.

54006 email : karampet@math.auth.gr

. , , . , . 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.

1.1 . , . (70-19 ..) , 9 .. , , . , , , ,

. , , , , , . (circumdare), , (circus), .

( 1)

( , Cod. Vat. lat. 3225).

: , A , . : A 0 x a , 0y ( )0,0 ( ),0a , 0,0y x a= .

(0,0) (a,0)

y

x

A

:

( )( ) ( )0

aJ y x y x dx=

( )( ) ( )( )20

1 'a

L y x y x dx= + = A , 19 . ( )( )J y x

[ ]1 ,C a b ( ) \ . (290-350..) . , . , (365.. -300.., (13 )), (287..-212.. , ) (200..-140.., ), , , (180 ..- 120.., (14 )), (160.. -90.., ) ... 5 ( ) ( ).

180.. ( ) . 5 :

1. ,

. 2. ,

. 3. .

:

4. , .

5. , .

6. , .

7. , .

8. , .

9. , .

( 150 . . 300 ..). , . , . ( ) Fermat (1601-1655). Fermat Bernoulli.

: ( )0, aA y ( ),b bB x y ( ) 0,0 by x x x= . ( ),0dD x ADB

,.

bx

( , )b bB x y( )0, aA y

( ), 0dD x :

( )( ) ( )( )20

1 'bx

J y x y x dx= + ( ) ( ) ( )0 , 0,a d b by y y x y x y= = = ( )y x dx x= . ,A D ,D B , D ( ) ( )1 2sin sin = .

bx

( , )b bB x y( )0, aA y

( ), 0dD x1 2

Galileo Galilei (1564-1642) 1638 Discorsi e dimostrazioni matematiche, intorno a due nuove scienze ( ) 2 . . (C CB) . . , Galileo , . Galileo , .

Galileo Galilei (1564-1642)

A

B

C

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

). 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, .

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 .

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 , :

, , 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.

1696, Acta Eruditorim, Johann Bernoulli (1667-1748) : , AMB , , .

Johann Bernoulli (1667-1748)

B(b,yf)

A(0,0)

y*(x)

x

y(x)

M

( ) 1*y x C

, .

Bernoulli . Bernoulli (brachistochrone problem) (brachistos)) (chronos).

( ) ( ) ( ) ( )21 22

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

+= = = + = :

( ) 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

4

Fermat Johann Bernoulli

.

sinsin rrv v =

sin dxds

=

1 1 rr r

dxdxv ds v ds

=

1 rr r

dxv ds

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

=

= = = + = = + = = + +

. Jacob Bernoulli . , , .

A

B

: ) , , ) /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, .

( )( ) ( ) ( ) ( ) ( ) ( )( )1: , , ,...,b na

J y x F x y x y x y x dx= :

( ) ( ) ( ) ( )2

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

J F x y x y x Q dx=

( ) ( ) ( )( )1, ,dQ L x y x y xdx =

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 . :

( ) ( ) ( ) ( ) ( ) ( )( )1 1: , , ,..., , ,...ba

J F x y x z x y x z x dx= ( ) ( ), ,...y x z x / /

( ) ( ) ( ) ( ) ( ) ( )( )1 1, , ,..., , ,... 0, 1,2,..,i x y x z x y x z x i m = = F

( ) ( ) ( )1 1 2 2:a m mF F l x l x l x= + + + + "

. Lagrange ( ) ( ), ,...y x z x . , :

( ) ( ) ( ) ( ) ( ) ( )( )1 1, , ,..., , ,... , 1, 2,..,b i ia

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) ,

. . 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)), .

2 : J A, ( ) J. [ ],a b \ [ ]1 ,a bC

( ) [ ]: ,y x a b \ .

( )( ) ( ) ( )( ), , 'ba

J y x f x y x y x dx= [ ]

1,a bC .

( )y x ( ) ( )( )J y x . \ \ , . , () . () , , :

( ) ( )( )

( ), ( ),

( ) ( ), ( ),

x t a x t u t t

y t c x t u t t

==

(1)

( ) ( ) ( )11 1

22 2

( )( ) ( )( )( ) ( )

, ,

( )( ) ( ) pn m

y tx t u ty tx t u t

x t u t y t

y tx t u t

= = = ## #

, . :

( ) ( ) ( )( )0

( ), , ,ft

f f tJ h x t t g x t u t t dt= + (2)

( )*u t , ( )*u t u ,

(1), ( )*x t , ( )*x t

x , (2). ( )*u t (optimal control), ( )*x t (optimal trajectory). , . .

( ) ( ) ( )( ) ( ) ( ) ( )( )( )

0 0

* * * *( ), , , ( ), , ,

( ) ,

f ft t

f f f ft t

u x

J h x t t g x t u t t dt h x t t g x t u t t dt

u t x t

= + +

(2), , , . , (), (). . 1 .

O e d

() : ( ) ( ) ( )d t a t b t= +

( )d t t , ( ) ( ),a t b t .

1 1( ) ( ) ( ) ( )x t d t x t d t= = 2 1( ) ( ) ( )x t d t x t= =

:

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

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 .

( ) ( )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= = ,

( ), . . (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)

( )* 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.

[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 .