On the hydrodynamics of optimum sculling propulsion of ships and on the linearized lifting surface theory

TECHNISCEE UNIVERS ITEIT S cheepshydromechaiai ca krchi ef Mekeiweg 2, 2628 D Deif t Tel: 015-786873/Fax:781836 ON THE HYDRODYNAMICS OF

OPTIMUM SCULLING PROPULSION OF SHIPS AND ON THE LINEARIZED LIFTING SURFACE THEORY

Stellingen behorende bij het proefschrift:

ON THE HYDRODYNAMICS OF

OPTIMUM SCULLING PROPUlSION OF SHIPS AN!) ON TI-TE LINEARIZED LIFTING SURFA THEORY

van Pieter Sijtsma

1. Een essentieel verschil tussen de optimaliseringsproblemen in Potze's proefschrift [1] en die in deel één en twee van het onderhavige proefschrift

is dat door Potze de dwarskracht niet wordt meegenomen in het optirnaliseringsproces. Bij Potze leidt dit tot randwaardeproblemen waarbij de normaalsnelheden op de wervelv!akken expliciet worden gegeven. In het tweedimensionale geval kan met behuip van Stelling 3.2 uit [2], blz.137 bewezen worden dat er een unieke potentiaal bestaat die aan hot

randwaardeprobleem voldoet. Deze potentlaal is oneindig vaak continu

differentieerbaar tot op de wervelvlakken.

W. Potze, On optimum scaling propulsion of ships, RUG, 1987.

O.A. Ladyzhenskaya en N.N. Ural'tseva, Linear and quasilinear elliptic equations, Academic press, 1968.

2. De mathematische behandeling van het optimaliseringsprobleem in deel één

van dit proefschrift zou eon stuk eenvoudiger, eleganter en sneller kunnen als

j-begrensde functies van de klasse H2(R) continu zouden zijn. Dit hoeft niet het geval te zijn, maar eon voorbeeld is bij de auteur niet bekend.

3. "Optimalisatie" in plaats van "optimalisering" is een veo! gemaakte

taalfout, ook door mensen die zich met het ondeiwerp bezighouden.

4. In hot veel voorkomende geval van een lineair affiene nevenconditie zijn de voorwaarden "X=P - P" en "X=P + Z" van de Thomasversie van de stelling van KuhnTucker (zie blz.35 van dit proefschrift) strikt zwakker dan die van Luenberger ([3], blz.249), die uitgaat van de conditie 2nint(P) 0, waarbij

2={Q(u0)+8Q(u0;h), hY}cZ.

B(0;e)cX, zodanig dat q+bEP voor alle bEB(0;e). We willen laten zien dat aan de voorwaarde van Thomas voldaan is, dus dat X = P - P en X = P + X. Daarvoor moeten we aantonen dat x eX impliceert XE P - P en x E P + Z.

Zij nu x een willekeurig element van X, dan bestaat er een A >0 en een waarvoor x=Ab. Aangezien q+beP, moet er dus een pEP bestaan,

zodanig dat p = q + b en dus x = Ap - Aq. Hieruit voigt dat XE P - P want q e P en verder dat XEP+Z want qEZ. Dit wilden we bewijzen, zodoende geldt tevens dat voldaan is aan de voorwaarden van Thomas.

Anderzijds zijn er gevallen waarin wel de condities van Thomas gelden, maar niet die van Luenberger. Een voorbeeld daarvan is het optimaliserings-probleem in deel één van dit proefschrift.

[3] D.C. Luenberger, Optimization by vector space methods, Wiley, 1969.

In het elital der Wetenschap is de Wiskunde een dienende speler. Indien zij zelf probeert te scoren is zij nutteloos.

Als het stadsbestuur van Groningen de binnenstad 's nachts echt veiliger wil maken, moet er een caféverbod na middernacht worden ingevoerd.

Het is een vooroordeel orn te veronderstellen dat jongens van nature meer aanleg hebben voor wiskunde dan meisjes. Andersorn is ook de veronderstelling dat ze evenveel aanleg hebben een vooroordeel.

S. Alle gemotoriseerde sporten rnoeten verboden worden, vanwege de negatieve

educatieve werking die daarvan uitgaat en het zinloos belasten van het milieu.

9. Beschouw de volgende kansen:

de kans dat de kleur van een auto zwart is,

de kans dat de bestuurder van een auto asociaal weggedrag vertoont.

TECHSCE LINIVERSITEIT Laboratorium voor Scheepshydromechan;ca Archief Mek&weg 2, 2628 CD D&ft 1ej 015.786873- ax 015-781833 RJJKSUNIVERSITEIT GRONINGEN ON THE HYDRODYNAMICS OF

OPTIMUM SCULLING PROPULSION OF SHIPS AND ON THE LINEARIZED LWrING SURFACE THEORY

PROEFSCHRIFT

ter verkrijging van het doctoraat in de

Wiskunde en Natuurwetenschappen

aan de Rijksuniversiteit Groningen

op gezag van de

Rector Magnificus Dr. S.K. Kuipers

in het openbaar te verdedigen op

vrijdag 14 februari 1992 des namiddags te 2.45 uur precies

door

PIETER SIJTSMA

geboren op 16 mel 1963 te Groningen

Eerste promotor: Prof. dr. J.A. Sparenberg Tweede promotor: Prof. dr. E.G.F. Thomas

CONTENTS

Contents i

Preface 4

PART I

On optimum sculling propulsion with an inequality constraint on the side force

Introduction 7

Description of the physical problem 10

Formulation of the mathematical problem 16

Existence, uniqueness and reformulation 25

An alternative version of the generalized Kuhn-Tucker Theorem 35

Applicability and application of the generalized Kuhn-Tucker Theorem 41

Determination of the solution 51

Numerical results 57

Notations 61

References 63

PART II

On optimum sculling propulsion with prescribed side force by wings of finite span

Introduction 65

Description of the mathematical model 68

How to solve the problem 74

The numerical method 80

Numerical results 83

Notations 93

References 94

PART III

On useful shapes of rigid wings for large amplitude sculling

propulsion

Introduction 95

Formulation of the mathematical problem 98

Short discussion of two known results 106

Numerical approach to the problem 108

Numerical results and conclusions 113

Appendix 123

Notations 124

Acknowledgements 125

References 125

PART iV

A functional analytic approach to the linearized theory of lifting surfaces using external force fields (inviscid and incompressible

fluid)

Introduction 127

Mathematical Preamble 130

2.1. Prerequisites of the theory of distributions 130 2.2. Prerequisites of the theory of partial differential equations 135

2.3. Prerequisites of vector analysis 136

2.4. The time-independent Poisson equation 139

2.5. The time-dependent Poisson equation 140

2.6. Representation of a vector field by its rotation and its

divergence 142

Velocity induced by external forces 143

3.1. The equations of motion 143

3.2. Solution of the equations of motion 144

3.3. Example: a singular "blow" 148

3.4. Locally time-integrable force fields 150 3.5. Forces starting an "infinitely long time ago at R = co" 152

Examples of external force fields 157

4.1. Singular force in the origin 157

4.2. Singular force moving in its own direction 157

4.3. Singular force moving perpendicularly to its own direction 160

4.4. The lifting surface 162

The normal velocity at a dipole surface layer 169

5.1. Some preliminaries 169

5.2. Continuity of the normal velocity 170

5.3. Limiting procedure for calculating the normal velocity 178

Application to lifting surface theory 182

6.1. The general procedure 182

6.2. "Stationary lifting surface theory" 183

Notations 188

References 190

Samenvatting 191

PREFACE

This thesis consists of four parts, three of which are motivated by the

idea of propelling a ship by wings. The fourth part is more general and gives some fundamental considerations about lifting surface theory.

In part one and part two, we consider sculling propulsion by one wing and by two wings, respectively. These wings are mounted vertically behind a ship and move sideways to and fro in a prescribed way. The sculling wings have varying angles of attack with respect to their directions of motion, such that the water is pushed backwards and hence a thrust is created.

From a hydrodynamical point of view, a sculling propeller seems to be more preferable than a conventional screw propeller, because of two reasons.

The first one is that the velocity of the wings is constant along their span,

while the velocity of a screw blade increases towards the tip. The second one is that the amount of water influenced by the sculling wings can be larger. By this, the induced velocities of the fluid will remain in general smaller,

hence the lost kinetic energy can possibly be less and the hydrodynamical efficiency can be higher.

From a mechanical point of view, a disadvantage of the sculling propeller may be the complexity of the mechanism needed to move the wings.

