Skip to main content
SpringerLink
Log in

Stochastic \(H_{2}/H_{\infty }\) Control for Mean-Field Stochastic Differential Systems with (x, u, v)-Dependent Noise

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper studies a continuous-time stochastic \(H_{2}/H_{\infty }\) control problem for mean-field stochastic differential systems, with random initial value and diffusion coefficients depending explicitly on the state, control and disturbance as well as their expectations. A mean-field stochastic bounded real lemma is first established, characterizing the equivalence between \(H_{\infty }\) robust stability and the solvability of two indefinite differential Riccati equations. Based on this extremely useful result, an equivalent condition for the existence of \(H_{2}/H_{\infty }\) controller is proposed by utilizing the solution of two sets of cross-coupled indefinite Riccati equations. Moreover, when an \(H_{2}/H_{\infty }\) controller exists, both the optimal control input and the corresponding worst-case disturbance admit linear feedback representations of the state and its mathematical expectation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andersson, D., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63, 341–356 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)

    Book  MATH  Google Scholar 

  3. Buckdahn, R., Djehiche, B., Li, J.: A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64, 197–216 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buckdahn, R., Li, J., Peng, S.: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Their Appl. 119(10), 3133–3154 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caun, R., Assuno, E., Teixeira, M.: \(H_2/H_\infty \) formulation of LQR controls based on LMI for continuous-time uncertain systems. Int. J. Syst. Sci. 52(3), 612–634 (2021)

    Article  MATH  Google Scholar 

  6. Chilali, M., Gahinet, P.: \(H_\infty \) design with pole placement constraints: an LMI approach. IEEE Trans. Autom. Control 41(3), 358–367 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, B., Zhang, W.: Stochastic \(H_2/H_\infty \) control with state-dependent noise. IEEE Trans. Autom. Control 49(1), 45–57 (2004)

    Article  MATH  Google Scholar 

  8. Elliott, R., Li, X., Ni, Y.H.: Discrete time mean-field stochastic linear-quadratic optimal control problems. Automatica 49(11), 3222–3233 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gao, M., Sheng, L., Zhang, W.: Stochastic \(H_2/H_\infty \) control of nonlinear systems with time-delay and state-dependent noise. Appl. Math. Comput. 266, 429–440 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Gao, M., Zhao, J., Sun, W.: Stochastic \(H_2/H_\infty \) control for discrete-time mean-field systems with Poisson jump. J. Frankl. Inst. 358(6), 2933–2947 (2021)

    Article  MATH  Google Scholar 

  11. Haddad, W.M., Lanchares, M., Chen, Y.: Mixed norm \(H_{2}/H_{\infty }\) and entropy covariance control: a convex optimisation approach. Int. J. Control. https://doi.org/10.1080/00207179.2020.1833251

  12. Hinrichsen, D., Pritchard, A.: Stochastic \(H_\infty \). SIAM J. Control Optim. 36(5), 1504–1538 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hosking, J.: A stochastic maximum principle for a stochastic differential game of a mean-field type. Appl. Math. Optim. 66, 415–454 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, J., Li, X., Yong, J.: A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Appl. Math. Optim. 70, 29–59 (2014)

    Article  MathSciNet  Google Scholar 

  15. Hou, T., Zhang, W., Ma, H.: Infinite horizon \(H_2/H_\infty \) optimal control for discrete-time Markov jump systems with \((x, u, v)\)-dependent noise. J. Global Optim. 57, 1245–1262 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khargonekar, P., Rotea, M.: Mixed \(H_2/H_\infty \) control: a convex optimization approach. IIEEE Trans. Autom. Control 36(7), 824–837 (1991)

    Article  MATH  Google Scholar 

  17. Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Limebeer, D., Anderson, B., Hendel, B.: A Nash game approach to mixed \(H_2/H_\infty \) control. IEEE Trans. Autom. Control 39(1), 69–82 (1994)

    Article  MATH  Google Scholar 

  19. Li, X., Sun, J., Yong, J.: Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probab. Uncertain. Quant. Risk. 1(1), 1–24 (2016)

    Article  MathSciNet  Google Scholar 

  20. Li, X., Wang, W., Xu, J., Zhang, H.: Stackelberg game approach to mixed \(H_2/H_\infty \) problem for continuous-time system. J. Syst. Sci. Complex. 32, 1324–1339 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, X., Xu, J., Wang, W., Zhang, H.: Mixed \(H_2/H_\infty \) control for discrete-time systems with input delay. IET Control Theory A 12(16), 2221–2231 (2018)

    Article  Google Scholar 

  22. Liu, Y., Hou, T.: Robust\(H_2/H_\infty \) fuzzy filtering for nonlinear stochastic systems with infinite Markov jump. J. Syst. Sci. Complex. 33, 1023–1039 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ma, H., Zhang, W., Hou, T.: Infinite horizon \(H_2/H_\infty \) control for discrete-time time-varying Markovian jump systems with multiplicative noise. Automatica 48(7), 1447–1454 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ma, L., Zhang, T., Zhang, W., Chen, B.: Finite horizon mean-field stochastic \(H_2/H_\infty \) control for continuous-time systems with \((x, v)\)-dependent noise. J. Frankl. Inst. 352, 5393–5414 (2015)

