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.

Though the Lyapunov function method is more efficient than the largest Lyapunov exponent method in evaluating the stochastic stability of multi-degree-of-freedom (MDOF) systems, the construction of Lyapunov function is a challenging task. In this paper, a specific linear combination of subsystems’ energies is proposed as Lyapunov function for MDOF nonlinear stochastic dynamical systems, and the corresponding sufficient condition for the asymptotic Lyapunov stability with probability one is then determined. The proposed procedure to construct Lyapunov function is illustrated and validated with several representative examples, where the influence of coupled/uncoupled dampings and excitation intensities on stochastic stability is also investigated.

The response and stability of a single degree-of-freedom (SDOF) viscoelastic system with strongly nonlinear stiffness under the excitations of wideband noise are studied in this paper. Firstly, terms associated with the viscoelasticity are approximately equivalent to damping and stiffness forces; the viscoelastic system is approximately transformed to SDOF system without viscoelasticity. Then, with application of the method of stochastic averaging, the averaged Itô differential equation is obtained. The stationary response and the largest Lyapunov exponent can be analytically expressed. The effects of different system parameters on the response and stability of the system are discussed as well.

The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method. First, a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay. Then, stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system, and after that the Lyapunov function of the averaged Itô equation is taken as the optimal linear combination of the corresponding independent first integrals in involution. Finally, the stability of the system is determined by using the largest eigenvalue of the linearized system. Two examples are used to illustrate the proposed procedure and the effects of delayed time on the Lyapunov stability are discussed as well.