Backstepping Design for Incremental Stability of Stochastic Hamiltonian Systems

Incremental stability is a property that ensures the uniform asymptotic stability of each trajectory rather than a fixed equilibrium point or trajectory. This makes it a stronger stability notion for dynamical systems. Here, we introduce a notion of incremental stability for stochastic control systems and provide its description in terms of a notion of incremental Lyapunov functions. Moreover, we provide a backstepping controller design scheme providing controllers along with corresponding incremental Lyapunov functions rendering a class of stochastic control systems, namely stochastic Hamiltonian systems, incrementally stable. Moreover, to illustrate the effectiveness of the design approach, we design a controller making a controlled spring pendulum system in a stochastic environment incrementally stable.