Stochastic stability of quasi-integrable Hamiltonian systems with time delay by using Lyapunov function method

Abstract

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. Lyapunov stability, Lyapunov function, time delay, stochastic averaging Citation:Ling Q, Jin X L, Huang Z L. Stochastic stability of quasi-integrable Hamiltonian systems with time delay by using Lyapunov function method. Sci

Observer-based robust adaptive control for uncertain stochastic Hamiltonian systems with state and input delays

“…So it often is viewed as a proper candidate of Lyapunov function for many physical systems. On account of this advantage, the Hamilton function approach has caused broad attention in the field of nonlinear controller design, as well as in practical control problems (see, e.g., [2][3][4] and the references therein). Recently, the authors obtain some results on time delay Hamiltonian systems (see, e.g., [5][6][7][8]).…”