Boundary control for stochastic nonlinear heat equation with and without time delay

In this paper, we consider boundary stabilization for stochastic nonlinear heat equations with and without time delay. The backstepping approach is adopted in the investigation. We first design a boundary controller for stochastic nonlinear heat equation without time delay and derive a criterion under which the controlled system achieves mean square exponential stability, utilizing the Lyapunov functional method, stochastic analysis technique and Wirtinger’s inequality. Secondly, we propose a stabilizing boundary controller for stochastic nonlinear heat equation with time delay and show its closed-loop system to be mean square exponentially stable. Some numerical simulations are presented to validate the results.