2_Από Τον Λογισμό Των Μεταβολών Στην Θεωρία Βέλτιστου...
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Transcript of 2_Από Τον Λογισμό Των Μεταβολών Στην Θεωρία Βέλτιστου...
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. , , . , . Abtsract. Optimal Control Theory has its roots in the development of a major field of Mathematics: the Calculus of Variations. The main scope of this work is twofold: a) to trace the development of the calculus of variations, and b) to refer to the primary target of the optimal control theory.
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: ( )0, aA y ( ),b bB x y ( ) 0,0 by x x x= . ( ),0dD x ADB
,.
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( , )b bB x y( )0, aA y
( ), 0dD x :
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Galileo Galilei (1564-1642) 1638 Discorsi e dimostrazioni matematiche, intorno a due nuove scienze ( ) 2 . . (C CB) . . , Galileo , . Galileo , .
Galileo Galilei (1564-1642)
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Galileo . Galileo . 1669 J. Jungius (1587-1657), . Jacob Bernoulli 1690, 1691 Acta eruditorum, Huygens (1629-1695), Leibniz Johann Bernoulli ( Jacob). (catenary ) ( ( ) ( ) ( )/ 2ax axy x e e a= + a
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). Huygens (catenary) Leibniz 1690.
. Pierre de Fermat (1601-1665), (1662). Fermat . Gallileo. Fermat Snell (1580-1626..) ( ). Ibn Sahl (940-1000.) 984... Thomas Harriot (1560-1621), Willebrord von Roijen Snel (1580-1626) . Descartes Snell . , . Huygens 1703, Fermat, .
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Pierre de Fermat (1601-1665)
( ) ( )1 1 2 2sin sin = 1 2 .
Fermat : n , il iv
, 1
ni
i i
ltv=
= . c i iv , . /i ic v = ( ),
1
1 ni i
it l
c
== .
t 1
n
i ii
S l=
= .
S d=
.
.. 0d
= . Snell .
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1685 Newton (1643-1727) Philosophiae naturalis principia mathematica ( Principia ), . , . Newton 1694. , . : ) Fermat Newton Jacob Bernoulli Euler ) ( ) . Newton , , . .
( )2
20
2 , 21 '
R xdxF K K uf x
= =+ ( )f x .
Isaac Newton (1643-1727)
Newton Principia
1659 Christiaan Huygens (1629-1695) Horologium oscillatorium , :
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, , AMB , .
Christiaan Huygens (1629-1695)
:
.
(cycloid- Galileo 1599) :
( ) ( )( ) ( ) ( )( )sin , 1 cosx t r t t y t r t= = r .
c/2
. Jacob Bernoulli ( 1690 Acta Eruditorum), Joseph Louis Lagrange Leonhard Euler.
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1696, Acta Eruditorim, Johann Bernoulli (1667-1748) : , AMB , , .
Johann Bernoulli (1667-1748)
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y(x)
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( ) 1*y x C
, .
Bernoulli . Bernoulli (brachistochrone problem) (brachistos)) (chronos).
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dsmu t mgy x u t gy xdt
= = = u(t) m , g=9.81. T s , :
( )( )
2 1/ 22
1/ 20 0 0 0
(1 )1 11 ( ( ))2 ( ) 2
T T T T y xdtT dt ds y x dx dxds gy x g y x
+= = = + = :
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( ) 1*y x C : ( )( ) ( )( )
2 1/ 2
1/ 20
(1 )12
T y xT y x dx
g y x+=
(cycloid). : Gottfried Wilhelm Leibnitz (1646-1716), Newton, Bernoulli Jacob Bernoulli (1654-1705), Ehrenfried Walter von Tschirnhaus (1651 1708) De LHospital (1661-1704).
Jacob Bernoulli (1654-1705)
Groningen Bernoulli
1695 1705.
Johann Bernoulli , Snell ( Fermat).
1
2 2
3 3
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Fermat Johann Bernoulli
.
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sinsin rrv v =
sin dxds
=
1 1 rr r
dxdxv ds v ds
=
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dx cvds
= 2 2 2ds dx dy= +
( )2 2 22 2 2 2 22 2 222
2 2 2 22 2
2
1 1 12 1 ,21 1
v gy
dx dx dxcv c v c vds ds dx dy
dyc v c gy y k kdy dx c gdydx dx
=
= = = + = = + = = + +
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: ) , , ) /ds dt y . . Jacob Brook Taylor (1715) Johann Bernoulli. Leonhard Euler (1707-1783) Johann Bernoulli, 1744, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti, .
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J y x F x y x y x y x dx= :
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21 2 ... 1 0n
nn n
F d F d F d Fy dx dx dxy y y
+ + =
Euler-Poisson 2n. Euler .. Euler. . Euler
( ) ( ) ( )( )1: , , ,ba
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( ) ( ) ( )( )1, ,dQ L x y x y xdx =
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Lagrange. Euler , . Euler 2 On the motion of bodies in a non-resisting medium, determined by the method of maxima and minima, , , Maupertuis. Lagrange, Poisson Jacobi, Hamilton . Joseph-Louis Lagrange (1736-1813) Euler , . Euler Lagrange, Lagrange. Lagrange Euler, . Mecanique Analytique . :
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x y x z x y x z x dx c i m = = Mayer. Lagrange ( ) ( ), ,...x t y t .
Leonhard Euler(1707-1783)
Joseph Louis Lagrange (1736-1813) 1786, Adrien-Marie Legendre (1752-1833) ( ). Legendre . Lagrange , Theories des functions analytiques 1797. Legendre, Carl Gustav Jacob Jacobi (1804-1851) 1836 (50 ), . , . William Rowan Hamilton (1805-1865) ,
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. . Jacobi 1838, Hamilton Hamilton-Jacobi Richard Ernest Bellman (1920-1984) 100 ( 1953).
