Backstepping control design for stochastic Hamiltonian systems

In this paper, the problem of adaptive tracking for a class of stochastic Hamiltonian control systems with unknown nonlinear drift and diffusion functions is considered. Some difficulties come forth: integral chain consists of vectors, unknown control gain is a definite-positive matrix, and control and tracking error are in different channels, which are rarely considered in stochastic nonlinear controls. To overcome these problems, a vector form of adaptive backstepping controller is designed such that the closed-loop stochastic Hamiltonian system has a unique solution that is globally bounded in probability and the L₄-norm of the tracking error converges to an arbitrarily small neighborhood of zero. As applications, an example from mechanical systems instead of numerical one is presented.