We say that the sculling propeller carries out an optimum motion, when

the kinetic energy that is created by the wings and that is left behind in the

water is

as small as possible. In order to obtain nontrivial

solutions, of

course some constraints have to be put on the action of the propeller. It will

always be required that a prescribed mean thrust is produced. Besides this, other constraints can be introduced, which will be discussed below. Optimum sculling propulsion has been investigated previously by Potze [1].

thrust, but also periodic side forces, acting at the stern of the ship. These forces can have a disturbing influence on the course of the ship. Hence it

seems desirable to put constraints on these side forces. The influence of such

constraints on the optimum efficiency is considered in the first two parts of

this thesis.

In part one, we consider a onewing sculling propeller by means of a

linear and twodimensional theory in which the wing is represented by a

lifting line of twosided infinite length. The existence and the uniqueness of

the optimum propulsion is investigated, under the condition that the side

force, exerted by the wing is smaller than a prescribed value. In order to obtain the optimum propulsion, an optimization problem with an inequality constraint has to be solved. For that purpose a new, unpublished version of

the generalized KuhnTucker Theorem is used, which is due to Thomas and Van der Meer. At the end of this part some numerical results are given.

In part two, sculling propulsion is considered for propellers consisting

of two wings of finite span. Here again a linear theory is used, while the

wings are represented by lifting lines of finite lengths. The possibility of an optimum propulsion is considered when the sam of the side forces delivered

by the two wings has a constant, prescribed value, for instance zero. Also a

nonzero value for the total side force can be useful when a ship has to

pursue a prescribed course in case of a side wind. In this part, an

optimization problem with an equality constraint has to be solved. It appears that, under this constraint, indeed an optimum propulsion exists, which can be calculated in a relatively simple manner. Results are compared with those

obtained by putting no constraint on the side force.

In part three of this thesis a problem is considered which is inspired by

sculling propulsion. Here we investigate the possibility of shapes of wings,

which can move in a nontrivial way through an inviscid and incompressible

fluid, without leaving free vorticity behind. For this purpose we consider

rigid, flat wings, that is, wings without thickness and without curvature,

which have special shapes. The manner in which these surfaces move is that they slide tangentially to an arbitrary cylindrical surface. Since they will not leave behind free vorticity, they do not experience induced resistance. In fact, when we prescribe only the lengths of the chords along the span, hence not their relative position, it is found by numerical means that a unique wing

shape with this property seems to exist. Wings of this shape and carrying out this motion can not produce a mean thrust, however, they can be used to create a "base motion" in the theory of sculling propulsion.

In part four, we solve, by means of the theory of distributions, the

threedimensional linearized equations of motion, in case of an incompressible

and inviscid fluid, for

a large

class of given external force fields. As

applications are treated some examples of timedependent force fields which

are confined to a set of zero volume. As a special case,

linearized lifting

surface theory is considered, in which the wing is represented by a moving

force field concentrated on a planform which is allowed to change its

curvature, its span and its chordwise dimensions. At the end of this part a

formula is given by which the normal velocity at

the planform can be

calculated, when the external force field at it, in other words the pressure jump over it, is given. This is done by showing that the Hadamard principal

value is valid also for general curved surfaces. Up to now this seemed, to the best knowledge of the author, only to have been proved for flat wings and for ship screws with a straight generator perpendicular to the axis of the screw.

Each part of this thesis is intended to be a separate article and can be read as such, without knowledge of the other parts. That is why there are

repetitions in the introductions of some of the parts. Furthermore, the nomenclature differs somewhat from part to part. Therefore we listed notations

after each part.

The approaches to the problems in part one and part four are by means of

functional analysis. In part two and part three a more applied mathematical treatment is given, without going into the matter of existence or uniqueness

of a solution.

[1] W. Potze, 0m optimum sculling propulsion of ships, Ph.D. thesis,

PART I

ON OPTIMUM SCULLING PROPULSION

WITH AN INEQUALITY CONSTRAINT ON THE SIDE FORCE

1. INTRODUCTION

We consider sculling propulsion by means of one wing, mounted vertically

behind a ship (see figure 1.1). The wing carries out a lateral motion, while

its angle of attack is adjusted such that a thrust is created.

Fig. 1.1: Stern of a ship, equipped with a one-wing 8CUlliflg propeller W.

The motion of the wing can be split into two motions. First, the motion of a chosen pivotal line of the wing, which motion is determined by the path

of this line and its velocity along that path. Second, an oscifiating motion

of the wing around the pivotal line.

We say that the wing carries out an optimum motion, when the kinetic energy that it creates and that is left behind in the water is as small as

possible, under the condition that it produces a prescribed mean thrust. Optimum sculling propulsion by one or two wings has been previously

investigated by Potze and Sparenberg ([14], [15], [16]), where the paths along

which the pivotal lines move were prescribed, while the oscillating motion of the wings was optimized.

In case of sculling propulsion by one wing, the ship is propelled in a way as fishes do. The tailfin of a fish oscillates in order to deliver a mean

thrust. In several studies the efficiency of fish tail propulsion (or

propulsion by one oscillating wing) is investigated (for instance [2], [3],

[4], [6], [7], [9] and [24]). In these papers the motion of the wing is chosen

in advance in

one way or another,

next the thrust coefficient and the

efficiency are calculated. All the abovementioned investigations of this

paragraph concern small lateral amplitudes of the wing motion. We also refer to [211, where it has been proved that, if besides the oscillating motion of

the wing also its large amplitude path is not prescribed, an optimum motion exists, when certain constraints are put on the shape of this path. A drawback

of a onewing propeller is that the fluctuating lateral forces acting at the

stern of the ship are large. These are of the same order of magnitude as the

propulsive forces and may be disturbing.

It may however be of interest to consider one wing instead of two wings, because the machinery needed to move one wing will be less complicated. In

this case it is not possible to demand the side force to be continually zero,

since then the wing can not produce thrust anymore. It is however possible to

demand the side force to be smaller than some prescribed value, if this

prescribed value is not too small.

In this part we will examine this possibility.

In order to discuss the

problem as much as possible in a thoroughly mathematical way, we use a simple,

linear and twodimensional theory, in which the wing is represented by a

lifting line (hence zero chord length) or bound vortex of twosided infinite

length. In chapter 2 we describe this mathematical model.

The mathematical model that we use leads to an optimization problem with inequality constraints. In order to compute the optimum motion belonging to

this model, we need the results of a generalized Kuhn-Tucker Theorem for

inequality constraints. Classical theorems on this matter, for instance [11],

p.249, are not applicable to our problem, since they require function spaces with a positive cone with non-empty interior (regularity condition). For the

function spaces that we use, this is an impossible demand.

In [131, p.128, we found a theorem that permits the positive cone to have

empty interior. We could have used this theorem for our situation, but it turned out that its conditions were difficult to verify. We chose to apply

another version of the Kuhn-Tucker Theorem, with other conditions, which can be checked more easily in our situation. This version, which has not yet been published, is due to Van der Meer and Thomas [12].

Before we apply this theorem, we will formulate the minimization problem

in a mathematically correct way (chapter 3). Having done this, we prove in

chapter 4 that this minimization problem has a unique solution, we derive some properties of this solution and, with the use of these properties, we

reformulate the problem, so that the Kuhn-Tucker Theorem can be applied to it.

We remark that, although we have tried to give a rigorous treatment of the

problem, we had to make some assumptions in the course of the analysis, which we could not justify mathematically. The numerical results, however, support the validity of these assumptions.

In

order to make this

article more self-contained, we reproduce in

chapter 5 the proof of the version of Thomas and Van der Meer of the

Kuhn-Tucker Theorem. In chapter 6 we prove that we can apply it to our situation. We will see in chapter 7 that, with the results of the Kuhn-Tucker

Theorem, the optimization problem leads to a boundary value problem. We will describe

a method by which we can solve

this boundary value problem

numerically.

In chapter 8 we give some numerical results. We compute the strength of the concentrated bound vortex in case of an optimum motion and the thrust and the side force that it produces. Furthermore, we give the strength of the free

vorticities, shed by the bound vortex and the hydrodynamical propulsive efficiency. Because our theory is two-dimensional, it follows that this

efficiency is without taking into account the losses caused by the tip vortex, the disturbances at the free water surface and the inhomogeneous flow behind a ship. It can be expected that the efficiency that we calculate will be higher

than in reality, which is also due to the fact that we assume the fluid to be inviscid. At the end of chapter 8, we show a picture with streamlines of the

wake.

