This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition.
IntroductionPort-Hamilton systems [1] have been developed as an extended Hamiltonian system, and it is one of most essential system representations for controlling complex physical systems. Port-Hamilton systems are defined by particular variable pairs of an input and an output. The pair is called port, and the variables are called effort and flow. The product of an effort and the corresponding flow has the physical dimension of power, i.e., the time derivative of energy. Thus, the sum of the products of all pairs is equivalent to the time derivative of the total energy of a given system, i.e., Hamiltonian. Indeed, ports can be derived from derivatives of a Hamiltonian. By using port-Hamiltonian systems, physical network systems interconnected through the ports can be described. Then, an energy balance equation can be defined on the terminal of the port-interconnections by the port variables. If the energy balance holds in any interaction with environments, this property can be used for stability analysis, because an energy of systems can be considered as a Lyapunov function. Thus, various control methods using energy flows on such a network, e.g., passivity-based controls [1,2], can be used. Port-Hamilton systems have been extended for systems governed by partial differential equations, and it is called a distributed port-Hamilton system [2, 3]. In the system, ports are defined on a boundary of a control system domain, and they are called boundary ports. Then, the balance equation can be augmented as that on the boundary. As a remarkable property of distributed port-Hamilton systems, Stokes the- *