Optimal control for a mathematical model of nerve impulse spreading

The article concerns the matter of existence of optimal control for the mathematical model set forward by R. Fitzhugh and J. M. Nagumo for modelling of nerve impulse spreading. The model belongs to the group of diffusion-reaction models simulating a wide range of processes such as chemical reactions with diffusion and nerve impulse spreading. In case, that there is an asymptotical stability of the studied model, and under an assumption that the rate of variation of one component is greatly higher than the other one, the said model could be reduced to a problem of optimal control of a Sobolev type semi-linear equation with Showalter–Sidorov initial condition. The article contents a demonstration of the only weak generalized solution for the model under discussion with Showalter–Sidorov initial condition and optimal control existence.