Stabilization of stochastic nonlinear systems driven by noise of unknown covariance

This paper poses and solves a new problem of stochastic (nonlinear) disturb ance attenuation where the task is to make the system solution bounded (in expectation, with appropriate nonlinear weighting) by a monotone function o f 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. Our development starts with a set of new global stochastic Lyapuno v theorems. For an exemplary class of stochastic strict-feedback systems wi th vanishing nonlinearities, where the equilibrium is preserved in the pres ence of noise, we develop an adaptive stabilization scheme (based on tuning functions) that requires no a priori knowledge of a bound on the covarianc e. Next, we introduce a control Lyapunov function formula for stochastic di sturbance attenuation. Finally, we address optimality and solve a different ial game problem with the control and the noise covariance as opposing play ers; for strict-feedback systems the resulting Isaacs equation has a closed -form solution.