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

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–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–Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension
$N$
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
$L^2$
-gain, whose elements and dimension depend on
$N$
. The value of
$N$
and the decay rate are obtained from the LMI. We prove that the LMI is always feasible provided
$N$
and the
$L^2$
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
$N$
implies its feasibility for
$N+1$
. Numerical examples demonstrate the efficiency of the method.