Lecture 24

Lecture – 24

Calculus of Variations : An Overview

Dr. Radhakant PadhiAsst. Professor

Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

Fundamental Theorems of Calculus

Theorem – 1:

Theorem – 2:

Theorem – 3:

( ) ( ) ( ), provided is continuousx

d f d f x f xdx

σ σ =∫

( ) ( )

provided , has continuous partial derivative /

f x yd f x y dy dydx x

f x y f x

∂ ∂

∫ ∫

( ) ( ) ( ) ( )2 2

( ) ( )2 1

2 1( ) ( )

,, , ( ) , ( )

provided , , have continuous partial derivatives with respect to

f x y d dd f x y dy dy f x x f x xdx x dx dx

ψ ψψ ψ

∂ ⎡ ⎤= + −⎢ ⎥∂ ⎣ ⎦∫ ∫

Calculus of Variations:Basic Concepts

Function (to each value of the independent variable, there is a corresponding value of the dependent variable)

Increment of a function

Functional (to each function, there is a corresponding value of the dependent variable)

Increment of a functional

( ) 32 3x t t t= + ( )( ) ( )

J x t x t dt

t t dt

( ) ( )x x t t x tΔ + Δ − ( ) ( )( ) ( )( )J J x t x t J x tδΔ + −

Example

( ) ( )

( )( )

( )( )( )( )

( ) ( ) ( )

2 ( ) ( ) 1 2 ( ) 1

2 ( ) 2 ( ) ( ) ( ) 1 2 ( ) 1

4 ( ) ( ) 2 ( )

J J x t x t J x t

x t x t dt x t dt

x t x t x t dt

x t x t x t x t x t dt

x t x t x t dt

Δ = + −

⎡ ⎤ ⎡ ⎤= + + − +⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤= + + + − +⎣ ⎦⎣ ⎦

⎡ ⎤= +⎣ ⎦

∫ ∫

22 ( ) 1ft

J x t dt⎡ ⎤= +⎣ ⎦∫

Function and its increment Functional and its increment

Reference: D. S. Naidu, Optimal Control Systems, CRC Press, 2002.

Differential of a function Variation of a functional

( ) ( )

: First deviation : Second deviation

lim lim

. . in general

df d f

f f t t f t

df d ft tdt dt

df d f

dfdf f tdt

i e df f t

Δ → Δ →

Δ = + Δ −

⎛ ⎞⎛ ⎞= Δ + Δ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= Δ = Δ⎜ ⎟

⎝ ⎠

( ) ( )

: First variation : Second variation

( ) ( ) ( )

J J x t x t J x t

J Jx xx x

Δ = + −

⎛ ⎞∂ ∂⎛ ⎞= + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

Result – 1:Derivative of variation = Variation of derivative

Result – 2:Integration of variation = Variation of integration

( ) ( ) ( )

( ) ( )

d dx t x t x tdt dt

dx t dx tdt dt

⎡ ⎤⎡ ⎤ = −⎣ ⎦ ⎣ ⎦

⎡ ⎤= ⎣ ⎦

( ) ( ) ( )

( ) ( )

x d x x d

x d x d

δ τ τ τ τ τ

τ τ τ τ

δ τ τ

⎡ ⎤= −⎣ ⎦

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

∫ ∫

Exercise

Evaluate the variation of :

( ) 2 ( ) 3 ( ) 4

Note: By 'variation', we mean 'first variation' (by default)

J x t x t x t dt⎡ ⎤= + +⎣ ⎦∫

Solution

( ) ( )

:( ) ( ) ( )

2( ( ) ( )) 3( ( ) ( )) 4 2 ( ) 3 ( ) 4

2 2 ( ) 3( ) 4 2 3 4

4 2( ) 3

4 3 Neglect

J J x t x t J x t

x t x t x t x t dt x t x t dt

x x x x x x x x dt

x x x x dt

δ δ δ

Δ = + −

⎡ ⎤ ⎡ ⎤= + + + + − + +⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤= + + + + + − + +⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤= + +⎣ ⎦

∫ ∫

Method - 1

( )ing the higher order term

2 ( ) 3 ( ) 4 ( )

(Variation of integral = Integral of variation)

x t x t dt x tx

x x x dtx

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠⎡ ⎤∂ ⎡ ⎤= + +⎢ ⎥⎣ ⎦∂ ⎢ ⎥⎣ ⎦

⎡ ⎤∂ ⎡ ⎤= + +⎢ ⎥⎣ ⎦∂⎢ ⎥⎣ ⎦

Method - 2 :

Solution

Boundary Conditions

Fixed End Point Problems

Free End Point Problems• Completely free

• May be required to lie on a curve

( )( )

0 0, ( ) : Specified

, ( ) : Specifiedf f

t x tt

( )x t

( )0 0, ( )t x t

( ), ( )f ft x t

( )tη

( )x t

( )0 0, ( )t x t

( )tη

( ), ( )f ft x t

Optimum of a Functional( )

( ) ( ) ( )

A functional is said to have a relative optimum at , if 0

such that for all functions which satisfy ,

the increment of has the "same sign".