    Article  MATH  Google Scholar 

  25. Ma, L., Zhang, W.: Output feedback \(H_{\infty }\) control for discrete time mean-field stochastic systems. Asian J. Control 17(6), 2241–2251 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ma, L., Zhang, T., Zhang, W.: \(H_{\infty }\) control for continuous time mean-field stochastic systems. Asian J. Control 18(5), 1630–1640 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mei, W., Zhao, C., Ogura, M., Sugimoto, K.: Mixed \(H_2/H_\infty \) control of delayed Markov jump linear systems. IET Control Theory Appl. 14(15), 2076–2083 (2020)

    Article  MathSciNet  Google Scholar 

  28. Moon, J., Basar, T.: Linear quadratic risk-sensitive and robust mean field games. IEEE Trans. Autom. Control 62(3), 1062–1077 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mukaidani, H.: \(H_2/H_\infty \) control of stochastic systems with multiple decision makers: a Stackelberg game approach. In: IEEE 52nd Annual Conference on Decision and Control, pp. 1750-1755 (2013)

  30. Ni, Y., Elliott, R.J., Li, X.: Discrete-time mean-field stochastic linear-quadratic optimal controls, II: infinite horizon case. Automatica 57, 65–77 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qiu, L., Zhang, B., Xu, G., Pan, J., Yao, F.: Mixed \(H_2/H_\infty \) control of Markovian jump time-delay systems with uncertain transition probabilities. Inf. Sci. 373, 539–556 (2016)

    Article  MATH  Google Scholar 

  32. Yong, J.: Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J. Control. Optim. 51(4), 2809–2838 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yong, J.: Linear-quadratic optimal control problems for mean-field stochastic differential equations-time-consistent solutions. Trans. Am. Math. Soc. 369, 5467–5523 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Scherer, C., Gahinet, P., Chilali, M.: Multiobjective output-feedback control via LMI optimization. IEEE Trans. Autom. Control 42(7), 896–911 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sun, J.: Mean-Field stochastic linear quadratic optimal control problems: open-loop solvabilities. ESAIM Control Optim. Calc. Var. 23(3), 1099–1127 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, M.: Stochastic \(H_2/H_\infty \) control with random coefficients. Chin. Ann. Math. Ser. B 34, 733–752 (2013)

    Article  MATH  Google Scholar 

  37. Wu, C., Chen, B., Zhang, W.: Multiobjective \(H_2/H_\infty \) control design of the nonlinear mean-field stochastic jump-diffusion systems via fuzzy approach. IEEE Trans. Fuzzy Syst. 27(4), 686–700 (2019)

    Article  Google Scholar 

  38. Wu, K., Guo, B., Zhang, W., Chen, B.: Mixed \(H_2/H_\infty \) control for linear infinite-dimensional systems. Int. J. Control Autom. Syst. 14(1), 128–139 (2016)

    Article  Google Scholar 

  39. Zhang, W., Huang, Y., Zhang, H.: Stochastic \(H_2/H_\infty \) control for discrete-time systems with state and disturbance dependent noise. Automatica 43(3), 513–521 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, W., Huang, Y., Xie, L.: Infinite horizon stochastic \(H_2/H_\infty \) control for discrete-time systems with state and disturbance dependent noise. Automatica 44, 2306–2316 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, W., Ma, L., Zhang, T.: Discrete-time mean-field stochastic \(H_2/H_\infty \) control. J. Syst. Sci. Complex. 30, 765–781 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, W., Zhang, H., Chen, B.: Stochastic \(H_2/H_\infty \) control with \((x, u, v)-\)dependent noise: finite horizon case. Automatica 42, 1891–1898 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work of third author was supported by the National Natural Science Foundation of China (Nos. 12271158 and 11871121), the Key Projects of Natural Science Foundation of Zhejiang Province, China (No. LZ22A010005) and the Natural Science Foundation of Zhejiang Province, China (No. LY21A010001).

Author information

Authors and Affiliations

  1. Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Meijiao Wang

  2. Department of Mathematical Sciences, Huzhou University, Zhejiang, 313000, China

    Qingxin Meng

  3. School of Risk and Actuarial Studies and CEPAR, University of New South Wales, Sydney, NSW, 2052, Australia

    Yang Shen

  4. School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA, Australia

    Peng Shi

Authors
  1. Meijiao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingxin Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peng Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingxin Meng.

Additional information

Communicated by Nikolai Osmolovskii.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, M., Meng, Q., Shen, Y. et al. Stochastic \(H_{2}/H_{\infty }\) Control for Mean-Field Stochastic Differential Systems with (x, u, v)-Dependent Noise. J Optim Theory Appl 197, 1024–1060 (2023). https://doi.org/10.1007/s10957-023-02220-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02220-5

Keywords

Mathematics Subject Classification

Search

Navigation