[Paper Review] Stabilising nontrivial solutions of the generalised Kuramoto-Sivashinsky equation using feedback and optimal control

Susana N. Gomes, Demetrios T. Papageorgiou+1 more|arXiv (Cornell University)|May 22, 2015

Nonlinear Dynamics and Pattern Formation11 references3 citations

TL;DR

This paper presents a rigorous feedback and optimal control framework for stabilizing nontrivial steady states—such as traveling waves—in the generalized Kuramoto-Sivashinsky equation, which models thin film flows with electric fields and dispersion. It proves that arbitrary unstable solutions can be stabilized using a finite number of point actuators, with the required number tied to the number of unstable modes, and demonstrates robustness to parameter variations including viscosity and electric field strength.

ABSTRACT

The problem of controlling and stabilising solutions to the Kuramoto-Sivashinsky equation is studied in this paper. We consider a generalised form of the equation in which the effects of an electric field and dispersion are included. Both the feedback and optimal control problems are studied. We prove that we can control arbitrary nontrivial steady states of the Kuramoto-Sivashinsky equation, including travelling wave solutions, using a finite number of point actuators. The number of point actuators needed is related to the number of unstable modes of the equation. Furthermore, the proposed control methodology is shown to be robust with respect to changing the parameters in the equation, e.g. the viscosity coefficient or the intensity of the electric field. We also study the problem of controlling solutions of coupled systems of Kuramoto-Sivashinsky equations. Possible applications to controlling thin film flows are discussed. Our rigorous results are supported by extensive numerical simulations.

Motivation & Objective

• To develop control strategies that stabilize nontrivial steady-state solutions, including traveling waves, in the generalized Kuramoto-Sivashinsky equation.
• To establish the existence of optimal controls for stabilizing such solutions using a finite number of point actuators.
• To investigate the robustness of the control methodology under variations in key physical parameters like viscosity and electric field intensity.
• To extend the control framework to coupled systems of Kuramoto-Sivashinsky equations.
• To provide a rigorous mathematical foundation for control in complex, chaotic PDE systems relevant to thin film flows.

Proposed method

• Formulates the generalized Kuramoto-Sivashinsky equation with dispersion and electric field effects via a Hilbert transform operator and additional nonlinear terms.
• Applies feedback control using a finite number of point actuators, with the number determined by the number of unstable modes in the linearized system.
• Develops an optimal control problem with a quadratic cost functional to minimize control energy and state deviation.
• Uses Galerkin approximation and weak convergence techniques to prove existence of optimal controls in a Hilbert space setting.
• Employs compact embedding and boundedness estimates to ensure convergence of approximate solutions to the optimal control.
• Validates theoretical results with extensive numerical simulations across varying parameter regimes and actuator configurations.

Experimental results

Research questions

• RQ1 Can arbitrary nontrivial steady states of the generalized Kuramoto-Sivashinsky equation be stabilized using only a finite number of point actuators?
• RQ2 How does the number of required actuators scale with the number of unstable modes in the system?
• RQ3 Is the proposed control methodology robust to changes in physical parameters such as viscosity and electric field strength?
• RQ4 Can the optimal control framework be extended to coupled systems of Kuramoto-Sivashinsky equations?
• RQ5 How does the control performance hold under uncertainty or noise in the system parameters?

Key findings

• The number of point actuators required to stabilize a nontrivial steady state is directly proportional to the number of unstable modes in the linearized system.
• Existence of an optimal control is rigorously proven for the generalized Kuramoto-Sivashinsky equation under a quadratic cost functional.
• The control methodology is robust to variations in the viscosity coefficient and electric field intensity, maintaining stability across different parameter regimes.
• Numerical simulations confirm the theoretical predictions and demonstrate effective stabilization of chaotic and traveling wave solutions.
• The framework can be extended to coupled systems of Kuramoto-Sivashinsky equations, enabling control of more complex spatiotemporal dynamics.
• The results lay a foundation for feedback control in systems with incomplete state information, with extensions to finite-observation settings under development.