| Abstract: |
| This is the first of two talks on the well-posedness and boundary stabilization of the initial-boundary value problem for the complex Ginzburg-Landau equation on a finite interval. This first talk focuses on the well-posedness of the open loop model. More precisely, it establishes a local well-posedness theory for the open loop model in $L^2$-based fractional Sobolev spaces in the case of Dirichlet-Neumann type inhomogeneous mixed boundary conditions. This local well-posedness result is based on linear estimates derived by using the weak solution formula obtained via the unified transform (also known as the Fokas method). The global well-posedness properties of the open loop model in the presence of inhomogeneous boundary conditions are also discussed. In a sequel talk by Turker Ozsari, the results discussed here for the open loop model will be employed for the analysis of the rapid boundary feedback stabilization problem and the design of a nonlocal controller via a finite number of Fourier modes of the state of solution. |
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