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Boundary controllability for the quasilinear wave equation. (English) Zbl 1185.93018
Summary: 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 can be geometrically characterized by a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, basing on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.

93B27Geometric methods in systems theory
35B35Stability of solutions of PDE
35L65Conservation laws
35L70Nonlinear second-order hyperbolic equations
Full Text: DOI arXiv
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