Brayton-moser formulation of infinite dimensional port-hamiltonian systems with application to boundary control

In this paper, for a class of distributed port-Hamiltonian systems defined on a one-dimensional spatial domain, an equivalent Brayton-Moser formulation is provided. The dynamic is expressed as a gradient equation with respect to a new storage function, the 'mixed-potential,' with the dimensions of power. The system is then passive with respect to a supply rate that is related to the reactive power, and that depends on the boundary port variables and on their time derivatives. This equivalent representation is the starting point for the development of boundary control laws able to shape the mixed-potential function. Differently from energy-balancing control schemes, this technique allows to deal with pervasive dissipation in the system in an effective way. The general theory is illustrated with the help of an example, the boundary stabilisation of a transmission line with internal dissipation.