**1.3 History of topic and relevant research**

Port-Hamiltonian systems are a framework for passivity-based controls. *Passivity* (Van der Schaft, 2000) is a property by which the energy supplied from the outside of systems through input/output variables can be expressed as a function of the stored energy. The storage function is equivalent to a Hamiltonian in dynamical systems. The collocated input/output variable pairs, called *port variables*, are defined systematically in terms of port-Hamiltonian systems, and they are used as controls and for making observations. Passivity-based controls consist of *shaping Hamiltonians* and *damping assignments*. The Hamiltonians of these systems can be changed by "connecting" them to other port-Hamiltonian systems by means of the port variables. The Hamiltonian of controlled systems is equal to the sum of those of the original system and controllers. Thus, if we can design such a changed Hamiltonian beforehand, the connections give the Hamiltonian of the original system "shaping". Such connected port-Hamiltonian systems with a shaped Hamiltonian can be stabilized to the minimum of the storage function by adding dissipating elements to the port variables.

The energy preserving properties of port-Hamiltonian systems can be described in terms of a *Dirac structure* (Van der Schaft, 2000; Courant, 1990), which is the generalization of symplectic and Poisson structures (Arnold, 1989). Dirac structures enable us to model complex systems as port-Hamiltonian representations, e.g., distributed parameter systems with nonlinearity (Van der Schaft and Maschke, 2002), systems with higher order derivatives (Le Gorrec et al., 2005; Nishida, 2004), thermodynamical systems (Eberard et al., 2007), discretized distributed systems (Golo et al., 2004; Voss and Scherpen, 2011) and their coupled systems. This chapter mainly uses the port-Hamiltonian representation of PDEs for boundary controls based on passivity, i.e., the DPH system. The boundary integrability of DPH systems is derived from a *Stokes-Dirac structure*(Van der Schaft and Maschke, 2002), which is an extended Dirac structure in the sense of Stokes theorem. Because of this boundary integrability, the change in the internal energy of DPH systems is equal to the energy supplied through port variables defined on the boundary of the system domain. Hence, passivity-based controls for distributed parameter systems can be considered to be boundary energy controls.
