The

ory

and

Impl

emen

tati

on o

f N

umer

ical

Met

hods

Bas

ed o

nR

unge

-Kut

ta I

nteg

rati

on fo

r So

lvin

g O

ptim

al C

ontr

ol P

robl

ems

by

Ada

m L

owel

l Sch

war

tz

S.B

. (M

assa

chus

etts

Ins

titut

e of

Tec

hnol

ogy)

198

9S.

M. (

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y) 1

989

Adi

sser

tatio

n su

bmitt

ed in

par

tial s

atis

fact

ion

of th

e

requ

irem

ents

for

the

degr

ee o

f

Doc

tor

of P

hilo

soph

y

in

Eng

inee

ring

Ele

ctri

cal E

ngin

eeri

ng a

nd C

ompu

ter

Scie

nces

in th

e

GR

AD

UA

TE

DIV

ISIO

N

of th

e

UN

IVE

RSI

TY

of

CA

LIF

OR

NIA

at B

ER

KE

LE

Y

Com

mitt

ee in

cha

rge:

Prof

esso

r E

lijah

Pol

ak, C

hair

Prof

esso

r Ja

mes

W.D

emm

elPr

ofes

sor

Shan

kar

Sast

ryPr

ofes

sor

And

rew

K.P

acka

rd

1996

The

ory

and

Impl

emen

tati

on o

f N

umer

ical

Met

hods

Bas

ed o

n R

unge

-Kut

ta I

nteg

rati

on fo

r So

lvin

gO

ptim

al C

ontr

ol P

robl

ems

Cop

yrig

ht

199

6

by

Ada

m L

owel

l Sch

war

tz

Abs

trac

t

TH

EO

RY

AN

D I

MP

LE

ME

NT

AT

ION

OF

NU

ME

RIC

AL

ME

TH

OD

S B

ASE

D

ON

RU

NG

E-K

UT

TA

INT

EG

RA

TIO

N F

OR

SO

LVIN

G O

PT

IMA

L C

ON

TR

OL

PR

OB

LE

MS

by

Ada

m L

owel

l Sch

war

tz

Doc

tor

of P

hilo

soph

yin

Ele

ctri

cal E

ngin

eeri

ng

Uni

vers

ity o

f C

alif

orni

a at

Ber

kele

y

Prof

esso

r E

lijah

Pol

ak, C

hair

Thi

s di

sser

tatio

n pr

esen

ts t

heor

y an

d im

plem

enta

tions

of

num

eric

al m

etho

ds f

or a

ccur

atel

y

and

effic

ient

ly s

olvi

ng o

ptim

al c

ontr

ol p

robl

ems.

The

met

hods

we

cons

ider

are

bas

ed o

n so

lvin

g

ase

quen

ce o

f di

scre

te-t

ime

optim

al c

ontr

ol p

robl

ems

obta

ined

usi

ng e

xplic

it, fi

xed

step

-siz

e

Run

ge-K

utta

int

egra

tion

and

finite

-dim

ensi

onal

B-s

plin

e co

ntro

l pa

ram

eter

izat

ions

to

disc

retiz

e

the

optim

al c

ontr

ol p

robl

em u

nder

con

side

ratio

n.O

ther

dis

cret

izat

ion

met

hods

suc

h as

Eul

ers

met

hod,

col

loca

tion

tech

niqu

es, o

r nu

mer

ical

impl

emen

tatio

ns, u

sing

var

iabl

e st

ep-s

ize

num

eric

al

inte

grat

ion,

of

spec

ializ

ed o

ptim

al c

ontr

ol a

lgor

ithm

s ar

e le

ss a

ccur

ate

and

effic

ient

tha

n di

s-

cret

izat

ion

by e

xplic

it, fi

xed

step

-siz

e R

unge

-Kut

ta f

or m

any

prob

lem

s.

Thi

sw

ork

pres

ents

the

first

theo

retic

al f

ound

atio

n fo

r R

unge

-Kut

ta d

iscr

etiz

atio

n.T

he th

eory

pro

vide

s co

nditi

ons

on th

e

Run

ge-K

utta

par

amet

ers

that

ens

ure

that

the

disc

rete

-tim

e op

timal

con

trol

pro

blem

s ar

e co

nsis

tent

appr

oxim

atio

ns to

the

orig

inal

pro

blem

.

Add

ition

ally

,we

deri

ve a

num

ber

of r

esul

ts w

hich

hel

p in

the

effic

ient

num

eric

al im

plem

en-

tatio

n of

thi

s th

eory

.T

hese

inc

lude

met

hods

for

refi

ning

the

dis

cret

izat

ion

mes

h, f

orm

ulas

for

com

putin

g es

timat

es o

f in

tegr

atio

n er

rors

and

err

ors

of n

umer

ical

sol

utio

ns o

btai

ned

for

optim

al

cont

rol

prob

lem

s, a

nd a

met

hod

for

deal

ing

with

osc

illat

ions

tha

t ar

ise

in t

he n

umer

ical

sol

utio

n

of s

ingu

lar

optim

al c

ontr

ol p

robl

ems.

The

se r

esul

ts a

re o

f gr

eat

prac

tical

im

port

ance

in

solv

ing

optim

al c

ontr

ol p

robl

ems.

We

also

pre

sent

, and

pro

ve c

onve

rgen

ce r

esul

ts f

or,a

fam

ily o

f nu

mer

ical

opt

imiz

atio

n al

go-

rith

ms

for

solv

ing

a cl

ass

of o

ptim

izat

ion

prob

lem

s th

at a

rise

fro

m t

he d

iscr

etiz

atio

n of

opt

imal

cont

rol p

robl

ems

with

con

trol

bou

nds.

Thi

s fa

mily

of

algo

rith

ms

is b

ased

upo

n a

proj

ectio

n op

er-

ator

and

a d

ecom

posi

tion

of s

earc

h di

rect

ions

int

o tw

opa

rts:

one

par

t fo

r th

e un

cons

trai

ned

sub-

spac

e an

d an

othe

r fo

r th

e co

nstr

aine

d su

bspa

ce.

Thi

s de

com

posi

tion

allo

ws

the

corr

ect

activ

e

-1

-

cons

trai

nt s

et t

o be

rap

idly

ide

ntifi

ed a

nd t

he r

ate

of c

onve

rgen

ce p

rope

rtie

s as

soci

ated

with

an

appr

opri

ate

unco

nstr

aine

d se

arch

dir

ectio

n, s

uch

as t

hose

pro

duce

d by

a l

imite

d m

emor

y qu

asi-

New

ton

or c

onju

gate

-gra

dien

t m

etho

d, t

o be

rea

lized

for

the

con

stra

ined

pro

blem

.T

he a

lgor

ithm

is e

xtre

mel

y ef

ficie

nt a

nd c

an r

eadi

ly s

olve

prob

lem

s in

volv

ing

thou

sand

s of

dec

isio

n va

riab

les.

The

the

ory

we

have

dev

elop

ed p

rovi

des

the

foun

datio

n fo

r ou

r so

ftw

are

pack

age

RIO

TS.

Thi

s is

a g

roup

of

prog

ram

s an

d ut

ilitie

s, w

ritte

n m

ostly

in

C a

nd d

esig

ned

as a

too

lbox

for

Mat

-

lab,

tha

t pr

ovid

es a

n in

tera

ctiv

e en

viro

nmen

t fo

r so

lvin

g a

very

bro

ad c

lass

of

optim

al c

ontr

ol

prob

lem

s.

Am

anua

l de

scri

bing

the

use

and

ope

ratio

n of

RIO

TS

is i

nclu

ded

in t

his

diss

erta

tion.

We

belie

ve R

IOT

S to

be

one

of t

he m

ost

accu

rate

and

effi

cien

t pr

ogra

ms

curr

ently

ava

ilabl

e fo

r

solv

ing

optim

al c

ontr

ol p

robl

ems.

