ingenieur- mathematik constantin carathéodory, bellman‘s equation and the maximum principle hans...
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Ingenieur-Mathematik
Constantin Carathéodory,
Bellman‘s Equation and the Maximum Principle
Hans Josef Pesch
University of Bayreuth, Germany
50 Years of Optimal Control, Bedlewo, Poland, Sept. 15-19, 2008
Ingenieur-Mathematik
Outline
• Carathéodory‘s Royal Road of the Calculus of Variations,
• „his“ Bellman‘s Equation,
• and his Precursor of the Maximum Principle
Hans Josef Pesch, Roland Bulirsch: The Maximum Principle, Bellman‘s Equation, and Carathéodory‘s WorkJ. of Optimization Theory and Applications, Vol. 80, No. 2, Feb. 1994
Ingenieur-Mathematik
Carathéodry‘s Royal Way in the Calculus of Variations
Relationship between
and
allows the reduction of
to
Hilbert‘s Independence Theorem
Hamilton-Jacobi Equation
Problems of the Calculus of Variations
Problems of Finite Optimization
Ingenieur-Mathematik
Formulation of an equaivalent variational problem (2)
Let
Then: integration along two curves yields
Thus
and therefore any line element where
can be passed by one and only one extremal curve
and
will be needed for the Legendre-Clebsch condition
Ingenieur-Mathematik
Existence of extremals for a special variational problem
If there exists
with
for all and all with
then there holds: The solutions of are extremals of
Ingenieur-Mathematik
Theorem: sufficient condition (Crathéodory, 1931)
If there exists
for which there hold
and
for sufficiently small , then the solutions of
yield
Ingenieur-Mathematik
Crathéodory‘s Fundamental Equations (1931)
Hence we have to determine the functions
such that
(as function of ) possesses a minimum for
with value
(Carathéodory, 1935)
That is the so-called Bellman Equation
or
Thus
No imbedding or extremal fields on Carathéodory‘s Royal Road
Ingenieur-Mathematik
Carathéodory‘s formulation of Weierstraß‘ Excess Function
Substituting the fundamental equations and replacing by yields
Hence we obtain the necessary condition of Weierstraß
Ingenieur-Mathematik
Carathéodory‘s precursor of the Maximum Principle (1926)
Introducing canonical variables
and solving this equation for yields
Defining the Hamiltonian
yields
Ingenieur-Mathematik
Lagrangian variational problems
Side conditions (Lagrangian problems)
Similarly
Introducing
the fundamental equations takes the form
(Carathéodory: 1926)
with
Defining
the Weierstraß necessary condition takes the form
Ingenieur-Mathematik
Carathéodory‘s precursor of the Maximum Principle (1926)
From
there is only a little step via
to
with
s.t.
Ingenieur-Mathematik
The Maximum Principle (precursor, 1926)
Constantin Carathéodory (Κωνσταντίνος Καραθεοδωρή)* Sept. 13, 1873 in Berlin; † Feb. 2, 1950, Munich
I will be glad if I have succeeded in impressing the idea that it is not only pleasant and entertainingto read at times the works of the old mathematicialauthors, but that this may occasionally be of usefor 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 haveseen that even under conditions which seem mostfavorable very important results can be discardedfor a long time and whirled away from the main streamwhich is carrying the vessel science. …
If their ideas are too far in advance of their time, andif the general public is not prepared to accept them, these ideas may sleep for centuries on the shelvesof our libraries … awaiting the arrival of the prince charming who will take them home. (C.C. 1937)
Ingenieur-Mathematik
• 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)
Constantin Carathéodory (1873 - 1950)
Ingenieur-Mathematik
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)
Ingenieur-Mathematik
Magnus Rudolph Hestenes (1906 – May 31, 1991)
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.
The Maximum Principle (first formulation, 1950)
Ingenieur-Mathematik
Richard Ernest Bellman (Aug. 26, 1920 – March 19, 1984)
Rufus Philip Isaacs (1914 – 1981)
The Maximum Principle (Bellman‘s & Isaacs‘ Equation, 1951+)
Ingenieur-Mathematik
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.
Ingenieur-Mathematik
Lev Semenovich Pontryagin (Лев Семёнович Понтрягин) (Sept. 3, 1908 – May 3. 1988)
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
Ingenieur-Mathematik
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.
Ingenieur-Mathematik
Thank you for your attention!
The referenced paper can be downloaded fromwww.uni-bayreuth.de/departments/ingenieurmathematik
Email: [email protected]