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A new set of boundary conditions has been derived by rigorous methods for the shallow water equations in a limited domain. The aim of this article is to present these boundary conditions and to report on numerical simulations which have been performed using these boundary conditions. The new boundary conditions which are mildly dissipative let the waves move freely inside and outside the domain. The problems considered include a one-dimensional shallow water system with two layers of fluids and a two-dimensional inviscid shallow water system in a rectangle.
In this article we apply the optimal and
the robust control theory to the sine-Gordon equation. In our case
the control is given by the boundary conditions and we work in a finite
time horizon. We present at the beginning the optimal control problem
and we derive a necessary condition of optimality and we continue by
formulating a robust control problem for which existence and uniqueness
of solutions are derived.
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