Optimization And Optimization And Differential EquatiDifferential Equationsons 最佳化與微分方最佳化與微分方

程程Peng-Jen Lai (Peng-Jen Lai ( 賴鵬仁賴鵬仁 ))

Department of Mathematics Department of Mathematics National Kaohsiung Normal UniversityNational Kaohsiung Normal University

(( 高雄師範大學數學系高雄師範大學數學系 ))2011110220111102

國立清華大學工業工程與工程管理系微分國立清華大學工業工程與工程管理系微分方程專題演講方程專題演講

ContentContent

有限維度與無窮維度之最佳化問題有限維度與無窮維度之最佳化問題 The calculus of variation The calculus of variation 變分問題之最變分問題之最佳解佳解

1. Some examples1. Some examples2. Review of calculus2. Review of calculus3. Euler-Lagrange Equation3. Euler-Lagrange Equation 工業應用之例子工業應用之例子

給一個實數會對應到一個實數值，這種映射關係叫函數給一個實數會對應到一個實數值，這種映射關係叫函數 fufunctionnction，，前一頁是求函數極值前一頁是求函數極值 (( 最佳值最佳值 )) 之例子。實數軸之例子。實數軸是是一維一維，，所以是在一維空間搜尋最佳解所以是在一維空間搜尋最佳解，，他的主要數學工他的主要數學工具是具是微積分微積分。。

給一個函數會對應到一個實數值，這種映射關係叫泛函 給一個函數會對應到一個實數值，這種映射關係叫泛函 ffunctional, functional operator(unctional, functional operator( 範函算子範函算子 ), operator ), operator 算子算子，，上面，上面， F F 就是一個泛函就是一個泛函，，上例是求泛函極值上例是求泛函極值 (( 最佳值最佳值 )) 之之例子。例子。 函數空間是 函數空間是無窮維無窮維，，所以是在所以是在無窮維無窮維空間搜尋最空間搜尋最佳解佳解，，他的主要數學工具是他的主要數學工具是泛函分析跟變分法泛函分析跟變分法。。

三維的函數極值三維的函數極值

Review of calculusReview of calculus c is a critical point c is a critical point

of f(x) if of f(x) if f ’(c)=f ’(c)=0 or 0 or

f ’(c)f ’(c) does not exist does not exist

(a singular point (a singular point 奇異點、奇異點、尖點尖點、、或不連續的點或不連續的點 ).).

Relative extrema may Relative extrema may occur at a singular point occur at a singular point or an end point.or an end point.

Remark:Remark: Larson Larson 那本書那本書要求 要求 relative extrema relative extrema

一定是內點一定是內點 ..

那些最佳化那些最佳化 (( 求極值求極值 )) 的問的問題會跟微分方程有關係題會跟微分方程有關係 ??

答案是，泛函算子的最佳化答案是，泛函算子的最佳化(( 求極值求極值 )) 問題會跟微分方問題會跟微分方程有關係程有關係

What is the calculus of variatioWhat is the calculus of variation (n ( 變分法變分法 ) ?) ?

The The calculus of variationcalculus of variation is a theory to is a theory to discuss how to find (the) optimal solutidiscuss how to find (the) optimal solutions to the following problem:ons to the following problem:

2

1

( ) ( , , ')x

xI u f x u u dx

min ( )u

I u

1 2 1 2:[ , ] , u( ) , ( ) .u x x x a u x b

2

1

( ) ( , , ')x

xI u f x u u dx

1 2 1 2:[ , ] , u( ) , ( ) .u x x x a u x b

min ( )u

I u

2

1

( ) ( , , ')x

xI u f x u u dx

1 2 1 2:[ , ] , u( ) , ( ) .u x x x a u x b

The shortest path (geodesic The shortest path (geodesic 測測地線地線 ) problem) problem

Find the shortest curve joinning A and B.Find the shortest curve joinning A and B. Mathematical Modelling 數學建模 :Mathematical formulation:

