EM"'""°"""'"'"""""""""""'1II
MIT LIBRARIES DUPL
3
TOflD
0D7?0fiTfl
3
*
/
1
OISV^Y
HD28
.M414
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Generalized
Picard-Lindelof
Iris
Marie Mack, Ph. D.
July 1991
WP
Theory
3317-91-MSA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Generalized
Iris
Picard-Lindelof
Marie Mack, Ph. D.
WP
July 1991
Forthcoming
in
Theory
SIAM
J.
3317-91-MSA
of Mathematical Analysis
Oe^w«y
Generalized Picard-Lindelof Theory
Abstract;
In
this paper, a constraint algorithm is developed for the
purpose of obtaining conditions for the existence and unidueness of
solutions to systems of differential/algebraic eouations. The
underlying
3
system
onnople
of our constraint theory is to conceptually
of differential/algebraic equations to a
prove the existence and uniqueness of
its solution
reduce
form so that we can
similar
in
manner
to that done via utilizing the Picard-Lindelof theory for ordinary
differential equations.
Key Words: Picard-Lindelof
theory, differential/algebraic equations,
reduction technique, constraint, Lagrangian dynamics.
INTRODUCTION
The theory developed
this paper,
in
which we will
call our
constraint theory, will be utilized to obtain conditions for the
existence and uniqueness of the solution to a system of
differential/algebraic equations as
equations.
is
done for ordinary differential
Recall that the Picard-Lindelof theory (or the method of
successive approximations)
under which an
initial
is
used to investigate the conditions
valued ordinary differential equation has a
solution and that that solution is unique. The underlying principle of
our constraint theor7
is to
conceptually reduce a system of
differentiai/algeDraic equations to a form so that
existence and uniqueness of
its
solution
in a
we
can prove
manner similar
to that
done via utilizing the Picard-Lindelof theory for ODE's.
Our constraint algorithm can De summarized as follows,
differentiate the system of differential/algebraic equations with
respect to the independent vanaDle
x,
study row spaces of certain
JacoDian matrices, perform various algebraic manipulations to
attempt to reduce the ill-posed system
of differential/algebraic
equations to a well-posea, overdetermined system; then invoke
theorems such as The Fredholm Alternative Theorem, The implicit
Function Theorem, and The Picard-Lindeldf Theorem, to obtain
conditions for the existence and uniqueness of the solution to the
system
of differential/algebraic equations.
Now we
of this
will
paper
Dnefly summarize what
is to
appear
in
section two
we
in
our constraint
will introduce our constraint theory and
present several important theorems
analysis
the remainder
Section one will contain a discussion of concepts from
Lagrangian Dynamics, some of which will be utilized
theory
m
-
which will be utilized
in
our
Sections three and four will contain a derivation of our
constraint algorithm for nonlinear systems of differential/algebraic
equations.
In
section five
we
constraint algorithm. Finally,
will present an application of our
in
section six,
we
will present a
discussion of an extension of our local results to semi-local results.
Motivation from Lagrangian Dynamics
I.
Our constraint theory
is
motivated by concepts from laqrangian
dynamics. For more details related to our snort digression on Lagrangian
dynamics, the interested reader
equations
of
is
referred to
[3]
and
The Lagrangian
[8].
motion for conservative mechanical systems can be written
as.
<3L
dt
^q,-
such tnat
L := L(t, a(t), £'(t)) is the
contains the
k
Lagrangian, and
dynamical coordinates,
Qj
= q,(t),
The second order system (1,1) represents
standard case
if q"j
is,
if
if
for
1
I
1
i k.
