A bang-bang optimal control for a nonlinear system modeling the Gate Control Theory of Pain

Gregorio Díaz; Jesús Ildefonso Díaz

doi:10.3934/dcdss.2025008

Article Contents

2025, Volume 18, Issue 12: 3776-3798. Doi: 10.3934/dcdss.2025008

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A bang-bang optimal control for a nonlinear system modeling the Gate Control Theory of Pain

Gregorio Díaz^2, and
Jesús Ildefonso Díaz^{1, 2, ,}

1.
Instituto Matemático Interdisciplinar (IMI), Univ. Complutenese de Madrid, Spain
2.
Análisis Matemático y Matemática Aplicada, Univ. Complutenese de Madrid, Spain

^*Corresponding author: Jesús Ildefonso Díaz
^*Corresponding author: Jesús Ildefonso Díaz

Received: July 2024

Revised: January 2025

Early access: February 8, 2025

Published online: February 8, 2025

The research of the authors was partially supported by the project PID2020-112517GB-I00 of the AEI (Spain). JID is also supported by the project PID2023-146754NB-I00, funded by MCIU/AEI/10.13039/501100011033 and FEDER, EU.

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Abstract

We consider a nonlinear system of coupled ordinary differential equations (representing the excitatory, inhibitory, and T-cell potentials) based on the Gate Control Theory of Pain, initially proposed by R. Melzack and P.D. Wall in 1965, and later mathematically modeled by N.F. Britton and S.M. Skevington in 1988. Our main results focus on an optimal control problem associated with this model, where the short frequency, understood as a bounded time-dependent function, is treated as the control variable. The cost function accounts for a person's pain at a given final time and incorporates additional criteria. We demonstrate the uniqueness of the optimal control and establish the bang-bang nature of the control. In a previous section, we extend the mathematical analysis of the model developed by Britton and Skevington by presenting a series of mathematical inequalities. These inequalities strengthen the model's alignment with the principal requirements for reproducing the core structure of Pain Theory.

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Mathematics Subject Classification: 34A34, 34C60, 34H05, 49J30, 49K15.

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Figure 1. Basic design of the pain gate

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Figure 2. Qualitative representation of the potentials $ {{\rm{V}}}_{ {{\rm{I}}}}, {{\rm{V}}}_{ {{\rm{E}}}} $ y $ {{\rm{V}}}_{ {{\rm{T}}}} $

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Figure 3. Phase plain for the potentials $ {{\rm{V}}}_{ {{\rm{I}}}} $ and $ {{\rm{V}}}_{ {{\rm{T}}}} $

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Figure 4. Monotone dependence of $ {{\rm{V}}}_{ {{\rm{T}}}} $ with respect to $ x_{s} $

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Figure 5. Monotone dependence of $ {{\rm{V}}}_{ {{\rm{T}}}} $ with respect to $ \alpha_{c {{\rm{I}}}} $

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Figure 6. Dependence of $ {{\rm{V}}}_{ {{\rm{I}}}} $ with respect to $ x_{s} $

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Figure 7. Dependence of $ {{\rm{V}}}_{ {{\rm{I}}}} $ with respect to $ \alpha_{c{\rm{I}}} $

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Figure 8. Representation of the potentials for time-depending frequencies

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Figure 9. Phase plane for some variable frequencies

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Figure 10. Other input types of frequencies

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References

[1]	H. Amann, Ordinary Differential Equations: An Introduction to Nonlinear Analysis, Walter de Gruyter, Berlin, 1990.
[2]	U. An der Heiden, Analysis of Neural Networks, Lecture Notes in Biomath., 35 Springer-Verlag, Berlin-New York, 1980.
[3]	A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, Springfield, USA. 2007.
[4]	N. F. Britton, M. A. J. Chaplain and S. M. Skevington, The role of N-methyl-D-aspartate (NMDA) receptors in wind-up: A mathematical model, J. Math. Appl. Med. Biol., 13 (1996), 193-205. doi: 10.1093/imammb/13.3.193.
[5]	N. F. Britton and S. M. Skevington, A mathematical model of the gate control theory of pain, J. Bio. Syst., 137 (1988), 91-105.
[6]	A. C. Casal and J. I. Díaz, On the principle of pseudo-linearized stability: Applications to some delayed nonlinear parabolic equations, Nonlinear Anal., 63 (2005), 997-1007. doi: 10.1016/j.na.2005.01.013.
[7]	W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975.
[8]	R. Melzack and P. D. Wall, Pain mechanisms: A new theory, Science, 150 (1965), 971-979. doi: 10.1126/science.150.3699.971.
[9]	R. Melzack and P. D. Wall, The Challenge of Pain, Penguin, Harmondsworth, UK. 1982.
[10]	P. W. Nathan, The gate control theory of pain: A critical review, Brain, 99 (1976), 123-158.
[11]	P. W. Nathan and P. Rudge, Testing the gate control theory of pain in man, J. Neurol. Neurosurg. Psych., 37 (1974), 1366-1372. doi: 10.1136/jnnp.37.12.1366.
[12]	E. Trélat, Contrôle Optimal: Théory et Applications, Vuibert, Collection Mathématiques Concrètes, Paris, 2005.
[13]	H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J., 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5.