EXPONENTIAL STABILITY OF SAINT-VENANT EQUATIONS IN SUPERCRITICAL FLOW

In this paper, the stability of a class of $2\times 2$ hyperbolic systems with the same sign propagation speeds is investigated under proportional feedback control. Semigroup theory is used to verify the well-posedness of the system. The exponential stability of the closed-loop system is analyzed by constructing a strictly weighted Lyapunov function. In addition, the characteristic equation and eigenfunctions of the system are deduced. For the special case where $\gamma$ is equal to 0, by using Schur-Cohn criterion, the sufficient and necessary stability conditions, which make the eigenvalues have the negative real part, is obtained.