In this work the stability analysis of a weakly hyperbolic system i.e. with non-diagonalizable principal part and nonuniform coefficients is addressed. We give a sufficient condition of exponential stability in H^1 x L^2 norm and express the optimal convergence rate that can be obtained. The used approach and results, based on Lyapunov analysis of cascade systems are inspired by (see references). Two cases will be treated independently: with and without the source terms. A numerical scheme is designed for several challenging test cases to illustrate the performance and robustness of the Lyapunov stability analysis.