Prof

esso

r E

lijah

Pol

ak

Dis

sert

atio

n C

omm

ittee

Cha

ir

-ii

-

For

Mom

and

Dad

We

are

gene

rall

y th

e be

tter

per

suad

ed b

y th

e re

ason

s w

e di

s-co

ver

ours

elve

s th

an b

y th

ose

give

n to

us

by o

ther

s.

Mar

cel P

rous

t

You

neve

r w

ork

so h

ard

asw

hen

you

re n

ot b

eing

pai

d fo

r it

.

Geo

rge

Bur

ns

The

rear

e th

ree

type

s of

peo

ple

in th

is w

orld

:T

hose

that

are

good

at m

ath

and

thos

e th

at a

ren

t.

-iii

-

Ack

now

ledg

men

ts

The

wor

k in

thi

s th

esis

wou

ld n

ot h

ave

been

pos

sibl

e w

ithou

t th

e in

valu

able

dis

cuss

ions

I

have

had

with

sev

eral

ind

ivid

uals

. T

hese

indi

vidu

als,

all

of w

hom

wer

e ve

ry g

ener

ous

with

the

ir

time,

inc

lude

Pro

f. D

imitr

i B

erts

ekas

, D

r.Jo

hn B

etts

, Pr

of.

Lar

ry B

iegl

er,

Prof

. C

arl

de B

oor,

Prof

. A

sen

Don

tche

v, P

rof.

Jos

eph

Dun

n, P

rof.

Rog

er F

letc

her,

Prof

. W

illia

m H

ager

,D

r. C

raig

Law

renc

e, P

rof.

Rog

er S

arge

nt,

Prof

. M

icha

el S

aund

ers,

Dr.

Osk

ar V

on S

tryk

, Pr

of.

And

reT

its,

Dr.

Step

hen

Wri

ght

and

the

help

ful

engi

neer

s at

the

Mat

hwor

ks.

Als

o,fo

r sh

arin

g w

ith m

e th

eir

prog

ram

min

g ex

pert

ise,

I w

ish

to t

hank

my

fello

wgr

adua

te s

tude

nts

Stev

e B

urge

tt an

d R

aja

Kad

iyal

a. T

wo

othe

r fe

llow

grad

uate

stu

dent

s, N

eil

Get

z an

d Sh

ahra

m S

hahr

uz, d

eser

vem

entio

n

for

the

enjo

yabl

e tim

e I

spen

t w

ith t

hem

dis

cuss

ing

and

form

ulat

ing

idea

s.T

hank

s al

so g

o to

Prof

. Ron

Fea

ring

for

kee

ping

me

empl

oyed

as

an in

stru

ctor

for

Sig

nals

and

Sys

tem

s.

For

the

gritt

y de

tails

of

adm

inis

trat

ion,

the

Cor

y H

all

staf

f, p

artic

ular

ly D

iann

a B

olt,

Mar

y

Byr

nes,

Chr

is C

olbe

rt,

Tito

Gat

chal

ian,

Hea

ther

Lev

ien,

Flo

ra O

vied

o, a

nd M

ary

Stew

art

enor

-

mou

sly

sim

plifi

ed m

y lif

e at

Ber

kele

y.

The

re is

no

over

stat

ing

the

impo

rtan

ce o

f th

eir

help

.

Iw

ould

lik

eto

rese

rve

spec

ial

ackn

owle

dgm

ent

for:

Car

los

Kir

jner

(m

y of

ficem

ate

with

who

m I

spe

nt m

ost

of m

y ho

urs)

for

ans

wer

ing

ques

tions

on

topi

cs r

angi

ng f

rom

fun

ctio

nal

anal

-

ysis

to

topo

logy

to

optim

izat

ion;

Pro

f. S

hank

ar S

astr

y w

ho p

rovi

ded

acce

ss t

o th

e co

mpu

ter

equi

pmen

t I

used

for

dev

elop

ing

my

soft

war

e as

wel

l as

enc

oura

gem

ent

and

a w

illin

gnes

s to

beco

me

invo

lved

in a

sub

ject

that

is r

emov

edfr

om h

is u

sual

are

a of

inte

rest

; Pro

f. J

ames

Dem

mel

who

spa

rked

my

inte

rest

in

num

eric

al i

nteg

ratio

n an

d is

res

pons

ible

for

my

unde

rsta

ndin

g of

num

eric

al i

nteg

ratio

n m

etho

ds;

Prof

. A

ndre

wPa

ckar

d, a

col

leag

ue w

hose

app

roac

h to

aca

dem

ia

is r

efre

shin

g an

d st

imul

atin

gI

have

tho

urou

ghly

enj

oyed

kno

win

g an

d w

orki

ng w

ith A

ndy;

and

mos

t im

port

antly

,m

yad

viso

r an

d m

ento

r,Pr

of.

Pola

k.T

he w

ork

desc

ribe

d in

thi

s th

esis

is

the

resu

lt of

my

colla

bora

tion

with

Pro

f. P

olak

and

any

sign

s of

exc

elle

nce

that

may

be

cont

aine

d

here

in a

re d

ue t

o th

e hi

gh l

evel

ofqu

ality

tha

t he

dem

ande

d of

me.

His

ins

iste

nce

on p

erfe

ctio

n

was

rele

ntle

ss a

nd o

ften

pai

nful

.B

ut h

is c

omm

itmen

t to

qua

lity

will

ser

veas

agu

ide

for

the

rest

of m

y lif

e.I

amgr

atef

ul f

or h

is d

eep

invo

lvem

ent i

n m

y w

ork.

Fina

lly,I

amgl

ad t

o m

entio

n th

e pe

ople

in

my

pers

onal

lif

e th

at m

ade

the

endl

ess

hour

s of

wor

k on

thi

s di

sser

tatio

n to

lera

ble.

The

se a

re m

y pa

rent

s St

an a

nd H

elen

e, m

y br

othe

r Jo

hn a

nd

his

wif

e C

arri

e (a

nd th

eir

bran

d-ne

wda

ught

er R

ebec

ca),

my

sist

er M

elis

sa, m

y gr

andp

aren

ts B

en-

jam

in, F

ranc

is, N

atha

n, P

aulin

e an

d L

illia

n, m

y ps

eudo

-aun

t Joa

nn L

omba

rdo,

my

vari

ous

hous

e-

mat

es o

ver

the

year

s M

itch

Ber

kson

, M

icha

el C

ohn,

Joh

n an

d To

mok

oFe

rgus

on,

Scot

t Sh

enk,

Dan

Vas

silo

vski

, C

olin

Wee

ks,

and

my

good

fri

ends

Law

renc

e C

ande

ll, J

ohn

Geo

rges

, G

ary

and

Lau

ra G

runb

aum

, Eal

on J

oels

on, A

lan

Sbar

ra, a

nd J

eff

Stei

nhau

er.

Im

ake

spec

ial m

entio

n of

my

beau

tiful

gir

lfri

end

Jess

ica

Dan

iels

who

has

bee

n ve

ry p

atie

nt, e

ncou

ragi

ng a

nd l

ovin

g. T

hesu

p-

port

, in

ever

y fo

rm, p

rovi

ded

to m

e by

thes

e pe

ople

was

indi

spen

sabl

e.

