Skip to main content
SpringerLink
Log in

Uniform Convergence of the Solutions of Riccati Equations for a Family of Optimal Control Problems

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

The uniform convergence of the solutions of a parameterized family of Riccati differential equations which arise in the context of optimal control problems for systems described by a family of linear time-varying evolution equations is considered on the infinite time interval in Hilbert space. Some sufficient conditions and an illustrative example are given for uniform convergence of the solutions of a family of the Riccati differential equations. The uniform convergence of the solutions of the Riccati differential equations with respect to parameters is important for applications to problems in adaptive control of stochastic evolution systems.

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. Lopez A and Zuazua E, Some new resultsrelated to the null controllability of the 1-d heat equation, Sem EDP, Ecole Polytechnique, 1998, VIII: 1–22.

    Google Scholar 

  2. Zuazua E, Boundary observability for the finite-difference space semi-discretization of the 2-D wave equation in the square, J. Math. Pures Appl., 1999, 78: 523–563.

    Article  MathSciNet  Google Scholar 

  3. Infante J A and Zuazua E, Boundary observability for the space semi-discretizations of the 1-D wave equation, M2AN Math. Model Numer. Anal., 1999, 33: 407–438.

    Article  MathSciNet  Google Scholar 

  4. Zuazua E, Controllability of partial differential equations and its semi-discrete approximation, Discrete Contin. Dyn. Syst., 2002, 8: 469–513.

    Article  MathSciNet  Google Scholar 

  5. Leon L and Zuazua E, Boundary controllability of the finite-difference space semi-discretization of the beam equation, ESAIM Control Optim. Calc. Var., 2002, 8: 827–862.

    Article  MathSciNet  Google Scholar 

  6. Negreanu M and Zuazua E, Uniform boundary controllability of a discrete 1-D wave equation, System Control Lett., 2003, 48: 261–279.

    Article  MathSciNet  Google Scholar 

  7. Tcheugoue Tebou L R and Zuazua E, Unoform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numer. Math., 2003, 95: 563–598.

    Article  MathSciNet  Google Scholar 

  8. Zuazua E, Optimal and approximate control of finite-difference approximation schemes for the 1-D wave equation, Rendiconti di Mathematica VIII, 2004, 24: 201–237.

    MathSciNet  MATH  Google Scholar 

  9. Zuazua E, Propagation, observation, control and numerical approximation of waves approximated by finite difference method, SIAM Rev., 2005, 47: 197–243.

    Article  MathSciNet  Google Scholar 

  10. Zuazua E, Control and numerical approximation of the wave and heat equations, International Congress of Mathematics, III, 2006, 1389–1417.

  11. Labbe S and Trelat E, Uniform controllability of semidiscrete approximations of parabolic control systems, System Control Lett., 2006, 55: 597–609.

    Article  MathSciNet  Google Scholar 

  12. Castro C and Micu S, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method, Numer. Math., 2006, 102: 413–462.

    Article  MathSciNet  Google Scholar 

  13. Boyer F, Hubert F, and Le Rousscau J, Discrete Carleman estimates for the elliptic operators and uniform controllability of semi discretized parabolic equations, Math. Pur. Appl., 2010, 93: 240–276.

    Article  MathSciNet  Google Scholar 

  14. Boyer F, Hubert F, and Le Rousscau J, Discrete Carleman estimates for the elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim., 2010, 48: 5357–5397.

    Article  MathSciNet  Google Scholar 

  15. Boyer F, Hubert F, and Le Rousscau J, Uniform null-controllability properties for space/time-discretized parabolic equations, Numer. Math., 2011, 118: 601–661.

    Article  MathSciNet  Google Scholar 

  16. Nguyen Thuy N T, Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces, Discrete and Continuous Dynamical Systems Series B, 2015, 20: 613–640.

    Article  MathSciNet  Google Scholar 

  17. Gibson J S, The Riccati integral equations for optimal control problems on Hilbert spaces, SIAM J. Control Optim., 1979, 17: 537–565.

    Article  MathSciNet  Google Scholar 

  18. Banks H T and Kunisch K, The linear regulator problem for parabolic systems, SIAM J. Control Optim., 1984, 22: 684–698.

