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By means of a direct and constructive method based on the theory of
semi-global
solution, the local exact boundary
observability is established for one-dimensional first order
quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the
exact boundary controllability and the exact boundary observability is then shown in the quasilinear case.
Several kinds of exact synchronizations and the generalized exact synchronization are introduced for a coupled system of 1-D wave equations with various boundary conditions and we show that these synchronizations can be realized by means of some boundary controls.
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