Constructive method for boundary control of stochastic 1D parabolic PDEs

Recently, qualitative methods for finite-dimensional boundary state-feedback control were introduced for stochastic 1D parabolic PDEs. In this paper, we present constructive and efficient design conditions for state-feedback control of stochastic 1D heat equations driven by a nonlinear multiplicative noise. We consider the Neumann actuation and apply modal decomposition with either trigonometric or polynomial dynamic extension. The controller design employs a finite number of comparatively unstable modes. We provide mean-square L²stability analysis of the full-order closed-loop system, where we employ Itô's formula, leading to linear matrix inequality (LMI) conditions for finding the controller gain and as large as possible noise intensity for the mean-square stabilizability. We prove that the LMIs are always feasible for small enough noise intensity. We further show that in the case of linear multiplicative noise, the system is stabilizable for noise intensities that guarantee the stabilizability of the stochastic finite-dimensional part of the closed-loop system. Numerical simulations illustrate the efficiency of our method.