Sampled-Data Control of 2-D Kuramoto–Sivashinsky Equation

This article addresses sampled-data control of 2-D Kuramoto–Sivashinsky equation over a rectangular domain $\Omega$. We suggest to divide the 2-D rectangular $\Omega$ into $N$ subdomains, where sensors provide spatially averaged or point state measurements to be transmitted through communication network to the controller. Note that, differently from 2-D heat equation, here, we manage with sampled-data control under point measurements. We design a regionally stabilizing sampled-data controller applied through distributed in space characteristic functions. Sufficient conditions ensuring regional stability of the closed-loop system are established in terms of linear matrix inequalities (LMIs). By solving these LMIs, we find an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. A numerical example demonstrates the efficiency of the results.