# American Institute of Mathematical Sciences

December  2017, 6(4): 559-586. doi: 10.3934/eect.2017028

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

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

Received  March 2017 Revised  May 2017 Published  September 2017

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.

Citation: Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028
##### References:
 [1] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, USA, 1993. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, 1999. doi: 10.1017/CBO9780511574795. Google Scholar [6] A. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, AMS, Providens, Rhode Island, 2000.Google Scholar [7] N. Krasovskii and A. Subbotin, Game-Theoretical Control Problems, Springer, Berlin, 1988. doi: 10.1007/978-1-4612-3716-7. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] V. Maksimov, On reconstruction of bundary controls in a parabolic equation, Advances in Differential Equations, 14 (2009), 1193-1211. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] Yu. S. Osipov, On the theory of differential games for systems with distributed parameters, Doklady Mathematics, 223 (1975), 1314-1317. Google Scholar [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. Google Scholar [20] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, AMS, Providence, Rhode Island, 2010. doi: 10.1090/gsm/112. Google Scholar

show all references

##### References:
 [1] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, USA, 1993. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, 1999. doi: 10.1017/CBO9780511574795. Google Scholar [6] A. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, AMS, Providens, Rhode Island, 2000.Google Scholar [7] N. Krasovskii and A. Subbotin, Game-Theoretical Control Problems, Springer, Berlin, 1988. doi: 10.1007/978-1-4612-3716-7. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] V. Maksimov, On reconstruction of bundary controls in a parabolic equation, Advances in Differential Equations, 14 (2009), 1193-1211. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] Yu. S. Osipov, On the theory of differential games for systems with distributed parameters, Doklady Mathematics, 223 (1975), 1314-1317. Google Scholar [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. Google Scholar [20] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, AMS, Providence, Rhode Island, 2010. doi: 10.1090/gsm/112. Google Scholar
 [1] Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027 [2] Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072 [3] John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851 [4] Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216 [5] Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203 [6] Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 [7] Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643 [8] Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1 [9] B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077 [10] Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639 [11] Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205 [12] Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150 [13] Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118 [14] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [15] Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101 [16] Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203 [17] Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441 [18] Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251 [19] Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 781-795. doi: 10.3934/dcdss.2020044 [20] Victoria Sadovskaya. Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2085-2104. doi: 10.3934/dcds.2013.33.2085

2018 Impact Factor: 1.048

## Metrics

• PDF downloads (18)
• HTML views (98)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]