Start control and final observation problem for the Fitz Hugh–Nagumo system for the Dirichlet–Showalter–Sidorov condition

The paper discusses the start control and the final observation of solutions of the Dirichlet–Showalter–Sidorov problem for the degenerate Fitz Hugh–Nagumo system of equations. This system refers to the class of reaction-diffusion equations and describes the propagation of waves in active biological media, such as the heart muscle, or brain tissue. On the one hand, the Fitz Hugh–Nagumo system of equations is the development of more familiar model of Kolmogorov–Petrovsky–Piskunov, and on the other hand, the simplification of the Hodgins–Huxley model. When constructing a mathematical model taking into account that the speed of one desired function of the Fitz Hugh–Nagumo system of equations significantly exceeds the speed of the other, it has been suggested to investigate the degenerate case. The studied task of the start control and final observation simulates the situation when, after a short-time control action, the expected result for a certain period of time, i.e. at the initial moment of time sends a high-power pulse is sent to a nerve system and the required state of the system is expected after a certain set time. On the basis of Galerkin's methods and compactness, the existence theorem of the problem of starting control and final observation in a weak generalized case is proved.