Finite-dimensional control of the Kuramoto-Sivashinsky equation under point measurement and actuation

Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under assumption that at least one of the observation or control operators is bounded. This paper suggests a constructive method for such controllers in the case where both the observation and control operators are unbounded. We consider boundary control of the 1D linear Kuramoto-Sivashinsky equation with in-domain point measurement. We employ a modal decomposition approach via dynamic extension, where we use the eigenfunctions of a Sturm-Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer's dimension N may be larger than this number. We suggest a direct Lyapunov approach to the full-order closedloop system and provide LMIs for finding N and the resulting exponential decay rate. We prove that the LMIs are always feasible, provided N is large enough. A numerical example demonstrates the efficiency of the method and shows that the resulting LMIs are non-conservative.