mechanical system of the
a
can be solved for the accelerations
it
can be expressed
in
terms
of
q^
and
q'j,
for
1
i
q",
u
(1
in
k,
i
i
i
k)
such a case,
the initial dynamical coordinates and velocities are prescribed, the
accelerations can be obtained, and hence the motion of the mechanical
system can be determined via integration
(1
1
)
is
such that
acceleration
q"j
(
we
1
mechanical system
But
are not able to solve for
i
is
i
i k),
this is a
said to be
if
the mechanical system
all k
components
nonstandard case
^ud^
subject to constraints
that
of the
the
This means
ir>3i '.ne
ayr.amicdi
time that
of
is,
coorainares ana velocities are
a functional relationship exists
Now suppose we consider systems with
k
riot
indepenoent Tuncnon'i
between
g_
g'
degrees of freedom
independent generalized coordinates). Given the Lagranqian
we
and
{^v
Lit,4(t),^'(t)),
define the following matrix of partial derivatives with respect to the
velocities:
2
a L
w(t4(t)4'(t))=
[w,
(1
^q,^q,
such that
1
1
ij
1 k
By expanding the first term of the Lagrangian equations of motion
and using the matrix defined by
2W,jqj
(
1
,2)
we
(ID
obtain equations of the form
1
=a,(t,a(t)4(t)),
'
T\
J)
j-t
such that
1
Lagrangian
if
W
is
i
is
U
k.
The accelerations can be uniquely solved for that
standard,
if
and only
if
the matrix
W
is of
is,
maximal rank
rank-deficient, the Lagrangian equations of motion do not yield
the
But
all
the accelerations as functions of the dynamical coordinates and velocities
m
.n [d] trie 3'.:^.^o''z exar^'ine
some
aeiaii ihe -aara.ngian eqoaticr'S
:"'
for the nonstandard case to understand tne nature of tne solution to
"'cvvp
tr,e
equations of motion
Let us
now examine
Its ^ank ? is less
the case
than k
W
where
in
(
12)
is
rank-deficient, that
is,
Therefore, there exist k-R iinearly indeoenoent
eigenvectors
null
V(m)(t4(t),4(t))
:=
iv,(m)(t,g(t),^'(t)),
...
,Vk(m)(t,g(t)4(t))jT
for tne matrix w, such t.hat
2
for
I
v,('^v)(t,(^(t)4Xt))
<
m
1
k-R and
W,,(t4(t)4(t))
M
relations involving the
j
i k.
Qj
=
(14)
0,
Equations (1.3) and (1.4) yield the following k-R
and
q'j:
k
'
2v,(f"Ht4(t),q'(t))a,(t4(t)4(t))
for
1
<
m
i
k-R,
=
The relations defined
(15)
0,
in (1.5)
are called
constraints
m
the Lagrangian sense. They are consequences of the equations of motion, and
place restrictions on the choice of the initial values of the dynamical
coordinates and velocities.
Introduction to Constraint Algorithm
2.
Suppose we consider general nonlinear overdetermined systems
of
differential/algebraic equations of tne form:
E(x,y(x),it(x))
=
0,
xe
(2.1)
[xo.Vir.j.
suDject to the initial conditions
such that
and
m
2 n.
To utilize our constraint theory to obtain conditions for the existence
and uniqueness of solutions to systems of the form
(2.
1
),
we
will find
necessary to assume certain algebraic and smoothness properties for
which will be valid
domain
we
(2.
of our
m some
problem
neighborhood around
a specified point
).
(2.
l
),
the
Given these algebraic and smoothness assumptions,
will obtain a local solution to a certain class of
1
m
it
systems
of the
form
These local results will then be extended to semi-local results over
larger subdomains of the domain of our problem - for which our algebraic
and smoothness assumptions also hold.
Given ce^iain proDiem-aeDendent matcninq or interface conc-Jcns, we
may
also oe aole to extend our semi-local results to glooal results for
systems
of differential/algeoraic equations
with discontinuities
"piecewise patching" our semi-local results together
interfaces of our domain
rank change,
it
is a
if
system
(2^
1
)
is
at the
)
The extension
discontinuous
such that dF/ay undergoes a
system with turning points (which
system with discontinuities
t)y
of
is
analogous to a
our semi-local results to
handle systems of differential/algeDraic equations with turning points will
not be discussed
in
any detail,
in
part because our motivating problems are
decoupled systems of differential/algebraic equations without turning
points.
Theoretical Tools
2.1
important tools, whicn
tne Fredholm Alternative
tne ^Mcara-L'.noeiof
!S 3 too]
used
;n
we
wiil utilize in our constraint analysis,
Tneorem
theorem
[i].