-iv

-

No

tati

on

Spac

es a

nd E

lem

ents

IRn

Euc

lidea

nn-

spac

e

rIR

mC

arte

sian

pro

duct

of

rco

pies

of

IRm

Lm

,2[0

, 1]

(Lm

[0, 1

],/ \,

\ /L

m 2[0

,1],

|||| L

m 2[0

,1])

Li N

finite

dim

ensi

onal

sub

spac

e of

Lm

,2[0

, 1],

i=

1, 2

Li N

time

sam

ples

of

elem

ents

inL

i N,

i=

1, 2

L(

)N

L(

)N

L

1 N,

-th o

rder

spl

ine

sub-

spac

e

L(

)N

splin

e co

effic

ient

s of

ele

men

ts in

L(

)N

.H

2IR

n

Lm 2

[0, 1

]H

,2

IRn

L

m ,2

[0, 1

]

H2

HN

IRn

L

1 Nor

IR

n

L2 N

,H

N

H

,2

HN

IRn

L

1 Nor

IR

n

L2 N

uk

(uk,

1,.

..,u

k,r)

IR

m

...

IRm

u(u

0,.

..,u

N1

)

LN

=(

,u)

H

,2

N

N=

(

,uN

)

HN

=(

,u)

H

N

(

1,..

.,

N+

1)

L

(

)N

.

Fun

ctio

ns

/ \,\ /

inne

r pr

oduc

t in

Hilb

ert s

pace

||||

norm

in H

ilber

t spa

ce

VA

,NV

A,N

:L

i N

Li N

,i=

1, 2

WA

,NW

A,N

:H

N

HN

,W

A,N

((

,u))

=(

,VA

,N(u

))

SN

,

SN

,

:L

(

)N

L

(

)N

k,it k

+c i

u[

k,i]

valu

e of

con

trol

sam

ple

at

k,iD

(

;h)

dire

ctio

nal d

eriv

ativ

ed u

f(

)de

riva

tive

off(

)with

res

pect

toth

e co

mpo

nent

s of

u=

VA

,n(u

)d

f(u

)de

riva

tive

off(

u)w

ith r

espe

ct to

the

com

pone

nts

of

=S

N,

(u)

F(x

, w)

Rig

ht h

and

side

of

diff

eren

ce e

qua-

tion

prod

uced

by

RK

dis

cret

izat

ion

Sets

BB

L

m ,2

[0, 1

]is

the

set o

n w

hich

all d

iffe

rent

ial o

pera

tors

are

defi

ned.

IN

{0,

1, 2

, . .

. }N

{d

n}

n=1

{0,

1,2,

...,

N

1}

11 co

lum

n ve

ctor

of

ones

.B

(v,

){

v

IRm

|||v

v|

| 2

}q

{1,

...,

q}

t Nt N

={

t k}

N1

k=0

is th

e di

scre

tizat

ion

mes

h, o

r ...

t Nt N

={

t k}

N+

1k=

+1is

a s

plin

e kn

otse

quen

ceA

Run

ge-K

utta

par

amet

ers

A=

[c,A

,b]

II

={

i 1,i

2,.

..,i

r}

={

i|c

j

c i,

ju

j k

UU

NU

N=

V1 A

,N(U

N)

L

,2

HIR

n

U

H

,2

HN

IRn

U

N

HN

HN

IRn

V

1 A,N

(UN

)

U(

)N

U(

)N

={

u

L(

)N

|

k

U}

Dif

fere

ntia

l and

Dif

fere

nce

Equ

atio

ns

x

(t)

solu

tion

at ti

me

tof

dif

fere

ntia

leq

uatio

n gi

ven

=(

,u):

initi

alco

nditi

on

and

con

trol

inpu

tux

kso

lutio

n at

tim

e st

epk

ofdi

ffer

ence

equ

atio

n, r

esul

ting

from

RK

dis

cret

izat

ion,

for

=(

,u):

initi

al c

ondi

tion

and

con

trol

sam

ples

ux

N kx

N k=

x

kwith

=W

A,N

(

N)

-v

-

Tabl

e of

Con

tent

s

AC

KN

OW

LE

DG

EM

EN

TS

v

NO

TA

TIO

N

vi

CH

AP

TE

R 1

: In

trod

ucti

on

1.1

Num

eric

alM

etho

ds f

or S

olvi

ng O

ptim

al C

ontr

ol P

robl

ems

......

......

......

......

......

1

1.2

Con

trib

utio

ns to

the

Stat

e-of

-the

-Art

......

......

......

......

......

......

......

......

......

......

......

5

1.3

Dis

sert

atio

nO

utlin

e ...

......

......

......

......

......

......

......

......

......

......

......

......

......

......

.....

6

CH

AP

TE

R 2

:C

onsi

sten

t A

ppro

xim

atio

ns B

ased

on

Run

ge-K

utta

Int

egra

tion

2.1

Intr

oduc

tion

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...

8

2.2

The

ory

of C

onsi

sten

t App

roxi

mat

ions

......

......

......

......

......

......

......

......

......

......

....

10

2.2.

1 O

verv

iew

ofco

nstr

uctio

n of

con

sist

ent a

ppro

xim

atio

ns...

......

......

......

...

13

2.3

Defi

nitio

nof

Opt

imal

Con

trol

Pro

blem

......

......

......

......

......

......

......

......

......

......

.. 16

2.4

App

roxi

mat

ing

Prob

lem

s ...

......

......

......

......

......

......

......

......

......

......

......

......

......

...20

2.4.

1 Fi

nite

Dim

ensi

onal

Ini

tial-

Stat

e-C

ontr

ol S

ubsp

aces

......

......

......

......

......

20

2.4.

2 D

efini

tion

of A

ppro

xim

atin

g Pr

oble

ms

......

......

......

......

......

......

......

......

.. 29

2.4.

3 E

pico

nver

genc

e ...

......

......

......

......

......

......

......

......

......

......

......

......

......

......

33

2.4.

4 Fa

ctor

s in

Sel

ectin

g th

e C

ontr

ol R

epre

sent

atio

n...

......

......

......

......

......

...

36

2.5

Opt

imal

ityFu

nctio

ns f

or th

e A

ppro

xim

atin

g Pr

oble

ms

......

......

......

......

......

......

. 38

2.5.

1 C

ompu

ting

Gra

dien

ts ..

......

......

......

......

......

......

......

......

......

......

......

......

....

38

2.5.

2 C

onsi

sten

cyof

App

roxi

mat

ions

....

......

......

......

......

......

......

......

......

......

...41

2.6

Coo

rdin

ate

Tra

nsfo

rmat

ions

and

Num

eric

al R

esul

ts...

......

......

......

......

......

......

...

48

2.7

App

roxi

mat

ing

Prob

lem

s B

ased

on

Splin

es...

......

......

......

......

......

......

......

......

....

53

2.7.

1 Im

plem

enta

tion

of S

plin

e C

oord

inat

e T

rans

form

atio

n ...

......

......

......

......

67

2.8

Con

clud

ing

Rem

arks

....

......

......

......

......

......

......

......

......

......

......

......

......

......

......

..70

CH

AP

TE

R 3

: P

roje

cted

Des

cent

Met

hod

for

Pro

blem

s w

ith

Sim

ple

Bou

nds

3.1

Intr

oduc

tion

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...

71

3.2

Alg

orith

mM

odel

for

Min

imiz

atio

n Su

bjec

t to

Sim

ple

Bou

nds

......

......

......

......

. 74

3.3

Com

puta

tiona

lRes

ults

....

......

......

......

......

......

......

......

......

......

......

......

......

......

......

90

3.4

Con

clud

ing

Rem

arks

....

......

......

......

......

......

......

......

......

......

......

......

......

......

......

..94

-vi

-

CH

AP

TE

R 4

:N

umer

ical

Iss

ues

4.1

Intr

oduc

tion

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...

96

4.2

Inte

grat

ion

Ord

er a

nd S

plin

e O

rder

Sel

ectio

n...

......

......

......

......

......

......

......

......

. 98

4.2.

1 So

lutio

ner

ror

for

unco

nstr

aine

d pr

oble

m...

......

......

......

......

......

......

......

. 99

4.2.