2. DESCRIPTION 0F THE PHYSICAL PROBLEM

We assume the whole threedimensional space to be filled with an

incompressible and inviscid fluid of density p, in which we have a Cartesian reference frame x,y,z. The fluid is assumed to be "at rest" with respect to

this reference frame for large values of lyl.

Fig. 2.1: The motion of the lifting line.

As has been stated in the introduction, we consider a twodimensional

model, in which we replace the propelling wing of finite span by a

concentrated bound vortex T of (twosided) infinite length in the zdirection. This vortex, or lifting line, moves along the path G such that its velocity

has a constant, prescribed component U in the positive xdirection, which is the velocity of the ship. The path G is given by the equation y=g(x), with

2T . .

g(x) = rszn(--x), where b is the distance passed through by the ship in one

period of time r = b/U.

We see that the motion is periodic in time and furthermore that the

motion of r is the same on portside as on starboardside. In other words, the

function g is bperiodic and furthermore g(x+b) = g(x) for each XER. We also

each XER.

We say that the concentrated vortex r has the strength

() at the time

that its xcoordinate equals . () is reckoned positive when the circulation

of the fluid around the lifting

line is in the direction as drawn in figure

2.1. We demand that r(x) is bperiodic. The velocity field of the fluid is

twodimensional:

ii(x,y,t) = (v1(x,y,t), v2(x,y,t)). (2.1)

Since we are dealing with a hydrodynamical problem, the velocity field has to be free of divergence:

av av

dzvv= + - =0.

_{ax ôy}

Fig. 2.2: Lift force per unit span, acting on the lifting line (wing).

The motion of the lifting line causes a lift force, acting on this line

(see figure 2.2). We denote by (x) that unit normal vector on G, which has a component in

the positive ydirection and by V(x)

the magnitude of the

velocity of r along G.

Although we consider a wing which does not only translate, but also rotates, it is allowed to use Joukowsky's Theorem. This is because the chord

length of the wing is zero in our model, since it is represented by a lifting

line. In fact, our theory is an approximation of the situation in which the product of the angular velocity and the chord length of the wing is

sufficiently small with respect to its velocity V(x). Therefore it is

necessary that the chord length of the wing is small with respect to the

radius of curvature of G.

Using Joukowsky's Theorem, the lift

force Z(x) per unit span of the

lifting line equals:

Z(x)=L(x)(x)=pF(x)V(x)(x)=pF(x)_cos0(x)U

(sim9(x)\

_eosO(x)J' (2.3)

where 8(x) is the angle between G and

the xaxis at

the place under

consideration.

Because the strength of the concentrated bound vortex F varies with time, it sheds free vorticity into the water. In (2.3) the disturbance velocity, caused by this free vorticity, is neglected. This is allowed if the strength

of the bound vortex is small, say F(x)= O(e). Then we neglect forces of O(e2). n+1

In the following we neglect quantities of Q(e ) with respect to quantities of O(e'), hence we develop a linearized theory.

We decompose Z(x) in a thrust component (i.e. the xcomponent) per unit

span FT(x) and a side force component (i.e. the ycomponent) per unit span

Fs(x). In the following we will often omit the addition "per unit of span".

Because tg8(x)=g'(x), we have:

pUF(x)g'(x),

Fs(x) =pUT(x). (2.4)

We demand that the mean value of the thrust FT(x) has the prescribed

value T. This gives:

IbFT(x)dx=

_PUÇ

_b F(x)g'(x)dx= _PUÇ_b F(x)g'(x)dx=T.

Furthermore, we demand, at any time, the side force Fs(x) to be in absolute

sense less than a prescribed value 3, hence:

VxEF: Fs(x)pU1(x)3.

(2.6)

If we optimize the propulsion (i.e. minimize the lost kinetic energy)

under the inequality constraint (2.6), then we must beware of taking 3 not too small, because otherwise the prescribed thrust T can not be delivered anymore.

In fact, by (2.4) we have FT(x)= F(x)g'(x) and fFT(x)dx =

43;

-J

_b0

Ig(x)Idx=_J'

_b b Hence if we define: bT 5'm2n ₄

then we must take in (2.6) 3; Smin. It is also easily seen that Smin is the

smallest value of 3; for which T still can be obtained. If 3;=8mifl then we must

have, in order to attain the prescribed thrust T:

{Purx=

Smjn, xE(O,b)U(*b,b),

But this means that P leaves behind concentrated vortices at the points x=

and x

= b. Since a concentrated vortex has, per unit of span, an infinite

amount of kinetic energy around it, 1' leaves behind per period of time and per

unit of span an infinite amount of kinetic energy. This is neither realistic

nor desirable, hence we will assume that 3;> Smzn.

Suppose we would optimize the propulsion with only the constraint of a prescribed mean thrust, hence no constraint on Fs(x), and the same values of

p, U, b, r and T as before. Then we obtain a solution, to which a side force belongs, of which the absolute value has a maximum, which we will call S.

Clearly, it makes no sense to take in (2.6), hence in the following we

assume:

Smin<S<Sx.

(2.10)

In realistic cases, when the amplitude of the motion r is not too large

with respect to the length period b, say r b, it turns out by numerical means rb

I F(x)g(x)dx

PUT(X)=Smjn, xE(b,b). (2.9)

(2.7)

that the bound vorticity 1(x) behaves approximately like a sinus. Because of (2.5) it must be about equal to:

bT 2irx cos irpUr

(h--)

Then the side force (2.4) has a maximum absolute value S, for which we

have:

S,PX=bT (2.12)

From (2.8), (2.10) and (2.12) it follows that in realistic cases the maximum

side force 3 can be reduced to about x10078.5 percent of the value S,

As we mentioned already, the strength of the concentrated bound vortex 1' varies with time, hence it sheds free vorticity into the water, which is parallel to

the zaxis.

In our linearized theory, in which the fluid is inviscid and the velocities are sufficiently small, the free vorticity is

assumed to stay at the position, where it is formed, hence it is distributed

ori the path G of F. Denoting the strength of the free vorticity by y(x), we

have the relation:

y(x)V1+g(x)2=

_-

_p1(x).

dx

Then the only vorticity that is present in the fluid is on G, hence the

fluid outside G is free of rotation:

-. av2 av

rotv= - - =0.

_{ax ay}

So outside G a velocity potential Ø(x,y,t) exists, for which we have by (2.2):

(i=gradç=

(a a

ax

2

a2

=o. (2.15)

Now we assume that the ship has "started to move at x = - " and, at the

(2.11)

(2.13)

moment under consideration, has "arrived at x = +". Then the velocity field of the fluid has become independent of time and b-periodic in the x-direction. It is only induced by the free vorticity y on G. When the functions g and -y are sufficiently smooth, we can prove ([18]) that a potential Ø(x,y) belonging to

this velocity field exists, with the following properties: Ø is b-periodic in the x-direction, the normal derivative on G, which is the normal velocity on G, exists and is continuous across G. This is in agreement with the absence of sources and sinks on G.

We define the jump [](x) over G by:

[çb]t(x)= lim{Ø(x,g(x)+e)-çb(x,g(x)-e)} c'i O

It is easily seen that: = 1(x).

The aim of this paper is to minimize the kinetic energy of the water per period b in the x-direction, left behind by the lifting line. Hence, in our

simple two-dimensional model we have to minimize the kinetic energy that is located in the region {(x,y,z)ER3;0<x<b, yELR, 0<z<1}:

rb

rbroc

E()=.pJ

J )(xy)Ii2dYdx=.PJ J grad4(x,y)I2 dydx. (2.18)

O_ao

This is the kinetic energy, per unit of span, that is created in one period of

time. Furthermore, we have the constraints z4 = 0, (2.5) and (2.6), of which the last two can be written by (2.17) as:

)g(x)dx= - bT (2.19)

-

_{pU pU} (2.20)

In the next chapter we translate this optimization problem

into a

mathematically well posed problem.

(2.16)

3. FORMULATION OF THE MATHEMATICAL PROBLEM

One of the questions to be answered in this chapter is how to choose the function space in which we will optimize the potential 4, as introduced below (2.14). This potential must be well defined outside the line G. Further, it is

prescribed to be periodic with period b in the xdirection.

The propelling wing leaves behind a finite amount of kinetic energy per unit of time. This means that the integral:

JaJ --(x,y)2+

-(x,y)

}dYdx

çCçJ ₂

(3.1)

is finite on each strip (a,c)x( cc,00), where co<a<c<co. Further we know that

the fluid velocity (- , decreases exponentially as y-. This is because

the integral of the free vorticity y(x), over one period b of motion, is zero.

