Part 1

Part 2:

Lecture 9: Strong and weak minima. Jacobi's sufficient condition. (1/3 Links to an external site., 2/3 Links to an external site., 3/3 Links to an external site.)

Lecture 10: Weierstrass necessary condition. (1/3 Links to an external site., 2/3 Links to an external site. , 3/3 Links to an external site.)

Lecture 11: Weierstrass sufficient condition. (1/4 Links to an external site., 2/4 Links to an external site., 3/4 Links to an external site., 4/4 Links to an external site.)

Suggested exercises Download Suggested exercises

Lecture 12: Hamilton-Jacobi theory, Bellman equation, Pontryagin maximum principle.  (1/2 Links to an external site., 2/2 Links to an external site.)

Lecture 13: Jacobi's equation again, Jacobi condition and a field of extremals. Global minimum and convexity. (1/4 Links to an external site., 2/4 Links to an external site., 3/4 Links to an external site., 4/4 Links to an external site.)

Lecture 14: Applications. (1/4 Links to an external site., 2/4 Links to an external site., 3/4 Links to an external site., 4/4 Links to an external site.)