Adaptive Boundary Control of the Kuramoto-Sivashinsky Equation Under Intermittent Sensing
Mohamed Camil Belhadjoudja, Mohamed Maghenem, Emmanuel Witrant, Christophe Prieur
TL;DR
The paper addresses boundary stabilization of the perturbed Kuramoto-Sivashinsky equation under intermittent sensing, formulating a two-subdomain model with an interface at $x=Y$ and adaptive boundary controllers at $x=0$ and $x=1$. A Lyapunov-based, adaptive design with a cubic feedback $\kappa$ and adaptation laws for $\hat{\theta}$ yields $L^2$-ISS when the perturbation bound is known, and $L^2$-global uniform ultimate boundedness when it is not; with full-state sensing the approach guarantees convergence to arbitrarily small neighborhoods of the origin even without a bound on the perturbation. The analysis reduces to a switched system between the two sensing intervals, and the results are complemented by a numerical simulation showing bounded controls and decay of the Lyapunov functions to a disturbance-insensitive bound. This framework offers a practical strategy for boundary control of PDEs under sensing limitations and perturbations, with potential extensions to more complex sensing patterns and uncertainties.
Abstract
We study in this paper boundary stabilization, in the L2 sense, of the perturbed Kuramoto-Sivashinsky (KS) equation subject to intermittent sensing. We assume that we measure the state on a given spatial subdomain during certain time intervals, while we measure the state on the remaining spatial subdomain during the remaining time intervals. We assign a feedback law at the boundary of the spatial domain and force to zero the value of the state at the junction of the two subdomains. Throughout the study, the equation's destabilizing coefficient is assumed to be unknown and possibly space dependent but bounded. As a result, adaptive boundary controllers are designed under different assumptions on the perturbation. In particular, we guarantee input-to-state stability (ISS) when an upperbound on the perturbation's size is known. Otherwise, only global uniform ultimate boundedness (GUUB) is guaranteed. In contrast, when the state is measured at every spatial point all the time (full state measurement), convergence to an arbitrarily-small neighborhood of the origin is guaranteed, even if the perturbation's maximal size is unknown. Numerical simulations are performed to illustrate our results.