    Article  MathSciNet  Google Scholar 

  19. Kappel F and Salamon D, An approximation theorem for the algebraic Riccati equation, SIAM J. Control Optim., 1990, 28: 1136–1147.

    Article  MathSciNet  Google Scholar 

  20. Banks H T and Ito K, Approximation in LQR problem for infinite dimensional systems with unbounded input operators, J. Math. Systems Estim. Control, 1997, 7: 119–122.

    MathSciNet  MATH  Google Scholar 

  21. Liu Z and Zhang S, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics, 1999, 398.

  22. Lasiecka I and Triggiani R, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract parabolic systems, Encyclopedia of Mathematics and Its Applications, vol. 74, Cambridge University Press, Cambridge, 2000.

    Book  Google Scholar 

  23. Ramdani K, Takahashi T, and Tucsnak M, Uniformly exponentially stable approximation for a class of second order evolution equations, ESAIM COCV, 2007, 13: 503–527.

    Article  Google Scholar 

  24. Hille E and Phillips R S, Functional Analysis and Semi-Groups, RI: American Mathematical Society, Providence, 1957.

  25. Duncan T E, Goldys B, and Parik-Duncan B, Adaptive control of linear stochastic evolution systems, Stochastics and Stochastics Reports, 1991, 35: 129–192.

    Article  MathSciNet  Google Scholar 

  26. Duncan T E, Mastowski B, and Parik-Duncan B, Adaptive boundary control of linear stochastic distributed parameter systems, Proceedings of the 32nd IEEE Conference on Decision and Control, 1993, 369–374.

  27. Da Prato G and Ichikawa A, Quadratic control for linear time varying systems, SIAM J. Control Optim., 1990, 28: 359–381.

    Article  MathSciNet  Google Scholar 

  28. Chojnowska-Michalik A, Duncan T E, and Pasik-Duncan B, Uniform operator continuity of the stationary Riccati equation in Hilbert space, Appl. Math. Optim., 1992, 25: 171–187.

    Article  MathSciNet  Google Scholar 

  29. Lasiecka I and Triggiani R, Uniformly convergence of the solutions to Riccati equations arising in boundary/point control problems, Proceedings of Stochastic Theory and Adaptive Control, Eds. by Duncan T E and Parik-Duncan B, University Kansas, 1992, 285–305.

  30. Lasiecka I, Approximations of the solutions of infinite dimensional algebraic Riccati equation with unbounded input operators, Numerical Funct. Anal. and Optimiz., 1990, 11: 303–378.

    Article  MathSciNet  Google Scholar 

  31. Lasiecka I and Triggiani R, Numerical approximation for abstract systems modeled by analytic semigroups and applications, Mathematics and Computation, 1991, 57: 639–662.

    Article  MathSciNet  Google Scholar 

  32. Lasiecka I and Triggiani R, Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Application Theory, Springer Verlag, Berlin, 1991.

    MATH  Google Scholar 

  33. Bensoussan A, Da Prato G, Delfour M C, et al., Representation and Control of Infinite-dimensional Systems, Birkhauser, Berlin, 2007.

    Book  Google Scholar 

  34. Niamsup P and Phat V N, Linear time-varying systems in Hilbert space: Exact controllability implies complete stabilizability, Thai Journal of Mathematics, 2009, 7: 189–199.

    MathSciNet  MATH  Google Scholar 

  35. Zabczyk J, Remarks on the algebraic Riccati equation in Hilbert space, Appl. Math. Optim., 1975, 2: 251–258.

    Article  MathSciNet  Google Scholar 

  36. Tanabe H, Equations of Evolution, Pitman, London, 1979.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Zhaoqiang Ge

Authors
  1. Zhaoqiang Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaoqiang Ge.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant No. 61174081.

This paper was recommended for publication by Editor LIU Yungang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ge, Z. Uniform Convergence of the Solutions of Riccati Equations for a Family of Optimal Control Problems. J Syst Sci Complex 33, 74–86 (2020). https://doi.org/10.1007/s11424-019-8062-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-019-8062-y

Keywords

Search

Navigation