Carathéodory‘s Royal Road of the Calculus of Variations: Missed Exits to the Maximum Principle

1. Carathéodory‘s Royal Road of the Calculus of Variations: Missed Exits to the Maximum Principle of Optimal Control Theory Hans Josef Pesch University of Bayreuth, Germany 14th Inter. Symp. on Dynamic Games 2010, Banff, Canada, June 20-23, 2010

2. Outline • Carathéodory‘s Royal Road of the Calculus of Variations: • his Bellman‘s Equation, • his Precursor of the Maximum Principle, • his closest approch to optimal control AND • Leitmann‘s equivalent approach Continuation of Ref.: Hans Josef Pesch, Roland Bulirsch: The Maximum Principle, Bellman‘s Equation, and Carathéodory‘s Work J. of Optimization Theory and Applications, Vol. 80, No. 2, Feb. 1994

3. Carathéodry‘s Royal Road in the Calculus of Variations Relationship between Hilbert‘s Independence Theorem and Hamilton-Jacobi Equations allows the reduction of Problems of the Calculus of Variations to Problems of Finite Optimization

4. Variations of Simple variational problems

5. Variation of the integral Legendre-Clebsch condition or Special variation

6. Formulation of equaivalent variational problems (1) Let then independent of

7. and and therefore any line element where will be needed for the Legendre-Clebsch condition can be passed by one and only one extremal curve Formulation of equaivalent variational problems (2) Let Then: integration along two curves yields Thus

8. for all and all with then there holds: The solutions of are extremals of Existence of extremals for a special variational problem If there exists with

9. for sufficiently small , then the solutions of Theorem: sufficient condition (Carathéodory, 1931) If there exists for which there hold and yield

10. or possesses a minimum for (as function of ) with value Thus (Carathéodory, 1935) Carathéodory‘s Fundamental Equations (1931) Hence we have to determine the functions such that No imbedding or extremal fields on Carathéodory‘s Royal Road That is the so-called Bellman Equation

11. Carathéodory‘s formulation of Weierstraß‘ Excess Function Substituting the fundamental equations and replacing by yields Hence we obtain the necessary condition of Weierstraß

12. Carathéodory‘s precursor of the Maximum Principle (1926) Introducing canonical variables and solving this equation for yields Defining the Hamiltonian yields Missed exit?

13. Introducing the Lagrange function the fundamental equations take the form (Carathéodory: 1926) Lagrangian variational problems Side conditions (Lagrangian problems) Similarly

14. on the sphere must be positive. Hence Lagrangian variational problems Legendre-Clebsch condition: The minimum of the quadratic form subject to the constraint

15. Lagrangian variational problems Because of the equations can be solved for

16. respectively Lagrangian variational problems Defining the Hamiltonian Carathéodory‘s closed approach to optimal control (from 1935) degree of freedom: control with the Hamiltonian degree of freedom: control? Missed exit?

17. we obtain as long as By means of the Euler-Lagrange equation canonical equations and because of Exit to the maximum principle (1) Defining the maximizing Hamiltonian

18. Hence, must have a maximum with respect to along a curve From here it is only a „little“ step to Exit to the maximum principle (2) Furthermore Missed exit?

19. The Maximum Principle (precursor, 1926) I will be glad if I have succeeded in impressing the idea that it is not only pleasant and entertaining to read at times the works of the old mathematicial authors, but that this may occasionally be of use for the actual advancement of science. Besides this there is a great lesson we can derive from the facts which I have just referred to. We have seen that even under conditions which seem most favorable very important results can be discarded for a long time and whirled away from the main stream which is carrying the vessel science. … If their ideas are too far in advance of their time, and if the general public is not prepared to accept them, these ideas may sleep for centuries on the shelves of our libraries … awaiting the arrival of the prince charming who will take them home. (C.C. 1937) Constantin Carathéodory (Κωνσταντίνος Καραθεοδωρή) * Sept. 13, 1873 in Berlin; † Feb. 2, 1950, Munich

20. Leitmann‘s Equivalent Problem Approach (1967, 2001, 2008) explained to the President of Austria, Heinz Fischer, May 2010

21. and a diffeomorphism with and the operator by Leitmann‘s Equivalent Problem Approach (1967, 2001, 2008) Ref.: Wagener, 2009 Take the set Define the set

22. are Leitmann equivalent by if there exist a -function with Idea: simplifies the variational problem Leitmann‘s Equivalent Problem Approach (1967, 2001, 2008) Definition: The functionals and The two functionals are Carathéodory equivalent if

