Stability Under Random Perturbations

Stability of stochastically perturbed systems is considered in Chap. 10. If the nonperturbed system has an asymptotically stable equilibrium, and a bounded neighborhood of this equilibrium is such that the system is “destroyed” if the perturbed trajectory leaves that neighborhood, then the stability of the system can be characterized by the exit time from this neighborhood. If the noise is small, the main term of the asymptotics of the exit time characterizes the stability. Using the results obtained in previous chapters, one can calculate the exit time up to its main term. If a problem of optimal stabilization is considered and the stochastic perturbations are small, one should maximize the main term of the exit time. Results of this type are considered in this chapter.