Scherpen, Jacquelien M.A.

Seslija, Marko

van der Schaft, Abraham

English

Reaction-diffusion systems model the evolution of the constituentsdistributed in space under the influence of chemicalreactions and diffusion [6], [10]. These systems arisenaturally in chemistry [5], but can also be used to modeldynamical processes beyond the realm of chemistry such asbiology, ecology, geology, and physics. In this paper, byadopting the viewpoint of port-controlled Hamiltonian systems[7] we cast reaction-diffusion systems into the portHamiltonianframework. Aside from offering conceptually aclear geometric interpretation formalized by a Stokes-Diracstructure [8], a port-Hamiltonian perspective allows to treatthese dissipative systems as interconnected and thus makestheir analysis, both quantitative and qualitative, more accessiblefrom a modern dynamical systems and control theorypoint of view. This modeling approach permits us to drawimmediately some conclusions regarding passivity and stabilityof reaction-diffusion systems.It is well-known that adding diffusion to the reaction systemcan generate behaviors absent in the ode case. Thisprimarily pertains to the problem of diffusion-driven instabilitywhich constitutes the basis of Turing’s mechanismfor pattern formation [11], [5]. Here the treatmentof reaction-diffusion systems as dissipative distributed portHamiltoniansystems could prove to be instrumental in supplyof the results on absorbing sets, the existence of the maximalattractor and stability analysis.Furthermore, by adopting a discrete differential geometrybasedapproach [9] and discretizing the reaction-diffusionsystem in port-Hamiltonian form, apart from preserving ageometric structure, a compartmental model analogous tothe standard one [1], [2] is obtaine

Proceedings - University of Groningen

Putting reaction-diffusion systems into port-Hamiltonian framework

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Putting reaction-diffusion systems into port-Hamiltonian framework

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Proceedings - University of Groningen