2 C

onst

rain

edPr

oble

ms

......

......

......

......

......

......

......

......

......

......

......

......

.....

103

4.3

Inte

grat

ion

Err

or a

nd M

esh

Red

istr

ibut

ion

......

......

......

......

......

......

......

......

......

...10

9

4.3.

1 C

ompu

ting

the

loca

l int

egra

tion

erro

r...

......

......

......

......

......

......

......

......

. 11

0

4.3.

2 St

rate

gies

for

mes

h re

finem

ent

......

......

......

......

......

......

......

......

......

......

...

112

4.4

Est

imat

ion

of S

olut

ion

Err

or...

......

......

......

......

......

......

......

......

......

......

......

......

....

120

4.5

Sing

ular

Con

trol

Pro

blem

s (P

iece

wis

e D

eriv

ativ

e V

aria

tion

of th

e C

ontr

ol)

.....

129

4.6

Oth

erIs

sues

....

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

....

142

4.6.

1 Fi

xed

vers

us V

aria

ble

Step

-Siz

e...

......

......

......

......

......

......

......

......

......

....

142

4.6.

2 E

qual

ityC

onst

rain

ts .

......

......

......

......

......

......

......

......

......

......

......

......

......

146

CH

AP

TE

R 5

:R

IOT

S U

ser

sM

anua

l

5.1

Intr

oduc

tion

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...

147

5.2

Prob

lem

Des

crip

tion

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

.15

0

Tra

nscr

iptio

n fo

r Fr

ee F

inal

Tim

e Pr

oble

ms

......

......

......

......

......

......

......

......

......

15

1

Tra

ject

ory

Con

stra

ints

......

......

......

......

......

......

......

......

......

......

......

......

......

......

....

152

Con

tinuu

m O

bjec

tive

Func

tions

....

......

......

......

......

......

......

......

......

......

......

......

...15

3

5.3

Usi

ngR

IOT

S ...

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...15

4

5.4

Use

rSu

pplie

d Su

brou

tines

......

......

......

......

......

......

......

......

......

......

......

......

......

...

167

5.5

Sim

ulat

ion

Rou

tines

.....

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...18

4

Impl

emen

tatio

n of

the

Inte

grat

ion

Rou

tines

......

......

......

......

......

......

......

......

......

. 19

3

5.6

Opt

imiz

atio

nPr

ogra

ms

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...20

6

Coo

rdin

ate

Tra

nsfo

rmat

ion

......

......

......

......

......

......

......

......

......

......

......

......

......

...21

1

Des

crip

tion

of th

e O

ptim

izat

ion

Prog

ram

s...

......

......

......

......

......

......

......

......

......

21

3

5.7

Util

ityR

outin

es ..

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

.23

2

5.8

Inst

allin

g,C

ompi

ling

and

Lin

king

RIO

TS

......

......

......

......

......

......

......

......

......

...24

2

CH

AP

TE

R 6

:C

oncl

usio

ns a

nd D

irec

tion

s fo

r F

utur

eR

esea

rch

249

AP

PE

ND

IX A

:P

roof

of

Som

e R

esul

ts in

Cha

pter

225

7

AP

PE

ND

IX B

:E

xam

ple

Opt

imal

Con

trol

Pro

blem

s 26

2

RE

FE

RE

NC

ES

266

-vi

i -

Cha

pter

1

INT

RO

DU

CT

ION

1.1

NU

ME

RIC

AL

ME

TH

OD

S F

OR

SO

LVIN

G O

PT

IMA

L C

ON

TR

OL

PR

OB

LE

MS

Num

eric

al m

etho

ds f

or s

olvi

ng o

ptim

al c

ontr

ol p

robl

ems

have

evo

lved

sig

nific

antly

ove

rth

e

past

thi

rty-

four

yea

rs s

ince

Pon

trya

gin

and

his

stud

ents

pre

sent

ed t

heir

cel

ebra

ted

max

imum

prin

cipl

e [1

].M

ost

earl

y m

etho

ds w

ere

base

d on

find

ing

a so

lutio

n th

at s

atis

fied

the

max

imum

prin

cipl

e, o

r re

late

d ne

cess

ary

cond

ition

s, r

athe

r th

an a

ttem

ptin

g a

dire

ct m

inim

izat

ion

of t

he

obje

ctiv

e fu

nctio

n (s

ubje

ct t

o co

nstr

aint

s) o

f th

e op

timal

con

trol

pro

blem

.Fo

rth

is r

easo

n, m

eth-

ods

usin

g th

is a

ppro

ach

are

calle

d in

dire

ct m

etho

ds.

Exp

lana

tions

of

the

indi

rect

app

roac

h ca

n be

foun

d in

[2-6

].

The

mai

n dr

awba

ck to

indi

rect

met

hods

is th

eir

extr

eme

lack

of

robu

stne

ss: t

he it

erat

ions

of

an in

dire

ct m

etho

d m

ust s

tart

clo

se, s

omet

imes

ver

y cl

ose,

to a

loca

l sol

utio

n in

ord

er to

sol

veth

e

two-

poin

t bo

unda

ry v

alue

sub

prob

lem

s.A

dditi

onal

ly,

sinc

e fir

st o

rder

opt

imal

ity c

ondi

tions

are

satis

fied

by m

axim

izer

s an

d sa

ddle

poi

nts

as w

ell a

s m

inim

izer

s, th

ere

is n

o re

ason

, in

gene

ral,

to

expe

ct s

olut

ions

obt

aine

d by

indi

rect

met

hods

to b

e m

inim

izer

s.

Bot

h of

the

se d

raw

back

s of

ind

irec

t m

etho

ds a

re o

verc

ome

by s

o-ca

lled

dire

ct m

etho

ds.

Dir

ect m

etho

ds o

btai

n so

lutio

ns th

roug

h th

e di

rect

min

imiz

atio

n of

the

obje

ctiv

e fu

nctio

n (s

ubje

ct

to c

onst

rain

ts)

of t

he o

ptim

al c

ontr

ol p

robl

em.

In t

his

way

the

opt

imal

con

trol

pro

blem

is

trea

ted

as

an

infin

ite

dim

ensi

onal

m

athe

mat

ical

pr

ogra

mm

ing

prob

lem

.T

here

ar

e tw

odi

stin

ct

appr

oach

es f

or d

ealin

g w

ith th

e in

finite

dim

ensi

onal

asp

ect o

f th

ese

prob

lem

s.T

he fi

rst a

ppro

ach

deve

lops

spe

cial

ized

conc

eptu

alal

gori

thm

s, a

nd n

umer

ical

im

plem

enta

tions

of

thes

e al

gori

thm

s,

for

solv

ing

the

mat

hem

atic

al p

rogr

ams.

Aco

ncep

tual

alg

orith

m i

s ei

ther

a f

unct

ion

spac

e an

alog

of a

fini

te d

imen

sion

al o

ptim

izat

ion

algo

rith

m o

r a

finite

dim

ensi

onal

alg

orith

m (

obta

ined

by

rest

rict

ing

the

cont

rols

to

a fin

ite d

imen

sion

al s

ubsp

ace

of t

he c

ontr

ol s

pace

) th

at r

equi

res

infin

ite

dim

ensi

onal

ope

ratio

ns s

uch

as th

e so

lutio

n of

dif

fere

ntia

l equ

atio

ns a

nd in

tegr

als.

An

impl

emen

-

tatio

n of

a c

once

ptua

l al

gori

thm

acc

ount

s fo

r er

rors

tha

t re

sult

whe

n re

pres

entin

g el

emen

ts o

f an

infin

ite

dim

ensi

onal

fu

nctio

ns

spac

e w

ith

finite

di

men

sion

al

appr

oxim

atio

ns

and

the

erro

rs

-1

-

Cha

p. 1

prod

uced

by

the

num

eric

al m

etho

ds u

sed

to p

erfo

rm i

nfini

te d

imen

sion

al o

pera

tions

.T

here

are

man

yex

ampl

es o

f co

ncep

tual

alg

orith

m f

or s

olvi

ng o

ptim

al c

ontr

ol p

robl

em,

som

e w

ith a

nd

som

e w

ithou

t im

plem

enta

tions

[7-3

1].

