Handbook of mathematics for engineers and scienteists part 147 pot

If ˆxt minimizes the functional J [x] in a strong resp., weak neigh-borhood of itself, then it is called a point of strong resp., weak minimum of the functional J [x].. If ˆx maximizes

990 SPECIALFUNCTIONS ANDTHEIRPROPERTIES

18.18.2-2 Generating function Fourier series expansions Integrals

The generating function is expressed as

2e xt

e t+1 ≡

∞



n=0

E n (x) t

n

n! (|t|< π).

This relation may be used as a definition of the Euler polynomials

Fourier series expansions:

E n (x) =4 n!

π n+1

∞



k=0

sin (2k+1)πx – 1

2πn

(2k+1)n+1 (n =0, 0< x <1; n >0, 0 ≤x≤ 1);

E2n (x) =4(–1)n(2n)!

π2n+1

∞



k=0

sin (2k+1)πx (2k+1)2n+1 (n =0, 0< x <1; n >0, 0 ≤x≤ 1);

E2n–1(x) =4(–1)n(2n–1)!

π2n

∞



k=0

cos (2k+1)πx (2k+1)2n (n =1, 2, , 0 ≤x≤ 1)

Integrals:

 x

a E n (t) dt =

E n+1(x) – E n+1(a)

 1

0 E m (t)E n (t) dt =4(–1)n(2m+n+2–1) m ! n!

(m + n +2)!B m+n+2,

where m, n =0,1, and Bnare Bernoulli numbers The Euler polynomials are

orthog-onal for even n + m.

Connection with the Bernoulli polynomials:

E n–1(x) = 2n

n



B n

x+1

2



– B nx

2



= 2

n



B n (x) –2n B

n

x

2



,

where n =1, 2,

References for Chapter 18

Abramowitz, M and Stegun, I A (Editors), Handbook of Mathematical Functions with Formulas, Graphs

and Mathematical Tables, National Bureau of Standards Applied Mathematics, Washington, D.C., 1964.

Bateman, H and Erd´elyi, A., Higher Transcendental Functions, Vol 1 and Vol 2, McGraw-Hill, New York,

1953.

Bateman, H and Erd´elyi, A., Higher Transcendental Functions, Vol 3, McGraw-Hill, New York, 1955 Gradshteyn, I S and Ryzhik, I M., Tables of Integrals, Series, and Products, Academic Press, New York,

1980.

Magnus, W., Oberhettinger, F., and Soni, R P., Formulas and Theorems for the Special Functions of

Mathematical Physics, 3rd Edition, Springer-Verlag, Berlin, 1966.

McLachlan, N W., Bessel Functions for Engineers, Clarendon Press, Oxford, 1955.

Polyanin, A D and Zaitsev, V F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd

Edition, Chapman & Hall/CRC Press, Boca Raton, 2003.

Slavyanov, S Yu and Lay, W., Special Functions: A Unified Theory Based on Singularities, Oxford University

Press, Oxford, 2000.

Weisstein, E W., CRC Concise Encyclopedia of Mathematics, 2nd Edition, CRC Press, Boca Raton, 2003 Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002.

Calculus of Variations and Optimization

19.1 Calculus of Variations and Optimal Control

19.1.1 Some Definitions and Formulas

19.1.1-1 Notion of functional

Let a class M of functions x(t) be given If for each function x(t)M there is a certain numberJ assigned to x(t) according to some law, then one says that a functional J = J [x]

is defined on M

Example 1 Let M = C1[t0, t1] be the class of functions x(t) defined on the interval [t0, t1 ] and continuously differentiable on this interval Then

J [x] =

 t1

t0

1+ [x  t (t)]2dt

is a functional defined on this class of functions Geometrically, this functional expresses the length of the

curve x = x(t) with endpoints A(t0, x(t0)) and B(t1, x(t1)).

Calculus of variations established conditions under which functionals attain their ex-trema

Suppose that a functionalJ = J [x] attains its minimum or maximum at a function ˆx.

A strong (zero-order) neighborhood of ˆx is the set of continuous comparison functions (or trial functions) x such that

|x (t) – ˆx(t)| < ε (t1 ≤t≤t2)

for a given ε >0 A weak (first-order) neighborhood of ˆx is the set of piecewise continuous

comparison functions x such that

|x (t) – ˆx(t)|+|x 

t (t) – ˆx  t (t)| < ε (t1≤t≤t2)

for a given ε >0 If ˆx(t) minimizes the functional J [x] in a strong (resp., weak)

neigh-borhood of itself, then it is called a point of strong (resp., weak) minimum of the functional

J [x] If ˆx maximizes the functional J [x] in a strong (resp., weak) neighborhood of itself,

then it is called a point of strong (resp., weak) maximum of the functional J [x] Any strong

extremum is also a weak extremum Strong and weak extrema are relative extrema The

extremum of the functional J [x] over the entire domain where it is defined is called an

absolute extremum An absolute extremum is also a relative extremum.