Then the potential ç5 tends to a constant value C1 as y- +m and to C2 as y- co. For reasons of symmetry we can choose C1 = C2 and without loss of generality, we can assume C1 = C2 = O. Then, because of the rapid decrease of the velocity, we find for any choice of finite a and e:

C

ç5(x,y)2dydx<co. (3.2)

a

-We can recapitulate the foregoing by stating that the potential Ø is a

periodic function with period b in the xdirection, well defined outside G and satisfying the following. Given arbitrary numbers a,cER, with co<a<c<c, let:

W(a;c)={(a,c)x(co,00)}\G, (3.3)

then ç5, and are all square integrable on W(a;c).

To make this more precise, we will recall some definitions. For open

Ac, the space D(A)

is defined as the space of all realvalued functions

cpeC(A), which have compact supports in A, that is supp()cA, where supp(ço) is by definition a closed set. These functions are called "test functions".

zero in D(A) for k-*ce, if there exists a compact set KcA, such that supp(p)cK,

for all k, and if Pk and each of its derivatives tend uniformly to zero for

k-cc.

We define the space of "distributions" D'(A) as the space of all

continuous, linear functionals D(A)-, hence as the dual space of D(A). The

image of VED(A) under a functional tLED(A) is denoted as v[y]. Differentiation of distributions is defined by transposition:

ôu

_{[ço]= u[],}

_(3.4)

multiplication with a function EC(l) is defined by:

cxu[]=u[cxp]. (3.5)

The space L2(A) consists of (equivalence classes of) realvalued, square integrable functions on A. L2(A) is a subspace of D'(A), since each function UEL2(A) defines a continuous, linear functional from D(A) to l by

u[ç]= <u, °>J(A) where <>(A) is the inner product on L2(A), defined by:

<Ui, U2>L2(A)J u1(x)U2(x)dx,

A

for u1,u2eL2(A).

The Sobolev spaces Hm(A), m=O,1,2,.., are defined by:

Hm(A)={UEL2(A); Vk,jkj m: DkuEL2(A)}. (3.7)

In (3.7), k is a multiindex: k=(k1,..,k), with

_{kl =Ik1I+..+lkI}

and

Dk= 81k1

o4'.

.. ax»

The differentiation in (3.7) is in the sense of distributions: def

U, _>(A)

(-1)<u,

_>(A),

(3.6)

(3.8)

for çoeD(A). The inner product on ll'"(A) is defined by:

<U, V>j(=

<D'eu, Dkv>(A). (3.10)

1k I Ñu

We also introduce the set:

D(A)={p; pcD(F)}.

(3.11)

When the boundary of A is sufficiently smooth (it should not contain "cusps"), it can be proved ([18], Th. 3.1, p.17) that D(À) is dense in Hm(A). Further we

define for open Ac the space H70() as the space of functions which are

locally of the class Hm(A), that is;

HToc(4){u: A-*i; \/çED(A): çEft"(A)}. (3.12)

We have the inclusions;

D(A)cD(A)cH1(A)cL2(A)cD(A). (3.13)

Now let us return to our optimization problem. A suitable space in which we can optimize the potential seems to be:

{: R2-*;

EH'(W(a;c)), for all finite a and c,

Ø is bperiodic in x_direction}, (3.14)

where W(a.;c) is as defined in (3.3). For the sake of uniqueness, we will impose the additional condition that:

VyR;

(b,y)=0.

(3.15)

This condition is not unfounded, because we may expect,

for reasons of

symmetry, that Ø(x,y) = Ø(bx,y). Hence, the space in which we will optimize

Y={ç5: -.R; EH'(W(a;c)), for all finite a and c,

is bperiodic in xdirection, satisfies (3.15))-. (3.16)

Equation (3.15) may have no meaning for E1V in the classical sense, since it

is defined on a region of zero measure, but in a generalized sense it can be

well defined. This will be discussed later on in this chapter.

In the space

i,

the hydrodynamic condition L.çt=

O is not taken into

account. But that is not objectionable, since this relation follows

automatically from the minimization process, as we wifi see later on. In fact, the relation div = =O is nothing but the Euler equation, when minimizing the kinetic energy. This means that the procedure that we follow here can not be carried out if we would have prescribed div _{= = f, where f} is not identically zero.

Because of the constraints (2.19) and (2.20), we need to know if and in what sense such functions aí have boundary values on G. Therefore we must look more closely at the space

i.

The space is a direct sum of í and , consisting of functions defined "above" G and functions defined "below" G, respectively. The space

is a subspace of H(?+), where Q

is the region

above G. The region below G wifi be denoted by IL. The space H(Th) is

isomorphic with the space H0(Ä+), where A is the open halfspace

x (0,).

The isomorphism Ht0(Th+)-*H0(A) is given by the mapping Ø(x,y)-(x,yg(x)). In

case of

this

halfspace A, it

turns out (see [10] or [20]) that

functions ueH1(A) have boundary values belonging to the function space

H(aA), which is isomorphic with the space H(l). (Spaces H8(), SEF, will be discussed below this paragraph.) Formally, this means that the mapping L,

which maps functions ço a D(A) onto their boundary values pIaA can be extended

uniquely

to a mapping L: H'(A)-.H(R),

which is linear, continuous and surjective. In other words, there exists a unique, surjective mapping

i

L: 11 (A+)-H2(), which is lmear and continuous and satisfies for

aD(A). The image Lu is called the "trace" of u on OA.

To understand this, we will define in the following the Sobolev spaces H(I) for saR. To this end, we need the concept of Fourier transforms in the

Schwartz space or space of rapidly decreasing functions. This space is defined by:

S()={:R'-C;

C(); Vk,10:

fxklJço(x)IO, _{if !IxlI-oo}, (3.17)} where k and I are multiindices and x E ; the "power" x means:

k k k

X =Xj'..x'

and k O means k1 O,. . , k,. O. In (3.17) we admitted complexvalued functions. This is done so that we can introduce the Fourier transform. Continuity on

n

S(R ) is defined by:

S

_*

ki

n

Vk,l0:

sup{xD(x)I; XE!R }-).0.

We define the space of tempered distributions S(l) as the dual space of

S(F'), that is the space of continuous, linear functionals

u: S(l)-C. The

space L2(l) is a subspace of S(l). Hence we have the inclusions:

D(P)cS(PJ) cH'()cL2(F)cS'(F?)cD'(lR.'). (3.20)

The Fourier transform T: S(l)-*S(U) is a continuous bijection, given by:

C -27ri<,x>

T(e)=

J p(x)dx. J R'

Following Schwartz [17] we can extend by transposition the domain of

definition of the Fourier transform to the space of tempered distributions

S'() by:

del

Tu[ço] = u[Tç], (3.22)

for unS'(l') and cpES(). The Fourier transform T: S(F)->S(PJ') is bijective

and leaves invariant the space L2(l"). The restriction T: L2(l)-.L2(P') is

precisely the classical FourierPlancherel transform. We define the conjugate Fourier transform : S(l")->S(l) by:

(3.18)

(3.19)

7p(x) f

(3.23)

to be extended analogous to 7: S'R")-*S'(ff). For uES'(R') we have:

7(Tu) = T(7u) = u, (3.24)

hence 7 is the inverse of T. We mention the following well known properties:

=

(2i)kT

D(Tu)=T((_21rix)ku),

= (_27rix)k7u,

Dk(7u) =7((27rie)ku). (3.25)

For ucL2(0) we have Plancherel's formula:

kLIIL2() = (3.26)

from which follows for u1,u2EL2(l):

<u1, U2>()= <1, TU2>L(p').

(3.27) Using the Fourier transform, we introduce the operator A3:

A3 =7(1+ 112 I2)3T. (3.28)

This is the same as the operator "T9 ", defined in [20], p.107. The operator

defines a continuous bijection A3: S(fl)-.S(R') by:

A3(x) =7((1+ 112 I2)3T,) (x). (3.29)

Since (1+lI27rlI2)3 is realvalued and even, it is easily seen that for S(F.') the following equality holds:

By (3.22), (3.5) and (3.29), the domain of definition of /18 can be extended to S'() by successive transposition. For

ES() and tES'() we can write:

A8ufço] =((1+

j27r2)Tu)

[] = (1+II2irJI2)48Tu[7ço]

=u[((1 + 1127 II28o)]= (3.31)

where we have also used (3.30). Hence the successive transposition is

equivalent with the direct transposition of A property of the operator A

is that:

Jl8A=/l8+t, (3.32)

for s,tLR. Consequently, since A is the identity, the inverse of A is given by:

i; =&.

