Optimal Control of a Nonlinear System with White Noise

Georges Kologo; Cédric Kpèbbèwèrè Some; Somdouda Sawadogo

doi:doi:10.11648/j.ajam.20251302.15

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Optimal Control of a Nonlinear System with White Noise

Georges Kologo^*

, Cédric Kpèbbèwèrè Some

, Somdouda Sawadogo

Published in American Journal of Applied Mathematics (Volume 13, Issue 2)

Received: 19 February 2025 Accepted: 10 March 2025 Published: 26 March 2025

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Abstract

In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control u(t) has a value in a convex space. Here, we propose sufficient minimum conditions on the Hamiltonian function to characterize the mean value of the cost function associated with the optimal control problem. The convexity of the Hamiltonian function is a sufficient condition for the existence of the optimal value of the control function. Under certain regularity assumptions, on the control system functional and on the non-differentiability criterion of Brownian motion, the existence and uniqueness results are established by the Cauchy-Lipschitz criteria. We also analyze the mathematical expectation stability of the system to check whether it will converge to an equilibrium point or not. For the study of this stability, the emphasis has been placed on root-mean-square stability. To highlight the results of our work, we apply this control problem to a SIRS-type epidemiological system for the coronavirus epidemic. To study the stability of this epidemiological system, we construct a Lyaunov function associated with the system and then use the results of Lyapunov's theorem to show the convergence of the system to a stable state.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 2)
DOI	10.11648/j.ajam.20251302.15
Page(s)	153-164
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Stochastic Optimal Control, Optimal Value, Epidemiological System, Stochastic Stability

References

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[6]	Filippo de Feo, Salvatore Federico, and Andrzej Swiech, optimal control of stochastic Delay diferential equations and applications to path-dependent financial and economic models, Papers 2302.08809, arXiv.org, SIAM J. CONTROL. OPTIM. VOL. 62 number 3, 2023.
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APA Style

Kologo, G., Some, C. K., Sawadogo, S. (2025). Optimal Control of a Nonlinear System with White Noise. American Journal of Applied Mathematics, 13(2), 153-164. https://doi.org/10.11648/j.ajam.20251302.15

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ACS Style

Kologo, G.; Some, C. K.; Sawadogo, S. Optimal Control of a Nonlinear System with White Noise. Am. J. Appl. Math. 2025, 13(2), 153-164. doi: 10.11648/j.ajam.20251302.15

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AMA Style

Kologo G, Some CK, Sawadogo S. Optimal Control of a Nonlinear System with White Noise. Am J Appl Math. 2025;13(2):153-164. doi: 10.11648/j.ajam.20251302.15

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@article{10.11648/j.ajam.20251302.15,
  author = {Georges Kologo and Cédric Kpèbbèwèrè Some and Somdouda Sawadogo},
  title = {Optimal Control of a Nonlinear System with White Noise},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {2},
  pages = {153-164},
  doi = {10.11648/j.ajam.20251302.15},
  url = {https://doi.org/10.11648/j.ajam.20251302.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251302.15},
  abstract = {In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control u(t) has a value in a convex space. Here, we propose sufficient minimum conditions on the Hamiltonian function to characterize the mean value of the cost function associated with the optimal control problem. The convexity of the Hamiltonian function is a sufficient condition for the existence of the optimal value of the control function. Under certain regularity assumptions, on the control system functional and on the non-differentiability criterion of Brownian motion, the existence and uniqueness results are established by the Cauchy-Lipschitz criteria. We also analyze the mathematical expectation stability of the system to check whether it will converge to an equilibrium point or not. For the study of this stability, the emphasis has been placed on root-mean-square stability. To highlight the results of our work, we apply this control problem to a SIRS-type epidemiological system for the coronavirus epidemic. To study the stability of this epidemiological system, we construct a Lyaunov function associated with the system and then use the results of Lyapunov's theorem to show the convergence of the system to a stable state.},
 year = {2025}
}

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TY  - JOUR
T1  - Optimal Control of a Nonlinear System with White Noise
AU  - Georges Kologo
AU  - Cédric Kpèbbèwèrè Some
AU  - Somdouda Sawadogo
Y1  - 2025/03/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251302.15
DO  - 10.11648/j.ajam.20251302.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 153
EP  - 164
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251302.15
AB  - In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control u(t) has a value in a convex space. Here, we propose sufficient minimum conditions on the Hamiltonian function to characterize the mean value of the cost function associated with the optimal control problem. The convexity of the Hamiltonian function is a sufficient condition for the existence of the optimal value of the control function. Under certain regularity assumptions, on the control system functional and on the non-differentiability criterion of Brownian motion, the existence and uniqueness results are established by the Cauchy-Lipschitz criteria. We also analyze the mathematical expectation stability of the system to check whether it will converge to an equilibrium point or not. For the study of this stability, the emphasis has been placed on root-mean-square stability. To highlight the results of our work, we apply this control problem to a SIRS-type epidemiological system for the coronavirus epidemic. To study the stability of this epidemiological system, we construct a Lyaunov function associated with the system and then use the results of Lyapunov's theorem to show the convergence of the system to a stable state.
VL  - 13
IS  - 2
ER  -