23. Constantin Carathéodory (1873 - 1950) • Born in Berlin to Greek parents, grew up in Brussels (father was the Ottoman ambassador) to Belgium • The Carathéodory family was well-respected in Constantinople (many important governmental positions) • Formal schooling at a private school in Vanderstock (1881-83); travelling with is father to Berlin, Italian Riviera; grammar school in Brussels (1985); high school Athénée Royal d'Ixelles, graduation in 1891 • Twice winning of a prize as the best mathematics student in Belgium • Trelingual (Greek, French, German), later: English, Italian, Turkish, and the ancient languages • École Militaire de Belgique (1891-95), École d'Application (1893-1896): military engineer • War between Turkey and Greece (break out 1897); British colonial service: construction of the Assiut dam (until 1900); Studied mathematics: Jordan's Cours d'Analyse a.o.; Measurements of Cheops pyramid (published in 1901)

24. Constantin Carathéodory (1873 - 1950) • Graduate studies at the University of Göttingen (1902-04) (supervision of Hermann Minkowski: dissertation in 1904 (Oct.) on Diskontinuierliche Lösungen der Variationsrechnung • In March 1905: venia legendi (Felix Klein) • Various lecturing positions in Hannover, Breslau, Göttingen and Berlin (1909-20) • Prussian Academy of Sciences (1919, together with Albert Einstein) • Plan for the creation of a new University in Greece (Ionian University) (1919, not realized due to the War in Asia Minor in 1922); the present day University of the Aegean claims to be the continuation • University of Smyrna (Izmir), invited by the Greek Prime Minister (1920); (major part in establishing the institution, ends in 1922 due to war • University on Athens (until 1924) • University of Munich (1924-38/50); Bavarian Academy of Sciences (1925) • C. played a remarkable opposing role together with the Munich „Dreigestirn“ (triumvirate) (Perron, Tietze) within the Bavarian Academy of Science during the Nazi terror in Germany

25. The Maximum Principle (first formulation, 1950) Thus, has a maximum value with respect to along a minimizing curve . Research Memorandum RM-100, Rand Corporation, 1950 I became interested in control theory in 1948. At that time I formulated the general control problem of Bolza …, and observed the maximum principle … is equivalent to the conditions of Euler-Lagrange and Weierstrass … It turns out that I had formulated what is now known as the general optimal control problem. Magnus Rudolph Hestenes (1906 – May 31, 1991)

26. The Maximum Principle (Bellman‘s & Isaacs‘ Equation, 1951+) Richard Ernest Bellman (Aug. 26, 1920 – March 19, 1984) Rufus Philip Isaacs (1914 – 1981)

27. Isaacs in 1973 about his Tenet of Transition of 1951 Once I felt that here was the heart of the subject ….. Later I felt that it … was a mere truism. Thus in (my book) Differential Games it is mentioned only by title. This I regret. I had no idea, that Pontryagin‘s principle and Bellman‘s maximal principle (a special case of the tenet, appearing a little later in the Rand seminars) would enjoy such a widespread citation.

28. The Maximum Principle (1956) This fact is a special case of the following general principle which we call maximum principle Doklady Akademii Nauk SSSR, Vol. 10, 1956 Lev Semenovich Pontryagin (Лев Семёнович Понтрягин) (Sept. 3, 1908 – May 3. 1988)

29. The Maximum Principle (1956) Vladimir G. Boltyanski Revaz V. Gamkrelidze

30. Boltyanski in 1991 about the Maximum Principle of 1956 By the way, the first statement of the maximum principle was given by Gamkrelidze, who has established (generalizing the famous Legendre Theorem) a sufficient condition for a sort of weak optimality problem. Then, Pontryagin proposed to name Gamkrelidze‘s condition Maximum Principle. … Finally, I understood that the maximum principle is not a sufficient, but only a necessary condition of optimality. Pontryagin was the Chairman of our department at the Steklov Mathematical Institute, and he could insist on his interests. So, I had to use the title Pontryagin‘s Maximum Principle in my paper. This is why all investigators in region of mathematics and engineering know the main optimization criterium as the Pontryagin‘s Maximum Principle.

31. Gamkrelidze in 2008 about Pontryagin My life was a series of missed opportunities, but one opportunity, I have not missed, to have met Pontryagin.* In respect hereof, the term “missed opportunity” has to be understood as reminiscence to that discussion. The author does not intend to use this term as if he would like to criticize that certain things should have be seen earlier. * at the Banach Center Conference on 50 Years of Optimal Control in Bedlewo, Poland, on September 15, 2008

32. Many Happy Returns, George! We all love You!

33. Thank you for your attention! The 1994 paper and a new version can be downloaded from www.ingmath.uni-bayreuth.de/ Email: hans-josef.pesch@uni-bayreuth.de