The paper deals with initial-boundary value problems for linear non-autonomous first order hyperbolic systems whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in L 2 as well as in C 1 under small bounded perturbations. To show this for C 1 , we prove a general smoothing result implying that the solutions to the perturbed problems become eventually C 1 -smooth for any L 2 -initial data.

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We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.