1) If 0, then is a r

x t x t x t

J J x J x J x

∈Ω − <

Δ = − ≥

( ) ( ) ( )

elative (local) "Minimum".

2) If 0, then is a relative (local) "Maximum".

Note: If the above relationships are satisfied for arbitrarily large 0,

then is a "global optimum".

J J x J x J x

Δ = − ≤

Optimum of a Functional

( )* : Optimum pathx t

( )x t

( ) ( ) ( )*Variation: x t x t x tδ = −*,x x

( )x tδ

Fundamental Theorem of Calculus of Variations

For to be a candidate for optimum, the following conditions hold good:

1) Necessary Condition: ( ), ( ) 0, admissible ( )

2) Sufficiency Condition:

J x t x t x tδ δ δ= ∀

0 (for minimum) 0 (for maximum)

Fundamental Lemma

( ) ( )

If for every continuous function

where the variation is continuous in , ,

g t x t dt

x t t t t

g t t t t

⎡ ⎤∈ ⎣ ⎦

⎡ ⎤= ∀ ∈⎣ ⎦

Necessary condition of optimalityProblem: ( ) ( ) ( )

Optimize , , by appropriate selection of .

Note: , are fixed.

J L x t x t t dt x t

⎡ ⎤= ⎣ ⎦∫

Solution: ( )Make sure 0 for arbitrary J x tδ δ=

1) Euler – Lagrange (E-L) Equation

2) Transversality (Boundary) Condition

0L d Lx dt x∂ ∂⎛ ⎞− =⎜ ⎟∂ ∂⎝ ⎠

L xxδ∂⎡ ⎤ =⎢ ⎥∂⎣ ⎦

Note: Part of this equation mayalready be satisfied by problem formulation

Necessary Conditions:

Proof:Hint: Use the necessary condition of optimality from the Fundamental theorem

( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ){ }0 0

Let , [ , ] : Optimumsolution

( ) : Some adjacent solution

, , , ,

x t t t t

x t x t

J J J L x t x t t dt L x t x t t dt

L x t x t t L x t x t t dt L dt

⎡ ⎤⎡ ⎤Δ = − = −⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤= − = Δ⎣ ⎦ ⎣ ⎦

∫ ∫

( )* : Optimum pathx t

( )x t

( ) ( ) ( )*Variation: x t x t x tδ = −*,x x

( )x tδ

( ) ( )

* * * *

However at every point ,

, , , ,

Assumption: is continuous and smooth in both and .

Then, in the limit,

f fx x x x

L L x x x x t L x t x t t

L Lx x HOTx x

L x xL LL L xx x

⎡ ⎤ ⎡ ⎤Δ = + + −⎣ ⎦ ⎣ ⎦∂ ∂

= + +∂ ∂

∂ ∂Δ → = +

∂ ∂. In that case,xδ⎡ ⎤

⎢ ⎥⎣ ⎦

L LJ J x x dtx x

δ δ δ∂ ∂⎡ ⎤Δ → = +⎢ ⎥∂ ∂⎣ ⎦∫

( ) ( )

L L d xx dt dtx x dt

L d x d L d xdt dt dtx dt dt x dt

L d Lx xdtx dt x

∂ ⎡ ∂ ⎤⎡ ⎤ ⎛ ⎞= ⎜ ⎟⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎝ ⎠⎣ ⎦

⎡ ⎤∂ ⎡ ∂ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

⎡ ∂ ⎤ ⎡ ∂ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∫ ∫

∫ ∫ ∫

ProofHowever,

0 (Necessary condition of optimali

L LJ x x dtx x

L L d Lxdt x xdtx x dt x

L d L Lxdt xx dt x x

δ δ δ

∂ ∂⎡ ⎤= +⎢ ⎥∂ ∂⎣ ⎦

⎡ ⎤ ⎡ ⎤∂ ∂ ∂⎡ ⎤ ⎛ ⎞ ⎛ ⎞= + −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∫ ∫

∫ty)

Hence,

Necessary Conditions

* Condition (1) must be satisfied regardless of the end condition.* Part of second equation may already be satisfied by the problem of specification. i.e. the amount of extra information contai

Note :

ned by thisequation varies with the boundary conditions specified.

