Transcript
  • .

    54006 email : [email protected]

    . , , . , . 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 .


Recommended