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. BuchholzH. EngelE. 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. CasasC. 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. MaksimovYu. 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. NestlerE. 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. BuchholzH. EngelE. 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. CasasC. 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. MaksimovYu. 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. NestlerE. 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

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