Adaptive Stabilization of the Kuramoto-Sivashinsky Equation Subject to Intermittent Sensing

We study in this paper the one-dimensional Kuramoto-Sivashinsky equation (KS), subject to intermittent sensing. Namely, we measure the state on a sub-interval of the spatial domain during certain intervals of time, and we measure the state on the remaining sub-interval of space during the remaining intervals of time. As a result, we assign an active control at the boundaries of the spatial domain, and we set a zero boundary condition at the junction of the two spatial subintervals. Under the assumption that the destabilizing coefficient is unknown, we design adaptive boundary controllers that guarantee global exponential stability (GES) of the trivial solution in the L
²
norm. Numerical simulations are performed to illustrate our results.