Asymptotic stability of forced equilibria for distributed port-Hamiltonian systems

The main contribution of this paper is an energy shaping procedure for the stabilization of forced equilibria for linear, lossless, distributed port-Hamiltonian systems via Casimir generation. Once inputs and outputs have been properly chosen to have a well-posed boundary control system, conditions for the existence of Casimir functions in closed-loop are given, together with their relation with the controller structure. These invariants suggest how to select the controller Hamiltonian to introduce a minimum at the desired equilibrium. Such equilibrium can be made asymptotically stable via damping injection, if proper “pervasive” damping injection conditions are satisfied. The methodology is illustrated with the help of a Timoshenko beam with constant non-zero force applied at one side of the spatial domain, and full-actuation on the other one.