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Boundary Controllability for the Quasilinear Wave Equation

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Abstract

We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.

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Authors and Affiliations

  1. Key Laboratory of Control and Systems, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Peng-Fei Yao

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  1. Peng-Fei Yao
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Correspondence to Peng-Fei Yao.

Additional information

Communicated by Irena Lasiecka.

This work is supported by the NNSF of China, grants no. 60225003, no. 60334040, no. 60821091, no. 60774025, and no. 10831007 and KJCX3-SYW-S01.

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Yao, PF. Boundary Controllability for the Quasilinear Wave Equation. Appl Math Optim 61, 191–233 (2010). https://doi.org/10.1007/s00245-009-9088-7

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  • DOI: https://doi.org/10.1007/s00245-009-9088-7

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