(3.33)

From (3.30) and (3.31) if follows that:

Au = (1 + II2Il2)8u = T(1+ )2irII )7tL, (3.34)

for uES(0), which implies:

( 1+11 2ir 112) =

J( i + 2ir) 2)s_

=(1+ 1127

eII2)2=A8.

(3.35)

If u is real-valued then we have that A = A8u, hence that il8u is real-valued

also. This means that the operator A maps real-valued distributions onto real

Fourier transform anymore, only the operator A3. Hence we may forget that we admitted complex-valued functions and distributions. From now on, we will assume all quantities to be real-valued.

Now we return to the Sobolev spaces, as defined by (3.7). With the help of (3.25) and (3.26), we can prove that the norm on Hm(P'), induced by (3.10), is equivalent with:

IIuIIjr)

11(1+II2ireII =IIhlmtLIIL2(R"). (3.36)

Equation (3.36) suggests a definition of more general Sobolev spaces JI8(R) for SEP, namely the following. We define for sEP the Sobolev spaces:

J18(F) ={uES'(R'): A3uL2(R')}, (3.37)

with inner products:

<, u2>J/s()= <A3u1, A3u2>(E'). (3.38)

From (3.36) it follows that this definition is in agreement with (3.7). It is

easily seen that Ht(P)cH(R7), for st and furthermore that:

IIuDHs(R) IIUIIJft(Rn) (3.39)

for uEHt(P2), where the norms are induced by (3.38). By (3.27), we can prove the property that:

<A3u1, u2>I(Rn)=<ul,A,u2>I(p"), (3.40)

for u1,u2EH3(P'), if s>0, or u1,u2EL2(P'), if s0. Since JE(P") is isomorphic to L2(P'3) by (3.37) and (3.38), we have the following.

Theorem 3.1: /t(P.') is a Hubert space for each SEP. U

(3.42)

Theorem 3.2: D(R) is dense in H'9(R"). U

We now again consider our halfspace:

ARx(0,co)={(x,y)ER2: y>O} (3.41)

(see below (3.16)). We identify its boundary ÔA with R. As mentioned before,

i I

from [20] it follows that a unique transformation C: H (A)-.H2(lR), a socalled "trace mapping", exists, which is linear, continuous and surjective and has

the property that £ç'=çc'IR for çoED(). The restriction _'I of ED(A) to R

(hence to ÔA) is defined as oIR(x)=lirnç(x,e). By definition of D(A), (3.11), we have that YlREt)().

Since we have at our disposal a trace theorem for the halfspace A+, it is not a big step to prove similar results for our function space , (3.16).

i

-

I

First, we remark that analogous a unique trace mapping L: JIz0(A)-*H?0(0)

exists and also unique trace mappings

L:

and L_: (see below (3.16)). Hence, the "jump" [Ø]i can be defined by:

+ del + def

{J(x)

= L(x) =

+ I

The mapping L.: Y-*H0(R) is however not surjective, because of the periodic

character of . Because []iEH$0(R)cL2(0,b) and gEd(R), the constraints

(2.19) and (2.20) are well defined as follows:

+ , bT

ÍJb[ø}(x)g(x)dx=

_<[p],

_{g >= -}

_{, (3.43)}

for almost every XER. (3.44)

In the space we take the inner product that corresponds with the

Hi_norm:

<øi, 2>Y

<øi,ø2>Rh(W(O;b))

2

+

With this inner product, we can prove that í is a Hubert space. We remark

that condition (3.15) can also be taken in the sense of a trace.

It can be

proved ([18], Th 3.7, p.31) that for OEi we have ç(*b,.)EL2(R). Hence this

condition is also well defined.

The minimization problem can be summarized as follows. We want to

minimize:

E(0) = P 00 II(W(ob))+ 0Ø 2 (3.46)

for OEY, where Ç is defined by (3.16) and (3.45), under the constraints (3.43) and (3.44). In the next chapter, we will prove that this minimization problem has a unique solution.

4. EXISTENCE, UNIQUENESS AND REFORMULATION

In this chapter, we will prove that the minimization problem, as stated at the end of the previous chapter has a unique solution. After having done this, we will derive some properties of this solution. Using some of these

properties, we will reformulate the minimization problem, at the end of this

chapter. We do this in such a way that we can apply the version of the

KuhnTucker Theorem that we will state in the next chapter.

The proof of the unique existence of the solution is based on the fact that a convex, closed subset of a Hilbert space has a unique element, which

minimizes the norm. Therefore, we must show that on í the "energy norm",

j-E(Ø)2, is equivalent with the norm II0l!', induced by (3.45). This means that

I I

there should exist constants C and C2, such that C1E(Ø)2 _{IIOIIC2E(0)2,} for all The first inequality follows immediately from (3.45) and (3.46), with

22

. .

C1= , the second inequality follows from the following theorem.

Theorem 4.1: There exists a constant C2>O, such that for all 0Eí:

Proof: We divide the region W(O;b) into two regions, B and B_, which are

defined by:

y>g(x), O<x<b},

y<g(x), O<x<b}. (4.2)

For ED(W) and i>O we have the equality:

tÇb

[(e(e)+11)+

9()+17)fl'(e)]de

from which the inequality follows:

+3bJ(,g()+de+3bK2ç,g(e)+)

03x

_o3Y

where K=max(Ig)I; O<<b}. Integrating with respect to x gives:

b

I

Jo +3b2' Jo (x,g(x)+1l)2dx+3b2K21' _{-(x,g(x)+i)2dx. (4.5)}

Integrating with respect to i gives:

2

+3b2K2 8Ç 2

IçoIpL2(w)<3bîw(b,g(b)+)d77+3b

_-

IIIIL2(W)8 2 IIlL.2(W)

Now suppose ØE and let D(W) be a sequence, such that II (pnI1(wy+O for

n-co. Relation (4.6) is valid for ço, and also for its limit ç5:

2

+3b2K2 2

2

-

IjIIL.(W+) IIjIIL2(W+)

where the first term of the right hand side has to be taken in the sense of

(4.6)

(4.7) (4.3)

the trace. But this term was assumed to be zero (see (3.15)), so that we have:

2 3Ø2 8Ø2

IIøIIH'(W) = I1L2(W+) +

!I-tI(w)

+ lII?L.(W)

<(3b2+1 2 +(3b2K2+l 2

-

) IlIL.(W+) ) Ií3IL.2(W)

<kII2

O2

\

-

( IIIIL2(W)J (4.8)

where k=max{3b2+1, 3b2K2+1}. For the region W_ we find the same result as (4.8). Summation of these results gives:

IlII2

_{IIIIH(W+)+ ¡llIH'(W)}

-

IIIL.2(W) +2 IITjlIL2(W+)2 + II(!L2(w) + IiIlL2(W.j/ =2 2 2iE(Ø). (4.9)

2K

This proves (4.1), where C2=2p..0

Now we can prove that the minimization problem has a unique solution. This is because of the following:

E() has to be minimized on the nonempty set

satisfies the constraints (3.43) and (3.44)}. (4.10)

It can be shown by an explicit construction that P is nonempty if > S,

(cl. (2.10)).

The set P is a convex and closed subset of the Hilbert space

C) The energy norm E()2 is a suitable norm for Y, since it _{is equivalent}

with the H1norm by Theorem 4.1. The corresponding inner product is:

i

'ø

2

+4

'ø2

<&,

2>EP<j

>L2(W(O;b))

P<j

j

>12(W(O;b))

It is well known (see e.g. [8},

p.144), that a closed, convex subset of a

Hilbert space possesses a unique element, which minimizes the norm. Hence we have in our case that a unique potential EP exists, such that:

E(0)=min{E(t.); Ø}.

(4.12)

This potential is the unique solution of our minimization problem.

In the following, we will derive some properties of .

Property i of

For each oED(iR2) we have:

2PJj2{aX(x,Y)+

0(x,y)(x,y))dydx

=0. (4.13)

Proof: First suppose tEí and [t]0. Then

+ei satisfies the constraints of (3.43) and (3.44) for all EEF,

hence E(0+e)>E(0) for each e0. We can

rewrite this as 2e< , _{> E}+ 2E(i) >0 for each e 0. This is only possible if

<GO,

E°

2 . 2

Now suppose peD(l ). We define :R \G- by:

(4.14)

for (x,y)R2\G. Then we have

i

and []0, hence from the previous paragraph it follows that:

ü<to,t>spj'J

O(x,

W(O;b)0X

y)(x,y)+ -

(x,y)(x,y)}dydx

)(x,y)

=PjjR2\Gi (x,y

+ -

(x,Y)(x,Y)}dydx.0

2 .

