Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems

Vyacheslav Maksimov

doi:10.3934/eect.2017028

Article Contents

2017, Volume 6, Issue 4: 559-586. Doi: 10.3934/eect.2017028

This issue Previous Article Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study Next Article Stability and instability of solutions to the drift-diffusion system

Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems

Vyacheslav Maksimov^1,2,

1.
Krasovskii Institute of Mathematics and Mechanics of UB RAS, Ekaterinburg 620990, Russia
2.
Ural Federal University, Ekaterinburg 620002, Russia

Received: February 28, 2017

Revised: April 30, 2017

Published: November 2017

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

A game control problems of the Schlögl and FitzHugh-Nagumo equations are considered. The problems are investigated both from the viewpoint of the first player (the partner) and of the second player (the opponent). For both players, their own procedures for forming feedback controls are specified.

Keywords:

Mathematics Subject Classification: Primary: 49J35, 91A245.

Citation:

Full Text(HTML)

Figure 1.

Download: Full-size image PowerPoint slide

Figure 2.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, USA, 1993.
[2]	T. Bretten and K. Kunisch, Riccati-based feedback control of the monodomian equations with the Fitzhugh-Nagumo model, SIAM J. Control and Optimization, 52 (2014), 4057-4081. doi: 10.1137/140964552.
[3]	R. Buchholz, H. Engel, E. Kanimann and F. Tröltzsch, On the optimal control of the Schlögl-model, Computatiomnal Optimization and Application, 56 (2013), 153-185. doi: 10.1007/s10589-013-9550-y.
[4]	E. Casas, C. Ryll and F. Tröltzsch, Sparse optimal control of the Schlögl and FitzHugh-Nagumo systems, Computational Methods in Applied Mathematics, 13 (2013), 415-442. doi: 10.1515/cmam-2013-0016.
[5]	H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, 1999. doi: 10.1017/CBO9780511574795.
[6]	A. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, AMS, Providens, Rhode Island, 2000.
[7]	N. Krasovskii and A. Subbotin, Game-Theoretical Control Problems, Springer, Berlin, 1988. doi: 10.1007/978-1-4612-3716-7.
[8]	I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part Ⅰ: Abstract Parabolic Systems, Cambridge University Press, 2000.
[9]	I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part Ⅱ: Abstract Hyperbolic-lice Systems over a Finite Time Horizon, Cambridge University Press, 2000. doi: 10.1017/CBO9780511574801.002.
[10]	V. Maksimov, Feedback minimax control for parabolic variational inequality, C.R.Acad.Sci., Paris, 328 (2000), 105-108. doi: 10.1016/S1287-4620(00)88424-0.
[11]	V. Maksimov, Yu. S. Osipov and L. Pandolfi, The robust boundary control: The case of Dirichlet boundary conditions, Dokl. Akad. Nauk, 374 (2000), 310-312.
[12]	V. Maksimov, On reconstruction of bundary controls in a parabolic equation, Advances in Differential Equations, 14 (2009), 1193-1211.
[13]	V. Maksimov, Game control problem for a phase field equation, Journal of Optimization Theory and Applocations, 170 (2016), 294-307. doi: 10.1007/s10957-015-0721-0.
[14]	C. McMillan and R. Triggiani, Min-max game theory and algebraic Riccati equations for boundary control problems with analytic semigroups: Part Ⅰ: The stable case, Lecture Notes in Pure and Applied Mathematics, 152 (1994), 757-780.
[15]	B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, Applicable Analysis, 90 (2011), 1075-1109. doi: 10.1080/00036811003735840.
[16]	B. S. Mordukhovich, Suboptimal minimax design of constrained parabolic systems with mixed boundary control, Applied Mathematics and Computations, 204 (2008), 580-588. doi: 10.1016/j.amc.2008.05.036.
[17]	P. Nestler, E. Scholl and F. Tröltzsch, Optimization of nonlocal time-delayed feedback controllers, Computational Optimization and Application, 64 (2016), 265-294. doi: 10.1007/s10589-015-9809-6.
[18]	Yu. S. Osipov, On the theory of differential games for systems with distributed parameters, Doklady Mathematics, 223 (1975), 1314-1317.
[19]	K. Rull, J. Lober, S. Martems, H. Engel and F. Tr¨oltzsch, Analytical, optimal, and Sparse optimal control of traveling wave solutions to reaction-diffusion systems, Control and selforganizing nonlinear systems (eds. F. Scholl, S. H. L. Klapp and P. Hovel), Springer, (2016), 189-210.
[20]	F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, AMS, Providence, Rhode Island, 2010. doi: 10.1090/gsm/112.