Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs

Ruijing Li

doi:doi:10.11648/j.cse.20240801.12

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Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs

Ruijing Li

Published in Control Science and Engineering (Volume 8, Issue 1)

Received: 4 January 2024 Accepted: 17 January 2024 Published: 24 January 2024

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Abstract

In this paper, we investigate the relaxed and strict near-optimality conditions for mean-field singular FBSDEs, where the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman’s optimality principle does not hold. The purpose of this paper is to establish necessary and sufficient conditions of near-optimality for relaxed and strict mean-field singular controls. For strict mean-field singular FBSDEs, whose wellposedness is ensured under the twice continuously differentiable assumptions of coefficients. Then, the moment estimations of variational processes as well as first- order and second-order adjoint processes are presented by using Burkholder-Davis-Gundy inequality. Further, by introducing Hamiltonian function via Ekeland’s variational principle, the necessary near-optimality conditions are established. For relaxed mean-field singular FBSDEs, we first give the definition of admissible set of relaxed singular controls, then use the mapping defined by Dirac measure, we prove that the near-optimal problem of strict singular controls is a particular case of the near- optimal problem of relaxed singular ones. Further, a well known chattering lemma is introduced. By virtue of this famous lemma addition with the stability of trajectories with respect to the control variable and dominated convergence theorem, necessary as well as sufficient near-optimality conditions for relaxed controls are established.

Published in	Control Science and Engineering (Volume 8, Issue 1)
DOI	10.11648/j.cse.20240801.12
Page(s)	13-28
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), 2024. Published by Science Publishing Group

Keywords

Near-optimal Singular Control, Mean-field SDE, Relaxed and Strict Control, Adjoint Equation, Ekeland’s Variational Principle

References

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APA Style

Li, R. (2024). Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs. Control Science and Engineering, 8(1), 13-28. https://doi.org/10.11648/j.cse.20240801.12

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ACS Style

Li, R. Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs. Control Sci. Eng. 2024, 8(1), 13-28. doi: 10.11648/j.cse.20240801.12

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AMA Style

Li R. Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs. Control Sci Eng. 2024;8(1):13-28. doi: 10.11648/j.cse.20240801.12

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@article{10.11648/j.cse.20240801.12,
  author = {Ruijing Li},
  title = {Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs},
  journal = {Control Science and Engineering},
  volume = {8},
  number = {1},
  pages = {13-28},
  doi = {10.11648/j.cse.20240801.12},
  url = {https://doi.org/10.11648/j.cse.20240801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.cse.20240801.12},
  abstract = {In this paper, we investigate the relaxed and strict near-optimality conditions for mean-field singular FBSDEs, where the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman’s optimality principle does not hold. The purpose of this paper is to establish necessary and sufficient conditions of near-optimality for relaxed and strict mean-field singular controls. For strict mean-field singular FBSDEs, whose wellposedness is ensured under the twice continuously differentiable assumptions of coefficients. Then, the moment estimations of variational processes as well as first- order and second-order adjoint processes are presented by using Burkholder-Davis-Gundy inequality. Further, by introducing Hamiltonian function via Ekeland’s variational principle, the necessary near-optimality conditions are established. For relaxed mean-field singular FBSDEs, we first give the definition of admissible set of relaxed singular controls, then use the mapping defined by Dirac measure, we prove that the near-optimal problem of strict singular controls is a particular case of the near- optimal problem of relaxed singular ones. Further, a well known chattering lemma is introduced. By virtue of this famous lemma addition with the stability of trajectories with respect to the control variable and dominated convergence theorem, necessary as well as sufficient near-optimality conditions for relaxed controls are established.},
 year = {2024}
}

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TY  - JOUR
T1  - Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs
AU  - Ruijing Li
Y1  - 2024/01/24
PY  - 2024
N1  - https://doi.org/10.11648/j.cse.20240801.12
DO  - 10.11648/j.cse.20240801.12
T2  - Control Science and Engineering
JF  - Control Science and Engineering
JO  - Control Science and Engineering
SP  - 13
EP  - 28
PB  - Science Publishing Group
SN  - 2994-7421
UR  - https://doi.org/10.11648/j.cse.20240801.12
AB  - In this paper, we investigate the relaxed and strict near-optimality conditions for mean-field singular FBSDEs, where the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman’s optimality principle does not hold. The purpose of this paper is to establish necessary and sufficient conditions of near-optimality for relaxed and strict mean-field singular controls. For strict mean-field singular FBSDEs, whose wellposedness is ensured under the twice continuously differentiable assumptions of coefficients. Then, the moment estimations of variational processes as well as first- order and second-order adjoint processes are presented by using Burkholder-Davis-Gundy inequality. Further, by introducing Hamiltonian function via Ekeland’s variational principle, the necessary near-optimality conditions are established. For relaxed mean-field singular FBSDEs, we first give the definition of admissible set of relaxed singular controls, then use the mapping defined by Dirac measure, we prove that the near-optimal problem of strict singular controls is a particular case of the near- optimal problem of relaxed singular ones. Further, a well known chattering lemma is introduced. By virtue of this famous lemma addition with the stability of trajectories with respect to the control variable and dominated convergence theorem, necessary as well as sufficient near-optimality conditions for relaxed controls are established.
VL  - 8
IS  - 1
ER  -

