Hamiltonian formulation of distributed-parameter systems with boundary energy flow

The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories

“…As we discuss below, we expect that our approach will be related to the Stokes-Dirac structures of van der Schaft and Maschke (Ref. 34) used in boundary control theory.…”

“…The latter are in turn related to the concept of Stokes-Dirac structures (Ref. 34). The precise link between Stokes-Dirac structures and infinite-dimensional Dirac structures involves Poisson reduction and was established in Ref.…”

“…We therefore denote the restriction of this bundle to C by C . If the annihilator of is given by (34), then an easy coordinate computation shows that…”

Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain

“…In this case the variable w(t, z) denotes the position with respect to equilibrium of the string at position z and the constant c denotes the propagation speed (celerity) c = T ρ with T Young's modulus and ρ the mass density. However, the physical formulation of the model of the string is given as a set of coupled conservation laws and induces the following choice of the state variables: the momentum x 1 (t, z) = ρ ∂w ∂t (t, z) and the elastic strain x 2 (t, z) = ∂w ∂z (t, z) [18,22,25]. It may be noticed that this formulation is more general than (2.1) as it remains valid in the case of a non-uniform string when Young's modulus T and the mass density ρ depend on z.…”

Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

“…In the last decade an approach based on the extension of the Hamiltonian formulation to open distributed parameter systems (van der Schaft & Maschke, 2002) has been developed for modeling and control. It has been shown that distributed port-Hamiltonian systems encompass a large class of physical systems, including mechanical, electrical, electro-mechanical, hydraulic and chemical systems to mention some.…”