[7],
tne imDlicit Function
Theorem
2'"e
[5],
and
Recall The Fredholm Alternative Theorem
descr'oing the solution to rectangular systems of I'near
eouations of the form
Ax
= D,
(2 2)
such that
A G
R^^'<^
and
D 6
R^
Theorem
2.1:
For any A and
(1)
(11)
Ax
=
z1a
H,
Fredholm Alternative Theorem
one and only one of the following systems has a solution'
&
= o^,
zi*o
D
The general version of the Implicit Function Theorem can De stated as
follows:
Theorem
Let
a c
2.2: Implicit
Rq+r
t5e
Function Theorem
an open set. Consider a general nonlinear system of
equations of the form
F(w,z)
=
Few,,
,Wq,2,,
,2^)
=
0,
(23:
such that
FA
-> Rr
we
Ra,
and
ZG Rr
Let F be a function of class 6"p(A).
Suppose aF/az.
-
neignDorhood U
Suppose (w^.Zo^ e A and F(w^,2o)
evaluated at (w^,z^)
c
- is of
maximal rank
Rq of wq, and a neighborhood V
c
"
Then there
^
is a
R^ of Zo, and a unique
function ^;Li -* V, such that
£(w,^(w))
for all
w
= 0,
(2.4)
Furthermore,
e U
^ is
of class 6'P.n
Once we have utilized the implicit Function Theorem
to reduce an
implicit system of differential equations to an explicit system,
we
can
invoke the Ptcard-Lindelof Theorem to prove existence and uniqueness of a
solution to the system of implicit differential equations. The following
definition
is
needed to understand the Picard-Lindeldf Theorem,
Definition
2.1:
Suppose
there exists a constant L
>
llf(x.yi)-I(x.y2)liiUly, -
f
is
defined
in a
domain A"
such that, for every
^l
(x,yi),
c
[xg.Xfin] x
R"
{x,^) e R"
(/S)
''
triHn
1
]i rdid to xati :.:y
This tact wil; by
(z:o, Lip) in
if
the
domain
fe
cer;:.*?^ jy
Lipschitz constant
fe
Lipschiiz condition
.3
wi
if,
addition ^
constant
L
of
c
a region ft
Ix -
wnere
a
.>
is 0;
:jr;;.tar;t L i?
Ciass C%,
K)
called
vve
'.ne
can vvnte
K
convex then an apDii cation
vi
iicif/^ivjl
m%
[6]
of the General"
shows that the
are sufficient for
!=
zed
e:<istence and
R"
Lip in
The Lipscn;;-
ii^f(>i,ii)/dv||.
The results
%
The
can be defined as
L
sup
.=
Jl"
;r; J?,",
R G
Derivative Mean-Value Theorem
joundedness
Lip In
[--h'X^ re'E.pHrt :c £;
of the Picard-Lindelof Thecrerri
Rr^+i
:<,;)1
0, b
,
all (x v
)
in
vvhich is defined as foliows:
a
i
can be deduced for
i!v
- v^ll
<
b'
But vve will only be interested
;
in the
application of the
Picard-Lindelof Theorem when applied to an antisyrnmietnc interval on the
j<-axis: that is, the
deduced for
results of applying the Picard-Lindelof Theorem will be
all (x,v) in a
region
% c
[xQ,Xfjf,]
x R",
which
is
defined as
follows:
R: (xo,Xo+a]
where
M
.=
ilv
a > 0, b > 0. if
sup
llfJL
f.
-yo'l
is of
^
^
class
r'3(Jl),
then lis bounded there. Let
11
and
€
.:
rmnia.lj/h)
vv'e dfi?
novv prepared to state the Ficard-Lindel.:!' theorern fGr an
antiiyrnrnetncal interval [xn^Xn+e]
Theorem
Suppose
2.3:
le
approKimations
$q(XJ
=
Picard-Lindelof Theorem
Cr'^iip) on
Jl
Conoidsrthe ccntinucus successive
^ which exist on
[xr,,Xo-*-e],
and are oefined as
(2.6^
V£),
*k+1^^^)=
'/£
*
r9 71
f'V3,lk('5))dS.
for
r.;
k
s
[01
-.4.-1
'^-^^
N.