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Author Information

Georges Kologo

Department of Mathematics, Joseph KI Zerbo University (UJKZ), Ouagadougou, Burkina Faso

Contact Email

http://orcid.org/0009-0009-6770-5113
Cédric Kpèbbèwèrè Some

Department of Mathematics, Joseph KI Zerbo University (UJKZ), Ouagadougou, Burkina Faso; Department of Mathematics, Virtual University of BURKINA FASO (UV-BF), Ouagadougou, Burkna Faso

Contact Email

http://orcid.org/0000-0003-4902-5033
Somdouda Sawadogo

Department of Mathematics, Joseph KI Zerbo University (UJKZ), Ouagadougou, Burkina Faso; Department of Mathematics, Ecole Normale Superieure Institute of Science and Technology (IST-ENS), Ouagadougou, Burkina Faso

Contact Email

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Plain Text BibTeX RIS

APA Style

Kologo, G., Some, C. K., Sawadogo, S. (2025). Optimal Control of a Nonlinear System with White Noise. American Journal of Applied Mathematics, 13(2), 153-164. https://doi.org/10.11648/j.ajam.20251302.15

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ACS Style

Kologo, G.; Some, C. K.; Sawadogo, S. Optimal Control of a Nonlinear System with White Noise. Am. J. Appl. Math. 2025, 13(2), 153-164. doi: 10.11648/j.ajam.20251302.15

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AMA Style

Kologo G, Some CK, Sawadogo S. Optimal Control of a Nonlinear System with White Noise. Am J Appl Math. 2025;13(2):153-164. doi: 10.11648/j.ajam.20251302.15

Copy | Download

@article{10.11648/j.ajam.20251302.15,
  author = {Georges Kologo and Cédric Kpèbbèwèrè Some and Somdouda Sawadogo},
  title = {Optimal Control of a Nonlinear System with White Noise},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {2},
  pages = {153-164},
  doi = {10.11648/j.ajam.20251302.15},
  url = {https://doi.org/10.11648/j.ajam.20251302.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251302.15},
  abstract = {In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control u(t) has a value in a convex space. Here, we propose sufficient minimum conditions on the Hamiltonian function to characterize the mean value of the cost function associated with the optimal control problem. The convexity of the Hamiltonian function is a sufficient condition for the existence of the optimal value of the control function. Under certain regularity assumptions, on the control system functional and on the non-differentiability criterion of Brownian motion, the existence and uniqueness results are established by the Cauchy-Lipschitz criteria. We also analyze the mathematical expectation stability of the system to check whether it will converge to an equilibrium point or not. For the study of this stability, the emphasis has been placed on root-mean-square stability. To highlight the results of our work, we apply this control problem to a SIRS-type epidemiological system for the coronavirus epidemic. To study the stability of this epidemiological system, we construct a Lyaunov function associated with the system and then use the results of Lyapunov's theorem to show the convergence of the system to a stable state.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Optimal Control of a Nonlinear System with White Noise
AU  - Georges Kologo
AU  - Cédric Kpèbbèwèrè Some
AU  - Somdouda Sawadogo
Y1  - 2025/03/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251302.15
DO  - 10.11648/j.ajam.20251302.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 153
EP  - 164
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251302.15
AB  - In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control u(t) has a value in a convex space. Here, we propose sufficient minimum conditions on the Hamiltonian function to characterize the mean value of the cost function associated with the optimal control problem. The convexity of the Hamiltonian function is a sufficient condition for the existence of the optimal value of the control function. Under certain regularity assumptions, on the control system functional and on the non-differentiability criterion of Brownian motion, the existence and uniqueness results are established by the Cauchy-Lipschitz criteria. We also analyze the mathematical expectation stability of the system to check whether it will converge to an equilibrium point or not. For the study of this stability, the emphasis has been placed on root-mean-square stability. To highlight the results of our work, we apply this control problem to a SIRS-type epidemiological system for the coronavirus epidemic. To study the stability of this epidemiological system, we construct a Lyaunov function associated with the system and then use the results of Lyapunov's theorem to show the convergence of the system to a stable state.
VL  - 13
IS  - 2
ER  -

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