| Abstract: |
| This is the second of two talks on well-posedness and boundary stabilization for the complex Ginzburg--Landau (CGL) equation on a finite interval. In the prequel talk (D. Matzavinos, Part I), the open-loop Dirichlet--Neumann initial-boundary value problem is shown to be locally (and, in suitable regimes, globally) well posed at low regularity via sharp spatiotemporal estimates obtained from a unified-transform (Fokas method) representation formula.
Building on those estimates, I present a rapid boundary feedback stabilization (chaos suppression) scheme using Neumann actuation: a nonlocal controller depending only on finitely many Fourier modes, based on a slow--fast decomposition into a finite-dimensional ``slow'' part and a rapidly decaying tail. We quantify (i) how many modes guarantee exponential stabilization at a prescribed decay rate, and (ii) the minimal number ensuring stabilization at some exponential rate. Closed-loop well-posedness follows locally by combining Part I spatiotemporal bounds with the feedback structure, while global energy solutions are obtained from the stabilization estimates. Uniqueness is derived by reducing to a homogeneous-boundary problem through a bounded, invertible Volterra-type integral transform on Sobolev spaces. Numerical simulations illustrating chaos suppression will also be discussed.
This research was supported by TUBITAK 1001 Grant 122F084. |
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