A Criterion for the Viability of Stochastic Semilinear Control Systems via the Quasi-Tangency Condition

TL;DR

This paper establishes a criterion for the viability of stochastic semilinear control systems using stochastic quasi-tangency, providing explicit conditions and generalizing deterministic results to stochastic settings.

Contribution

It introduces a stochastic quasi-tangency criterion for viability, linking approximate and actual viability, and extends deterministic viability results to stochastic systems.

Findings

Approximate viability equals viability for linear systems

Nagumo's stochastic theorem is established

Explicit criteria for viability of smooth sets and the unit ball are provided

Abstract

In this paper we study a criterion for the viability of stochastic semilinear control systems on a real, separable Hilbert space. The necessary and sufficient conditions are given using the notion of stochastic quasi-tangency. As a consequence, we prove that approximate viability and the viability property coincide for stochastic linear control systems. We obtain Nagumo's stochastic theorem and we present a method allowing to provide explicit criteria for the viability of smooth sets. We analyze the conditions characterizing the viability of the unit ball. The paper generalizes recent results from the deterministic framework.