Network Modeling and Control of Physical Systems , DISC Theory of Port-Hamiltonian systems Chapter 3 : Compositional modelling of distributed-parameter systems

The Hamiltonian formulation of distributed-parameter systems has been a challenging reserach area for quite some time. (A nice introduction, especially with respect to systems stemming from fluid dynamics, can be found in [26], where also a historical account is provided.) The identification of the underlying Hamiltonian structure of sets of p.d.e.’s has been instrumental in proving all sorts of results on integrability, the existence of soliton solutions, stability, reduction, etc., and in unifying existing results, see e.g. [11], [24], [18], [17], [25], [14]. Recently, there has been also a surge of interest in the design and control of nonlinear distributed-parameter systems, motivated by various applications. At the same time, it is well-known from finite-dimensional nonlinear control systems [35], [32], [6], [21], [28], [27], [34] a Hamiltonian formulation is helpful in the control design, and the same is to be expected in the distributed-parameter case. However, in extending the theory as for instance exposed in [26] to distributed-parameter control systems a fundamental difficulty arises in the treatment of boundary conditions. Indeed, the treatment of infinite-dimensional Hamiltonian systems in the literature is mostly focussed on systems with infinite spatial domain, where the variables go to zero for the spatial variables tending to infinity, or on systems with boundary