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Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the one dimensional (1D) heat equation, under the assumption that one of the observation or control operators is bounded. This article suggests a constructive method for such controllers for 1D parabolic partial differential equations (PDEs) with both observation and control operators being unbounded. We consider the Kuramoto&#x2013;Sivashinsky equation under either boundary or in-domain point measurement and boundary actuation in the presence of disturbances in the PDE and measurement. We employ a modal decomposition approach via dynamic extension, using eigenfunctions of a Sturm&#x2013;Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> may be larger. We suggest a direct Lyapunov approach to the full-order closed-loop system, which results in a linear matrix inequality (LMI), for input-to-state stabilization (ISS) and guaranteed <inline-formula><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula>-gain, whose elements and dimension depend on <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>. The value of <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and the decay rate are obtained from the LMI. We prove that the LMI is always feasible provided <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and the <inline-formula><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> or ISS gains are large enough, thereby obtaining guarantees for our approach. Moreover, for the case of stabilization, we show that feasibility of the LMI for some <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> implies its feasibility for <inline-formula><tex-math notation="LaTeX">$N+1$</tex-math></inline-formula>. Numerical examples demonstrate the efficiency of the method.

ieee transactions on automatic control

Finite-Dimensional Boundary Control of the Linear Kuramoto-Sivashinsky Equation Under Point Measurement With Guaranteed <inline-formula><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula>-Gain

Finite-Dimensional Boundary Control of the Linear Kuramoto-Sivashinsky Equation Under Point Measurement With Guaranteed $L^2$-Gain