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سلسله مراتب بهینه سازی: V-1 Maximum and Minimum of Functions V-2 Maximum and Minimum of Functionals V-3 The Variational Notation V-4 Constraints and Lagrange Multiplier 04/03/ :413
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Dr. Wang XingboDr. Wang Xingbo
FallFall ,, 20052005
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
1/3308/05/23 23:12
:عناوینمقدمه ای ازبهینه سازی معادالت جبری، 1.
chapter5_Calculus-of-Variations 3 اسالید 4 تا 1صفحات Functional Euler's equation.
فانکشنال و مسئله بهینه سازی دینامیکی: 2..1 مطلب فایل ترکیبی 10 تا 3صفحات
، خانم نجیمی؟Matlab فایل m.نگاهی به 3.
ادامه با استفاده از اسالید جاری.4.
و چند strictتعمیم، بعینه سازی قطعی، 5. 1 فایل ترکیبی و 18 تا 16متغیره صفحات
Calculus of Variations - PowerPoint Presentation به بعد16صفحه 08/05/23 23:12 2
:سلسله مراتب بهینه سازی
• V-1 Maximum and Minimum of Functions• V-2 Maximum and Minimum of Functionals• V-3 The Variational Notation• V-4 Constraints and Lagrange Multiplier
08/05/23 23:12 3
Functional and Calculus of Variation Functional and Calculus of Variation
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of Variations Introduction to Calculus of Variations
P=(a,y(a)) Q=(b,y(b))
Find the shortest curve connecting P = (a, y(a)) and Q = (b, y(b)) in XY plane
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The arclength isThe arclength is
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
dxxyb
a 2)]('[1
The problem is to minimize the above integral
5/3308/05/23 23:12
A function like J is actually called a functional . y(x) A function like J is actually called a functional . y(x) is call a permissible functionis call a permissible function
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
dxxyxyxFxyJb
a ))('),(,()]([
A functional can have more general form
))('),(,()]([ xyxyxfxyJ
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We will only focus on functional with integral We will only focus on functional with integral
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
A increment of y(x) is called variation of y(x), denoted as δy(x):
P variation of y(x):δy(x)
y(x)
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of y(Variation x) y(x)
consider the increment of J[y(x)] consider the increment of J[y(x)] caused by δy(x)caused by δy(x)
ΔJ [y(x)]= J [y(x)+δy(x)]- J [y(x)]ΔJ [y(x)]= J [y(x)+δy(x)]- J [y(x)]ΔJ [y(x)]=ΔJ [y(x)]=L[y(x), δy(x)]+β[y(x),δy(x)]L[y(x), δy(x)]+β[y(x),δy(x)]••max|δy(x)| max|δy(x)|
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
If β[y(x),δy(x)] is a infinitesimal of δy(x), then L is called variation of J[y(x)] with the first order, or simply variation of J[y(x)],denoted by δJ[y(x)]
8/3308/05/23 23:12
1.Rules for permissible functions1.Rules for permissible functions yy((xx) and variable ) and variable xx..
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
)()( ydxd
dxdy
δx=dxδx=dx
9/3308/05/23 23:12
Rules for functional Rules for functional J.J.
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
δ2 J =δ(δJ), …, δkJ =δ(δk-1J)δ(J1+ J 2)= δJ 1+δJ 2
δ(J 1 J 2)= J 1δJ 2+ J 2δJ 1
δ(J 1/ J 2)=( J 2δJ 1- J 1δJ 2)/ J 22
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Rules for functional Rules for functional JJ and and FF
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
( , , ') ( , , ')b b
a aJ F x y y dx F x y y dx
''
F FJ F y yy y
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If J[y(x)] reaches its maximum (or minimum) at yIf J[y(x)] reaches its maximum (or minimum) at y00(x), (x), then δthen δJJ[[yy00((xx)]=0. )]=0.
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
Let Let J J be a functional defined on be a functional defined on CC22[a,b] with [a,b] with JJ[y(x)] given by [y(x)] given by
dxxyxyxFxyJb
a ))('),(,()]([
How do we determine the curve How do we determine the curve yy((xx) which produces such a ) which produces such a minimum (maximum) value for minimum (maximum) value for JJ? ?
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The Euler-Lagrange Equation
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
dxxyxyxFxyJb
a ))('),(,()]([
( ) 0'
F d Fy dx y
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To find y(x) to minimize J of:
The following equation must be satisfied:
Let M(x) be a continuous function on the interval [a,b], Suppose that for any continuous function h(x) with h(a) = h(b) = 0 we have :
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Fundamental principle of variations Fundamental principle of variations
Then M(x) is identically zero on [a, b] .
b
adxxhxM 0)()(
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Choose h(x) = -M(x)(x - a)(x - b) ,Then M(x)h(x)≥0 on [a, b]
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Fundamental principle of variations Fundamental principle of variations
0 = M(x)h(x) = [M(x)]2[-(x - a)(x - b)] M(x)=0
If the definite integral of a non-negative function is zero, then the function itself must be zero .
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Example : Prove that the shortest curve connecting planar point P and Q is the straight line connected P and Q.
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
dxxyLb
a 2)]('[1
16/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
2/32
2/32
22
2
2/1222
2'
))]('[1()("
))]('[1()(")]('[))]('[1(
)]('[1))]('[1)((")]('[)(")]('[1
))]('[1
)('(0
xyxy
xyxyxyxy
xyxyxyxyxyxy
xy
xydxdF
dxdF
y y
0)(" xy y(x)=ax+b 17/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Beltrami Identity.
