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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This research was partly supported by MURST Research Grant, Teoria del Controllo dei Sistemi Dinamici.
Communicated by R. Conti
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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
- Conjugate points
- Jacobi condition
- necessary and sufficient optimality conditions