The

con

cept

ual

algo

rith

m a

ppro

ach

for

solv

ing

optim

al c

ontr

ol p

robl

ems

has

seri

ous

draw

-

back

s.

Firs

t,cu

stom

ized

sof

twar

e fo

r co

ntro

lling

the

err

ors

prod

uced

in

the

num

eric

al a

ppro

xi-

mat

ions

of

infin

ite d

imen

sion

al f

unct

ions

and

ope

ratio

ns m

ust b

e in

corp

orat

ed in

to th

e im

plem

en-

tatio

n of

a c

once

ptua

l al

gori

thm

.M

ore

seri

ousl

y,be

caus

e fu

nctio

n ev

alua

tions

are

per

form

ed

only

app

roxi

mat

ely

the

func

tion

grad

ient

s us

ed b

y m

athe

mat

ical

pro

gram

min

g so

ftw

are

will

not

be c

oord

inat

ed w

ith t

hose

sam

e fu

nctio

ns.

Tha

t is

, th

e gr

adie

nts

will

onl

y be

app

roxi

mat

ions

to

the

deri

vativ

esof

the

func

tions

.T

his

mea

n, f

or e

xam

ple,

tha

t it

is p

ossi

ble

that

the

neg

ativ

e of

a

func

tion

grad

ient

may

not

be

a di

rect

ion

of d

esce

nt f

or t

he a

ppro

xim

atio

n of

tha

t fu

nctio

n.T

his

prob

lem

is

exac

erba

ted

as a

sta

tiona

ry p

oint

is

appr

oach

ed.

Are

late

d pr

oble

m i

s th

at a

cer

tain

amou

nt o

f pr

ecis

ion

in t

he f

unct

ion

eval

uatio

ns i

s re

quir

ed t

o en

sure

suc

cess

ful

line

sear

ches

.

Toge

ther

,th

ese

fact

s m

ean

that

, in

pra

ctic

e, h

igh

prec

isio

n in

num

eric

al o

pera

tions

suc

h as

int

e-

grat

ion

is r

equi

red

even

inea

rly

itera

tions

of

the

optim

izat

ion

proc

edur

e.Si

nce

high

pre

cisi

on i

n

earl

y ite

ratio

ns d

oes

not

cont

ribu

te t

o th

e ac

cura

cyof

the

final

sol

utio

n, t

his

requ

irem

ent

mak

es

the

impl

emen

tatio

n of

con

cept

ual a

lgor

ithm

inef

ficie

nt f

or m

ost p

robl

ems.

An

alte

rnat

e di

rect

met

hod

appr

oach

is

one

whi

ch w

e te

rm c

onsi

sten

t ap

prox

imat

ions

.In

the

cons

iste

nt a

ppro

xim

atio

ns a

ppro

ach,

the

opt

imal

con

trol

is

obta

ined

by

solv

ing

a se

quen

ce o

f

finite

dim

ensi

onal

, di

scre

te-t

ime

optim

al c

ontr

ol p

robl

ems

that

are

inc

reas

ingl

y ac

cura

te r

epre

-

sent

atio

ns o

f th

e or

igin

al, c

ontin

uous

-tim

e pr

oble

m.

The

sol

utio

ns o

f th

e ap

prox

imat

ing,

dis

cret

e-

time

optim

al c

ontr

ol p

robl

ems

can

be o

btai

ned

usin

g st

anda

rd,

finite

dim

ensi

onal

mat

hem

atic

al

prog

ram

min

g te

chni

ques

.U

nder

sui

tabl

e co

nditi

ons,

sol

utio

ns o

f th

e ap

prox

imat

ing

prob

lem

s

conv

erge

toa

solu

tion

of t

he o

rigi

nal

prob

lem

.In

thi

s se

nse,

suc

h di

scre

te-t

ime

optim

al c

ontr

ol

prob

lem

s ar

e ca

lled

cons

iste

nt a

ppro

xim

atio

nsto

the

orig

inal

pro

blem

.

The

firs

t ri

goro

us d

evel

opm

ents

of

algo

rith

ms

base

d on

sol

ving

fini

te d

imen

sion

al a

ppro

xi-

mat

ing

prob

lem

s us

ed E

uler

sm

etho

d an

d pi

ecew

ise

cons

tant

con

trol

rep

rese

ntat

ions

(w

hich

resu

lts in

a fi

nite

dim

ensi

onal

con

trol

par

amet

eriz

atio

n) to

dis

cret

ize

the

orig

inal

pro

blem

(se

e th

e

intr

oduc

tion

to C

hapt

er 2

for

ref

eren

ces)

.Fr

om a

num

eric

al a

naly

sts

poin

t of

view

, the

cho

ice

of

Eul

ers

met

hod

may

see

m s

tran

ge s

ince

Eul

ers

met

hod

is a

n ex

trem

ely

inef

ficie

nt m

etho

d fo

r

solv

ing

diff

eren

tial

equa

tions

.B

ut

ther

e ar

e re

ason

s fo

r ch

oosi

ng

Eul

ers

met

hod

as

a

Spe

akin

g m

ore

accu

rate

ly,t

he d

iscr

etiz

ed p

robl

ems

need

not

be

a di

scre

te-t

ime

optim

al c

ontr

ol p

robl

ems.

For

inst

ance

, if

the

cont

rols

are

rep

rese

nted

as

finite

dim

ensi

onal

B-s

plin

es, t

he d

ecis

ion

vari

able

s of

the

dis

cret

ized

pro

blem

s ar

e sp

line

coef

ficie

nts,

not

cont

rol v

alue

s at

dis

cret

e tim

es.

-2

-

Cha

p. 1

disc

retiz

atio

n pr

oced

ure

for

optim

al c

ontr

ol p

robl

ems.

Firs

t, up

unt

il th

is w

ork,

ther

e ha

s be

en n

o

theo

ry s

uppo

rtin

g th

e us

e of

iter

ativ

e hi

gher

-ord

er in

tegr

atio

n m

etho

ds in

the

cons

truc

tion

of c

on-

sist

ent

appr

oxim

atio

ns.

Seco

nd,

only

rec

ently

has

it

been

dem

onst

rate

d th

at t

here

can

be

an

adva

ntag

e to

usi

ng h

ighe

r-or

der

disc

retiz

atio

n m

etho

ds f

or s

olvi

ng o

ptim

al c

ontr

ol p

robl

ems.

The

use

of h

ighe

r-or

der

disc

retiz

atio

n m

etho

ds f

or s

olvi

ng o

ptim

al c

ontr

ol p

robl

ems

rem

ains

an

activ

e

area

of

rese

arch

.It

is

diffi

cult

to d

emon

stra

te a

the

oret

ical

adv

anta

ge t

o us

ing

high

er o

rder

met

h-

ods

rath

er t

han

Eul

ers

met

hods

whe

n so

lvin

g ge

nera

l, co

nstr

aine

d op

timal

con

trol

pro

blem

s.

How

ever

, man

yop

timal

con

trol

pro

blem

s th

at a

rise

in p

ract

ice

are,

in f

act,

solv

ed m

uch

mor

e ef

fi-

cien

tly w

ith h

ighe

r-or

der

met

hods

.

