Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms
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We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.
KeywordsSystem of balance laws Shallow water equations Telegraph equation Finite-time stability Dynamical boundary conditions Exponential stability Decay rate
Mathematics Subject Classification35L50 35L60 76B75 93D15
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