Asymptotic stabilisation of distributed port-hamiltonian systems by boundary energy-shaping control

This paper illustrates a general synthesis methodology of asymptotic stabilising, energy-based, boundary control laws, that is applicable to a large class of distributed port-Hamiltonian systems. Similarly to the finite dimensional case, the idea is to design a state feedback law able to perform the energy-shaping task, i.e. able to map the open-loop port-Hamiltonian system into a new one in the same form, but characterised by a new Hamiltonian with a unique and isolated minimum at equilibrium. Asymptotic stability is then obtained via damping injection on the boundary, and is a consequence of the La Salle's Invariance Principle in infinite dimensions. The general theory is illustrated with the help of a simple concluding example, i.e. The boundary stabilisation of a transmission line with distributed dissipation.