The successive approxinnations
^ converge uniformly on
unique solution $, such that ^Cxq)
Thus
if
le
=
[xQ,Xo+e] to the
v^.D
ro(Jl) and satisfies a Lipschitz condition with respect to v on
the Picard-Lindelof
solution to (2.1)
Theorem asserts
that there exists a unique local
'R,
Derivation of Constraint Algorithnri
3.
iDur
wdu
constraint t.heoru
f
I'BQ u 3 r s y s 1 1?
1
s of di
'n
f
?'
dh .spplieu tj variable ano constant coHff^:;^nt
? p n t -5
,•
i
a g t- b r a c ^ a u a t o n s
i
i
;
.
h o vv e v r
i?
,
vv
e vv 1
1
Drespnt cur ccnstraint thporu for rpctanquiar noniinpar sust^rns of the
on y
i
]
•'orrn
But for a derivation of our algorithm applied to constant and variable
(.2.1.)
coefficient systems, see
Suppose we consider
c
set Dq
R-ri+i
,
a
[4].
nonlinear function
and takes on values
F_,
V'/hich is
in Ri^- that is, a
defined on an open
function of the form
such that
and
m
>
n.
The open set Dn contains the convex set
D
.= [.:<n.x^^,]
X Rn X
R^'.
Now suppose we temporarily assume
1^
that there exists a
smooth function.
such that
and the following is true:
F(x.i^(x),z(x)) = 0,
subiect to the initial conditions
(.5
2)
.
,- .-.
i
*
m
t
^ n.
y/e are interested in nonlinear
systems
of the
form
(3 2),
such that one
of the lollovv'ing three situations is true:
rn = n.
(i)
on ti
c
such
D,
:=
vi.f
c
Rri
m
> n,
(lii)
m
>
n.
n.
IdF/.?z] is
rank-deficient and of constant ranK
th.^t
X
vv,,.
(n)
rank on
and the Jacobian
Wv
IS a
A
'rt'r,
neighborhood of
v.-,
and
VV,
c
R^ is a neighborhood of
ana the Jacobian IdF/.izj is of maximal rank on n.
and the Jacobian [dF/'^zJ
i^
rank-deficient and of constant
Zq.
Definition of Constraints
3.1
The following definiticn
Definition
is
made
for notational convenience.
The zeroth order constraints for the system
3. 1:
(3 2)
are defined as
FtOkxxZ) =F(xx2)
= 0.
(3,3)
cuch that
(xxD
e D,
and
\
n
f
fprAOtiai'inn
dPfoVdx
=
'=:V'=;'''='m
[aFioi/azJr
The system
(3.4),
*
("^ '^)
W^t^h rP'^nert- fn v N/ipldQ
[dPtoi/aj^z
*
[dPtoi/ax] =
along with the substitution z
o.
= ^',
(34)
can be written as
(3.5)
[0]
a
aF
[0]
az
such that
fttoi :=
a'o'(xxz)
=
-([aEioi/ayJz
*
[aF(oi/ax]).
(3.6!
we Win aenote
rank and nuility of [dFloi/dzj on
tr^e
n as the constants
/?([dF:o;/a2j)= R2.0
and
//([aF!o]/az]) = No.o = ^-^2,0
respectively
•
Also, let us denote the nullity of [dROi/aTjT on
n as the
constant
A/(iaFfOI/azJ^) = N2,o =
where m[0]
:=
rri[0] -
m.
Because the matrix [dFj^l^dzJ^
N2,o
on
>
R2,o,
n. The matrix
is
rank-oef icient on
n and/or m
>
n,
[aFto'/azJ^ has N2,o linearly independent null
vectors of the form
=
[vo/'UxxZ).
.
,
Vo.m('HxxZ)F.
(3.7)
cweh that
[y^u)]T[^Ft01/a2] = 0^,
for
1
<
<
1
N2.0, and (x,y,z) e
Definition 3.2:
system
(3 8)
n.D
The f/rst order constraints
(3 2) are defined as
Fm*,:=Fm*,(xxz):=[V'^ra[01 =0,
f
or
1
i
of the nonlinear
)
i
N2.0
,
and (x^Z) e
tt.
