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Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates

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Sobolev Spaces in Mathematics III

Part of the book series: International Mathematical Series ((IMAT,volume 10))

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Abstract Uniform stabilization with nonlinear boundary feedback is asserted for classes of hyperbolic and Petrowski type multidimensional partial differential equations with variable coefficients (in space), as a consequence of the continuity (boundedness) of the corresponding purely Boundary Control → Boundary observation open-loop map of dissipative character, of interest in its own right. The interior is assumed inaccessible. There are explicit hyper-bolic/Petrowski type dynamical PDE classes where such a property holds and classes where it fails. When available, it has a number of attractive and unexpected consequences. In particular, when accompanied by exact controllability of the corresponding open-loop linear model, it implies uniform stabilization with optimal decay rates—when a nonlinear function of the boundary observation closes up the loop, to generate the corresponding boundary feedback dissipative problem.

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    Irena Lasiecka & Roberto Triggiani

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    Victor Isakov

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Lasiecka, I., Triggiani, R. (2009). Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates. In: Isakov, V. (eds) Sobolev Spaces in Mathematics III. International Mathematical Series, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85652-0_5

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