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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