(3.9)
D
VVh rn]] lyt £'*'
Definition 3.3:
uenote the overdeterrrnriHij s^HtHm of
ijiffprential/dlgpbrau: equations which oonsists of the zeroth order
constraints (3.3), and the first ofinr constraints
overdetermined system
o:
(3.9): that is,
an
the form
(I
i.'>
such that
and
mil]
:=
N2o.n
rnlO] +
To further Ciassify our
first order constraints,
we
vYill
need
to define
the following Jacobian matrices:
ji '^
= J1
dvj
(3.
J2,0
- J2.,0(^ ^^2) := li^FfOj/dZJ
(3.
i
1
&)
(3.
1
1
c)
^:<:ti)
=
[i^Fj-'l'
1
1
a)
and
J3,0 3 J3,0(v^v^z)
for
(xil) s
n
We
=
[.J1
.0!J2,0)
will also
assume tnat
Ri
,o
•=
mJ^'°) and Rzfi
.-
aTj^.o)
are constants on ti.
We
will
now define
nonlinear system
of
(3.2).
the three types of first order constraints of the
Because the matrices defined
in (3.1 1) are
functions
(xxz), our definitions of the three types of first order constraints will
be pointwise: that
is,
will be for each h<:i^z) e
n.
First Order Consirdints of Type
Definition 34:
It
«!iFm+i
/az.is not
constraint of type
v:\
the
dFm+1 idz}^
row space
ipace
d*'
J^-'^
,
Fm+i
v-.
j :'ir;t
First Order Constraints of Type
-n
the
order
row ipacs
of
J-'*^,
B
uut [oFrn+i /^vj^Fm-M /cizlis not in
of ^^'^ Fm+i is a first order constraint of type B
,
Definition 3.6:
if
.-c.vv
AD
Definition 3.5;
if
Vne
A
First Order Constraints of Type
i'3Fm+i /fX'kiFrn+i I'h-A i^ in
constraint of type
the
row soace
of
^'''^
•
n
C
Fm+i is a
nrjt
'jT'LcV
CD
Let there be
first order constraints of type A,
of type u.
and
of type C, for all
(3.1) IS
such that
Although
\:a}j_,z)
e
n
n.y^^'^, n^^^^_.
we have ^\\'[iX
We
and
will
assume that the function defined Dy
/?,tciare
constants on
order constraints of type L(for L
they nnay not be linearly independent. Also,
constraints of type L
may be
n. with
some
=
A,B) on
of the first order
linearly dependent on the zeroth order
n
rorrr.t.raint.i.
Thyn?iorc'. int triHre be rii^i
order cunstraints of type L on J\
zeroth order constraints, fort
number
of linearly
=
•'I'J-iii
nearly indepenijent fir;!
.vTnch are ai?o linearly mdepenijenl of
A and
B^
Also, let
,•?)•'-
1
i/:ii'-i
independent first order constraints of type
t.ne
denote the
C.
These
first order constraints are such that
.
and
n^
+
n^
lit
n,
i
+
f!^
^
w.
(L = A,E;:) are
constants on n.
Vve vvlll not discard any of our redundant or linearly
constraints because our classification
difficult to numiencally detect
\\:,\i^'^
is
dependent first ordei
pointwise. Decuase
it
may
dependence, and because there
notational advantage in perservinq the size of
f.^iJ.
be
is a
i^Jow y^/8 will
dsiine higher of-dsr cnnstrsmts,
constraint theory
,2ncl
tor;
•J
|.<
i
1
J
1
1
:
,
n ; ij
Consider tne overdetermmed system
to the mitia) conditions
?f.j|?ipct
=
^^^'^o^
iii:..
uch that
'i
3.
1
3-3)
(3.13b)
No
No
„ :=
p
1
<
s
<
n,
6 {0}
<
r<
2,)
;=
m[/?]-R2^,
=
1
- IX, V
U
M,
ml/?).
NiS-fi
)
= n -
Ro ^
I
m
for
i.7
=
:f
>
r
r
''lV
-
]
"l
m[^-ij
+
N2^-i
for
we
Also, for noldt.ional convenience,
define the followlnqJacotiidn
matr'x:
-'
1
-
"^
1
s
i
'^
'-'
'
'
...i_'.s.'
-
1
''-'rj;
-
I
'-i
"^
-'
1
^
V-'.
1,
I
•-L/
n..
