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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.

MSC:

93B05 Controllability
93B27 Geometric methods
35B35 Stability in context of PDEs
35L65 Hyperbolic conservation laws
35L70 Second-order nonlinear hyperbolic equations
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