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Linearized stability of partial differential equations with application to stabilization of the Kuramoto-Sivashinsky equation. (English) Zbl 1387.35045

Summary: Linearization is a useful tool for analyzing the stability of nonlinear differential equations. Unfortunately, the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations. General results giving conditions for when stability (or instability) of the linearized equation implies the same for the nonlinear equation are given here. These results are applied to stability and stabilization of the Kuramoto-Sivashinsky equation, a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibrium solutions depends on the value of a positive parameter \(\nu\). It is shown that if \(\nu>1\), then the set of constant equilibrium solutions is globally asymptotically stable. If \(\nu<1\), then the equilibria are unstable. It is also shown that stabilizing the linearized equation implies local exponential stability of the equation. Stabilization of the Kuramoto-Sivashinsky equation using a single distributed control is considered and it is described how to use a finite-dimensional approximation to construct a stabilizing controller. The results are illustrated with simulations.

MSC:

35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory
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