Sufficient maximum principle for partially observed mean-field stochastic optimal control problems with delays

Heping Ma; Yu Shi; Weifeng Wang

doi:10.3934/eect.2024035

Article Contents

2024, Volume 13, Issue 6: 1436-1459. Doi: 10.3934/eect.2024035

This issue Previous Article Next Article The global existence of the solution to the stage-structured SIR system for rumor spreading and its regularity

Sufficient maximum principle for partially observed mean-field stochastic optimal control problems with delays

1.
School of Science, Hubei University of Technology, China
2.
School of Science, Wuhan University of Technology, China
3.
School of Mathematics and Statistics, South-Central Minzu University, China

^*Corresponding author: Yu Shi
^*Corresponding author: Yu Shi

Received: September 2023

Revised: May 2024

Early access: June 2024

Published: December 2024

The first author is supported by [the National Natural Science Foundation of China (Grant No. 11801154)].
The second author is supported by [the National Natural Science Foundation of China (Grant No. 12371480) and Fundamental Research Funds for the Central Universities project of China (Grant No. 104972024KFYjc0069)].
The third author is supported by [the Natural Science Foundation of Hubei Province (Grant No. 2023AFC006)]

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Abstract

This paper deals with partially observed optimal control of mean-field system with noisy memory and discrete delay. By means of the maximum principle, a sufficient optimality condition is established with an assumption of concavity. The main features of this paper include the introduction of a unified adjoint equation. Three examples that shed light on the theoretical results are established in this paper. In particular, a pension fund problem is solved, and an explicit expression for the optimal consumption rate is obtained by means of the maximum principle. Finally, analytic optimal controls are given in the paper and the numerical simulation shows the effect of delay on the optimal solution.

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Mathematics Subject Classification: Primary: 93E03, 93E20; Secondary: 93C41, 49N05.

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Figure 1. Curve of $ P_1(t) $

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Figure 2. Curve of $ \Gamma(t) $

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Figure 3. Curve of $ u(t) $

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Figure 4. Curve of $ \Delta(t) $

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Figure 5. The solution $ P_1(t) $ with different time delays $ \delta $

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