochastically parametric excitations is proposed. First, the motion equations of controlled systems are reduced to lower-dimensional averaged Itô stochastic differential equations by using the stochastic averaging method. Second, a dynamic programming equation for the averaged system with an appropriate performance index (with undetermined parameters in cost function) is established based on the dynamic programming principle, and the optimal control law is derived from a minimization condition with respect to control. Third, the Lyapunov function method is adopted to evaluate the stability boundary of asymptotic stability with probability one for the uncontrolled/controlled systems. Finally, the parameters in cost function are selected to guarantee the sufficient stability of the controlled systems. Numerical results for a nine-dimensional mathematical system and a three-dimensional practical system, which describes a structure including viscoelastic element, illustrate the effectiveness of the feedback control strategy, and stability domains can be obviously enlarged when imposing the feedback controls on the original systems.