Property 2 of o: (ER\G) and LìP =O on l \G. Hence outside G, the velocity

field,

=grad0(x,y),

is free of divergence. This shows that there is no

objection to omit this property in the formulation of the mathematical minimization problem (see below (3.16)).

Proof: Suppose 97ED(R2) and supp(ç')nG=ø, then we have, using the previous property: a a a a , _{>L2(I?2\G)+ <-a-3:7} ,

82

= - < () o,

>(R2\G) <()

O, >(G)

= - <Z0, Ç0>I(p2\G)O. (4.16)

So LIP,J= O on l2\G in the sense of distributions. But is a hypoelliptic

differential operator, so OEC(I2\G) and =O in the classical sense. N

Fig. 4.1: The support of .

Property 3 of

o: Suppose çoeD(2) has its support as drawn in figure 4.1,

then:

71

J{(x,g(x) +e)°(x,g(x)

+e)

0

,g(x)_e)}.V+g(x)2dx=O.

ç(x,g(x)e)--- (x

_0m

This means that, in a generalized sense, the normal derivative of , which is the component of the fluid velocity normal to G, is continuous across G. In

other words, neither fluid is created nor annihilated at the line G. Hence the velocity field, i=grad0(x,y), is also free of divergence on the line G.

Proof: We consider two parts A and BE of supp('), as drawn in figure 4.2.

Then we have by Property i of

{U(X,y)(X,y)+

-

(x,y)(x,y)}dxdy

lim(

_{E+O +5f} )

JJAE B {

(x,y)(x,y)+

(x,y)(x,y)}dxdy=O.

n

Fig. 4.2: The regions A5 and B.

The integration over A5 can be rewritten as:

ifA5 + }ixtiy=ff {div(ço grad

_A}

= 1f div(çogrado)dxdy=f --° ds J A5 8A5 2 =

_f (x,g(x)+e)°(x,g(x)+e)V +g'(x) dx.

Analogously: fi, supp(ço)\G (4.18) (4.19)

ri (e0 0ço

₊ 0.o 0co

-

_Jdxdy

z

=J

(x,g(x)-a)°(x,g(x)-e)

8m

Vi

+g(x) dx.

Substitution of (4.19) and (4.20) into (4.18) proves the property. U

Property 4 of

0(x+b,-y)=0(x,y).

Proof: Let *(x,y)=1o(x+.b, -y), then it satisfies the constraints

(3.43), (3.44)

and E(0)=E(*). Hence

is also a solution of the minimization problem. By the uniqueness of this solution we have 45 _{= o.}

Corollary 4.1: [0](x+b)= -[](x). U

(4.22)

Property 5 of

0(b-x,y)= -0(x,y).

(4.23)

Proof: Let *(x,y)

_{= -0(b--x,y), then}

_{=o by the same}

_{reasoning as} above. U

Corollary 4.2: [0]ii('b-x)= -{0]i(x). U

(4.24)

Defining the following "symmetry properties" for Øeí and vEH0(IR):

Ø(x+b, -y) =çt(x,y), (4.25)

qb-x,y)= -Ø(x,y),

(4.26)

v(x+b)=-v(x),

(4.27)

(4.28)

we can recapitulate the last two properties of c5 by stating that cP satisfies (4.25) and (4.26)

and that

the

"jump" [}

satisfies (4.27) and (4.28). Herewith, we can reduce the space , in which we optimize the potential

, to

the space Yci, given by:

(4.20)

Y=:IR2-.IR; 0E111(W(a;c)), for all finite a and e,

Ø satisfies (4.25) and (4.26)).. (4.29)

Because of (4.26), functions 0Y automatically satisfy (3.15). Since Y is a closed subspace of the Hubert space i, it is a Hubert space too. As inner

product on the space Y, we choose the inner product that corresponds with the energy norm:

<01, 02>Y= <01, 02>E

> L.(W(O;b)) , > L2(W(O;b))

802

+p<8O1 °02

(4.30)

+ I I

The jump [O]_ will be an element of the space i1(LR)cH0R), which is defined

as:

l-i; veH0(R), y satisfies (4.27) and

(4.28)).. (4.31) I

If vEH(F), then (4.27) and (4.28) imply that y is even:

v(x)=v(x).

(4.32)

+ I

It can be proved that the mapping £:Y-*H(ll), defined by (3.42), is now

surjective. We can minimize E(0), (3.46), under the constraints (3.43) and (3.44), just as well on the space Y. Then the same unique solution will be found.

It is also possible to weaken the constraint (3.43) by:

jb [0](x)g'(x)dx<_o bTU.

That this is allowed can be seen as follows. Suppose that in (4.33) the strict

inequality "<" holds for the solution of the new minimization problem. Then some factor <1 exists, such that (4.33) still holds for

2=t1. The

(4.33)

potential 2 satisfies also the constraint (3.44) and furthermore E(2) <E(q'i1). This contradicts the assumption that cP is the solution of the minimization

problem.

Unfortunately, we can not directly apply the KuhnTucker Theorem that we

will state in the next chapter, to our minimization problem, but we have to

reformulate it by making two assumptions. For that purpose, we introduce the

sets I_,IcF by:

s

fI;

[o](x)=jJ

[oI(x)

[](x)

26 Fig. 4.3: Assumed shape of

First, we assume that these sets can be written

as the

disjoint union of

closed intervals:

uI= U {kb-6 kb+6],

OD

II+=kU[(k+2)b6, (k+)b+6],

for some 6<b (see figure 4.3). Now we weaken the constraint (3.44) as:

(4.34)

Jo

X

(4.38)

f

4[](x) for almost every xEL,

l[J1(x)

j,

for almost every xEI. (4.36) Because of the symmetry properties (4.27) and (4.28) of the "jumps"

we can equivalently replace the two inequalities of (4.36) by:

for almost every xE[-6,6]. (4.37)

Under the constraints (4.33) and (4.37), it can be proved in the same way as in the beginning of this chapter that a unique solution exists.

Our second assumption is that this solution also satisfies (3.43) and

(3.44). Then it has to be equal to . Summarizing, we assume that Zi is also

the solution of minimizing E() for ØEY, under the constraints (4.33) and

(4.37). It can be checked, after having numerically found the solution,

whether (3.43) and (3.44) hold. By the same reasoning as below (4.33), the

solution cP satisfies the equality:

jb+

bT

o U.

In the next chapter, we will state the version of Thomas and Van der Meer of the KuhnTucker Theorem and we give its proof. In the chapter thereafter we prove that we can apply this theorem to our minimization problem, using the

last formulation of it. That is: minimize E(), (3.46), for ØEY, (4.29), under

the constraints:

+ bT

{](x)g(x)dx-for almost every x[S,6].

(4.39)

5. AN ALTERNATIVE VERSION OF THE GENERALIZED KUHN-TUCKER

THEOREM

In this chapter we consider optimization with inequality constraints. This means that we have a problem as follows. Suppose there are spaces Y and X, a functional E:Y-e.LR and a mapping Q:Y.X (see figure 5.1). We want to find an element u0cY, for which:

E(u0)=min{E(u), ueY; Q(u)O}. (5.1)

The inequality in (5.1) is formally defined as:

Q(u)O Q(u)EP, _(5.2)

where PcX is some cone, called a "positive cone".

Fig. 5.1: Illustration of the problem.

In the case of equality constraints, one can use the Lagrange multiplier

method to determine the solution of the minimization problem. In case of

inequality constraints, one has to apply a generalization of this method. That

we can use, in the latter case, a method with (generalized) Lagrange

multipliers, when certain conditions are fulfilled, follows from a theorem in

[11], p.249. This is called the "generalized Kuhn-Tucker Theorem". One of the

conditions of this theorem is the "regularity condition", which states that the positive cone P must have a non-empty interior. This demand can not be

fulfilled when, for example, X is the function space L2(a,b), with norm:

2

Iu(

(fb2)l

U

where a possible positive cone P is defined by:

P={ueL2(a,b); u()O, for almost every eE(a,b)}. (5.4)

This P has no interior points. In our minimization problem, as described in

I

the previous chapters, we consider the space H2(l). For this space, we have

the same complication, hence the theorem, as stated in [11] is not useful for us.