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Author Information

Ruijing Li

School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, China

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Plain Text BibTeX RIS

APA Style

Li, R. (2024). Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs. Control Science and Engineering, 8(1), 13-28. https://doi.org/10.11648/j.cse.20240801.12

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ACS Style

Li, R. Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs. Control Sci. Eng. 2024, 8(1), 13-28. doi: 10.11648/j.cse.20240801.12

Copy | Download

AMA Style

Li R. Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs. Control Sci Eng. 2024;8(1):13-28. doi: 10.11648/j.cse.20240801.12

Copy | Download

@article{10.11648/j.cse.20240801.12,
  author = {Ruijing Li},
  title = {Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs},
  journal = {Control Science and Engineering},
  volume = {8},
  number = {1},
  pages = {13-28},
  doi = {10.11648/j.cse.20240801.12},
  url = {https://doi.org/10.11648/j.cse.20240801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.cse.20240801.12},
  abstract = {In this paper, we investigate the relaxed and strict near-optimality conditions for mean-field singular FBSDEs, where the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman’s optimality principle does not hold. The purpose of this paper is to establish necessary and sufficient conditions of near-optimality for relaxed and strict mean-field singular controls. For strict mean-field singular FBSDEs, whose wellposedness is ensured under the twice continuously differentiable assumptions of coefficients. Then, the moment estimations of variational processes as well as first- order and second-order adjoint processes are presented by using Burkholder-Davis-Gundy inequality. Further, by introducing Hamiltonian function via Ekeland’s variational principle, the necessary near-optimality conditions are established. For relaxed mean-field singular FBSDEs, we first give the definition of admissible set of relaxed singular controls, then use the mapping defined by Dirac measure, we prove that the near-optimal problem of strict singular controls is a particular case of the near- optimal problem of relaxed singular ones. Further, a well known chattering lemma is introduced. By virtue of this famous lemma addition with the stability of trajectories with respect to the control variable and dominated convergence theorem, necessary as well as sufficient near-optimality conditions for relaxed controls are established.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs
AU  - Ruijing Li
Y1  - 2024/01/24
PY  - 2024
N1  - https://doi.org/10.11648/j.cse.20240801.12
DO  - 10.11648/j.cse.20240801.12
T2  - Control Science and Engineering
JF  - Control Science and Engineering
JO  - Control Science and Engineering
SP  - 13
EP  - 28
PB  - Science Publishing Group
SN  - 2994-7421
UR  - https://doi.org/10.11648/j.cse.20240801.12
AB  - In this paper, we investigate the relaxed and strict near-optimality conditions for mean-field singular FBSDEs, where the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman’s optimality principle does not hold. The purpose of this paper is to establish necessary and sufficient conditions of near-optimality for relaxed and strict mean-field singular controls. For strict mean-field singular FBSDEs, whose wellposedness is ensured under the twice continuously differentiable assumptions of coefficients. Then, the moment estimations of variational processes as well as first- order and second-order adjoint processes are presented by using Burkholder-Davis-Gundy inequality. Further, by introducing Hamiltonian function via Ekeland’s variational principle, the necessary near-optimality conditions are established. For relaxed mean-field singular FBSDEs, we first give the definition of admissible set of relaxed singular controls, then use the mapping defined by Dirac measure, we prove that the near-optimal problem of strict singular controls is a particular case of the near- optimal problem of relaxed singular ones. Further, a well known chattering lemma is introduced. By virtue of this famous lemma addition with the stability of trajectories with respect to the control variable and dominated convergence theorem, necessary as well as sufficient near-optimality conditions for relaxed controls are established.
VL  - 8
IS  - 1
ER  -

Copy | Download