If then the Euler-Lagrange equation is simplified and is equivalent to:
0Fx
''
FF y Cy
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The Brachistochrone Problem
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Introduction to Calculus of VariationsIntroduction to Calculus of Variations
Find a path that wastes the least time for a bead travel from P to Q
P Q
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Let a curve y(x) that connects P and Q represent the wire :
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Brachistochrone Problem
b
a vdsxyF )]([
21 | '( ) | , '( )ds y x dx v y x
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By Newton's second law ,we obtain:
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Brachistochrone Problem
))()(()]([21 2 xyaymgxvm
b
adx
xyaygxyxyF
))()((2|)('|1)]([2
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• Euler-Lagrange Equation
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Brachistochrone Problem
Cxgyxy
xyxgyxy
xgyxy
)()(')
)]('[1)(2)(('
21
)(2)]('[1
2
2
22
2
21)())]('[1( k
gCxyxy
22/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Brachistochrone Problem
The solution of the above equation is a cycloid curve
)cos1(21)(
)sin(21)(
2
2
ky
kx
23/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Integration of the Euler-Lagrange Equation Integration of the Euler-Lagrange Equation
Case 1. F(x, y, y’) = F (x)
Case 2. F (x, y, y’) = F (y) :F y(y)=0
Case 3. F (x, y, y’) = F (y’) :
0))(( ' yFdxd
y CyFy )'(' ' 1y C
1 2y C x C 24/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Integration of the Euler-Lagrange EquationIntegration of the Euler-Lagrange Equation
Case 4. F (x, y, y’) = F (x, y)
Fy (x, y) = 0 y = f (x)
Case 5. F (x, y, y’) = F (x, y’)
0))(( ' yFdxd
y 1)',(' CyxFy )1,(' Cxfy
dtCtfCy )1,(225/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Integration of the Euler-Lagrange EquationIntegration of the Euler-Lagrange EquationCase 6 F (x, y, y’) = F (y, y’)
)',()',(' yyFyyFdxd
yy
yy FyFy '''
)'('")''( ''' yyy FyFyFy
'"'' yy FyFyF
')''( ' FFy y
CFFy y ''
26/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Euler-Lagrange Equation of The Euler-Lagrange Equation of VariationalVariational Notation Notation
b
aFdxJ
'' yy FyyFF
b
adxyyxF 0)',,(
b
a yy dxyyxFyyyxyF 0))',,(')',,(( '
27/3308/05/23 23:12
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Lagrange Multiplier Method for the Calculus The Lagrange Multiplier Method for the Calculus of Variations of Variations
dxxyxyxFxyJb
a ))('),(,()]([
ldxxyxyxGb
a ))('),(,( BbyAay )(,)(
Conditions Conditions
28/3308/05/23 23:12
The minimize problem of following functional is equal to the conditional ones.
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Lagrange Multiplier Method for the Calculus The Lagrange Multiplier Method for the Calculus of Variationsof Variations
where λ is chosen that y(a)=A, y(b)=B
b
a
b
adxxyxyxGdxxyxyxF ))('),(,())('),(,(
ldxxyxyxGb
a ))('),(,(
0)()( '' yyyy GFGFdxd
29/3308/05/23 23:12
ExampleExample
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Lagrange Multiplier Method for the Calculus of VariationsThe Lagrange Multiplier Method for the Calculus of Variations
E-L equation is
2
0)(][ dyxyyJ
2
0 2 3))('(1 dxxyunder
1)'1
'(2
y
ydxd
1'1
'2
cxy
y
Leads to 22 )1(
)1('cx
cxy
222 )1()2( cxcy 30/3308/05/23 23:12
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Variation of Multi-unknown functions Variation of Multi-unknown functions
b
adxxyxyxyxyxFxyxyJ ))('),('),(),(,()](),([ 212121
0)',',,,( 2121 dxyyyyxFb
a 0)',',,,( 2121 dxyyyyxF
b
a
0)'( '
2
1
dxFyFyii yi
b
ai
yi
0}{2
1'
i
b
a yyi dxFdxdFy
ii
31/3308/05/23 23:12
The Euler-Lagrange equation for a functional The Euler-Lagrange equation for a functional with two functions with two functions yy11((xx),),yy22((xx) are ) are
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Variation of Multi-unknown functionsVariation of Multi-unknown functions
2,1,0' iFdxdF
ii yy
32/3308/05/23 23:12
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Higher Derivatives Higher Derivatives
b
adxxyxyxyxFxyJ ))("),('),(,()]([
0"2
2
' yyy FdxdF
dxdF
33/3308/05/23 23:12
What is the shape of a beam which is bent and which isWhat is the shape of a beam which is bent and which is
clamped so thatclamped so that y y (0) = (0) = yy (1) = (1) = yy’ (0) = 0 ’ (0) = 0 and yand y’ (1) = 1.’ (1) = 1.
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Example Example
1
0
2)"(][ dxykyJ 0"2 2
2
ydxdk
dcxbxaxy 23 3 2y x x
34/3308/05/23 23:12
Class is Over! Class is Over!
See you!See you!
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
35/3308/05/23 23:12