Skip to main content
SpringerLink
Log in

A uniqueness result for the isaacs equation corresponding to nonlinear H∞ control

  • Published:
Mathematics of Control, Signals and Systems Aims and scope Submit manuscript

Abstract

The dynamic programming equation (DPE) corresponding to nonlinear H∞ control is considered. When the cost grows quadratically in the state, it is well known that there may be an infinite number of viscosity solutions to the DPE. In fact, there may be more than one classical solution when a classical solution exists. For the case of fixed feedback control, it is shown that there exists a unique viscosity solution in the class of solutions meeting a certain growth condition, and a representation in terms of available storage is obtained. For the active control case, where the H∞ problem is represented by a differential game, a similar representation result is obtained under the assumption of existence of a suboptimal feedback control.

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. J. A. Ball and J. W. Helton, H∞ control for nonlinear plants: connections with differential games,Proc. 28th IEEE CDC, 1989, pp. 956–962.

  2. J. A. Ball and J. W. Helton, Nonlinear H∞ control theory for stable plants.Math. Control Signals Systems,5 (1992), 233–261.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Bardi and F. Da Lio, On the Bellman equation for some unbounded control problems,Nonlinear Differential Equations Appl.,4 (1997), 499–510.

    Google Scholar 

  4. T. Basar and P. Bernhard,H ∞-Optimal Control and Related Minimax Design Problems, Birkhäuser, Boston, First edition, 1991, Second edition, 1995.

    Google Scholar 

  5. A. Bensoussan, J. Frehse, and H. Nagai, Some results on risk-sensitive control with full observation,Proc. 34th IEEE CDC, 1995, pp. 1658–1661.

  6. C. D. Charalambous and R. J. Elliott, Finite-dimensional output feedback dynamic games andL 2-gain for sector nonlinearities,IEEE Trans. Automat. Control, submitted.

  7. M. G. Crandall and P.-L. Lions, On existence and uniqueness of solutions of Hamilton— Jacobi equations,Nonlinear Anal. TMA.,10 (1986), 353–370.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Dupuis and W. M. McEneaney, Risk-sensitive and robust escape criteria,SIAM J. Control Optim.,35 (1997), 2021–2049.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. J. Elliott and N. J. Kalton,The Existence of Value in Differential Games, Memoirs of the American Mathematical Society, Vol. 126, Providence, RI, 1972.

  10. L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations,Indiana Univ. Math. J.,33 (1984), 773–797.

    Article  MathSciNet  Google Scholar 

  11. W. H. Fleming and M. R. James, The risk-sensitive index and theH 2 andH ∞ norms for nonlinear systems,Math. Control Signals Systems,8 (1995), 199–221.

    Article  MATH  MathSciNet  Google Scholar 

  12. W. H. Fleming and W. M. McEneaney, Risk-sensitive optimal control and differential games, inStochastic Theory and Adaptive Control (T. E. Duncan and B. Pasik-Duncan, eds.), Lecture Notes in Control and Information Sciences, Vol. 184, Springer-Verlag, Berlin, 1992, 185–197.

    Google Scholar 

  13. W. H. Fleming and W. M. McEneaney, Risk-sensitive control with ergodic cost criteria,Proc. 31st IEEE CDC, 1992, pp. 2048–2052.

  14. W. H. Fleming and W. M. McEneaney, Risk sensitive control on an infinite time horizon,SIAM J. Control Optim.,33 (1995), 1881–1915.

    Article  MATH  MathSciNet  Google Scholar 

  15. W. H. Fleming and H. M. Soner,Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1992.

    Google Scholar 

  16. H. Ishii, Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations,Indiana Univ. Math. J.,33 (1984), 721–748.

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Ishii, Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above, preprint (1997).

  18. A. Isidori, Robust regulation of nonlinear systems,MTNS Abstracts (1991), Kobe, Japan.

  19. M. R. James, Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games,Math. Control Signals Systems,5 (1992), 401–417.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. R. James, A partial differential inequality for dissipative nonlinear systems,Systems Control Lett.,21 (1993), 315–320.

    Article  MATH  MathSciNet  Google Scholar 

  21. W. M. McEneaney, Uniqueness for viscosity solutions of nonstationary HJB equations under some a priori conditions (with applications),SIAM J. Control Optim.,33 (1995), 1560–1576.

    Article  MATH  MathSciNet  Google Scholar 

  22. W. M. McEneaney, Robust control and differential games on a finite time horizon,Math. Control Signals Systems,8 (1995), 138–166.

    Article  MATH  MathSciNet  Google Scholar 

  23. W. M. McEneaney, Connections between Risk-Sensitive Stochastic Control, Differential Games and H∞ Control: the Nonlinear Case, Ph.D. thesis, Brown University, 1993.

  24. W. M. McEneaney and M. V. Day, Characteristic characterization of viscosity supersolutions corresponding to H∞ control,Proc. IFAC 13th World Congress, 1966, Vol. E, pp. 401–406.

    Google Scholar 

  25. W. M. McEneaney and P. Dupuis, A risk-sensitive escape criterion and robust limit,Proc. 33rd IEEE CDC, 1994, pp. 4195–4197.

  26. W. M. McEneaney and K. Ito, Infinite time-horizon risk sensitive systems with quadratic growth,Proc. 36th IEEE CDC, 1997, pp. 3413–3418.

  27. W. M. McEneaney and K. D. Mease, Nonlinear H∞ control of aerospace plane ascent,Proc. 34th IEEE CDC, 1995, pp. 3994–3995.

  28. P. Soravia, H∞ control of nonlinear systems: differential games and viscosity solutions,SIAM J. Control Optim.,34 (1996), 1071–1097.

    Article  MATH  MathSciNet  Google Scholar 

  29. A. J. van der Schaft,L 2-gain analysis of nonlinear systems and nonlinear state feedback H ∞ control,IEEE Trans. Automat. Control,37 (1992), 770–784.

    Article  MATH  MathSciNet  Google Scholar 

  30. P. Whittle, A risk-sensitive maximum principle,Systems Control Lett.,15 (1990), 183–192.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, 27695-8205, Raleigh, North Carolina, USA

    William M. McEneaney

Authors
  1. William M. McEneaney
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to William M. McEneaney.

Additional information

This research was partially supported by AFOSR under Grant F49620-95-1-0296 and by ONR under Grant N0014-96-1-0267.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McEneaney, W.M. A uniqueness result for the isaacs equation corresponding to nonlinear H∞ control. Math. Control Signal Systems 11, 303–334 (1998). https://doi.org/10.1007/BF02750395

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02750395

Key words

Search

Navigation