December  2018, 7(4): 571-585. doi: 10.3934/eect.2018027

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

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

* Corresponding author: Luca Di Persio

Received  May 2018 Revised  July 2018 Published  September 2018

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.

Citation: 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
References:
[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.   Google Scholar

[2]

S. AlbeverioL. 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.  Google Scholar

[3]

S. AlbeverioL. Di PersioE. 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.  Google Scholar

[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.  Google Scholar

[5]

V. Barbu, Analysis and control of nonlinear infinite-dimensional systems, Mathematics in Science and Engineering, 190. Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[6]

V. BarbuF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. BonaccorsiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. CasasC. 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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Vol. 152. Cambridge university press, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H. W. Diehl, The theory of boundary critical phenomena, International Journal of Modern Physics B, 11 (1997), 3503-3523.  doi: 10.1142/S0217979297001751.  Google Scholar

[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.  Google Scholar

[24]

R. Fitz Hugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

C. MarinelliL. 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.  Google Scholar

[30]

J. NagumoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.   Google Scholar

[2]

S. AlbeverioL. 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.  Google Scholar

[3]

S. AlbeverioL. Di PersioE. 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.  Google Scholar

[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.  Google Scholar

[5]

V. Barbu, Analysis and control of nonlinear infinite-dimensional systems, Mathematics in Science and Engineering, 190. Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[6]

V. BarbuF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. BonaccorsiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. CasasC. 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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Vol. 152. Cambridge university press, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H. W. Diehl, The theory of boundary critical phenomena, International Journal of Modern Physics B, 11 (1997), 3503-3523.  doi: 10.1142/S0217979297001751.  Google Scholar

[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.  Google Scholar

[24]

R. Fitz Hugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

C. MarinelliL. 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.  Google Scholar

[30]

J. NagumoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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