With

in t

he c

ateg

ory

of d

irec

t m

etho

ds b

ased

on

the

idea

of

cons

iste

nt a

ppro

xim

atio

ns, t

here

is a

fur

ther

sub

-cla

ssifi

catio

n th

at h

elps

to

esta

blis

h w

here

our

wor

k st

ands

in

rela

tion

to o

ther

met

hods

. T

his

sub-

clas

sific

atio

n sp

ecifi

es h

owth

e di

scre

tizat

ion

of a

n op

timal

con

trol

pro

blem

into

a fi

nite

dim

ensi

onal

app

roxi

mat

ing

prob

lem

is a

ccom

plis

hed:

via

col

loca

tion

(or

mor

e ge

ner-

ally

,a

Gal

erki

n ap

prox

imat

ion)

or

via

itera

tive

inte

grat

ion.

C

urre

ntly

,th

e m

ost

popu

lar

dis-

cret

izat

ion

sche

me

is

base

d on

co

lloca

tion

and

met

hods

si

mila

r in

sp

irit

to

collo

catio

n [1

6-18

,32-

41].

In c

ollo

catio

n m

etho

ds, t

he s

yste

m o

f di

ffer

entia

l equ

atio

ns d

escr

ibin

g

the

dyna

mic

sys

tem

is

repl

aced

by

a sy

stem

of

equa

tions

tha

t re

pres

ent

collo

catio

n co

nditi

ons

to

be s

atis

fied

at a

fini

te n

umbe

r of

tim

e po

ints

.T

he r

esul

ting

mat

hem

atic

al p

rogr

am i

nvol

ves

not

only

the

con

trol

par

amet

ers

as d

ecis

ion

vari

able

s bu

t al

so a

lar

ge n

umbe

r of

add

ition

al v

aria

bles

that

rep

rese

nts

the

valu

e of

sta

te v

aria

bles

at

mes

h po

ints

.C

ollo

catio

n sc

hem

es o

ffer

sev

eral

adva

ntag

es o

ver

itera

tive

inte

grat

ion

sche

mes

:

1.

Itis

eas

ier

to p

rove

con

verg

ence

and

ord

er o

f co

nver

genc

e re

sults

.

2.

Som

ere

sults

for

the

ord

er o

f er

ror,

asa

func

tion

of t

he d

iscr

etiz

atio

n le

vel,

betw

een

solu

-

tions

of

the

appr

oxim

atin

g pr

oble

ms

and

solu

tions

of

the

orig

inal

pro

blem

(na

mel

y,fo

r

unco

nstr

aine

d op

timal

con

trol

pro

blem

s) a

re s

uper

ior

to o

ther

sch

emes

[36]

.

3.

Cer

tain

diffi

culti

es in

here

nt to

som

e op

timal

con

trol

pro

blem

s, s

uch

as s

tiff

diff

eren

tial e

qua-

tions

and

hig

hly

unst

able

dyn

amic

s, a

re g

reat

ly m

itiga

ted

in c

ollo

catio

n sc

hem

es.

4.

Sim

ple

boun

ds o

n st

ate

vari

able

s tr

ansl

ate

into

sim

ple

boun

ds o

n th

e de

cisi

on v

aria

bles

of

the

mat

hem

atic

al p

rogr

am.

5.

Func

tion

grad

ient

s ar

e ea

sier

to

com

pute

sin

ce t

hey

dono

t re

quir

e th

e de

riva

tive

ofth

e st

ate

with

res

pect

to th

e co

ntro

ls.

How

ever

, rel

ativ

e to

itera

tive

inte

grat

ion,

col

loca

tion

sche

mes

hav

e se

riou

s dr

awba

cks

as w

ell:

1.

The

appr

oxim

atin

g pr

oble

ms

are

sign

ifica

ntly

larg

er a

t a g

iven

disc

retiz

atio

n le

veld

ue to

the

incl

usio

n of

sta

te v

aria

bles

as

deci

sion

par

amet

ers.

-3

-

Cha

p. 1

2.

The

appr

oxim

atin

g pr

oble

ms

are

sign

ifica

ntly

har

der

to s

olve

beca

use

of t

he a

dditi

on o

f a

larg

e nu

mbe

r of

(no

nlin

ear)

equ

ality

con

stra

ints

that

rep

rese

nt th

e co

lloca

tion

cond

ition

s.

3.

The

accu

racy

ofso

lutio

ns o

btai

ned

by s

olvi

ng th

e ap

prox

imat

ing

prob

lem

s ca

n be

som

ewha

t

inac

cura

te d

ue to

the

pres

ence

of

the

collo

catio

n co

nstr

aint

s.

4.

Ifth

e nu

mer

ical

alg

orith

m f

or s

olvi

ng th

e ap

prox

imat

ing

prob

lem

s is

term

inat

ed p

rem

atur

ely

the

solu

tion

may

not

be

usef

ul s

ince

the

collo

catio

n co

nditi

ons

will

not

be

satis

fied.

Bec

ause

of

thes

e di

sadv

anta

ges,

sol

utio

ns o

btai

ned

usin

g a

collo

catio

n sc

hem

e of

ten

have

to

be

subs

eque

ntly

refi

ned

usin

g an

indi

rect

sol

utio

n m

etho

d[4

].

The

wor

k in

thi

s th

esis

is

base

d on

dis

cret

izin

g op

timal

con

trol

pro

blem

s us

ing

expl

icit,

fixed

ste

p-si

ze R

unge

-Kut

ta i

nteg

ratio

n te

chni

ques

.T

he a

dvan

tage

of

this

sch

eme

over

collo

ca-

tion

sche

mes

is

that

the

app

roxi

mat

ing

prob

lem

s th

at r

esul

t ca

n be

sol

ved

very

effi

cien

tly a

nd

accu

rate

ly.

On

the

othe

r ha

nd,

som

e of

the

fea

ture

s lis

ted

abov

e as

adva

ntag

es a

ssoc

iate

d w

ith

collo

catio

n ar

e sa

crifi

ced.

Spec

ifica

lly,

conv

erge

nce

resu

lts a

re m

ore

diffi

cult

to p

rove

for

the

Run

ge-K

utta

met

hod

and,

in th

e ca

se o

f un

cons

trai

ned

prob

lem

s, th

e or

der

of e

rror

for

sol

utio

n of

the

appr

oxim

atin

g pr

oble

ms

is lo

wer

(se

e[4

2] a

nd P

ropo

sitio

n 4.

6.2)

.A

lso,

it is

qui

te c

onve

nien

t

from

a p

rogr

amm

ing

poin

t of

view

that

sta

te v

aria

ble

boun

ds b

ecom

e bo

unds

on

the

deci

sion

var

i-

able

s of

the

mat

hem

atic

al p

rogr

am (

adva

ntag

e 4)

.H

owev

er, t

his

adva

ntag

e is

mor

e th

an o

ffse

t by

the

addi

tion

of th

e sy

stem

of

equa

lity

cons

trai

nts

repr

esen

ting

the

collo

catio

n co

nditi

ons.

Fina

lly,

the

diffi

culti

es o

f so

lvin

g pr

oble

ms

with

hig

hly

unst

able

dyn

amic

s ca

n al

so b

e ha

ndle

d w

hen

usin

g ex

plic

it R

unge

-Kut

ta in

tegr

atio

n. A

met

hod

for

doin

g so

is d

iscu

ssed

in th

e C

hapt

er 6

.

As

far

as w

e kn

ow,

the

wor

k re

port

ed i

n th

is t

hesi

s re

pres

ents

the

onl

y w

ork

on c

onsi

sten

t

appr

oxim

atio

n sc

hem

es u

sing

Run

ge-K

utta

inte

grat

ion.

Thu

s,at

the

very

leas

t, ou

r w

ork

com

ple-

men

ts t

he w

ork

of o

ther

aut

hors

tha

t de

al w

ith c

ollo

catio

n sc

hem

es.

