43 research outputs found
Reduction of Stokes-Dirac structures and gauge symmetry in port-Hamiltonian systems
Stokes-Dirac structures are infinite-dimensional Dirac structures defined in
terms of differential forms on a smooth manifold with boundary. These Dirac
structures lay down a geometric framework for the formulation of Hamiltonian
systems with a nonzero boundary energy flow. Simplicial triangulation of the
underlaying manifold leads to the so-called simplicial Dirac structures,
discrete analogues of Stokes-Dirac structures, and thus provides a natural
framework for deriving finite-dimensional port-Hamiltonian systems that emulate
their infinite-dimensional counterparts. The port-Hamiltonian systems defined
with respect to Stokes-Dirac and simplicial Dirac structures exhibit gauge and
a discrete gauge symmetry, respectively. In this paper, employing Poisson
reduction we offer a unified technique for the symmetry reduction of a
generalized canonical infinite-dimensional Dirac structure to the Poisson
structure associated with Stokes-Dirac structures and of a fine-dimensional
Dirac structure to simplicial Dirac structures. We demonstrate this Poisson
scheme on a physical example of the vibrating string
Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems
Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac
structure, capturing the topological laws of the system, are defined on
simplicial manifolds in terms of primal and dual cochains related by the
coboundary operators. These finite-dimensional Dirac structures offer a
framework for the formulation of standard input-output finite-dimensional
port-Hamiltonian systems that emulate the behavior of distributed-parameter
port-Hamiltonian systems. This paper elaborates on the matrix representations
of simplicial Dirac structures and the resulting port-Hamiltonian systems on
simplicial manifolds. Employing these representations, we consider the
existence of structural invariants and demonstrate how they pertain to the
energy shaping of port-Hamiltonian systems on simplicial manifolds
Reaction-Diffusion Systems as Complex Networks
The spatially distributed reaction networks are indispensable for the
understanding of many important phenomena concerning the development of
organisms, coordinated cell behavior, and pattern formation. The purpose of
this brief discussion paper is to point out some open problems in the theory of
PDE and compartmental ODE models of balanced reaction-diffusion networks.Comment: A discussion paper for the 1st IFAC Workshop on Control of Systems
Governed by Partial Differential Equation
Port-Hamiltonian systems on discrete manifolds
This paper offers a geometric framework for modeling port-Hamiltonian systems
on discrete manifolds. The simplicial Dirac structure, capturing the
topological laws of the system, is defined in terms of primal and dual cochains
related by the coboundary operators. This finite-dimensional Dirac structure,
as discrete analogue of the canonical Stokes-Dirac structure, allows for the
formulation of finite-dimensional port-Hamiltonian systems that emulate the
behaviour of the open distributed-parameter systems with Hamiltonian dynamics.Comment: MATHMOD 2012 - 7th Vienna International Conference on Mathematical
Modellin
Putting reaction-diffusion systems into port-Hamiltonian framework
Reaction-diffusion systems model the evolution of the constituents distributed in space under the influence of chemical reactions and diffusion [6], [10]. These systems arise naturally in chemistry [5], but can also be used to model dynamical processes beyond the realm of chemistry such as biology, ecology, geology, and physics. In this paper, by adopting the viewpoint of port-controlled Hamiltonian systems [7] we cast reaction-diffusion systems into the portHamiltonian framework. Aside from offering conceptually a clear geometric interpretation formalized by a Stokes-Dirac structure [8], a port-Hamiltonian perspective allows to treat these dissipative systems as interconnected and thus makes their analysis, both quantitative and qualitative, more accessible from a modern dynamical systems and control theory point of view. This modeling approach permits us to draw immediately some conclusions regarding passivity and stability of reaction-diffusion systems. It is well-known that adding diffusion to the reaction system can generate behaviors absent in the ode case. This primarily pertains to the problem of diffusion-driven instability which constitutes the basis of Turing’s mechanism for pattern formation [11], [5]. Here the treatment of reaction-diffusion systems as dissipative distributed portHamiltonian systems could prove to be instrumental in supply of the results on absorbing sets, the existence of the maximal attractor and stability analysis. Furthermore, by adopting a discrete differential geometrybased approach [9] and discretizing the reaction-diffusion system in port-Hamiltonian form, apart from preserving a geometric structure, a compartmental model analogous to the standard one [1], [2] is obtaine