In this paper we study the global exponential stability in the $L^{ 2}$ norm of semilinear $1$-$d$ hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the $L^{ 2}$ norm with respect to both interior and boundary disturbances.
Categories: math.AP math.OC