The function classes conventionally used in calculus of variations:

1 The class C[t0, t1] of continuous functions on the interval [t0, t1] with the norm

x(t)0 = max

t1 ≤t≤t2|x (t)|.

2 The class C1[t0, t1] of functions with continuous first derivative on the interval [t0, t1] with the normx(t)1= maxt

1 ≤t≤t2|x (t)|+ max

t1 ≤t≤t2|x 

t (t)|.

991

992 CALCULUS OFVARIATIONS ANDOPTIMIZATION

3 The class C n [t0, t1] of functions with continuous nth derivative on the interval [t0, t1] with the normx(t) n= n

k=1tmax1 ≤t≤t2|x(k)

t (t)|.

When stating variational problems, one should indicate which kind of extremum is to

be found and specify the function class in which it is sought

Remark 1. The class P C1[t0, t1 ] of continuous functions with piecewise continuous derivative on the

interval [t0, t1] is equipped with the norm indicated in item 1 The class P C n [t0, t1 ] of continuous functions

with piecewise continuous nth derivative on the interval [t0, t1 ] is equipped with the norm indicated in item 2.

Remark 2 We do not specify function classes whenever the results are valid for an arbitrary normed space.

19.1.1-2 First variation of functional

The difference

of two functions x(t) and x0(t) in a given function class M is called the variation (or increment) of the argument x(t) of the functional J [x].

The difference

ΔJ ≡ΔJ [x] = J [x + δx] – J [x] (19.1.1.2)

is called the increment of the functional J [x] corresponding to the increment δx of the

argument

First definition of the variation of a functional If the increment (19.1.1.2) can be

represented as

ΔJ = L[x, δx] + β[x, δx]δx, (19.1.1.3) whereL[x, δx] is a functional linear in δx and β[x, δx] →0asδx →0, then the linear partL[x, δx] of the increment of the functional is called the variation of the functional and

is denoted by δ J In this case, the functional ΔJ [x] is said to be differentiable at the

point x(t).

Second definition of the variation of a functional Consider the functional F(α) =

J [x + αδx] The value

δ J = F 

of its derivative with respect to the parameter α at α = 0 is called the variation of the functional J [x] at the point x(t).

If the variation of a functional exists as the principal linear part of its increment, i.e., in the sense of the first definition, then the variation understood as the value of the derivative

with respect to the parameter α at α =0also exists and these definitions coincide

Remark The second definition of the variation of a functional is somewhat wider than the first: there exist functionals whose increments do not have principal linear parts, but their variations in the sense of the second definition still exist.

19.1.1-3 Second variation of functional

A functional J [x, y] depending on two arguments (that belong to some linear space) is

said to be bilinear if it is a linear functional of each of the arguments If we set y≡xin a bilinear functional, then the resulting functionalJ [x, x] is said to be quadratic A bilinear

functional in a finite-dimensional space is called a bilinear form.

A quadratic functionalJ [x, x] is said to be positive definite if J [x, x] >0for all x≠ 0.

A quadratic functionalJ [x, x] is said to be strongly positive if there is a constant C >0 such thatJ [x, x]≥Cx2for all x.

t0

 t1

t0

K (s, t)x(s)y(t) ds dt,

where K(s, t) is a fixed function of two variables, is a bilinear functional on C[t0, t1 ].

First definition of the second variation of a functional If the increment (19.1.1.2) of a

functional can be represented as

ΔJ = L1[δx] + 1

whereL1[δx] is a linear functional, L2[δx] is a quadratic functional, and β →0asδx →0, then the quadratic functionalL2[δx] is called the second variation (second differential) of the functionalJ [x] and is denoted by δ2J

Second definition of the second variation of a functional Consider the functional

F(α) = J [x + αδx] The value

of its second derivative with respect to the parameter α at α =0is called the second-order variation of the functional J [x] at the point x(t).

The first and second variations of a functional permit stating necessary conditions for the minimum or maximum of a functional

Necessary conditions for the minimum or maximum of a functional:

1 The first variation must be zero, δ J =0

2 The second-order variation must be nonnegative, δ2J ≥ 0, in the case of minimum; and

nonpositive, δ2J ≤ 0, in the case of maximum

19.1.2 Simplest Problem of Calculus of Variations

19.1.2-1 Statement of problem

The extremal problem

J [x]≡ t1

t0

L (t, x, x  t ) dt → extremum, x = x(t), x (t0) = x0, x (t1) = x1 (19.1.2.1)

is called the simplest problem of calculus of variations The function L(t, x, x  t) is called

the Lagrangian, and the functional J is referred to as a classical integral functional.