But

fur

ther

,we

belie

ve t

hat

our

appr

oach

has

sig

nific

ant

theo

retic

al a

nd p

ract

ical

adv

anta

ges

that

will

mak

eit,

with

suf

ficie

nt

deve

lopm

ent,

a le

adin

g ap

proa

ch to

sol

ving

opt

imal

con

trol

pro

blem

s.

-4

-

Cha

p. 1

1.2

CO

NT

RIB

UT

ION

S T

OT

HE

STA

TE

-OF

-TH

E-A

RT

The

ori

gina

l go

al o

f th

is r

esea

rch

was

sim

ply

to d

evel

op a

fas

t an

d ac

cura

te s

oftw

are

pack

-

age

for

solv

ing

optim

al c

ontr

ol p

robl

ems

usin

g ex

plic

it R

unge

-Kut

ta i

nteg

ratio

n.

Inth

e pr

oces

s

of w

ritin

g th

is s

oftw

are

we

have

,by

nece

ssity

,de

v elo

ped

a st

rong

the

oret

ical

fou

ndat

ion

for

our

disc

retiz

atio

n ap

proa

ch a

s w

ell c

onst

ruct

ing

seve

ral n

ewal

gori

thm

s fo

r va

riou

s ty

pes

of c

ompu

ta-

tion.

The

follo

win

g is

a c

onci

se s

umm

ary

of th

e co

ntri

butio

ns p

rovi

ded

by th

is w

ork

to th

e st

ate-

of-t

he-a

rt in

num

eric

al m

etho

ds f

or s

olvi

ng o

ptim

al c

ontr

ol p

robl

ems:

Pr

ovid

es t

he fi

rst

conv

erge

nce

anal

ysis

and

im

plem

enta

tion

theo

ry f

or d

iscr

etiz

atio

n m

etho

ds

base

d on

Run

ge-K

utta

int

egra

tion.

Sp

ecifi

cally

,co

nditi

ons

on t

he p

aram

eter

s of

the

Run

ge-

Kut

ta m

etho

d ar

e pr

esen

ted

that

ens

ure,

for

ins

tanc

e, t

hat

stat

iona

ry p

oint

s of

the

dis

cret

ized

prob

lem

s ca

n on

ly c

onve

rge

tost

atio

nary

poi

nts

of th

e or

igin

al p

robl

em.

D

eriv

esa

non-

Euc

lidea

n m

etri

c ne

eded

for

the

finite

-dim

ensi

onal

opt

imiz

atio

n of

the

appr

oxi-

mat

ing

prob

lem

s an

d pr

esen

ts a

coo

rdin

ate

tran

sfor

mat

ion

whi

ch a

llow

s a

Euc

lidea

n m

etri

c to

be u

sed.

With

out

this

met

ric,

ser

ious

ill-

cond

ition

ing

can

be i

ntro

duce

d in

to t

he d

iscr

etiz

ed

prob

lem

.

Im

prov

esup

on t

he p

revi

ousl

y kn

own

boun

d fo

r th

e er

ror

in t

he s

olut

ion

of t

he a

ppro

xim

atin

g

prob

lem

s as

a f

unct

ion

of t

he d

iscr

etiz

atio

n le

vel

for

RK

4 (t

he m

ost

com

mon

fou

rth-

orde

r

Run

ge-K

utta

inte

grat

ion

met

hod)

whe

n so

lvin

g un

cons

trai

ned

optim

al c

ontr

ol p

robl

ems.

Thi

s

resu

lt, a

long

with

the

alr

eady

kno

wn

boun

ds f

or a

firs

t, se

cond

and

thi

rd o

rder

Run

ge-K

utta

met

hod

are

exte

nded

to

the

case

whe

re t

he fi

nite

dim

ensi

onal

con

trol

s ar

e re

pres

ente

d by

splin

es.

Pr

esen

ts a

new

, ver

y ef

ficie

nt a

nd r

obus

t nu

mer

ical

alg

orith

m, b

ased

on

the

proj

ecte

d N

ewto

n

met

hod

of B

erts

ekas

, for

sol

ving

a c

lass

of

mat

hem

atic

al p

rogr

amm

ing

prob

lem

s w

ith s

impl

e

boun

ds o

n th

e de

cisi

on v

aria

bles

.

D

ev el

ops

a ne

wm

etho

d fo

r co

mpu

ting

accu

rate

est

imat

es o

f th

e er

ror

betw

een

the

solu

tions

com

pute

d fo

r th

e ap

prox

imat

ing

prob

lem

s an

d so

lutio

ns o

f th

e or

igin

al p

robl

em.

Thi

s es

ti-

mat

e do

es n

ot r

equi

rea

prio

rikn

owle

dge

of e

rror

bou

nds

and

wor

ks f

or p

robl

ems

with

sta

te

and

cont

rol c

onst

rain

ts.

D

ev el

ops

a co

mpl

etel

y ne

wm

etho

d fo

r nu

mer

ical

ly s

olvi

ng s

ingu

lar

optim

al c

ontr

ol p

rob-

lem

s.

Thi

sm

etho

d is

des

igne

d to

elim

inat

e un

desi

rabl

e os

cilla

tions

tha

t oc

cur

in n

umer

ical

solu

tions

of

sing

ular

con

trol

pro

blem

s.

Pr

esen

ts o

ur s

oftw

are

pack

age

calle

d R

IOT

S, b

ased

on

the

theo

ry i

n co

ntai

ned

in t

his

thes

is,

for

solv

ing

optim

al c

ontr

ol p

robl

ems.

Alth

ough

the

re a

re m

any

impr

ovem

ents

tha

t ca

n be

mad

e to

RIO

TS,

it

is a

lrea

dy o

ne o

f th

e fa

stes

t, m

ost

accu

rate

and

eas

iest

to

use

prog

ram

s

avai

labl

e fo

r so

lvin

g op

timal

con

trol

pro

blem

s.

-5

-

Cha

p. 1

1.3

DIS

SE

RTA

TIO

N O

UT

LIN

E

The

org

aniz

atio

n of

thi

s di

sser

tatio

n fo

llow

s a

prog

ress

ion

lead

ing

from

bas

ic t

heor

etic

al

foun

datio

ns o

f di

scre

tizin

g op

timal

con

trol

pro

blem

s to

the

impl

emen

tatio

n of

a s

oftw

are

pack

age

for

solv

ing

a la

rge

clas

s of

opt

imal

con

trol

pro

blem

s.T

he t

heor

etic

al f

ound

atio

n is

pre

sent

ed i

n

Cha

pter

2.

Cha

pter

2 b

egin

s w

ith a

dis

cuss

ion

of t

he c

once

pt o

f co

nsis

tent

app

roxi

mat

ions

as

defin

ed b

y Po

lak

[43]

. Po

lak

sde

finiti

on o

f co

nsis

tent

app

roxi

mat

ions

ext

ends

ear

lier

defin

ition

s,

nam

ely

that

of

Dan

iels

[44]

, tha

t wer

e co

ncer

ned

only

with

con

verg

ence

of

glob

al s

olut

ions

of

the

appr

oxim

atin

g pr

oble

ms

to g

loba

l so

lutio

ns o

f th

e or

igin

al p

robl

em.

The

ear

lier

defin

ition

s w

ere

ther

efor

e of

lim

ited

use

sinc

e op

timiz

atio

n al

gori

thm

s co

mpu

te s

tatio

nary

poi

nts,

not

glo

bal

solu

-

tions

. Po

lak

sde

finiti

on o

f co

nsis

tenc

yde

als

with

sta

tiona

ry p

oint

s an

d lo

cal

min

ima

as w

ell

as

glob

al s

olut

ions

.T

he th

eory

of

cons

iste

nt a

ppro

xim

atio

ns is

use

d to

dev

elop

a f

ram

ewor

k fo

r di

s-

cret

izin

g op

timal

con

trol

pro

blem

s w

ith R

unge

-Kut

ta i

nteg

ratio

n.

