Robust feed-back control of travelling waves in a class of reaction–diffusion distributed biological systems

Reaction–Diffusion (RD) mechanisms can describe many biological phenomena such as neuron firing in the brain, the heartbeat, cellular organization activities or even biological disorders such as fibrillation. The FitzHugh–Nagumo (FHN) model is a particular case of RD systems. It is able to capture the key features of many biological processes and since it is relatively simple it has been widely employed during recent years. Some examples of its predictive capabilities include the representation of the normal behavior of some physiological phenomena, related to a travelling plane wave, as well as biological disorders associated with spiral or irregular fronts. The objective of this work is to design a control law that is able to stabilize complex behaviors (travelling plane wave) in biological systems using the FHN model as a case study. Since, in biological systems there usually exists a lack of detailed information on the system structure, our control law will be designed to be robust, i.e., it must be able to reach the predefined reference regardless the presence of structural uncertainties. To this purpose, we will extend some classical results on the finite-dimensional robust control theory to RD systems by means of order reduction techniques, in particular the Proper Orthogonal Decomposition method.