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We study global finite-dimensional observer-based stabilization of asemilinear 1D heat equation with globally Lipschitz semilinearity in the statevariable. We consider Neumann actuation and point measurement. Using dynamicextension and modal decomposition we derive nonlinear ODEs for the modes of thestate. We propose a controller that is based on a nonlinear finite-dimensionalLuenberger observer. Our Lypunov $H^1$-stability analysis leads to LMIs, whichare shown to be feasible for a large enough observer dimension and small enoughLipschitz 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 leadsto a nonlinear closed-loop ODE system, coupled with nonlinearinfinite-dimensional tail ODEs. We provide LMIs for $H^1$-stability and provethat for any $r>0$, the LMIs are feasible provided $M$ and $N$ are large enoughand the Lipschitz constant is small enough. Numerical examples demonstrate theefficiency of the proposed approach.

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