2{ : [1,6]

|(1) , (6) }

min arc length of differentiable

y y

u

u A u B

2

12

{ : [1,6]0|

(1) , (6) }

min 1 ( '( ))differentiable

y y

u

u A u B

u x dx

:[1,6] , u(1) , (6) .y yu A u B

2 2( , , ') 1 ( '( )) , ( ( , , ) 1 )f x u u u x f x y z z

6 6 2

1 1( ) ( , , ') 1 ( '( ))I u f x u u dx u x dx

min ( )u

I u

The brachistochrone problemThe brachistochrone problem最速降線問題最速降線問題

Among all smooth curves in a vertical plane join a given point A to a given lower point B not directly below it, find that particular curve along which a particle will slide down from A to B in the shortest possible time.

重力下的最快下降曲線 國立中央大學物理演示實驗1, 2, 3

:[0,1] , u(0) , (1) .y yu A u B

2 21 ( '( )) 1( , , ') , ( ( , , ) )

2 ( ) 2

u x zf x u u f x y z

gu x gy

21 1

0 0

1 ( ')( ) ( , , ')

2

uT u f x u u dx dx

gu

min ( )u

T u

Theorem Theorem Suppose f, u to be twice differentiable.Suppose f, u to be twice differentiable. If u minimizes , then u satisfies the E-L equatiIf u minimizes , then u satisfies the E-L equati

onon

2

1

( ) ( , , ')x

xI u f x u u dx

( )I u

1 2 1 2:[ , ] , u( ) , ( ) .y yu x x x A u x B

Euler-Lagrange Euler-Lagrange EquationEquation

Solve the Brachistochrone proSolve the Brachistochrone problem by the E-L equationblem by the E-L equation

2

1

21 ( ')( ) ( , , ') , ( , , ')

x

x

xI x f y x x dy f y x x

y

222 2 1 ( )1 ( )1 1 1

2 2 2 2C C C C

dxdydx dyds dydxT dx dy

y ygy g y g g

21 ( ')( ) 0 0 0

( ') ( ')

f d f d x

x dy x dy x y

2

' 10 [ ]

1 ( ')

d x

dy yx

2

1

21 ( ')( ) ( , , ') , ( , , ')

x

x

xI x f y x x dy f y x x

y

21 ( ')( ) 0 0 0

( ') ( ')

f d f d x

x dy x dy x y

2

' 10 [ ]

1 ( ')

d x

dy yx

2

1

21 ( ')( ) ( , , ') , ( , , ')

x

x

xI x f y x x dy f y x x

y

21 ( ')( ) 0 0 0

( ') ( ')

f d f d x

x dy x dy x y

擺線之模

擬

工業上的應

工業上的應

用用

工業上的應

工業上的應

用用

數學建模本來就無所不在數學建模本來就無所不在

Conclusion: the relation betweeConclusion: the relation between optimization and differential n optimization and differential

equationequation

變分法與微分方程之求解是雙向的變分法與微分方程之求解是雙向的

ReferencesReferences 1. P. Neittaanmäki, D. Tiba, Optimal control of nonlinear parabolic systems, Marcel Dekker 1994. 2. 徐長發 , 科技應用中的微分變分模型 , 華中科大出版 2004. 3. G.F. Smmons, Differential equations with applications and historical notes 1991. 4. J. Jost, Calculus of variation, Cambridge 1998. 5. J. Jost, Postmodern analysis, Springer 1998. 6. Russak, Calculus of Variations & Solution Manual ch2, 2002. 7. Sasane, Calculus of Variations & Optimal Control 2004. 8. R. Weinstock, Calculus Of Variations, With Applications To Physics And Engineering 1974. 9. Bernard Dacorogna, INTRODUCTION TO THE CALCULUS OF VARIATIONS. 10. Byerly, Introduction To The Calculus Of Variations 1917.

Thank you for your Thank you for your attention!attention!

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