•and
(x.V
We
Z^'
e
D.
n]]] .alio aviurne that
R,^. ^i^^'h^^f^z^^ '^z^> ^--^ ^re constants on
n
V'/e
see that the overdeternriined system (3.12) is obtained fronn the
-
zeroth to (/?-l)3t order constraints
;7th order constraints F^[^.,]+,
system (3.12) with respect
dF^ 1/dx
:=
.J2^
z' +
J'
^
(I
i
i
that
<
is, fronn
Ft-^-H - and
from the
^2^). Differentiating the nonlinear
to x yields
Z + aFU^ l/ax =
(3
1
4)
The system
(3.14), i3long wit.n the .-.ubstitution z
^
v
,
can De written as
i
=
!,.>]
i'7
i=;i
o
ic,
that
;.uch
if [J^'..^F is
rank-deficient and/or mix?]
>
n on
n, [J^^F
has N-.
,
>
linearly independent null vectors of the form
1
"7
I
for
i
1
i
1
^2^,
i?
::
f
and (xxi) « ** The first ^2^-1 null basis vectors
1,
in
(3.17) can be chosen to be isomorphic to the vectors
(3.18)
for
1
1
i
J
^2^,-1
[yjW]Tj2^
for
i
1
N-.
i
1
the following
,
jp
>
1
,
and
(x,v,2,^
e tt
By definition of
(3.
1
7), V'/e
; qT,
,,
(3 19)
and (x,y,z) « n. From systems (3 15) and (3.19)
,
N2^
nave
we
obtain
expressions:
i'"^
for
1
<
1
<
Now we
J^2Jf
,
"Of
and (xxi) « **
are prepared to pointwise define the three types of higher order
constraints for rectangular nonlinear systems of differential/algebraic
Hquatiins of thH form (3
2):
tnat
(//-^n^t orcer constraint:; of typps A, S,
]i.
and C for (3.2), such that ^:<:tz) ^
^
Definition 3.7: {p* I)st Order Constraints of Type
if
not in the ror/ jpacd
jFr^f^^ ]+,/ jz. li
constraint of
^'^rnti']+!''^Iis
not in the
i+, i:.
a (^ +
(p+ })st Order Constraints of Type
ifi
row space
*''S '"''vv
of J-v^
space
F^^f,, 1+,
of J2y?, but [JF^f_^i+,/^y.l
(x' +
is a
If (aFr„(_^]+,/3y.!
dFn^(^]+,/3zj is in the
order constraint of type C
B
dF^,.,^^
Dst order constraint
Definition 3.9: (p* })st Order Constraints of Type
{p^'-Vn'':
Dit order
typii a.H!
Definition 3.8;
'1"
oi J-^v^, F^.r,,
A
row space
]+,/3zJ is
of type
3Z
C
of J^^, Fn,(^]+, is a
iH
Let there be
U7 + i)st order constraints of type A,
of type B, and
of type C, for all
(xxz) e
n
V/e will
(3.1) is such that >7^+i^*l '^p^\^^\. ^^d
assume that tne function defined oy
-'/p^]^^^'
are constants on
n,
vvith
Thsre
rTidij
e:-:i':.t
constrairit:. of tupn
'mpar deDHrnJence amongit 'he
v/^ +
l
some
,:ri
n.
L
for
L may
order constraints of type
=
A and
B.
Also,
)st ordt'r
1
be linearly dependent on
of tne (/"+!
some
through /'th order constraints. Therefore, vve nave ru+,^
independent
(/7 +
1)st order constraints of type
L
i
)i-.
of tne zero'.r;
''i^+i^ linearly
on tl, which
also
ctrf;
linearly independent of the 2eroth through /'th order constraint;, such :nai
L
=
(>
A, B.
+
Also, let
n.,+,['^l
denote the number of linearly independent
Dst order constraints
of type C on 9\
These linearly independent
+
(/7
i
)st
order constraints are such tnat
n,,r^Uu^^^{B].n,,,l^Un2^,
and n^^+/iJ (L
if /7ji^ > 0,
U n,IU
and
+
vvill
is
constant on n. By construction, for
<
it
=
A and
sj^,
then
n^^
=
on
n.
not discard any of our redundant or linearly dependent
Dst order constraints because our classification
because
L
we have
if //Ji^ - 0,
We
(/":•
- A, B,
(3.21b)
may
because there
is
pointwise,
be difficult to numerically detect linear dependence, and
is a notational
advantage
in
preserving the size of
F.U''+1j.
