Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable

Francesco Cordoni; Luca Di Persio

doi:10.3934/eect.2018027

Article Contents

2018, Volume 7, Issue 4: 571-585. Doi: 10.3934/eect.2018027

This issue Previous Article Observability of wave equation with Ventcel dynamic condition Next Article Some partially observed multi-agent linear exponential quadratic stochastic differential games

Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable

Francesco Cordoni and
Luca Di Persio^,

University of Verona - Department of Computer Science, Strada le Grazie 15, 37134, Verona, Italy

* Corresponding author: Luca Di Persio
* Corresponding author: Luca Di Persio

Received: April 30, 2018

Revised: June 30, 2018

Published: September 2018

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Abstract

In the present paper we derive the existence and uniqueness of the solution for the optimal control problem governed by the stochastic FitzHugh-Nagumo equation with recovery variable. Since the drift coefficient is characterized by a cubic non-linearity, standard techniques cannot be applied, instead we exploit the Ekeland's variational principle.

Keywords:

Stochastic FitzHugh-Nagumo equation,
tochastic optimal control,
Ekeland's variational principle,
stochastic partial differential equations.

Mathematics Subject Classification: Primary: 34K35, 35R60, 49J53, 49K99, 60H15, 65K10, 93E03.

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[1]	S. Albeverio and L. Di Persio, Some stochastic dynamical models in neurobiology, Recent Developments Europena Communications in Mathematical and Theoretical Biology, 14 (2011), 44-53.
[2]	S. Albeverio, L. Di Persio and E. Mastrogiacomo, Small noise asymptotic expansions for stochastic PDE's, I. The case of a dissipative polynomially bounded non linearity, Tohoku Mathematical Journal, 63 (2011), 877-898. doi: 10.2748/tmj/1325886292.
[3]	S. Albeverio, L. Di Persio, E. Mastrogiacomo and B. Smii, A Class of Lévy driven SDEs and their explicit invariant measures, Potential Analysis, 45 (2016), 229-259. doi: 10.1007/s11118-016-9544-3.
[4]	V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics. Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.
[5]	V. Barbu, Analysis and control of nonlinear infinite-dimensional systems, Mathematics in Science and Engineering, 190. Academic Press, Inc., Boston, MA, 1993.
[6]	V. Barbu, F. Cordoni and L. Di Persio, Optimal control of stochastic FitzHugh-Nagumo equation, International Journal of Control, 34 (2016), 746-756. doi: 10.1080/00207179.2015.1096023.
[7]	V. Barbu and M. Iannelli, Optimal control of population dynamics, Journal of Optimization Theory and Applications, 102 (1999), 1-14. doi: 10.1023/A:1021865709529.
[8]	V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer Science & Business Media, 2012. doi: 10.1007/978-94-007-2247-7.
[9]	V. Barbu and M. Rckner, On a random scaled porous media equation, Journal of Differential Equations, 251 (2011), 2494-2514. doi: 10.1016/j.jde.2011.07.012.
[10]	S. Bonaccorsi, C. Marinelli and G. Ziglio, Stochastic FitzHugh-Nagumo equations on networks with impulsive noise, Electron. J. Probab, 13 (2008), 1362-1379. doi: 10.1214/EJP.v13-532.
[11]	S. Bonaccorsi and E. Mastrogiacomo, Analysis of the stochastic FitzHugh-Nagumo system, Inf. Dim. Anal. Quantum Probab. Relat. Top., 11 (2008), 427-446. doi: 10.1142/S0219025708003191.
[12]	T. Breiten and K. Kunisch, Riccati-based feedback control of the monodomain equations with the Fitzhugh-Nagumo model, SIAM Journal on Control and Optimization, 52 (2014), 4057-4081. doi: 10.1137/140964552.
[13]	E. Casas, C. Ryll and F. Trltzsch, Sparse optimal control of the Schlgl and FitzHugh-Nagumo systems, Computational Methods in Applied Mathematics, 13 (2013), 415-442. doi: 10.1515/cmam-2013-0016.
[14]	F. Cordoni and L. Di Persio, Small noise asymptotic expansion for the infinite dimensional Van der Pol oscillator, International Journal of Mathematical Models and Method in Applied Sciences, 9 (2015).
[15]	F. Cordoni and L. Di Persio, Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions, Journal of Mathematical Analysis and Applications, 451 (2017), 583-603. doi: 10.1016/j.jmaa.2017.02.008.
[16]	F. Cordoni and L. Di Persio, Gaussian estimates on networks with dynamic stochastic boundary conditions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 20 (2017), 1750001, 23pp. doi: 10.1142/S0219025717500011.
[17]	G. Da Prato, Kolmogorov Equations for Stochastic PDEs. Advanced Courses in Mathematics, CRM Barcelona. Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7909-5.
[18]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Vol. 152. Cambridge university press, 2014. doi: 10.1017/CBO9781107295513.
[19]	G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210.
[20]	G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.
[21]	L. Di Persio and G. Ziglio, Gaussian estimates on networks with applications to optimal control, Netw. Heterog. Media, 6 (2011), 279-296. doi: 10.3934/nhm.2011.6.279.
[22]	H. W. Diehl, The theory of boundary critical phenomena, International Journal of Modern Physics B, 11 (1997), 3503-3523. doi: 10.1142/S0217979297001751.
[23]	I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications, 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[24]	R. Fitz Hugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.
[25]	M. Fuhrman and C. Orrieri, Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift, SIAM Journal on Control and Optimization, 54 (2016), 341-371. doi: 10.1137/15M1012888.
[26]	M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control, Annals of Probability, 30 (2002), 1397-1465. doi: 10.1214/aop/1029867132.
[27]	A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500-544.
[28]	K. Kunisch and M. Wagner, Optimal control of the bidomain system (Ⅲ): Existence of minimizers and first-order optimality conditions, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1077-1106. doi: 10.1051/m2an/2012058.
[29]	C. Marinelli, L. Di Persio and G. Ziglio, Approximation and convergence of solutions to semilinear stochastic evolution equations with jumps, Journal of Functional Analysis, 264 (2013), 2784-2816. doi: 10.1016/j.jfa.2013.02.020.
[30]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the Institute of Radio Engineers, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.
[31]	S. Pitchaiah and A. Armaou, Output Feedback Control of the FitzHugh-Nagumo Equation Using Adaptive Model Reduction, Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, 2010. doi: 10.1109/CDC.2010.5717497.
[32]	G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stochastic Analysis and Applications, 14 (1996), 461-486. doi: 10.1080/07362999608809451.
[33]	H. C. Tuckwell, Random perturbations of the reduced Fitzhugh-Nagumo equation, Physica Scripta, 46 (1992), 481-484. doi: 10.1088/0031-8949/46/6/001.