Incremental stability analysis of random nonlinear systems with an application to random Lagrange systems

This paper is intended to investigate the incremental stability analysis of random nonlinear systems with colored noise and its application to random Lagrange systems. Under an analysis of the dynamical features of the considered random nonlinear systems, the definitions of incremental noise-to-state stability and incrementally asymptotic stability with respect to preliminary conditions and/or random processes are proposed. By introducing two extended forms of the original system, relevant incremental stability conditions are established based on Lyapunov functions with time-varying derivatives. We further provide an easily verifiable incremental noise-to-state stability result by using quadratic Lyapunov functions. To explain the application value of our devised results, an incremental stability control problem of random Lagrange systems is solved with the aid of a vector backstepping approach. We at last demonstrate the efficacy of our work through numerical simulations.