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Finite-dimensional observer-based controller design for PDEs is a challengingproblem. Recently, such controllers were introduced for the 1D heat equation,under the assumption that one of the observation or control operators isbounded. This paper suggests a constructive method for such controllers for 1Dparabolic PDEs with both (observation and control) operators being unbounded.We consider the Kuramoto-Sivashinsky equation (KSE) under either boundary orin-domain point measurement and boundary actuation. We employ a modaldecomposition approach via dynamic extension, using eigenfunctions of aSturm-Liouville operator. The controller dimension is defined by the number ofunstable modes, whereas the observer dimension $N$ may be larger than thisnumber. We suggest a direct Lyapunov approach to the full-order closed-loopsystem, which results in an LMI whose elements and dimension depend on $N$. Thevalue of $N$ and the decay rate are obtained from the LMI. We extend ourapproach to internal stabilization with guaranteed $L^2$-gain andinput-to-state stabilization. We prove two crucial properties of the derivedLMIs. First, We prove that the LMIs are always feasible provided $N$ and the$L^2$ or ISS gains are large enough, thereby obtaining guarantees for ourapproach. Moreover, for the case of stabilization, we show that feasibility ofthe LMI for some $N$ implies its feasibility for $N+1$ (i.e., enlarging $N$ inthe LMI cannot deteriorate the resulting decay rate of the closed-loop system).Numerical examples demonstrate the efficiency of the method.

Finite-dimensional boundary control of the linear Kuramoto-Sivashinsky equation under point measurement with guaranteed $L^2$-gain