Physical systems such as gas networks are usually operated in a state of equilibrium and one is interested in stable systems, where small perturbations are damped over time. This talk is devoted to the design of boundary controls of systems that are described by hyperbolic balance laws. An underlying tool for the study of these problems are Lyapunov functions that yield upper bounds on the deviation from steady states in suitable norms. Furthermore, numerical approaches and limits of stabilizability are discussed.