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

O. V. Gavrilova

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 3, pp. 12-18

Résumé

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.

Détail

Keywords: semilinear Sobolev type equations, Showalter–Sidorov problem, the start control and final observation problem, weak generalized solution, the Fitz Hugh–Nagumo system of equations.

Bibliographie

[1] R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane”, Biophysical Journal, 1:6 (1961), 445–466 | DOI

[2] J. Nagumo, S. Arimoto, S. Yoshizawa, “An active pulse transmission line simulating nerve axon”, Proceedings of the IRE, 50:10 (1962), 2061–2070 | DOI

[3] D. Wu, S. Zhu, “Stochastic resonance in Fitz Hugh-Nagumo system with time-delayed feedback”, Physics Letters A, 372:32 (2008), 5299–5304 | DOI | MR | Zbl

[4] V.I. Nekorkin, A.S. Dmitrichev, J.M. Bilbault, S. Binczak, “Polymorphic and regular localized activity structures in a two-dimensional two-component reaction diffusion lattice with complex thresholdex citation”, Physica D: Nonlinear Phenomena, 239:12 (2010), 972–987 | DOI | MR | Zbl

[5] S. Weber, M. Porto, “Multicomponent reaction-diffusion processes on complex networks”, Physical Review E, 74:4 (2006), 046108 | DOI | MR

[6] Borina M. Yu., Polezhaev A. A., “Spatial-temporal patterns in a multidimensional active medium formed due to polymodal interaction near the wave bifurcation”, Izvestiya VUZ. Applied nonlinear dynamics, 20:6 (2012), 15–24 | Zbl

[7] Bokareva T.A, Sviridyu G. A., “Whitney assemblies of phase spaces of certain semilinear equations of Sobolev type”, Mathematical Notes, 55:3 (1994), 237–242 | DOI | MR | Zbl

[8] A.B. Al'shin, M.O. Korpusov, A.G. Sveshnikov, Blow-up in nonlinear Sobolev-type equations, Walter de Gruyter GmbH and Co. KG, Berlin–N.Y., 2011, 648 pp. | DOI | MR

[9] Sviridyuk G. A., Manakova N. A., “The phase space of the Cauchy-Dirichlet problem for the Oskolkov equation of nonlinear filtration”, Russian Mathematics (Izvestiya VUZ. Matematika), 47:9 (2003), 33–38 | MR | Zbl

[10] Sagadeeva M. A., “On the problem of optimal measurement of dynamically distorted signals with allowance for the multiplicative effect”, Mathematical methods in engineering and technology, 2016, no. 2(84), 13–15 (in Russ.)

[11] Zamyshlyaeva A. A., Tsyplenkova O. N., “The Optimal Control over Solutions of the Initial-finish Value Problem for the Boussinesque-Love Equation”, Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming Computer Software, 5(264):11 (2012), 13–24 (in Russ.)

[12] Manakova N. A., Gavrilova O. V., “Optimal Control for a Mathematical Model of Nerve Impulse Spreading”, Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming Computer Software, 8:4 (2015), 120–126 | DOI | Zbl

[13] Gavrilova O. V., “The task of starting control and final observation of the Showalter-Sidorov problem for the Fitz Hugh-Nagumo model”, Proc. of the international conference “Voronezh Winter Mathematical School of S.G. Kerin-2016”, Nauchnaya kniga Publ., Voronezh, 2016, 115–118 (in Russ.)

[14] Lions J.-L., Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris, 1969, 554 pp. (in French) | MR | Zbl

[15] N.A. Manakova, E.A. Bogatyreva, “Mathematical model of the start control of electric field potential in conducting medium without dispersion considering relaxation”, 2016 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM) (Chelyabinsk, 2016), 1–5 | DOI

[16] Podval'ny S., Provotorov V. V., “Start control of parabolic systems with distributed parameters on the graph”, Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 2015, no. 3, 126–142 (in Russ.)

[17] N.A. Manakova, K.V. Vasiuchkova, “Numerical Investigation for the Start Control and Final Observation in Model Of- and I-beam Deformation”, Journal of Computational and Engineering Mathematics, 4:2 (2017), 26–40 | DOI | MR