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Sensitivity analysis of optimization problems in Hilbert space with application to optimal control

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Abstract

A family of optimization problems in a Hilbert space depending on a vector parameter is considered. It is assumed that the problems have locally isolated local solutions. Both these solutions and the associated Lagrange multipliers are assumed to be locally Lipschitz continuous functions of the parameter. Moreover, the assumption of the type of strong second-order sufficient condition is satisfied.

It is shown that the solutions are directionally differentiable functions of the parameter and the directional derivative is characterized. A second-order expansion of the optimal-value function is obtained. The abstract results are applied to state and control constrained optimal control problems for systems described by nonlinear ordinary differential equations with the control appearing linearly.

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References

  1. Fiacco AV (1983) Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York

    MATH  Google Scholar 

  2. Haraux A (1977) How to differentiate the projection on a convex set in Hilbert space. Some application to variational inequalities. J Math Soc Japan 29:615–631

    Article  MathSciNet  MATH  Google Scholar 

  3. Jittorntrum K (1984) Solution point differentiability without strict complementarity in nonlinear programming. Math Programming Stud 21:127–138

    Article  MathSciNet  MATH  Google Scholar 

  4. Lee EB, Marcus L (1967) Foundations of Optimal Control Theory. Wiley, New York

    Google Scholar 

  5. Lempio F, Maurer H (1980) Differential stability in infinite-dimensional nonlinear programming. Appl Math Optim 6:139–152

    Article  MathSciNet  MATH  Google Scholar 

  6. Malanowski K (1978) On regularity of solutions to optimal control problems for systems linear with respect to the control variable. Arch Automat Telemech 23:227–242

    MathSciNet  MATH  Google Scholar 

  7. Malanowski K (1984) Differential stability of solutions to convex, control constrained optimal control problems. Appl Math Optim 12:1–14

    Article  MathSciNet  MATH  Google Scholar 

  8. Malanowski K (1987) Stability and sensitivity of solutions to optimal control problems for systems with control appearing linearly. Appl Math Optim 16:73–91

    Article  MathSciNet  MATH  Google Scholar 

  9. Malanowski K (1987) Stability of Solutions to Convex Problems of Optimization. Lecture Notes in Control and Information Sciences, Vol 93. Springer-Verlag, Berlin

    Google Scholar 

  10. Malanowski K (1988) On stability of solutions to constrained optimal control problems for systems with control appearing linearly. Arch Automat Telemech 33:483–498

    MATH  Google Scholar 

  11. Malanowski K, Sokołowski J (1986) Sensitivity of solutions to convex constrained optimal control problems for distributed parameter systems. J Math Anal Appl 120:240–263

    Article  MathSciNet  MATH  Google Scholar 

  12. Maurer H (1979) Differential stability of optimal control problems. Appl Math Optim 5:63–82

    Article  MathSciNet  Google Scholar 

  13. Maurer H (1982) First- and second-order sufficient optimality conditions in mathematical programming and optimal control. Math Programming Stud 14:163–177

    Article  MathSciNet  MATH  Google Scholar 

  14. Maurer H, Zowe J (1979) First- and second-order sufficient optimality conditions for infinitedimensional programming problems. Math Programming 16:98–110

    Article  MathSciNet  MATH  Google Scholar 

  15. Mignot F (1976) Côntrol dans les inequations variationelles. J Funct Anal 22:130–185

    Article  MATH  Google Scholar 

  16. Outrata JV, Schindler Z (1981) An augmented Lagrangian method for a class of convex continuous optimal control problems. Problems Control Inform Theory 10:67–81

    MathSciNet  MATH  Google Scholar 

  17. Robinson SM (1980) Strongly regular generalized equations. Math Oper Res 5:43–62

    Article  MathSciNet  MATH  Google Scholar 

  18. Rockafellar RT (1984) Directional differentiability of the optimal value function in a nonlinear programming problem. Math Programming Stud 21:213–226

    Article  MathSciNet  MATH  Google Scholar 

  19. Shapiro A (1984) Second-order sensitivity analysis and asymptotic theory of parametrized nonlinear programs. Math Programming 33:280–299

    Article  MathSciNet  MATH  Google Scholar 

  20. Sokołowski J (1985) Differential stability of control constrained optimal control problems for distributed parameter systems. In: Kappel F, Kunisch K, Schappacher W (eds) Distributed Parameter Systems. Lecture Notes in Control and Information Sciences, Vol 75. Springer-Verlag, Berlin

    Google Scholar 

  21. Sokołowski J (1987) Sensitivity analysis of control constrained optimal control problems for distributed parameter systems. SIAM J Control Optim 25:1542–1556

    Article  MathSciNet  MATH  Google Scholar 

  22. Sokołowski J (1988) Shape sensitivity analysis of boundary optimal control problems for parabolic systems. SIAM J Control Optim 26:763–787

    Article  MathSciNet  MATH  Google Scholar 

  23. Wierzbicki A, Kurcyusz S (1977) Projection on a cone, penalty functionals and duality theory for problems with inequality constraints in Hilbert space. SIAM J Control Optim 15:25–56

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447, Warszawa, Poland

    Kazimierz Malanowski

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  1. Kazimierz Malanowski
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Communicated by J. Stoer

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Malanowski, K. Sensitivity analysis of optimization problems in Hilbert space with application to optimal control. Appl Math Optim 21, 1–20 (1990). https://doi.org/10.1007/BF01445154

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  • DOI: https://doi.org/10.1007/BF01445154

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