A single actuator vs. multi-actuator design of an input-feedback control for the generalized Kuramoto–Sivashinsky equation

• Published in: Nonlinear dynamics Vol. 111; no. 20; pp. 19371 - 19385
• Main Authors: Al Jamal, R., Smaoui, N.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2023; Springer Nature B.V
• Subjects: Actuator design; Automotive Engineering; Boundary conditions; Classical Mechanics; Control; Control systems design; Controllers; Dynamical Systems; Engineering; Feedback control; Mechanical Engineering; Nonlinear control; Nonlinear differential equations; Nonlinearity; Original Paper; Partial differential equations; Vibration; Input-feedback control; Generalized Kuramoto–Sivashinsky equation; Spillover; Approximation; Partial differential equations; Stabilization; Fréchet differentiabilty; Stability of nonlinear systems; Exponential stability; Linearization
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-023-08861-5

The paper is devoted to the input-feedback control design for a class of reaction-diffusion systems governed by the generalized Kuramoto–Sivashinsky (GKS) equation (a nonlinear partial differential equation that is first order in time, fourth order in space, and with a high-order nonlinearity subject to periodic boundary conditions). First, we show that the stability and instability of the equilibria depend on the value of the parameters of the GKS equation. Second, we demonstrate that stabilizing the linearized GKS equation implies local exponential stability of the nonlinear controlled system without having a spillover effect. That is, the use of a controller design based on a finite dimensional approximation will locally stabilize the solution of the infinite dimensional system of the GKS equation. Then, a single actuator and multi-actuator bounded input-feedback controllers are designed to control the GKS equation to any desired constant solution. Finally, numerical simulations that illustrate the proposed approach are presented where it is shown that the use of a single actuator input-feedback design controller is successful to drive the state of the system to any desired state, but with a slower convergence rate as compared to the multi-actuator design controller.