One usually assumes that the Lagrangian is jointly continuous in the arguments and has continuous partial derivatives of order≤ 3 The extremum in the problem is sought in the set

of continuously differentiable functions x(t)C1[t0, t1] on the interval [t0, t1] satisfying

the boundary (or endpoint) conditions x(t0) = x0and x(t1) = x1 Such functions are said to

be admissible.

An admissible function ˆx(t) provides a weak local minimum (resp., maximum) in prob-lem (19.1.2.1) if it provides a local minimum (resp., maximum) in the space C1[t0, t1], i.e.,

if there exists a δ >0such that the inequality

J [x]≥J [ˆx] (resp., J [x]≤J [ˆx]) (19.1.2.2)

holds for any admissible function x(t) C1[t

0, t1] such thatx(t) – ˆx(t)1< δ.

994 CALCULUS OFVARIATIONS ANDOPTIMIZATION

The classical calculus of variations deals not only with weak extrema, but also with

strong extrema In the latter case, one considers functions x(t) in the class P C1[t0, t1]; i.e., the extremum is sought in the class of piecewise continuously differentiable functions satisfying the endpoint conditions

An admissible function ˆx(t)  P C1[t0, t1] provides a strong local minimum (resp.,

maximum) in problem (19.1.2.1) if there exists a δ >0such that the inequality

J [x]≥J [ˆx] (resp., J [x]≤J [ˆx]) (19.1.2.3)

holds for any admissible function x(t)P C1[t0, t1] such thatx – ˆx0 < δ.

Since the set of admissible functions is wider for a strong extremum than for a weak

extremum, the following assertion is true: if ˆx(t)C1[t0, t1] provides a strong extremum, then it also provides a weak extremum It follows that, for such functions, a necessary condition for weak extremum is a necessary condition for strong extremum, and a sufficient condition for strong extremum is a sufficient condition for weak extremum

19.1.2-2 First-order necessary condition for extremum Euler equation

For the simplest problem of calculus of variations, the variation δ J of the classical

func-tionalJ [x] has the form

δ J [x, h] =

 t1

t0

∂L

∂x – d

dt

∂L

∂x  t



where h(t) is an arbitrary smooth function satisfying the conditions h(t0) = h(t1) =0and

d

dt = ∂

∂t + x  t ∂

∂x + x  tt ∂

∂x 

t.

A necessary condition for admissible function x(t) to provide a weak extremum in problem (19.1.2.1) is that δ J =0, i.e., that the function x(t) satisfies the Euler equation

∂L

∂x – d

dt

∂L

∂x  t

Here we assume that the functions L, ∂L/∂x, ∂L/∂x  t are continuous as functions of

three variables (t, x, and x  t ), and ∂L/∂x  t C1[t0, t1] The solutions of the Euler

equa-tion (19.1.2.5) are called extremals Admissible funcequa-tions satisfying the Euler equaequa-tion are called admissible extremals.

The Euler equation (19.1.2.5) in expanded form reads

L x – L tx 

t – x  t L xx 

t – x  tt L x 

t x 

From now on, the subscripts t, x, and x  tindicate the corresponding partial derivatives If

L x 

t x 

t≠ 0, then equation (19.1.2.6) is a second-order differential equation, so that its general solution depends on two arbitrary constants The values of these constants are in general

determined by the boundary conditions x(t0) = x0 and x(t1) = x1 The boundary value problem

L x– dt d L x 

t =0, x = x(t), x (t0) = x0, x (t1) = x1

does not always have a solution; if a solution exists, it may be nonunique

The Euler equation (19.1.2.5) for a classical functional is a second-order differential

equation, and so every solution x(t) of this equation must have the second derivative x  tt (t).

But sometimes a function on which a classical functionalJ [x] attains an extremum is not

twice differentiable

An extremal also satisfies the equation

d

dt L – x  t L x 

t

The points of an extremal x(t) at which L x 

t x 

t ≠ 0are said to be regular If all points of

an extremal are regular, then the extremal itself is said to be regular (or nonsingular) For

regular extremals, the Euler equation can be reduced to the form

x 

tt = f (t, x, x  t).

Remark 1. An extremal x(t) can have a break point only if L x 

t x 

t = 0

Remark 2 Extremals, i.e., functions “suspected for extremum,” should not be confused with functions that actually provide an extremum.

19.1.2-3 Integration of Euler equation

1◦ The Lagrangian L is independent of x 

t ; i.e., L(t, x, x  t ≡L (t, x).

