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On the sensitivity and relaxability of optimal control problems governed by nonlinear evolution equations with state constraints

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Abstract

In this work we examine the relation between the sensitivity (well posedness) and relaxability for nonlinear distributed parameter systems. We introduce the notion of “strong calmness” which describes the dependence of the value of the problem on the state constraints, and we show that it is equivalent to “relaxability”. We also present an alternative control free description of the relaxed problem. An example of a nonlinear parabolic optimal control problem of Lagrange type is worked out in detail.

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References

  1. Ahmed, N. U., Teo, K. L.: Optimal Control of Distributed Parameter Systems. New York: North-Holland. 1981.

    Google Scholar 

  2. Balder, E.: Lower semicontinuity of integral functionals with nonconvex integrands by relaxation-compactification. SIAM J. Control and Optim.19, 531–541 (1981).

    Google Scholar 

  3. Balder, E.: A general denseness result for relaxed control theory. Bull. Austral. Math. Soc.30, 463–475 (1984).

    Google Scholar 

  4. Cesari, L.: Optimization — Theory and Applications. New York: Springer. 1983.

    Google Scholar 

  5. Choquet, G.: Lectures on Analysis. Vol. I. New York: Benjamin. 1969.

    Google Scholar 

  6. Clarke, F.: Admissible relaxation in variational and control problems. J. Math. Anal. Appl.51, 557–576 (1975).

    Google Scholar 

  7. Clarke, F.: Optimization and Nonsmooth Analysis New York: Wiley. 1983.

    Google Scholar 

  8. De Blasi, F., Pianigiani, G.: Remarks on Hausdorff continues multifunctions and selections. Comm. Math. Univ. Carol.24, 553–562 (1983).

    Google Scholar 

  9. Edgar, G.: Measurability in a Banach space II. Indiana Univ. Math. J.28, 559–579 (1979).

    Google Scholar 

  10. Ioffe, A., Tichomirov, V.: Theory of Extremal Problems. Amsterdam: North Holland. 1979.

    Google Scholar 

  11. Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer. 1971.

    Google Scholar 

  12. Michael, E.: Continuous selections. Annals of Math.63, 361–383 (1956).

    Google Scholar 

  13. Moreau, J.-J.: Intersection of moving convex sets in a normed space. Math. Scand.36, 159–173 (1975).

    Google Scholar 

  14. Pagageorgiou, N. S.: Properties of the relaxed trajectories of evolution equations and optimal control. SIAM J. Control and Optim.27 (2) (1989) 267–289.

    Google Scholar 

  15. Papageorgiou, N. S.: Existence of optimal controls for nonlinear distributed parameter systems. Funkc. Ekvacioj. To appear.

  16. Papageorgiou, N. S.: Optimal control of evolution inclusions. J. Optim. Theory and Appl. To appear.

  17. Rockafellar, R. T.: Conjugate Duality and Optimization. Philadelphia: SIAM Publ. 1973.

    Google Scholar 

  18. Tanabe, H.: Equations of Evolution. London: Pitman. 1979.

    Google Scholar 

  19. Wagner, D.: Survey of mesurable selection theorems. SIAM J. Control and Optim.15, 857–903 (1977).

    Google Scholar 

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Authors and Affiliations

  1. School of Technology Mathematics Division, University of Thessaloniki, 54006, Thessaloniki, Greece

    Evgenios P. Avgerinos

  2. Department of Mathematics, University of California, 95616, Davis, CA, U.S.A.

    Nikolaos S. Papageorgiou

Authors
  1. Evgenios P. Avgerinos
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  2. Nikolaos S. Papageorgiou
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Research supported by N.S.F. Grant D.M.S.-8802688.

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Avgerinos, E.P., Papageorgiou, N.S. On the sensitivity and relaxability of optimal control problems governed by nonlinear evolution equations with state constraints. Monatshefte für Mathematik 109, 1–23 (1990). https://doi.org/10.1007/BF01298849

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  • DOI: https://doi.org/10.1007/BF01298849

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