  A boundary feedback stabilisation problem of non-uniform linear hyperbolic
systems of balance laws with additive disturbance is discussed. A continuous
and a corresponding discrete Lyapunov function is defined. Using an
input-to-state-stability (ISS) $ L^2- $Lyapunov function, the decay of
solutions of linear systems of balance laws is proved. In the discrete
framework, a first-order finite volume scheme is employed. In such cases, the
decay rates can be explicitly derived. The main objective is to prove the
Lyapunov stability for the $L^2$-norm for linear hyperbolic systems of balance
laws with additive disturbance both analytically and numerically. Theoretical
results are demonstrated by using numerical computations.