The Euler equation (19.1.2.5) becomes

In general, the solution of this finite equation does not satisfy the boundary conditions

x (t0) = x0and x(t1) = x1 There exists an extremal only in the exceptional cases in which

the curve (19.1.2.8) passes through the boundary points (t0, x0) and (t1, x1)

2◦ The Lagrangian L depends on x 

t linearly; i.e., L(t, x, x  t ≡M (t, x) + N (t, x)x  t The Euler equation (19.1.2.5) becomes

where derivative M x and function N are evaluated at x = x(t) In general, the curve

determined by equation (19.1.2.9) does not satisfy the boundary conditions, and hence, as

a rule, the variational problem does not have a solution in the class of continuous functions

If equation (19.1.2.9) is satisfied identically in a domain D on the plane OXY , then

L (t, x, x  t ) dt = M (t, x) dt + N (t, x) dx is an exact differential; consequently, J [x] is

inde-pendent of the integration path and has a constant value for all x In this case, the variational

problem becomes meaningless

3◦ The Lagrangian L is independent of x; i.e., L(t, x, x 

t ≡L (t, x  t)

The Euler equation (19.1.2.5) becomes

d

dt L x 

whence it follows that

L x 

Equation (19.1.2.11) is a first-order differential equation By integrating this equation, we obtain the extremals of the problem

996 CALCULUS OFVARIATIONS ANDOPTIMIZATION

Remark. Relation (19.1.2.11) is called the momentum conservation law.

4◦ The Lagrangian L is independent of t; i.e., L(t, x, x 

t ≡L (x, x  t)

The Euler equation (19.1.2.6) becomes

L x – x  t L xx 

t – x  tt L x 

t x 

whence we readily obtain a first integral of the Euler equation:

L – x  t L x 

Remark. Relation (19.1.2.13) is called the energy conservation law.

5◦ The Lagrangian L depends only on x 

t ; i.e., L(t, x, x  t ≡L (x  t)

The Euler equation (19.1.2.6) becomes

x 

tt L x 

t x 

In this case, the extremals are given by the equations

where C1and C2are arbitrary constants

Example 1 Let us give an example in which there exists a unique admissible extremal that provides a

global extremum.

Let

J [x] =

 1

0 (x  t)2dt → min, x = x(t), x( 0 ) = 0 , x( 1 ) = 1 The Euler equation becomes 2x  tt= 0 The extremals are given by the equation x(t) = C1t + C2 The unique

admissible extremal is the function ˆx(t) = t, which provides the global minimum Suppose that x  t (t)C1[ 0 , 1 ],

x( 0 ) = 0, x(1 ) = 1 Then

J [x] = J [ˆx + h] =

 1

0 ( 1+ h  t)2dt=J [ˆx] +

 1

0 (h  t)2dt≥J [ˆx]

for an arbitrary function h(t) such that h(0 ) = 0and h(1 ) = 0

Example 2 Let us give an example in which there exists a unique admissible extremal that provides a

weak extremum but does not provide a strong extremum.

Let

J [x] =

 1

0

(x  t)3dt → min; x = x(t), x( 0 ) = 0 , x( 1 ) = 1 The Euler equation becomes 6x  t x  tt= 0 The extremals are given by the equation x(t) = C1t + C2 The unique

admissible extremal is the function ˆx(t) = t, which provides a weak local minimum Indeed,

J [x] = J [ˆx + h] =

 1

0 ( 1+ h  t)3dt=J [ˆx] +

 1

0 (h  t)2( 3+ h  t ) dt for an arbitrary function h(t) such that h(0) = h(1 ) = 0 Thus, ifh1 ≤ 3 , then 3+h  t> 0 and henceJ [x]≥J [ˆx].

Consider the sequence of functions

g (t) =

–√

n, t [ 0 , 1/n),

0 , t [ 1/n, 1/2 ],

2/ √

n, t ( 1/2 , 1 ].

h n (t) =

 t

0

g (τ ) dτ (n =2 , 3, ).

Obviously, h n( 0) = h n( 1 ) = 0 andh(t)0→0as n → ∞ We set xn (t) = h  t (t) + h n (t) and obtain a sequence

of functions x n (t) such that x n( 0 ) = 0, x n( 1 ) = 1, x n (t) → ˆx(t) in C[0 , 1 ], and

J [xn] =J [ˆx + hn] =

 1

0 ( 1+ h  t)3dt= 1 + 3 1

0

g2n dt+

 1

0

g n3dt

= 1 +

 1/n

0 3n – n3/2

dt+

 1

1/2

12

n – 8

n √ n



dt= –√

n + O(1 ).

In this caseJ [xn]→ –∞ as n → ∞.

is denoted... case of minimum; and

nonpositive, δ2J ≤ 0, in the case of maximum

19.1.2 Simplest Problem of Calculus of Variations

19.1.2-1 Statement of problem... that, for such functions, a necessary condition for weak extremum is a necessary condition for strong extremum, and a sufficient condition for strong extremum is a sufficient condition for weak