Abstract
There is considered a system of conservation laws with non-standard boundary conditions. At a certain level of linearization, a bilinear controlled system of functional equations is associated by integrating the Riemann invariants of the system along its characteristics. For this associated system the basic theory (existence, uniqueness and smooth data dependence) is developed. Then some invariant set accounting for the positiveness of the variables with physical significance is obtained. Further, its equilibria are shown to be stable but not asymptotically stable as suggests the Stability Postulate of N. G. Četaev. Feedback asymptotic stabilization is obtained by using a suitably designed Lyapunov functional. Using the representation formulae for the solutions, all properties and results thus obtained are projected back on the boundary value problem with bilinear control at the boundaries. This shows a way to obtain stability by the first approximation for the linearized conservation laws with nonlinear boundary conditions.
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