Geodesic
Hessian of the solution of the Hamilton–Jacobi equation in the theory of extremal problems
Sbornik. Mathematics, Tome 195 (2004) no. 6, pp. 819-831 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An optimal control problem with separated conditions at the end-points is studied. It is assumed that there exists on the manifold of left end-points (as well as on the manifold of right end-points) a field of extremals containing the fixed extremal. A criterion describing necessary and sufficient conditions of optimality in terms of these two fields is proved. The sufficient condition is the positive-definiteness of the difference of the solutions of the corresponding matrix Riccati's equations and the necessary one is its non-negativity. The key part in the proof of the criterion is played by a formula relating the solution of Riccati's equation and the Hessian of the Bellman function. 
@article{SM_2004_195_6_a2,
     author = {M. I. Zelikin},
     title = {Hessian of the solution of the {Hamilton{\textendash}Jacobi} equation in the theory of~extremal problems},
     journal = {Sbornik. Mathematics},
     pages = {819--831},
     year = {2004},
     volume = {195},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2004_195_6_a2/}
}
TY  - JOUR
AU  - M. I. Zelikin
TI  - Hessian of the solution of the Hamilton–Jacobi equation in the theory of extremal problems
JO  - Sbornik. Mathematics
PY  - 2004
SP  - 819
EP  - 831
VL  - 195
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2004_195_6_a2/
LA  - en
ID  - SM_2004_195_6_a2
ER  - 
%0 Journal Article
%A M. I. Zelikin
%T Hessian of the solution of the Hamilton–Jacobi equation in the theory of extremal problems
%J Sbornik. Mathematics
%D 2004
%P 819-831
%V 195
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2004_195_6_a2/
%G en
%F SM_2004_195_6_a2
M. I. Zelikin. Hessian of the solution of the Hamilton–Jacobi equation in the theory of extremal problems. Sbornik. Mathematics, Tome 195 (2004) no. 6, pp. 819-831. http://geodesic.mathdoc.fr/item/SM_2004_195_6_a2/

[1] Zelikin M. I., Odnorodnye prostranstva i uravnenie Rikkati v variatsionnom ischislenii, Faktorial, M., 1998 | Zbl

[2] Zelikin M. I., Borisov V. F., Theory of chattering control with applications to astronautics, robotics, economics, and engineering, Birkhäuser, Boston, 1994 | MR | Zbl

[3] Osmolovskii N. P., Lempio F., “Transformation of quadratic forms to perfect squares for broken extremals”, Set-Valued Anal., 10 (2002), 209–232 | DOI | MR | Zbl

[4] Osmolovskii N. P., Lempio F., “Uslovie Yakobi i uravnenie Rikkati dlya lomanoi ekstremali”, Sovremennaya matematika i ee prilozheniya. Tematicheskie obzory, 60, VINITI, M., 1999, 187–215