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Putting reaction-diffusion systems into port-Hamiltonian framework

Seslija, M., van der Schaft, A. & Scherpen, J. M. A., 30-Mar-2010, p. 19. 1 p.

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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 obtained
Original languageEnglish
Pages19
Number of pages1
Publication statusPublished - 30-Mar-2010
Event29th Benelux Meeting on Systems and Control - Heeze, Netherlands
Duration: 30-Mar-20101-Apr-2010

Conference

Conference29th Benelux Meeting on Systems and Control
CountryNetherlands
CityHeeze
Period30/03/201001/04/2010

Event

29th Benelux Meeting on Systems and Control

30/03/201001/04/2010

Heeze, Netherlands

Event: Conference

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