In [13], p.128 a version of the generalized KuhnTucker Theorem is found

which does not require the

positive cone to have interior

points. This

theorem, however, requires other conditions to be fulfilled, which are very

hard to verify in our situation. Therefore, we chose to present a more

convenient version of the generalized KuhnTucker Theorem, due to Thomas and

Van der Meer [12]. Since this theorem has not been published, we will also

reproduce its proof.

Sufficient (but not necessary) demands will now be

that Q:Y-X is

surjective and that each xEX can be written as x=p1p2, for p1,p2EP. A

shorthand notation of this is X=P - P. The latter condition holds for the

positive cone (5.4). A restriction of the ThomasVan der Meer version is that the mapping Q must be affine linear,

but in a large class

of "practical"

problems this is the case. In fact, their theorem was motivated by problems like this.

Before we present the main result of

this chapter, we give

in the

following some definitions and propositions.

Definition 5.1 (Cone): A subset P of a linear space X is called a cone if P

satisfies the following:

OeP,

xcP and i>O imply

We call P a convex cone if P is a cone and also a convex set.

Proposition 5.1: Suppose P is a convex cone. Then a,beP implies a+bEP. On the other hand, if P is a cone with this property, then P is convex.

Proof: (1) Suppose P is convex and a,baP. Then, by definition of a cone,

2a,2beP and by the convexity a+b=..2a+-.2bP.

(2) Suppose a,baP implies a+bEP. If a,bcP then by definition of_{a cone \aEP}

and (lÀ)bEP for each

Àa[O,1J,

_{hence )a+(l))baP, which shows}

_the

convexity.

Definition 5.2 (ordering): A partially ordered set will be a set M on which

there is defined a _{partial ordering, that is a binary relation which is}

written

""

and satisfies the conditions:

xx

(reflexivity),

xy, yz

. xz

(transitivity).

"Partially" emphasizes that M may contain elements a and b for which neither

ab nor b<a holds.

Usually, one demands also the property of antisymmetry, that is

_{x y and y x}

imply x=y, but we do not need this in our problem. In fact, Definition 5.2 is

the definition of a socalled "preordering".

If X is a linear space and PcX is a convex cone, we can define a partial ordering on X by:

xy

yxEP.

_(5.5)

This relation satisfies the conditions of Definition 5.2. Condition (ii) is

fulfilled because of Proposition 5.1. Such a cone is called a "positive cone".

Definition 5.3 (Gateaux variation): If E is a functional, then we define its

Gateaux _{variation or directional derivative &E(y;v), in a point y with} direction y, by:

E(y+ev)E(y)

aE(y;v)=lzm_5Q EP.

We say that the Gateaux variation in y, with respect to y,

exists,

if the

limit in (5.6) exists. U

Definition 5.4 (minimization problem): Let Y be a linear space, X a complete

and metrizable topological vector space (e.g. a Banach space or a Hilbert

space) and PcX a closed convex cone. We write xO if xEP. Suppose we have a functional E:Y-.R (something to minimize) and a mapping Q:Y-*X (which will be

used to define the constraint). We assume that the Gateaux variation E of E

exists in each point yEY and in each direction vEY and that h*8E(u;h) is a

linear transformation from Y to , for each uEY. For Q we assume that it is an

affine linear transformation (that is Q=B+b, where B is a linear

transformation and b is

a fixed element of X), so

its Gateaux variation

8Q(u;h) exists for each LEY, it does not depend on u, it is linear in h and

8Q(u;h)=B(h)=(Q(v+th)Q(v)) for arbitrary VEY and tEO\{O}.

We call U0EY a solution of the minimization problem if U0EQ'(P) and

E(u0)=inf{E(u); uEQ1(P)}. In other words, if Q(u0)O and E(u0)E(u) for all

UEY with Q(u) O. In the following we assume that such a u0 exists. U

Furthermore we define the linear space:

heY,

heY, (5.7)

We shall eventually assume that the conditions X=PP and

X=P+Z are

fulfilled.

Proposition 5.2: The following conditions are equivalent:

X=P+Z,

VXEX: aznZ: xz.

Proof: (i) Suppose X= P + Z and x e X. Because also -XEX, there exists a p e P and a

and ZXEP, which is the same as xz.

(ii) Suppose VxEX: zEZ: xz and xEX. Because also XEX, we canfind a ZEZ

such that xz. This means that we can find a pEP such that z(x)=z+x =p.

Hence XrrpZEP+Z.

Remark 5.1: If Q:YX is surjective (see above Definition 5.1) then X=P+Z. Proof: Choose an arbitrary XEX. Then, by the surjectivity ofQ, UEY exists for

which Q(u) =x. Since Q is affine linear we can rewrite Q(u) =Q(a0)+OQ(u0;h),

for =i and h=uu0, hence we have that x=Q(u)EZ. Thus we can find for each

xEX a ZEZ, such that xz, namely z=x. By Proposition 5.2 we conclude that

x=P+z.

u

Now we come to the announced generalized KuhnTucker Theorem, which is as follows.

Theorem 5.1: Consider the minimization problem as defined by Definition 5.4.

Suppose further that the conditions X=PP and X=P+Z are fulfilled. Then a

functional L:X-R (the generalized Lagrange multiplier) exists, which is linear and continuous on X and positive on P and satisfies:

L(Q(u0))=O,

VhEY: L(8Q(u0;h))=aE(u0;h). u

In the remaining part of this chapter, we will give the proof of Theorem 5.1. This proof is based upon:

some lemmas,

Proposition 5.3: an extension theorem, Proposition 5.4: a theorem about continuity.

Lemma 5.1: If Q(u0)+OQ(u0;h)O for some heY, then Q('a0)+tOQ(u0;h)O for

o<t<1.

Proof: Q(u0)+tâQ(u0;h) can be written as a convex combination of two elements of P: Q(u0)+tOQ(u0;h)=(it)Q(u0)+t(Q(u0)+OQ(u0;h)). U

Lemma 5.2: If Q is affine linear and Q(u0)+OQ(u0;h)O for some hEY, then 8E(u0;h)O.

Proof: Since Q is affine linear and because of Lemnia 5.1 we obtain

Q(u0+th)=Q(u0)+taQ(u0;h)O for Ot1. By definition of o (see Definition

5.4), it follows

that E(u0+th)E(u0) for Otl, hence &E(u0;h)O. U

Lemma 5.3: If Q is affine linear and tQ(u0)+8Q(u0;h)O for_{some tEll and heY,} then ÔE(u0;h)O.

Proof: (i) If t5>O then Q(u0)+OQ(u0;)O and Lemma 5.2 yields ÔE(u0;h)= O.

(ii) If O then, by Proposition 5.1:

Q(u0)+8Q(u0;h)= (zQ(uo)+aQ(uo;h))+((1_)Q(uo))o. (5.8)

Hence, by Lemma 5.2, 8E(u0;h)O.

Lemma 5.4: líQ is affine linear and Q(u0)+8Q(u0;h)=O for somet9e andhEY,

then c3E(u0;h)=0.

Proof: (i) tQ(u0)+3Q(u0;h)0 and hence (Lemma 5.3) 8E(u0;h)0.

(ii) _{--Q(u0)-äQ(u0;h)=-Q(u0)+aQ(u0;-h)0 and hence (Lemma 5.3)}

ÔE(u0;-h)O, which means 3E(u0;h)O. U

Proposition 5.3: Let X be a real linear space, PcX a convex cone, Z a linear subspace of X, such that X=P+Z. Suppose f:Z-*P is a linear functional with f( x) O for all x e ZnP. Then there exists a linear extension 7:X - IR of f, satisfying:

VxeZ: 7(x)=f(x),

VxeP: 7(x)O.

A proof of this proposition can be found in [5], p.269.

space and let P be a closed convex cone in X, suchthat X=P - P. Suppose L: X - P

is a linear functional with domain 2(L)=X, which is positive on P (VxP:

LxO), then L is continuous.

For a proof we again refer to [5], p.295.

Finally, the proof of the generalized Kuhn-Tucker Theorem becomes:

Proof of Theorem 5.1: We define a functional l:Z-.i (where Z is given by (5.7)) by:

1(3Q(uo)+3Q(uo;h)) =3E(u0;h). (5.9)

It is well defined since i1Q(u0)+aQ(u0;h1) =2Q(u0)+aQ(u0;h2) implies -2)Q(u0)+ aQu0 ;h1 - h2) =O and hence, by Lenmia 5.4, aE(u0;h1 - h2) = OE(u0;h1)- âE(u0;h2)=O from which follows: 1(iiQ(uo)+8Q(uo;hi)) =

1('2Q(uo)+0Qeuo;h2)). By the linearity of h-OQ(u0;h) and h-8E(u0;h), I is

linear. We see that the functional ¿ satisfies (a) of the theorem, by choosing

in (5.9)

t=

i and h=O. Property (b) follows from the choice tS&=O.

