Finite-Dimensional Observer-Based Boundary Control for a 1-D Stochastic Heat Equation

In this article, we investigate the finite-dimensional observer-based boundary control for a stochastic heat equation with nonlinear multiplicative noise and non-local sensing mea-surement. We adopt the modal decomposition which decomposes the system into two subsystems: one is unstable with finite positive eigenvalues and the other is essentially stable. We design the controller for the unstable subsystem by trigonometric extension and show that the proposed controller actually leads to the resulting closed-loop system to be well-posed and to be the mean square exponentially stable by finding a suitable Lyapunov functional. Finally, numerical simulation is included to illustrate the effectiveness of the proposed method.