Optimal control over solutions of a multicomponent model of reaction-diffusion in a tubular reactor

This article studies a mathematical model of reaction-diffusion in a tubular reactor based on degenerate equations of reaction-diffusion type defined on a geometric graph. It is precisely the degenerate case that is studied, since when building the mathematical model it is taken into account that the speed of one sought function is significantly higher than the speed of the other. This model belongs to a wide class of semilinear Sobolev-type equations. We give sufficient conditions for the simplicity of the phase manifold of the abstract Sobolev-type equation in the case of $s$-monotone and $p$-coercive operator; we prove the existence and uniqueness of a solution to the Showalter–Sidorov problem in the weak generalized sense, and the existence of optimal control over weak generalized solutions to this problem. On the basis of the abstract theory, we find sufficient conditions for the existence of optimal control for a mathematical model of neural signal transmission.