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Stochastic H 2/H ∞ control with random coefficients

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Abstract

This paper is concerned with the mixed H 2/H ∞ control for stochastic systems with random coefficients, which is actually a control combining the H 2 optimization with the H ∞ robust performance as the name of H 2/H ∞ reveals. Based on the classical theory of linear-quadratic (LQ, for short) optimal control, the sufficient and necessary conditions for the existence and uniqueness of the solution to the indefinite backward stochastic Riccati equation (BSRE, for short) associated with H ∞ robustness are derived. Then the sufficient and necessary conditions for the existence of the H 2/H ∞ control are given utilizing a pair of coupled stochastic Riccati equations.

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References

  1. Basar, T. and Bernhar, P., H ∞-Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Birkhäuser, Boston, 1995.

    Book  MATH  Google Scholar 

  2. Chen, S. and Yong, J., Stochastic linear quadratic optimal control problems with random coefficients, Chin. Ann. Math. Ser. B, 21(3), 2000, 323–338.

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, B. and Zhang, W., Stochastic H 2/H ∞ control with state-dependent noise, IEEE Trans. Automat. Control, 49, 2004, 45–57.

    Article  MathSciNet  Google Scholar 

  4. Costa, O. L. V. and Marques, R. P., Mixed H 2/H ∞ control of discrete-time Markovian jump linear systems, IEEE Trans. Automat. Control, 43, 1998, 95–100.

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, X. and Zhou, K., Multiobjective H 2/H ∞ control design, SIAM J. Control Optim., 40, 2001, 628–660.

    Article  MathSciNet  MATH  Google Scholar 

  6. Hinrichsen, D. and Pritchard, A. J., Stochastic H ∞, SIAM J. Control Optim., 36, 1998, 1504–1538.

    Article  MathSciNet  MATH  Google Scholar 

  7. Hu, Y. and Zhou, X., Indefinite stochastic Riccati equations, SIAM J. Control Optim., 42, 2003, 123–137.

    Article  MathSciNet  MATH  Google Scholar 

  8. Khargonekar, P. P., and Rotea, M. A., Mixed H 2/H ∞ control: A convex optimization approach, IEEE Trans. Automat. Control, 36, 1991, 824–837.

    Article  MathSciNet  MATH  Google Scholar 

  9. Kohlmann, M. and Tang, S., Optimal control of linear stochastic systems with singular costs, and the mean variance hedging problem with stochastic market conditions, 2000, Preprint.

    Google Scholar 

  10. Kohlmann, M. and Tang, S., Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging, Stochastic Process. Appl., 97, 2002, 255–288.

    Article  MathSciNet  MATH  Google Scholar 

  11. Kohlmann, M. and Tang, S., Multidimensional backward stochastic Riccati equations and applications, SIAM J. Control Optim., 41, 2003, 1696–1721.

    Article  MathSciNet  MATH  Google Scholar 

  12. Kohlmann, M. and Tang, S., Minimization of risk and linear quadratic optimal control theory, SIAM J. Control Optim., 42, 2003, 1118–1142.

    Article  MathSciNet  MATH  Google Scholar 

  13. Limebeer, D. J. N., Anderson, B. D. O. and Hendel, B., A Nash game approach to mixed H 2/H ∞ control, IEEE Trans. Automat. Control, 39, 1994, 69–82.

    Article  MathSciNet  MATH  Google Scholar 

  14. Lim, A. E. B. and Zhou, X., Mean-variance portfolio selection with random parameters in a complete market, Math. Oper. Res., 27, 2001, 101–120.

    Article  MathSciNet  Google Scholar 

  15. Lim, A. E. B., Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Math. Oper. Res., 29, 2004, 132–161.

    Article  MathSciNet  MATH  Google Scholar 

  16. Peng, S., Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 30, 1992, 284–304.

    Article  MathSciNet  MATH  Google Scholar 

  17. Sweriduk, G. D. and Calise, A. J., Differential game approach to the mixed H 2/H ∞ problem, Journal of Guidance Control, and Dynamics, 20, 1997, 1229–1234.

    Article  MATH  Google Scholar 

  18. Tang, S., General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations, SIAM J. Control Optim., 42, 2003, 53–75.

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, W. and Chen, B., State feedback H ∞ control for a class of nonlinear stochastic systems, SIAM J. Control Optim., 44, 2006, 1973–1991.

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, W., Zhang, H. and Chen, B., Stochastic H 2/H ∞ control with (x, u, v)-dependent noise: Finite horizon case, Automatica, 42, 2006, 1891–1898.

    Article  MATH  Google Scholar 

  21. Zhang, W., Chen, B. and Tang, H., Some remarks on stochastic H ∞ control of linear and nonlinear Itô-type differential systems, Proceedings of the 30th Chinese Control Conference, 2011.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Finance and Stochastic Control, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Meijiao Wang

  2. Faculty of Science, Ningbo University, Ningbo, 315211, Zhejiang, China

    Meijiao Wang

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  1. Meijiao Wang
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Correspondence to Meijiao Wang.

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Wang, M. Stochastic H 2/H ∞ control with random coefficients. Chin. Ann. Math. Ser. B 34, 733–752 (2013). https://doi.org/10.1007/s11401-013-0790-z

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  • DOI: https://doi.org/10.1007/s11401-013-0790-z

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