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Boundary Predictive Control of Kuramoto-Sivashinsky Equation

Stevan Dubljevic, Cardiology Department, Cardiovascular Research Laboratories David, Geffen School of Medicine, UCLA, 675 Charles E. Young Drive South Box 951760 Room#3645 MRL, Los Angeles, CA 90095-1760

This paper focuses on boundary predictive control of higher order linear dissipative PDEs, and in particular on predictive control of a dissipative fourth-order equation given by the Kuramoto-Sivashinsky equation, which describes a variety of physicochemical phenomena like long-wave motions of the liquid thin film over a vertical plate or the evolution of laminar fronts [1]. The problem of stabilization of Kuramoto-Sivashinsky equation has been addressed within the output feedback formulation by Christofides and Armaou [2] and within optimal control setting by Lee and Tran [3]. However, an important aspect of optimal control synthesis in which actuation is applied at the boundary has not be addressed.

In this work, modal model predictive control synthesis for control of Kuramoto-Sivashinsky equation has been developed. The evolution of a linear dissipative PDE is initially given by an abstract evolution equation in an appropriate Sobolev space. Modal decomposition technique is used to decompose the infinite dimensional system into an interconnection of a finite-dimensional (slow) subsystem with an infinite-dimensional (fast) subsystem. The predictive controller synthesis is then formulated in a way that the construction of the cost functional accounts only for the weighted evolution of slow (finite-dimensional) states, while in the state constraints a high-order (finite-dimensional) approximation of fast states is utilized. As an example of the proposed controller synthesis methodology, the optimal stabilization of spatially-uniform unstable steady state of Kuramoto-Sivashinsky equation subject to variety of boundary conditions is considered. Simulation results demonstrate successful application of the proposed predictive control technique within infinite-dimensional closed-loop system setting.

[1] L. H. Chen and H. C. Chang, "Nonlinear waves on liquid film surface-II. Bifurcation analysis of the long-wave equation," Chemical Engineering Science, vol. 41, pp. 2477-

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2] A. Armaou and P. D. Christofides, "Feedback control of the Kuramoto-Sivashinsky equation," Physica D., vol. 137, pp. 49-61, 1999.

[3] P. D. Christofides and A. Armaou, "Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control," Syst. & Contr. Lett., vol. 39, pp. 283-294, 2000.

[4] C. H. Lee and H. T. Tran, "Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation," Journal of Computational and Applied Mathematics, vol. 173, pp. 1-19, 2005.