Stochastic nonlinear Lyapunov stabilization and inverse optimality

While the current robust nonlinear control toolbox includes a number of methods for systems affine in deterministic bounded disturbances, the problem when the disturbance is unbounded stochastic noise has hardly been considered. We present a control design which achieves global asymptotic (Lyapunov) stability in probability for a class of strict-feedback nonlinear continuous time systems driven by white noise. We then address the classical question of when is a stabilizing (in probability) controller optimal and show that for every system with a stochastic control Lyapunov function it is possible to construct a controller which is optimal with respect to a meaningful cost functional. Finally, we design an optimal backstepping controller whose cost functional includes penalty on control effort and which has an infinite gain margin. A reader of this paper needs no prior familiarity with techniques of stochastic control.