Geodesic
Convex functions occurring in variational problems and the absolute minimum problem
Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 191-208 
A. D. Ioffe. Convex functions occurring in variational problems and the absolute minimum problem. Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 191-208. http://geodesic.mathdoc.fr/item/SM_1972_17_2_a1/
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     title = {Convex functions occurring in variational problems and the absolute minimum problem},
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Voir la notice de l'article provenant de la source Math-Net.Ru

For the minimum problem of the functional $\int_{(a,\,x^0)}^{(b,\,x^1)}f(t,x(t),\dot x(t))\,dt$ (where $f(t,x,u)\colon T\times R^n\times R^n\to(-\infty,\infty)$, and the case $f=\infty$ corresponds to some constraints imposed on $x$ and $u$) we consider the problem of the existence of a function $\varphi(t,x)\colon T\times\ R^n\to R$ which has the following property: if $x_m(t)$ is a minimizing sequence, then, for any $\alpha$ and $\beta$ wich $a\leqslant\alpha<\beta\leqslant b$, and for any $x(t)$, \begin{multline*} \widetilde\varphi(\beta,x(\beta))-\varphi(\alpha,x(\alpha))-\int_\alpha^\beta f(t,x(t),\dot x(t))\,dt\\ \leqslant\varliminf\biggl[\varphi(\beta,x_m(\beta))-\varphi(\alpha,x_m(\alpha))-\int_\alpha^\beta f(t,x_m(t),\dot x_m(t))\,dt\biggr] \end{multline*} (every function $\varphi$ which has this property yields a necessary condition for the absolute minimum). We prove existence criterions for an arbitrary and continuous function $\varphi$. Bibliography: 9 titles.

[1] L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders, Philadelphia, 1969 | MR | Zbl

[2] A. D. Ioffe, V. M. Tikhomirov, “Dvoistvennost vypuklykh funktsii i ekstremalnye zadachi”, Uspekhi matem. nauk, XXIII:6(144) (1968), 51–117 | MR

[3] V. F. Krotov, “Metody variatsionnogo ischisleniya, osnovannye na dostatochnykh usloviyakh absolyutnogo minimuma”, Avtomatika i telemekhanika, 24:12 (1962), 1571–1583 ; 25:5 (1963), 581–598 | MR | MR

[4] A. D. Ioffe, “Preobrazovaniya variatsionnykh zadach”, Izv. AN SSSR, Tekhn. kibernetika, 1967, no. 4, 15–21 | MR

[5] V. F. Krotov, V. 3. Bukreev, V. I. Gurman, Novye metody variatsionnogo ischisleniya v dinamike poleta, Mashinostroenie, Moskva, 1969

[6] V. I. Gurman, Dissertatsiya, Moskva, 1969

[7] A. D. Ioffe, V. M. Tikhomirov, “Rasshireniya variatsionnykh zadach”, Trudy Mosk. matem. ob-va, XVIII, 1968, 188–246

[8] A. D. Ioffe, “Obobschennye resheniya sistem s upravleniem”, Diff. uravneniya, 5:6 (1969), 1010–1017 | MR | Zbl

[9] N. Burbaki, Topologicheskie vektornye prostranstva, IL, Moskva, 1959