A single‐input single‐output versus multiple‐input multiple‐output feedback controllers for the Korteweg‐de Vries‐Kuramoto‐Sivashinsky equation

Abstract

SummaryThe dynamics and control of the Korteweg‐de Vries‐Kuramoto‐Sivashinsky (KdVKS) equation subject to periodic boundary conditions are considered. First, the dynamics of the KdVKS equation is analyzed using the Fourier Galerkin reduced‐order method. This reduced‐order method is used to generate a finite‐dimensional approximation of the KdVKS equation. Then, a multiple‐input multiple‐output (MIMO) and a single‐input single‐output (SISO) controllers are designed to stabilize the finite‐dimensional approximation of the linearized KdVKS equation. We show that both the MIMO and SISO controllers which are designed to stabilize the linearized system will locally stabilize the infinite‐dimensional KdVKS equation. The Frechét differentiability of the semigroup generated by the closed‐loop system is crucial to prove the local stability results of the controllers. Furthermore, we show that the condition on the number of output measurements that must equal to the total number of unstable and critically stable eigenvalues of the linearized KdVKS equation is not necessary for the local exponential stability of the KdVKS equation. Finally, numerical simulations based on the two designed controllers are presented and compared to illustrate the developed theory.