1. 0 (Euler-Lagrange Equation)

2. 0 (Transversality Condition)f

L d Lx dt x

∂ ∂⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎡ ∂ ⎤⎛ ⎞ =⎜ ⎟⎢ ⎥∂⎝ ⎠⎣ ⎦

Example – 1Problem: ( ) ( ) ( )

Minimize with 0 2, 1 3J x x dt x x= + = =∫

Solution:

1) E-L Equation:

( )2L x x= +

1 210 1 2 0,2 4

L d L tx x x t c t cx dt x∂ ∂⎛ ⎞− = ⇒ − = = ⇒ = + +⎜ ⎟∂ ∂⎝ ⎠

2) Boundary condition: ( ) 20 2x c= =

11 2 341 314 4

3Hence, 24 4

t tx t

= + + =

= − =

= + +0

Transversality condition is automatically satisfied, since

0fx xδ δ= =

Example – 2Problem: ( ) ( )

Minimize with 0 2,J x x dt x= + =∫

Solution:

1) E-L Equation:

( )2L x x= +

1 210 1 2 0,2 4

L d L tx x x t c t cx dt x∂ ∂⎛ ⎞− = ⇒ − = = ⇒ = + +⎜ ⎟∂ ∂⎝ ⎠

2) Boundary condition: ( ) 20 2x c= =

0, 0 0

12 0,2 2

Hence, 24 2

f ft t

L Lx xx x

tx c c

t tx t

∂ ∂= ⇒ = ≠

∂ ∂

= + = ⇒ = −

= − +

( )1 : Freex

Transversality Condition

0ff tt

L Lx L x tx xδ δ⎡ ⎤∂ ∂⎡ ⎤ ⎧ ⎫+ − =⎨ ⎬⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎩ ⎭⎣ ⎦

Special Cases:

( ) ( )0 01) Fixed End Points: , and , are fixed.

No additional information!f ft x t x

02) and are fixed (free initial and final states)

L xxδ∂⎡ ⎤ =⎢ ⎥∂⎣ ⎦

General condition:

Transversality ConditionSpecial Cases:

0 03) , are fixed (free final time, free final state)

f ft t

L Lx L x tx x

δ δ∂ ∂⎧ ⎫+ − =⎨ ⎬∂ ∂⎩ ⎭

0 04) , and are fixed (free final time)

LL xx∂⎧ ⎫− =⎨ ⎬∂⎩ ⎭

0 05) , and are fixed (free final state)

Transversality ConditionSpecial Cases:

( ) ( ) ( )0 06) , is fixed; , is constrained to lie on a given curve

0. However,

Hence, the transversality condition is

f f f f f ft tt

t x t x t

L L dx L x t x t tx x dt

L LL xx x

ηδ δ δ δ η δ

∂ ∂⎧ ⎫+ − = = =⎨ ⎬∂ ∂⎩ ⎭

∂ ∂+ −

∂ ∂( )0 0

t tδ δ⎡ ⎤⎧ ⎫⎢ ⎥ = ≠⎨ ⎬

⎩ ⎭⎢ ⎥⎣ ⎦

( ) 0ft

η ∂⎧ ⎫+ − =⎨ ⎬∂⎩ ⎭Finally:

ExampleProblem: ( ) ( ) ( ) ( )

Minimize 1 with 0 0 and , lie on 5 15f fJ x dt x t x y t t= + = = − +∫Solution:

1) E-L Equation:

21L x= +

L d Lx dt x

d xdt x

∂ ∂⎛ ⎞− =⎜ ⎟∂ ∂⎝ ⎠∂⎛ ⎞− + =⎜ ⎟∂⎝ ⎠

212 10 0

x xx x xx

+ −⎛ ⎞ += ⇒ =⎜ ⎟⎜ ⎟ ++⎝ ⎠

+( ) 22x x x− ( ) 0=

Example

1) E-L Equation: ( ) 1 20 0L d L x x t c t cx dt x∂ ∂⎛ ⎞− = ⇒ = ⇒ = +⎜ ⎟∂ ∂⎝ ⎠

2) Boundary condition:

( ) ( )

