  Incremental stability is a property of dynamical systems ensuring the uniform
asymptotic stability of each trajectory rather than a fixed equilibrium point
or trajectory. Here, we introduce a notion of incremental stability for
stochastic control systems and provide its description in terms of existence of
a notion of so-called 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 with jumps,
incrementally stable. To illustrate the effectiveness of the proposed approach,
we design a controller making a spring pendulum system in a noisy environment
incrementally stable.
