Control against escape through a separatrix for multistable stochastic systems

The paper develops optimal control procedure for separatrix crossing motions in a perturbed near-Hamiltonian system. Away from the separatrix, control problems are considered using the well-known stochastic averaging procedure, which fails, however, in the vicinity of the separatrix. The motion near the separatrix is presented as a sequence of encirclements over the separatrix lobe between two consecutive vertices, and the problem of avoiding the escape is reduced to minimization of the mean energy during one encirclement. It is shown that the energy difference during one encirclement can by approximated by the stochastic Melnikov integral. This allows us to extend the stochastic Melnikov method to optimal control problems. The asymptotic solution is constructed as a stationary feedback, which is proved to be a near-optimal control for the original nonstationary problem. The requisite asymptotic estimation is given. Possible extensions to problems of controlling chaos are briefly discussed.