| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
13 Sep 2005 12:50:00 AM |
| Object: |
Control Invariance Generalization of Noether |
From Osher Doctorow
COPYRIGHT NOTICE
Control Invariance Generalization of Noether
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
My recent threads on Invariance-Intersection and even Black Holes
isolated simplicity of (Optimal) Control, especially of simultaneous
forces in multiple different directions, as key to (Probable)
Causation/Influence.
This conclusion in somewhat different terminology is similar to recent
work from another direction, the Optimal Control mathematicians in
Front for the Mathematics arXiv, especially Delfim F. M. Torres of U.
of Aveiro, Aveiro, Portugal, Paulo D. F. Gouveia of the same
Mathematics Dept. (both are in Control Theory Group (cotg)), I. K.
Gogodze of Applied Mathematics, U. Tbilisi, Russia, earlier work of D.
S. Djukic (1973). One reference of Torres' 2002 paper is J. (John) C.
Baez and J. W. Gilliam (1994) on discrete mechanics ("An algebraic
approach to discrete mechanics," Lett. Math. Phys. 31(3): 205-212).
Gouveia and Torres' most recent paper on the topic is "Automatic
computation of conservation laws in the calculus of variations and
optimal control," math.OC/0509140 v1 7 Sep 2005, which refers most
directly back to Torres' math.OC/0206230 v1 21 Jun 2002. Torres has
very many papers. OC refers to Optimal Control.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Control Invariance Generalization of Noether |
13 Sep 2005 02:02:25 AM |
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From Osher Doctorow
Optimal control problems are related by duality to the Kalman
filter-predictor, which readers may be familiar with as an important
way of tracking and controlling and even predicting artificial
satellite and spacecraft motion. It is exceptionally difficult
mathematically, physically, and in regarding to engineering. The main
relevant theorem at the foundational level is the Pontryagin/Pontrjagin
maximum principle, a generalization of the classical Euler-Lagrange and
Weierstrass optimality necessary condition of calculus of variations,
which involves 3 steps 2 of which are exceptionally difficult if not
impossible to implement. Torres' et al's way is to find conservation
laws preserved along the extremals of the problem to simplify the
problem (remember my threads' emphasis on simplicity). In trying to
find out how to get these conservation laws, the classic results of
Emmy Noether in the calculus of variations turn out to have generalized
into optimal control by D. S. Djukic (Internat. J. Control, 1 (1973)
No. 18, 667-672, by I. K Gogodze, Proc. of extended sessions of seminar
of the Vekua Institute of Applied Mathematics, Tbilisi U., Tbilisi,
Russia (1998), No. 3, 39-42, and by D. F. M. Torres "On the Noether
theorem for optimal control," European Journal of Control 8 (3003), No.
1, 56-63, which somewhat overlaps Torres' 2002 math.OC/0206230 v1 21
Jun 2002, "Caratheodory-Equivalence, Noether theorems, and Tonelli
full-regularity in the calculus of variations and optimal control."
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Control Invariance Generalization of Noether |
13 Sep 2005 02:19:52 AM |
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From Osher Doctorow
Torres came up with a technique different from the classical one, in
which one does not need to use transversality conditions, and therefore
in which one does not know boundary conditions. The idea is to
generalize quasi-invariance (which I'll try to discuss later) in a
Noether theorem with no transformation of the time variable, from one
argument to two transformation arguments. The one argument case
implies constancy of an expression in time t along every Pontryagin
extremal, and the theorems which generalize Noether's theorems to
optimal control involve the two argument genralization of the above.
Osher Doctorow
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