B,
24
Termination of Recursive Process
4.
Our recursive
croce.r.j -
inspections of row spaces
are of type
C
for
('a;i,z)
Dnrridnly involvinq diii'erentidtjnns and
-
terminates
n. The proof
s
if all
the u'
of tne
f
+
Dst order constraints
olloning
lemma
verifies that
our recursive process terminates at some finite stage.
Lemma
4.1:
There exists
a
nonnegative integer
constraints are of type C on
exists a nonnegative integer
nonnegative integer
Proof of
Because
R?^
^
Lemma
..i^v?
w
k,
;= Vv.
;<
k.
p/'
such that
such that R?.k=
is
such that
x
p"
,
P>jk =
all
the (k +
)st
1
order
or equivalently, there
'-'j.k+i
i^3,k+' then
cm W.
Also,
if
the
k > kD
4.1:
has only 2n columns,
^3^-' is
bounded aoove by
2n: that is.
^ 2n
Information about the number of type A and B constraints on
utliiized to express
Ri^ ,Rz,u*
W
can be
as follows:
*^^
Z'/'i
(4 )a)
I
1=1
Because
ni^'^l
and
ni^^l
are nonnegative sequences,
we
can conclude from
(41a) that
R3^
<
(4. lb)
R3/>+i,
we have reached
a
stage
constraints are of type
C.
that
unless
in
is,
our recursive process where
all
the
irom above by
Sirica r'?/-*! IS bouncJed
nonneqative intPQer
k,
;ijch tnat f,= k.
5y DOinVvVise invoMng Lprnrna
some stage
IstTninatt'S at
determine
i'=
k.
4.1
,
2n, vve
know
that Ihers Hxiit
,3
and
we
knovv
t.hat
our rHCursive Df"0Ci?ss
When our process terminates
the initial conditions are consistent, and also
11
j
vve mi.jst
must
c;';ecK
for
one of the following four situations on M:
(1)
andR3k-^2,k<'^2..k•
R2,k = n,
R2
(il)
n,
<
)<.
(iii)
R2
(iv)
Ryy
k
=
<
'"'
n,
Situations
and R?
'3fid
- i^2.k ^ 1^2
R^jr - R2,k =
and R^
(i)
j,
and
-
R2
i^
(ii),
,k-
1^2 k-
k = R2.k-
which will be referred
categorizations, are the most likely ones to occur.
categorizations occur,
we
to as our
If
primary
our primary
will have utilized our constraint theory to
transform (3 2) to an explicit ordinary differential equations
problem. Situations
(lii)
and
(iv),
which will be referred
categorizations, are desirable, but special cases
categorizations occur,
transform (3.2) to
We
a
we
If
to as our secondary
these secondary
will have utilized our constraint theory to
nonlinear algebraic system.
will utilize the necessary conditions for a solution to
form (3
2),
initial value
systems
which were obtained via our nonlinear constraint analysis,
of the
to
cerive iuflirient conijitions lor j
coriflraint .anal
is, £^1^]
i!3
si? as to
as defined in (3
how wy snouid
12.)
12) IS well-defined on
:.oiutiori to \Z l)
define
We have assumed
n. This assumption
ij^l
that
is
'rirt
.varf'
ieo oy cur
on a traiectury
:.ne fjnr.i.Min
;n
L that
defined oy
exp'icitiy stated in the
following definition:
Definition
M.
if
4.1:
The function
F'''^^,
defined
in
i.3.
12), is
well-defined
on
the following is true:
(i)
(ii)
For
= k,
>.7
For/'
= k,
the vector
For
L
-
A, B, C and
tt,
<
p
Let us extend the definition of
domain Dq
c
Fji<](x,v.2).
x.
the Jacobian matrices defined in (3.13) are smooth, and of
constant rank and nullity on
(iii)
a smiooth function of
[^F^i^l/.^x] is
R^n+i.
mat
is, let
^
k +
Fl''-
,
1
n^^
is a constant,
for arbitrary
x.
