Almost restoring problem of deterministic asymptotic stability against additive noises

This paper proposes a disturbance attenuation strategy for nonlinear controlled systems against the addition of additive Gaussian white noises. To achieve the purpose, we firstly revisit the following stochastic stability notions for stochastic systems: uniform almost sure asymptotic stability (UASAS), exponential p-stability, and finite-time stability in probability. Combining the previous notions, we propose new asymptotic stability properties and new stochastic Lyapunov functions, p-stability with respect to (w.r.t.) EV, and quasi almost Lyapunov functions (quasi-ALFs) for nonlinear systems with stochastic disturbance terms, the values of which are nonzero at the origins. Subsequently, we propose a strategy of noisy surface control to obtain sufficient conditions so that stochastic systems almost restore their smooth trajectories of the time when they were not vibrated by additive noises. The effectiveness of the control strategy is confirmed by the consideration of disturbance attenuation problems for stochastic linear systems with linear quadratic (LQ) controls.