Is Almost Sure Asymptotic Stabilizability Sufficient for Input Delay Tolerability of Nonlinear Stochastic Systems?

This article considers the problem of input delay tolerance (IDT) for stochastic control systems that are almost surely asymptotically stabilizable but inherently nonlinear (e.g., neither locally Lipschitz continuous nor linear growth and non-smoothly stabilizable). Using the notions of almost sure forward completeness [42] and weighted homogeneity, together with the existence of the solutions of continuous time-delay stochastic systems, we prove under Lyapunov-like conditions that : i) almost sure asymptotic stabilizability by nonsmooth homogeneous state feedback and dominated homogeneity of degree zero imply input delay tolerability of stochastic nonlinear systems, and ii) almost sure asymptotic stabilizability by homogeneous output feedback also implies input delay tolerability of the same class of multi-input/multi-output stochastic nonlinear systems. Our results contain not only the existing IDT results for the global Lipschitz or linear growth stochastic systems as special cases, but also important applications to significant classes of stochastic control systems with unstabilizable and undetectable linearization.