dc.contributor.author Al Jamal, Rasha dc.contributor.author Morris, Kirsten dc.date.accessioned 2018-01-22 15:53:31 (GMT) dc.date.available 2018-01-22 15:53:31 (GMT) dc.date.issued 2018-01-05 dc.identifier.uri https://doi.org/10.1137/140993417 dc.identifier.uri http://hdl.handle.net/10012/12911 dc.description This is a final draft of a work, prior to publisher editing and production, that appears in Siam J. Control Optim. Vol. 56, No 1, pp 120-147. http://dx.doi.org/10.1137/140993417. en dc.description.abstract 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. en dc.description.sponsorship Natural Sciences and Engineering Research Council of Canada (NSERC) en dc.language.iso en en dc.publisher Society for Industrial and Applied Mathematics en dc.subject Stability en dc.subject Control en dc.subject Stabilization en dc.subject Kuramoto-Sivashinsky en dc.subject Partial differential equations en dc.subject Linearized stability en dc.title Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation en dc.type Article en dcterms.bibliographicCitation Jamal, R. al, & Morris, K. (2018). Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation. SIAM Journal on Control and Optimization. https://doi.org/10.1137/140993417 en uws.contributor.affiliation1 Faculty of Mathematics en uws.contributor.affiliation2 Applied Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Reviewed en uws.scholarLevel Faculty en
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