Global stabilization of a 1D semilinear heat equation via modal decomposition and direct Lyapunov approach

In this paper we consider state-feedback global stabilization of a semilinear 1D heat equation with a nonlinearity exhibiting a linear growth bound. We study both non-local and boundary control via a modal decomposition approach. For both cases, we suggest a direct Lyapunov method applied to the full-order closed-loop system. The nonlinear terms are compensated by using Parseval's inequality, leading to efficient and constructive linear matrix inequality (LMI) conditions for obtaining the controller dimension and gain. For non-local control we provide sufficient conditions that guarantee global stabilization for any linear growth bound via either linear or nonlinear controller, provided the number of actuators is large enough. We prove that the nonlinear controller achieves at least the same performance as the linear one. For the case of boundary control, we employ a multi-dimensional dynamic extension, whereas in the numerical example we manage with a larger linear growth bound. The introduced direct Lyapunov approach gives tools for a variety of robust control problems for semilinear parabolic PDEs.