From Lemma 5.3 it follows that I is positive on ZnP, hence we can apply

Proposition 5.3, which states

that a linear

extension L:X-*P of ¿ exists,

satisfying:

VXEZ: L(x)=I(x),

VxEP: L(x)O.

Since X=P-P, from Proposition 5.4 it follows that L is continuous.

6. APPLICABILITY AND APPLICATION OF THE GENERALIZED KUHN-TUCKER

THEOREM

In this chapter, we will

prove that we can apply the

generalized

Kuhn-Tucker Theorem (Theorem 5.1), as stated in the previous chapter, to the modified minimization problem, formulated at the end of chapter 4.

The space Y of Theorem 5.1 (see Definition 5.4) is in our case defined by

space:

I

X = l X H(l), (6.1)

I

where the space H(R) consists of functions vEH2(l) which are even:

v( -) = v(). As inner product on X we define:

«Pi, /ii), (À2, 2)»x=À1À2+ </1k, /12>H(R), (6.2)

where )eR and /iEH(), i= 1,2. The space X is a Hubert space, hence it is

indeed a complete metrizable topological vector space, as required. The

functional E that we have to minimize is

the energy E(') of

(3.46). The

mapping Q that determines the constraints will be:

Q()= (-

Jo1b[]+()g,(e)d bT

Since [']()+

rj H(F<), we multiplied it in (6.3) with an even test function cxED(l), for which:

VE{-6,S]: cx()=1,

(6.4)

where the number 6 is the same as introduced in (4.35). Herewith Q is well

defined. We have in particular by (4.38):

Q(o)

= (o,

() ([o]L) +

(6.5)

The mapping Q is, in agreement with Definition 5.4, affine linear:

(jb

[])g()d, a(e)[J))

+(_ bT

(6.6)

We define the cone PcX, needed to determine the inequality, by:

P=ll+xP*, _(6.7)

where:

={); )O},

* I

P ={,uEH(F); for almost every E[-6, ]: z()O}.

With the definitions (6.3) and (6.7), the constraint Q(ç)0 or Q()EP is for

ØeY equivalent with (4.39) and (4.40).

Both E and Q are Gateaux differentiable in the whole space Y. Their

Gateaux derivatives are given by:

aE(c;h)=2<,h>E=PJ'I

_Jw(o;b)(aØoh_ae

+

did,

8Q(;h)=

(Jb+

a(e)[hJ)

J

Both Gateaux derivatives are linear with respect to h. The space Z, as defined by (5.7), is in our case:

z={

(5b

[hJ)g'()d,

J; hEY,

We can apply Theorem 5.1

if we satisfy the conditions "X=PP"

and "X=P+Z". Before we verify these conditions, we will state a very useful

theorem:

Theorem 6.1:

/iEH(l)

implies

EH).

The proof of this theorem is quite technical and is postponed to the end of

this chapter.

Using this theorem, we can easily prove that X=PP. Suppose XEX, say

x=(À,

), with )l and ¡iEH), then x can be written as:

x=\,

i)(A-, p), (6.13) (6.8) (6.9) (6.10) (6. 11) (6.12)

where:

/1(e)(Iu(e)I +«)),

-/1 (e)-11 Iit(C)Iii(C)).

-I

Because of Theorem 6.1, we have that

!I(l). It follows that (), p.1.)

and (À_, t_) are in the cone P, (6.7). This proves that X=PP.

Second we prove that X=P+Z holds. Consider an arbitrary xX, say

x

= (,

By Proposition 5.2 it is necessary and sufficient to prove that

we can find a zEZ, say _{pi), such that xz or, equivalently, such that}

I

zxEP. Because p

is an even function, there exists a pEH(lR) (cf. (4.31) and (4.32)), such that:

for almost every Ce[-6, 6]. (6.15)

Since the trace mapping £: hi.{h], (3.42),

is a surjection from Y onto

there exists a h1nY, such that:

[h1J(e)=(e), for almost every

CELR. (6.16)

Then h2EY can be chosen, such that:

[h2] i(C)=O, for almost every C[-6, 6], (6.17)

and also that:

f[h+h]()g'()dÀ

(6.18)

The latter is possible, since g O on the interval (6, ib). Now we define ) and p by:

Ç

fLz()=(5)[h1+h2]). (6.19)

We see that (Ar,p)aZ, (6.12), with =O and h=h1+h2. By (6.18) we have that By (6.19), (6.4), (6.17), (6.16) and (6.15), we can write:

/(e)

=cx()[h1+h2](e)=[h1+h2]() =

=uí(e)=[L(e), for almost every

a[-6, 6].

(6.20)

This implies that:

(e)(e), for almost every

E[-6, 6]. (6.21)

Hence, by definition of the cone P, we conclude that (À, p)i7, j) This

had to be proved, so that also the condition "X=P+Z" holds.

Thus we have proved that we can apply the generalized Kuhn-Tucker Theorem (Theorem 5.1) to our modified problem, as stated at the end of

chapter 4.

The theorem states that a functional L exist, which is linear, continuous and positive on P, such that:

L(Q('0))=O, (6.22)

VhaY: L(t9Q(0;h))=i3E(0;h). (6.23)

Since X is a Hilbert space, there exists a unique z0 a X, such that:

VxaX: L(x)=«x0, X»X, (6.24)

I

(6.31)

VÀE, E

H): L(A,j)=A0À+

<,

_(6.25)

The functional L is positive on P, (6.7). This means that for all _(.\,u)EP:

))+ <hg, /1>Jj4(p)O. Hence À0O and

VhEP*:

_<Po/->H(R)O.

_(6.26)

If /iHR) has

its support outside the interval [-6, 6], then by definition

of P', (6.9), both ji

and -

are in P* _{Hence (6.26) implies:}

supp(,u)n[-6, 6]=ø <jis, _h>H(R)=O,

where 0 denotes the empty set.

By (6.3) and (6.25), the "constraint" (6.22) implies:

[b

bT

-

I

[0])g()d

ij]+</1o,

a([o]i+-U)>H(R)=0.

We know by (4.38) that the first term in the left hand side of (6.28) is zero.

Moreover we have by definition of 6, (4.35), that []()+ =0, for almost

every E [-6,6]. Hence

x ([] +

has its support outside the interval [-6, 6]. Then from (6.27) it

follows that also the second term in the left

hand side of (6.28) is zero. On the other hand, if a potential satisfies:

Ijb

[]()g'(e)de

-77=0,bT

i[])+

=0, for almost every

e[-6, 6],

then L(Q())=0.

Next, we elaborate (6.23). The left hand side of it becomes by (6.11) and (6.25): b

L(aQ(0;h))= -À0f {h])g()d+

<jis,

c4h]±>jjp.

(6.27) (6.28) (6.29) (6.30)

We remark that, because of (6.27), the second term in the right hand side of (6.31) is independent of the choice of a. The right hand side of (6.23) is by (6.10) equal to:

OE(o;h)=pJj

(a0ah

808h

W(O;b)1Q

Q

+ - -J

Hence _o, À and z0 satisfy:

VhEY: _of [h]g'()d+

<,

=pff(a0ah

W(O;b)l

Q

+ _drid,

I

where À0 and /10EH(l) satisfies (6.26) and (6.27). We have to solve ,

and from (6.33), together with the constraints (6.29) and (6.30). In the

next chapter, we will describe how this can be done, by numerical means. Now we return to Theorem 6.1, which still has to be proved. The proof of

it will be given in the remaining part of this chapter. Therefore we need the

concept of absolute continuity:

Definition 6.1 (absolute continuity): A function mm on a finite interval [a,b}

is said to be absolutely continuous if for given e>0, there exists 6>0, such

that for any collection {[a,b]} (finite or not) of non-overlapping subintervals of [a,b]:

if

(b-a)<6.

_(6.34)

Lemma 6.1: If u is absolutely continuous on [a,b] then so is lui.

Proof: This follows directly from Definition 6.1

and from the

inequality

-dimmi du

Lemma 6.2: Suppose uH'(R), then also u EHi(R) and

d iIR= IiiIL2(R).

(6.32)