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Jacobi condition for elliptic forms in Hilbert spaces

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Abstract

In this paper, we give the definitions of conjugate and semi-conjugate points for a quadratic elliptic form in a Hilbert space, and we state the corresponding necessary and sufficient conditions (the Jacobi conditions) for the form to be positive (nonnegative). We apply the abstract results to a one-dimensional problem in the calculus of variations where both endpoints are allowed to vary. The conditions that we obtain complete and generalize previously known results.

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References

  1. Hestenes, M. R.,Applications of the Theory of Quadratic Forms in Hilbert Spaces to the Calculus of Variations, Pacific Journal of Mathematics, Vol. 1, pp. 525–581, 1951.

    Google Scholar 

  2. Dennemeyer, R.,Conjugate Surfaces for Multiple Integral Problems in the Calculus of Variations, Pacific Journal of Mathematics, Vol. 30, pp. 621–638, 1969.

    Google Scholar 

  3. Mikami, E. Y.,Focal Points in a Control Problem, Pacific Journal of Mathematics, Vol. 35, pp. 473–485, 1970.

    Google Scholar 

  4. Zeidan, V., andZezza, P.,Coupled Points in the Calculus of Variations and Applications to Periodic Problems, Transactions of the American Mathematical Society, Vol. 1, pp. 323–335, 1989.

    Google Scholar 

  5. Zeidan, V., andZezza, P.,Coupled Points in Optimal Control, IEEE Transactions on Automatic Control, Vol. 36, pp. 1276–1281, 1991.

    Google Scholar 

  6. Possehl, G.,Coupled Points for Infinite-Dimensional Control Problems, Preprint, 1989.

  7. Berquier, F.,Point Focaux, Problème de Bolza, et Controle Optimal, Comptes Rendus des Séances de l'Academie des Sciences, Paris, Serie Matématique, Vol. 309, pp. 687–690, 1989.

    Google Scholar 

  8. Dmitruk, A. V.,Jacobi-Type Conditions for the Problem of Bolza with Inequalities, Matematicheskie Zametki, Vol. 35, pp. 813–827, 1984.

    Google Scholar 

  9. Gregory, J.,Quadratic Form Theory and Differential Equations, Academic Press, New York, New York, 1980.

    Google Scholar 

  10. Cesari, L.,Optimization: Theory and Applications, Springer-Verlag, New York, New York, 1983.

    Google Scholar 

  11. Mawhin, J., andWillem, M.,Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, New York, 1989.

    Google Scholar 

  12. Zeidan, V., andZezza, P.,An Extension of the Conjugate Point Condition to the Case of Variable Endpoints, Proceedings of the 27th IEEE Conference on Decision and Control, Austin, Texas, pp. 1187–1191, 1988.

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Authors and Affiliations

  1. Dipartimento di Sistemi e Informatica, University of Florence, Florence, Italy

    P. Zezza (Professor)

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  1. P. Zezza
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Additional information

Communicated by R. Conti

This research was partly supported by MURST Research Grant, Teoria del Controllo dei Sistemi Dinamici.

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Zezza, P. Jacobi condition for elliptic forms in Hilbert spaces. J Optim Theory Appl 76, 357–380 (1993). https://doi.org/10.1007/BF00939612

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  • DOI: https://doi.org/10.1007/BF00939612

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