Stabilization of stochastic nonlinear systems driven by noise of unknown covariance

The paper poses and solves a new problem of stochastic (nonlinear) disturbance attenuation where the task is to make the system solution bounded (in probability) by a monotone function of the supremum of the covariance of the noise. This is a natural stochastic counterpart of the problem of input-to-state stabilization in the sense of Sontag (1989). Our development starts with a set of new global stochastic Lyapunov theorems and a control Lyapunov function formula for stochastic disturbance attenuation, followed by a design for an exemplary class of stochastic strict-feedback systems. We then address optimality and solve a differential game problem with the control and the noise covariance as opposing players; for strict-feedback systems the resulting Isaacs equation has a closed-form solution. Finally, we address the problem of vanishing nonlinearities where the equilibrium is preserved in the presence of noise, and develop an adaptive stabilization scheme (based on tuning functions) that requires no a priori knowledge of a bound on the covariance