Approximate Controllability of Stochastic Hemivariational Control problem in Hilbert spaces
Bholanath Kumbhakar, Deeksha, Dwijendra Narain Pandey
TL;DR
This work addresses approximate controllability for systems governed by stochastic evolution hemivariational inequalities in Hilbert spaces, formulated as $q'(t) in A q(t) + B u(t) + partial F(t,q(t)) + Sigma(t,q(t)) dW/dt$ on $I=[0,a]$ with $q(0)=x_0$. It develops a framework combining weak solutions, Clarke subdifferential calculus, and a multivalued fixed-point approach to obtain controllability results and demonstrates applicability via a stochastic heat propagation example. The main contributions include a constructive fixed-point scheme that yields controls driving the state arbitrarily close to a target under suitable hypotheses and the extension of deterministic hemivariational control theory to stochastic infinite-dimensional settings. The results provide rigorous tools for analyzing nonsmooth energy models under randomness, with potential applications in thermal control and diffusion processes.
Abstract
In this paper, we discuss the approximate controllability for control systems governed by stochastic evolution hemivariational inequalities in Hilbert spaces. The interest in studying this type of equation comes from its application in some physical models like the thermostat temperature control or the diffusion through semi-permeable walls. Firstly, we introduce the concept of weak solutions for hemivariational inequalities. Then, the controllability is formulated by utilizing stochastic analysis techniques and properties of Clarke subdifferential operators, as well as applying multivalued fixed point theorem. Finally, we conclude this article by an application in stochastic heat propagation problem.