Advertisement

Applied Mathematics and Optimization

, Volume 31, Issue 3, pp 297–326 | Cite as

Optimality, stability, and convergence in nonlinear control

  • A. L. Dontchev
  • W. W. Hager
  • A. B. Poore
  • Bing Yang
Article

Abstract

Sufficient optimality conditions for infinite-dimensional optimization problems are derived in a setting that is applicable to optimal control with endpoint constraints and with equality and inequality constraints on the controls. These conditions involve controllability of the system dynamics, independence of the gradients of active control constraints, and a relatively weak coercivity assumption for the integral cost functional. Under these hypotheses, we show that the solution to an optimal control problem is Lipschitz stable relative to problem perturbations. As an application of this stability result, we establish convergence results for the sequential quadratic programming algorithm and for penalty and multiplier approximations applied to optimal control problems.

Key words

Sufficient optimality conditions Stability Sensitivity Sequential quadratic programming Penalty/multiplier methods 

AMS classification

49K40 49M30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alt W (1989) Stability of solutions for a class of nonlinear cone constrained optimization problems, Part 1: Basic theory. Numer Funct Anal Optim 10:1053–1064Google Scholar
  2. 2.
    Alt W (1990) Stability of solutions to control constrained nonlinear optimal control problems. Appl Math Optim 21:53–68Google Scholar
  3. 3.
    Alt W (1991) Parametric optimization with applications to optimal control and sequential quadratic programming. Bayreuth Math Sehr 35:1–37Google Scholar
  4. 4.
    Dontchev AL, Hager WW (1993) Lipschitzian stability in nonlinear control and optimization. SIAM J Control Optim 31:569–603Google Scholar
  5. 5.
    Dontchev AL, Hager WW (1994) Implicit functions, Lipschitz maps, and stability in optimization. Math Oper Res 19:753–768Google Scholar
  6. 6.
    Dunn JC, Tian T (1992) Variants of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions. SIAM J Control Optim 30:1361–1384Google Scholar
  7. 7.
    Hager WW (1985) Approximations to the multiplier method. SIAM J Numer Anal 22:16–46Google Scholar
  8. 8.
    Hager WW (1990) Multiplier methods for nonlinear optimal control. SIAM J Numer Anal 27:1061–1080Google Scholar
  9. 9.
    Hager WW, Ianculescu G (1984) Dual approximations in optimal control. SIAM J Control Optim 22:423–465Google Scholar
  10. 10.
    Ito K, Kunisch K (1992) Sensitivity analysis of solutions to optimization problems in Hilbert spaces with applications to optimal control and estimation. J Differential Equations 99:1–40Google Scholar
  11. 11.
    Malanowski K (1988) On stability of solutions to constrained optimal control problems for systems with control appearing linearly. Arch Automat Telemech 33:483–497Google Scholar
  12. 12.
    Malanowski K (1992) Second-order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimizations problems in Hubert spaces. Appl Math Optim 25:51–79Google Scholar
  13. 13.
    Malanowski K (preprint) Two norm approach in stability and sensitivity analysis of optimization and optimal control problemsGoogle Scholar
  14. 14.
    Maurer H (1981) First- and second-order sufficient optimality conditions in mathematical programming and optimal control. Math Programming Stud 14:163–177Google Scholar
  15. 15.
    Maurer H, Zowe J (1979) First- and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math Programming 16:98–110Google Scholar
  16. 16.
    Orrell K, Zeidan V (1988) Another Jacobi sufficient criterion for optimal control with smooth constraints. J Optim Theory Appl 58:283–300Google Scholar
  17. 17.
    Reid WT (1972) Riccati Differential Equations. Academic Press, New YorkGoogle Scholar
  18. 18.
    Robinson SM (1976) Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems. SIAM J Numer Anal 13:497–513Google Scholar
  19. 19.
    Robinson SM (1980) Strongly regular generalized equations. Math Oper Res 5:43–62Google Scholar
  20. 20.
    Rudin W (1966) Real and Complex Analysis. McGraw-Hill, New YorkGoogle Scholar
  21. 21.
    Ursescu C (1975) Multifunctions with closed convex graph. Czechoslovak Math J 25:438–441Google Scholar
  22. 22.
    Yang B (1991) Numerical Methods for Nonlinear Optimal Control Problems with Equality Control Constraints. PhD dissertation, Department of Mathematics, Colorado State University, Fort Collins, COGoogle Scholar
  23. 23.
    Zeidan V (1984) Extended Jacobi sufficiency criterion for optimal control. SIAM J Control Optim 22:294–301Google Scholar
  24. 24.
    Zeidan V (1984) First- and second-order sufficient conditions for optimal control and the calculus of variations. Appl Math Optim 11:209–226Google Scholar
  25. 25.
    Zeidan V (1993) Sufficient conditions for variational problems with variable endpoints: coupled points. Appl Math Optim 27:191–209Google Scholar

Copyright information

© Springer-Verlag New York Inc 1995

Authors and Affiliations

  • A. L. Dontchev
    • 1
  • W. W. Hager
    • 2
  • A. B. Poore
    • 3
  • Bing Yang
    • 4
  1. 1.Mathematical ReviewsAnn ArborUSA
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA
  3. 3.Department of MathematicsColorado State UniversityFort CollinsUSA
  4. 4.Department of MathematicsWest Virginia TechMontgomeryUSA

Personalised recommendations