Global finite-dimensional observer-based stabilization of a semilinear heat equation with large input delay

We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lyapunov H¹-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a constant input delay r>0. To compensate the delay, we introduce a chain of M sub-predictors that leads to a nonlinear closed-loop ODE system, coupled with nonlinear infinite-dimensional tail ODEs. We provide LMIs for H¹-stability and prove that for any r>0, the LMIs are feasible provided M and the observer dimension N are large enough and the Lipschitz constant is small enough. Numerical examples demonstrate the efficiency of the proposed approach.