Robust Stability of D-symmetrizable Hyperbolic Systems

The systems under consideration are governed by a set of first-order linear partial differential hyperbolic equations together with boundary conditions. The Lyapunov method is used to verify the stability of the initial-boundary value problem. Necessary and sufficient conditions for stability are obtained under the assumption that the matrix coefficients in the differential equations and in the boundary conditions are D-symmetrizable. The considered systems have an interesting property: Hurwitz type stability and Schur type stability occur in one system simultaneously. The stability of the conditions type system is a stability of wave propagation. The stability of the discrete type system is a stability of the boundary feedback and the boundary reflections. Necessary and sufficient conditions for the robust stability of an initial-boundary value problem are obtained for the case where the matrix coefficients belong to a convex hull of stable and D-symmetrizable matrices.