Adrien-Marie Legendre (1752-1833)
Carl Gustav Jacob Jacobi (1804-1851)
Karl Wilhelm Theodor Weierstrass (1815-1897) Oskar Bolza (1857-1942), Gilbert A. Bliss (1876-1951), (1873-1950) McShane (1904-1989). , Morse (Marston Morse (1892-1977)), (2 Field Jesse Douglas 1936 Enrico Bombieri ), ( Hamilton, Richard Feynman (1918-1988)), .
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J y x f x y x y x dx= [ ]
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( ) ( )( )
( ), ( ),
( ) ( ), ( ),
x t a x t u t t
y t c x t u t t
==
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22 2
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, ,
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y tx t u ty tx t u t
x t u t y t
y tx t u t
= = = ## #
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f f tJ h x t t g x t u t t dt= + (2)
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(1), ( )*x t , ( )*x t
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0 0
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( ) ,
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f f f ft t
u x
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u t x t
= + +
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() : ( ) ( ) ( )d t a t b t= +
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:
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( )( )
1 1 1
2 2 2
1 1
2 2
0 1 0 00 0 1 1
;
x t x t u tx t x t u t
x t d t u t a tx t d t u t b t
= + = =
2( ) ( ) ( ) ( ) ( ) ( )d t a t b t x t a t b t= + = + . ( ) ft e , :
( )( )
( )( )
( )( )
( )( )
11 0 0
2 0 0 2
0;
0 0f f
f f
x t d tx t d t ex t d t x t d t
= = = =
e ft ( )x t :
( ) ( )1 20 , 0x t e x t , , 21 / secM m 22 / secM m . , :
( ) ( )1 1 2 20 , 0u t M M u t .. G .
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( ) ( )2x t d t= ( ) ( )1u t a t= , :
( ) ( )0
1 2 2 1ft
tk x t k u t dt G+
:
0
0
ft
ft
J t t dt= = ( )*u t ( )*x t ,
0
ft
t
J dt= . . M, :
M
t0 tf(1/2)(t0+tf)
a*(t)
-M
t0 tf(1/2)(t0+tf)
b*(t)
t0 tf(1/2)(t0+tf)
x2*(t)
t0 tf(1/2)(t0+tf)
x1*(t)e
( ) ( ) ( ) ( )11 2u t u t e t+ = ( ) ( ) ( )1 2,x t e x t e t= = ,
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( ), . . (time optimal control problem) ( )* uu t ( )0x t ( )fx t . :
0
0
ft
ft
J t t dt= = (terminal control problem) ( )* uu t ( )fx t ( )fr t . :
( ) ( ) ( ) ( )2 21
n
i f i f f fi
J x t r t x t r t= = =
( ) ( ) ( ) ( )Tf f f fJ x t r t H x t r t = H . . ( )x t ft ,
( )fr t . (tracking problem)
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( )* uu t ( )x t ( )r t . :
[ ] [ ]0 0
2( ) ( ) ( ) ( ) ( ) ( )f ft tT
Qt tJ x t r t Q x t r t dt x t r t dt = =
Q . . ( )x t ( )r t . (regulator problem) . ( )* uu t ( )x t . :
0
2 2( ) ( )ft
f QH tJ x t x t dt = +
Q , . (minimum control effort problem) ( )* uu t ( )0x t ( )fx t
. :
( ) ( ) ( )0 0
2f ft tTRt t
J u t Ru t dt u t dt = = R .
-
. (, , ) . , ( )* uu t ( )x t ( )r t , . :
0
2 2( ) ( ) ( )ft
Q RtJ x t r t u t dt = +
Q,R . . ( )x t ( )r t , .
[1] ., 1994, , . . [2] ., 2007,
, .
[3] Anderson, B., & Moore, J., 1971, Linear optimal control. Englewood CliHs, NJ: Prentice-Hall.
[4] M. Athans and P. Falb, 1966, Optimal Control : An Introduction to the Theory and its Applications, McGraw Hill Book Company, New York, NY.
[5] Bryson Jr., A., 1996, Optimal Control1950 to 1985. IEEE Control Systems Magazine, 6, 2633.
[6] S.Cuomo, 2000, , .
[7] Goldstine, H., 1981, A history of the calculus of variations from the 17th to the 19th century. New York: Springer.
[8] R.E. Kalman, 1960, Contributions to the Theory of Optimal Control, Bol. Soc. Mat. Mexicana, Vol.5, pp.102-119.
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[9] D.E. Kirk, 1970, Optimal Control Theory, Prentice Hall, Englewood Cliffs, NJ.
[10] F. L. Lewis and V. L. Syrmos, 1995, Optimal Control, Second Edition, John Wiley & Sons, New York, NY.
[11] E.J. McShane. 1989, The calculus of variations from the beginning through optimal control theory, SIAM J. Con. Optim., Vol.27, No.5, pp.916-939.
[12] D. S. Naidu, 2002, Optimal Control Systems, CRC Press LLC. [13] L. S. Pontryagin, 1961, Optimal Regulation Processes, Uspekhi Mat.
Nauk, USSR, Vol.14, 1959, pp.3-20) English Translation : Amer. Math. Society Trans., Series 2, Vol.18, 1961, pp.321-339.
[14] L. S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E. F. Mishchenco, 1962, The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, NY.
[15] Sussmann, H., & Willems, J., 1997, 300 years of optimal control: From the Brachystochrone to the Maximum Principle. IEEE Control Systems Magazine, 6, 3244.
[16] The MacTutor History of Mathematics Archive, web page of the University of St. Andrews, Scotland, http://www-history.mcs.st-andrews.ac.uk/index.html .