Stabilisation of parabolic semilinear equations. (English) Zbl 1367.93528
Summary: We design here a finite-dimensional feedback stabilizing Dirichlet boundary controller for the equilibrium solutions to parabolic equations. These results extend that ones in V.Barbu [Boundary stabilization of equilibrium solutions to parabolic e1uations, IEEE Transactions on Automatic Control, 58(9), 2416-2420 (2013)], which provide a feedback controller expressed in terms of the eigenfunctions \(\phi_j\) corresponding to the unstable eigenvalues \(\{\lambda_j\}^N_{j=1}\) of the operator corresponding to the linearised equation. In [loc. cit.], the stabilizability result is conditioned by the require of linear independence of \(\{\frac{\partial}{\partial\nu}\phi_j\}^N_{j=1}\), on the part of the boundary where control acts. In this work, we design a similar control as in Barbu (2013), and show that it assures the stability of the system. This time, we drop the requirement of linear independence and any other additional hypothesis. Some examples are provided in order to illustrate the acquired results. More exactly, boundary stabilization of the heat equation and the Fitzhugh-Nagumo equation is proved.
MSC:
| 93D15 | Stabilization of systems by feedback |
| 93C20 | Control/observation systems governed by partial differential equations |
| 35K91 | Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
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