Stabilization of inhomogeneous patterns in a diffusion–reaction system under structural and parametric uncertainties

Abstract

Many phenomena such as neuron firing in the brain, the travelling waves which produce the heartbeat, arrythmia and fibrillation in the heart, catalytic reactions or cellular organization activities, among others, can be described by a unifying paradigm based on a class of nonlinear reaction-diffusion mechanisms. The FitzHughNagumo (FHN) model is a simplified version of such class which is known to capture most of the qualitative dynamic features found in the spatiotemporal signals. In this paper, we take advantage of the dissipative nature of diffusion-reaction systems and results in finite dimensional nonlinear control theory to develop a class of nonlinear feed-back controllers which is able to ensure stabilization of moving fronts for the FHN system, despite structural or parametric uncertainty. In the context of heart or neuron activity, this class of control laws is expected to prevent cardiac or neurological disorders connected with spatiotemporal wave disruptions. In the same way, biochemical or cellular organization related with certain functional aspects of life could also be influenced or controlled by the same feed-back logic. The stability and robustness properties of the controller will be proved theoretically and illustrated on simulation experiments.

Robust feed-back control of distributed chemical reaction systems

“…Following the methodology employed in Vilas et al (2006), we define the subsets of natural numbers N a and N b , with N a being a finite subset of arbitrary numbers and N b = N \ N a its complement. These sub-sets will allow us to partition the eigenset (S, Λ,…”

“…Christofides and coworkers -see, for example (Christofides and Daoutidis, 1996;Shi et al, 2006)-employed this approach to derive stabilizing controllers based on feed-back linearization and applied it to chemical systems such as tubular reactors or particulate processes, among others. This methodology has been widely employed by many authors in the context of control of chemical reactors -see (Hoo and Zheng, 2001;Alonso et al, 2004b) and references therein-or regarding biological systems (Smagina et al, 2002;Vilas et al, 2006).…”

“…This formalism was applied in Alonso et al (2004a) to explore connections between physics of dissipative systems and nonlinear robust control. The results of this paper were extended in Vilas et al (2006) to the robust stabilization of complex behaviors such as limit cycles in reaction-diffusion (RD) systems.…”

“…To that purpose, we extend the methodology developed in Alonso et al (2004a) and Vilas et al (2006) to RDC systems. In this regard, the possibility of using the Kirchhoff transform is explored and the stability properties of the control law are proved by taking advantage of the structure of the transformed system.…”

Neuston. Experimento 1. Explorando a interface. Uma experiência de diálogo entre ciência e arte no âmbito marinho

“…A partir do trabalho de Carmen G. Sotelo sobre a identificação molecular e a filogenia de tunídeos (Quinteiro et al 1998), Xavier Queipo pesquisou nas linguagens da ciência e da literatura, propondo um exercício de interpretação de um poema em prosa (Árbores de Atuns, 2007), baseando-se em notas científicas. Álvarez Cáccamo comparou os sistemas de reação-difusão, descritos no trabalho do investigador Antonio A. Alonso (Vilas et al 2006), com a génese de um poema, cujo modelo ou equação subjacente desconhecia e cuja norma, assinalou, a interrogação. O escultor Miguel Vázquez refletiu na sua obra (Víricos, 2003) a capacidade dos vírus, sobre os quais investigava Antonio Figueras (Novoa et al 2006), para exercer como vetores de genes entre espécies, que o artista interpretou nas suas esculturas como diferentes materiais "poluentes" do material cerâmico.…”