(1) 0 0

5 1 0 5 1 0 1/5

Hence, /5 To find , / 5 5 15 75/ 26

f f f f

x c x t c t

LL y xx

x t tt t t t

= = ⇒ =

∂⎡ ⎤+ − =⎢ ⎥∂⎣ ⎦

− + = ⇒ − + = ⇒ =

= − + ⇒ =

Variational Problems in Multiple Dimensions: Without Constraints

Dr. Radhakant PadhiAsst. Professor

Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore

Multiple Dimension Problems without constraints

Problem: ( ) ( ) ( )0

Optimize , , by appropriate selection of .ft

J L X t X t t dt X t⎡ ⎤= ⎣ ⎦∫

Solution: ( )Make sure 0 for arbitrary J X tδ δ=

1) Euler – Lagrange (E-L) Equation

2) Transversality (Boundary) Condition

0L d LX dt X∂ ∂⎛ ⎞− =⎜ ⎟∂ ∂⎝ ⎠

Necessary Conditions:

[ ]1 2where TnX x x x

0f ft tT

L LX L X tX X

δ δ⎡ ⎤ ⎡ ⎤⎧ ⎫∂ ∂⎛ ⎞ ⎛ ⎞+ − =⎢ ⎥ ⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎩ ⎭⎢ ⎥ ⎣ ⎦⎣ ⎦

Variational Problems with Constraints

[ ] [ ]

1 2 1 2

Optimize : = , ,

Subject to : , , 0

T Tn n

J L X X t dt

X x x x ϕ ϕ ϕ

( ) ( )

Lagrange's Existence Theorem:

( ) : The above constrained optimizationproblem leads to the same solution as the followingunconstrained cost functional

= , , , ,

Let , ,

J L X X t X X t dt

L X X t L

⎡ ⎤+ Φ⎣ ⎦

( ) ( ), , , ,TX X t X X tλ+ Φ

Necessary Conditions of Optimality:(1) E-L Equations:

(a) 0 ( n equations)

L d LX dt X

L d Ldtλ λ

⎡ ⎤∂ ∂− =⎢ ⎥∂ ∂⎣ ⎦

⎡ ⎤∂ ∂− ⎢ ⎥∂ ∂⎣ ⎦

0 ( equations)

Note: 0 as there is no term in

L Lλλ

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

(2) Transversality Conditions:

L LX L X tX X

⎡ ⎤ ⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂⎪ ⎪⎢ ⎥ + − =⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎥ ⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭⎣ ⎦⎣ ⎦

⎛ ⎞∂⎜ ⎟∂⎝ ⎠

LLδλ λλ

⎡ ⎤ ⎛ ⎞∂⎢ ⎥ + − ⎜ ⎟⎢ ⎥ ∂⎝ ⎠⎣ ⎦0;

tδ⎡ ⎤⎧ ⎫⎪ ⎪⎢ ⎥ =⎨ ⎬⎢ ⎥⎪ ⎪⎩ ⎭⎣ ⎦

( ) ( )

E-L Equations:

1) (a) 0

(b) , , 0 (same constraint equation)

2)Transversality Conditions: , fixed, , free

L d LX dt X

L X X t

t X t X

L LX L XX

⎛ ⎞ ⎛ ⎞∂ ∂− =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂= Φ =⎜ ⎟∂⎝ ⎠

⎛ ⎞∂ ∂+ −⎜ ⎟∂ ∂⎝ ⎠

0 ( equations)

(b) 0 However is free 0

so 0 ( 1 equation)

t f f f

L t t t

⎡ ⎤⎛ ⎞=⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

= ⇒ ≠

Varaibles: 1 ( ) ( ) ( )

Boundary Conditions : 1f

n nX t

Constraint Equations

Nonholonomic constraints

Isoperimetric constraints

( ), , 0X X tΦ =

One way to get rid of Isoperimetric constraints is to convertthem into Nonholonomic constraints.

q X X t dt k=∫

Isoperimetric Constraints( )

Define: , ,

( ) ( )

Choose one of ( ) or ( ) and fix the other

Let ( ) 0 ( )

x q X X t

x dt k

x t x t k

x t x t

x tx t k

The following additional non-holonomicconstraint is introduced:

with boundary conditions: ( ) 0 ( )

x q X X t

x tx t

Summary :

The original problem is augmentedwith this information and solved.

Isoperimetric Constraints

References

T. F. Elbert, Estimation and Control Systems, Von Nostard Reinhold, 1984.

D. S. Naidu, Optimal Control Systems, CRC Press, 2002.

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