The

mai

n re

sults

in

Cha

pter

2

show

that

the

app

roxi

mat

ing

prob

lem

s ar

e co

nsis

tent

app

roxi

mat

ions

to

the

orig

inal

opt

imal

con

-

trol

pro

blem

if

the

Run

ge-K

utta

met

hod

satis

fies

cert

ain

cond

ition

s in

add

ition

to

the

stan

dard

cond

ition

s ne

eded

for

con

sist

ent i

nteg

ratio

n of

dif

fere

ntia

l equ

atio

ns.

Onc

e th

e co

nsis

tenc

yre

sult

is e

stab

lishe

d, th

e co

nver

genc

e re

sults

pro

vide

d by

the

theo

ry o

f co

nsis

tent

app

roxi

mat

ions

can

be

invo

ked.

In

the

proc

ess

of c

onst

ruct

ing

cons

iste

nt a

ppro

xim

atio

ns b

ased

on

Run

ge-K

utta

dis

-

cret

izat

ion,

we

show

that

a n

on-E

uclid

ean

inne

r-pr

oduc

t an

d no

rm,

depe

ndin

g on

the

bas

is u

sed

for

the

finite

dim

ensi

onal

con

trol

sub

spac

es,

mus

t be

use

d fo

r th

e sp

ace

of c

ontr

ol c

oeffi

cien

ts

upon

whi

ch t

he fi

nite

dim

ensi

onal

mat

hem

atic

al p

rogr

ams

that

res

ults

fro

m t

he d

iscr

etiz

atio

n ar

e

defin

ed.

With

out

this

non

-Euc

lidea

n m

etri

c, s

erio

us i

ll-co

nditi

onin

g ca

n re

sult.

We

also

sho

w

how

aco

ordi

nate

tra

nsfo

rmat

ion

can

be u

sed

to e

limin

ate

the

need

for

the

non

-Euc

lidea

n in

ner-

prod

uct a

nd n

orm

.T

he r

esul

ts a

re th

en e

xten

ded

to c

ontr

ol r

epre

sent

atio

ns b

ased

on

splin

es.

In C

hapt

er 3

, we

pres

ent a

ver

y ef

ficie

nt a

nd r

obus

t opt

imiz

atio

n al

gori

thm

for

sol

ving

fini

te

dim

ensi

onal

mat

hem

atic

al p

rogr

amm

ing

prob

lem

s th

at i

nclu

de s

impl

e bo

unds

on

the

deci

sion

vari

able

s.

Such

prob

lem

s ar

ise

from

the

dis

cret

izat

ion

of o

ptim

al c

ontr

ol p

robl

ems

with

con

trol

boun

ds.

InC

hapt

er 4

, ot

her

impo

rtan

t nu

mer

ical

iss

ues

are

addr

esse

d.T

hese

iss

ues

incl

ude

(i)

obta

inin

g bo

unds

on

the

erro

r of

sol

utio

ns to

the

appr

oxim

atin

g pr

oble

ms

base

d on

spl

ine

con-

trol

s,(i

i)de

velo

ping

heu

rist

ics

for

sele

ctin

g th

e in

tegr

atio

n or

der

and

cont

rol

repr

esen

tatio

n

orde

r,(i

ii)

prov

idin

g m

etho

ds f

or r

efini

ng t

he d

iscr

etiz

atio

n m

esh,

(iv)

prov

idin

g a

com

puta

ble

erro

r es

timat

e fo

r so

lutio

ns o

f th

e ap

prox

imat

ing

prob

lem

s an

d(v

)de

alin

g w

ith t

he n

umer

ical

diffi

culti

es t

hat

aris

e w

hen

solv

ing

sing

ular

opt

imal

con

trol

pro

blem

s.W

e al

so p

rese

nt n

umer

ical

data

to

supp

ort

our

clai

m t

hat

impl

emen

tatio

ns o

f co

ncep

tual

alg

orith

ms

are

inef

ficie

nt c

ompa

red

to th

e co

nsis

tent

app

roxi

mat

ions

app

roac

h to

sol

ving

opt

imal

con

trol

pro

blem

s.

-6

-

Cha

p. 1

The

nex

t ch

apte

r,C

hapt

er 5

, con

tain

s th

e us

ers

man

ual

for

RIO

TS.

RIO

TS

is o

ur s

oftw

are

pack

age,

dev

elop

ed a

s a

tool

box

for

Mat

lab

,fo

r so

lvin

g a

very

bro

ad c

lass

of

optim

al c

ontr

ol

prob

lem

s. T

his

clas

s in

clud

es p

robl

ems

with

mul

tiple

obj

ectiv

e fu

nctio

ns, fi

xed

or f

ree

final

tim

e

prob

lem

s, p

robl

ems

with

var

iabl

e in

itial

con

ditio

ns a

nd p

robl

ems

with

con

trol

bou

nds,

end

poin

t

equa

lity

and

ineq

ualit

y co

nstr

aint

s, a

nd t

raje

ctor

y co

nstr

aint

s.T

he u

ser

sm

anua

l in

clud

es a

mat

hem

atic

al d

escr

iptio

n of

the

clas

s of

pro

blem

s th

at c

an b

e ha

ndle

d, a

ser

ies

of s

ampl

e se

ssio

ns

with

RIO

TS,

a c

ompl

ete

refe

renc

e gu

ide

for

the

prog

ram

s in

RIO

TS,

exp

lana

tions

of

impo

rtan

t

impl

emen

tatio

n de

tails

, an

d in

stru

ctio

ns f

or i

nsta

lling

RIO

TS.

C

hapt

er6,

pre

sent

s ou

r co

nclu

-

sion

s an

d id

eas

for

futu

re r

esea

rch.

Fina

lly,

ther

e ar

e tw

oap

pend

ices

. T

hefir

st c

onta

ins

the

proo

fs o

f so

me

of th

e re

sults

in C

hapt

er 2

and

the

seco

nd d

escr

ibes

som

e ex

ampl

e op

timal

con

trol

prob

lem

s th

at w

e us

e, p

rim

arily

in C

hapt

er 4

, for

num

eric

al e

xper

imen

ts.

M

atla

b is

a s

cien

tific

com

puta

tion

and

visu

aliz

atio

n pr

ogra

m d

esig

ned

by T

he M

athW

orks

, Inc

.

-7

-

Cha

p. 1

Cha

pter

2

CO

NS

IST

EN

T A

PP

RO

XIM

AT

ION

S F

OR

OP

TIM

AL

CO

NT

RO

L

PR

OB

LE

MS

BA

SE

D O

N R

UN

GE

-KU

TTA

INT

EG

RA

TIO

N

2.1

INT

RO

DU

CT

ION

In t

his

Cha

pter

,we

esta

blis

h th

e th

eore

tical

fou

ndat

ion

of o

ur m

etho

d fo

r nu

mer

ical

ly s

olv-

ing

optim

al c

ontr

ol p

robl

ems.

Spec

ifica

lly,

we

cons

ider

app

roxi

mat

ions

to

cons

trai

ned

optim

al

cont

rol

prob

lem

s th

at r

esul

t fr

om n

umer

ical

sol

ving

the

dif

fere

ntia

l eq

uatio

ns d

escr

ibin

g th

e sy

s-

tem

dyn

amic

s us

ing

Run

ge-K

utta

int

egra

tion.

W

esh

owth

at t

here

is

a cl

ass

of h

ighe

r or

der,

expl

icit

Run

ge-K

utta

(R

K)

met

hods

tha

t pr

ovid

eco

nsis

tent

app

roxi

mat

ions

to t

he o

rigi

nal

prob

-

lem

, w

ith c

onsi

s