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 
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/
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     title = {Hessian of the solution of the {Hamilton{\textendash}Jacobi} equation in the theory of~extremal problems},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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