A Wiener filter, state-space flux-optimal control against escape from a potential well

Escape from a potential well, often identified with system failure, is reformulated for a class of two-dimensional dynamical systems as exiting the safe region of the system state space bounded by the separatrix derived from the potential well. A control against system failure based on minimizing the state-space flux out of this safe region is considered. For systems subject to weak external forcing with available control power exceeding a certain threshold, the finite lag, state-space flux-optimal control against failure, is shown to involve a Wiener filter. A non-Wiener solution is proposed for available power below this threshold. These results hold for wide-sense stationary forcing with any statistics and any spectrum.