Hessian of the solution of the Hamilton–Jacobi equation in the theory of extremal problems

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.