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Subgradients of the Value Function to a Parametric Optimal Control Problem

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Abstract

This paper studies the first-order behavior of the value function of a parametric optimal control problem with linear constraints and nonconvex cost functions. By establishing an abstract result on the Fréchet subdifferential of the value functions of a parametric mathematical programming problem, a new formula for computing the Fréchet subdifferential of the value function to a parametric optimal control problem is obtained.

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References

  1. 1.

    Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control, Consultants Bureau. New York and London (1987)

  2. 2.

    Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

  3. 3.

    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)

  4. 4.

    Cernea, A., Frankowska, H.: A connection between the maximum principle and dynamic programming for constrained control problems. SIAM J. Control Optim. 44, 673–703 (2005)

  5. 5.

    Cesari, L.: Optimization Theory and Application. Springer, New York (1983)

  6. 6.

    Clarke, F.H.: Method of Dynamic and Nonsmooth Optimization. SIAM, Philadelphia (1989)

  7. 7.

    Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)

  8. 8.

    Dacorogna, B.: Introduction to the Calculus of Variations. Imperial College Press, London (2004)

  9. 9.

    Ioffe, A.D., Tihomirov, V.M.: Theorey of Extremal Problems. North-Holand, Amsterdam (1979)

  10. 10.

    Ioffe, A.D.: Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. Trans. Anc. Monum. Soc. 349, 2871–2900 (1997)

  11. 11.

    Kien, B.T., Liou, Y.C., Wong, N.-C., Yao, J.-C.: Subgradients of value functions in parametric dynamic programming. Eur. J. Oper. Res. 193, 12–22 (2009)

  12. 12.

    Mordukhovich, B.S.: Metric approximations and necessary optimality conditions for general classes of extremal problems. Sov. Math., Dokl. 22, 526–530 (1980)

  13. 13.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I, Basis Theory. Springer, New York (2006)

  14. 14.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II, Applications. Springer, New York (2006)

  15. 15.

    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009)

  16. 16.

    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Fréchet subdifferential calculus and optimality conditions in nondifferetiable programming. Optimization 55, 685–708 (2006)

  17. 17.

    Mordukhovich, B.S., Nam, N.M.: Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30, 800–816 (2005)

  18. 18.

    Moussaoui, M., Seeger, A.: Sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets. SIAM J. Optim. 4, 659–675 (1994)

  19. 19.

    Moussaoui, M., Seeger, A.: Epsilon-maximum principle of Pontryagin type and perturbation analysis of convex optimal control problems. SIAM J. Control Optim. 34, 407–427 (1996)

  20. 20.

    Penot, J.-P.: Differetiability properties of optimal value functions. Can. J. Math. 56, 825–842 (2004)

  21. 21.

    Rockafellar, R.T., Wolenski, P.R.: Convexity in Hamilton-Jacobi theory I: dynamics and duality. SIAM J. Control Optim. 39, 1323–1350 (2000)

  22. 22.

    Rockafellar, R.T., Wolenski, P.R.: Convexity in Hamilton-Jacobi theory II: envelope representation. SIAM J. Control Optim. 39, 1351–1372 (2000)

  23. 23.

    Rockafellar, R.T.: Hamilton-Jacobi theory and parametric analysis in fully convex problems of optimal control. J. Glob. Optim. 248, 419–431 (2004)

  24. 24.

    Seeger, A.: Subgradient of optimal-value function in dynamic programming: the case of convex system without optimal paths. Math. Oper. Res. 21, 555–575 (1996)

  25. 25.

    Vinter, R.B.: Optimal Control. Birkhäuser, Boston (2000)

  26. 26.

    Vinter, R.B., Zheng, H.: Necessary conditions for optimal control problems with state constraints. Trans. Anc. Monum. Soc. 350, 1181–1204 (1998)

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Correspondence to B. T. Kien.

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Toan, N.T., Kien, B.T. Subgradients of the Value Function to a Parametric Optimal Control Problem. Set-Valued Anal 18, 183–203 (2010). https://doi.org/10.1007/s11228-009-0125-0

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Keywords

  • Parametric optimal control
  • Marginal function
  • Value function
  • Fréchet normal cone
  • Fréchet subgradient
  • Fréchet subdifferential
  • Coderivative

Mathematics Subject Classifications (2000)

  • 47J20
  • 49J40
  • 49J53
  • 90C33