%
on
and
z
n
on the open
us consider a function of the form
(4-2)
such that
and mik]
in (3. 13)
>
n.
The corresponding Jacobian matrices are as previously defined
4.1
Termination
Maximal Rank Differential
of Process:
Equations Case
rirsT vvP
wni ch
-'
k
cur"
vvili
ijiscusi nrin!iri9ar systarns
rscuriive procesi terminates with
~ ^-2
k
-2
^'""1
k
^
''''''t'
'''iii
initial conditions, vve also
smoothness properties
that
'See
nave R2
^ = n,
we
hold, then
11
'orm
of the
in
9.2
^^
(3.2) and (3. '2). ''or
- n ariu
addition to having consistent
R3 j^ - R2
k
'
'^2,k..
be able to obtain
vviii
certain
^'"''^
a
unique
'.ocai
solution to nonlinear syst.?ms of differential/algebraic liquations of the
formi (3 2).
The criteria, which give us conditions for the existence and
uniQueness of
a local
solution to rectangular nonlinear systems of
differential/algebraic equations of the form (3.2), are stated
in
the ne^t
two theorems.
Theorem
Let
W„ c
respectively
4.1:
i:<Q,Xvi„]
and V/^
D.
Rf^
Suppose there exists
R2.k = ^[^'^^^o,)b'Zo^^ -
(xo,^,2o) G
c
f"'
L^^
^
=
Suppose the function
well-defined function
Ft'<l,
be neighborhoods of Xn and
a Zq s
'''':<
L
'^
w.
'^i'^
c
R",
'-
Vj^,
such that
^^^^
neighborhood of
'^'^'l
defined by (3.1), and the corresponding
defined by (4.2), exist on
n and are of class
ji'J
Then
Zq can be un'.que'wj i'.'.pfiHiC
Proof of Theorem 4
lerrr.i of Xo
and
;ifj
G
1
C:.nsMjrr the follov'/inq bai; abuut
We claim
ir":
Zj-,
of raaiuj p, ruch that p
>
o that
i;
tnat Ff^'CxQ^i^j.z) s 0, for z s B(Zri,p) - \Zq}, or equival^ntly,
i!Ft^'(Xo.^.z)ll
>
0. for z ^ P(Zo.p) - iZol
Utilizing the fact that the function
Taylor's expansion of
L-^-J(xri..Vch^
is of
F^^'J
class r^lrt), and taking the
about the point
(xq^Vj^.z^i)
yields
Et^'(xo.^..z) = F[i<Kxo.Vo.=o3 * [3F5^Kxo.yo.=o)/^lI(i-io^ ' 0(||z-ZolF).
Taking norms of the previous expansion, utilizing the reverse triangle
inequality, and utilizing the definitions of the order syrnbol and singular
values, yield?
;l[aF[k](xo.,iJ3.,Zo)/^zi(z-Zo;ii - Gdiz-Zoii^)
!!£^'^Kxo.,Vo.z)|i ,
^
i|[aF[M(xo,xo.Zo)/az](z-Zo
2
sJIz-Zoll - c||z-Zol|2
-il
- c||z-Zoi!2
= llz-Zol|{s, * c(-liz-zcil)}
V'/e
K
know
that
!|z-Zul! >
* c(-llzrZoll)}
Thus,
>
if !|z-Zoll
we have proven
iuch that.
for z s
that
<
5(z,j,,p) -
{Zj:,}.
s^/c: that is, if p
iiE^''^(xQ,i!:o,z)|| >
We
<;
also
know
that
s^^/c.
0, for all
ze
3 r-
^"i
'-
^
/.-'I
is the smi3i lest E.ingu'.ar
••//here $„ >
defined via the Taylor's evpansion of
Thus,
if
the hypothesis of
value of
V^l}^My.rj,)i^j,ZQ)/dZj,
and
c
>
i5
E^^^i'<o.';lo-^' ^^^^'-^^^ '''Vi.'&'iij'' i^
Theorem
4.
i
is satisfied,
vve can
prove that a
'ocaliy unique set of consistent initial conditions for the initial value
proDlem (3 2) exists. We will now obtain conditions for the existence
of a
unique local solution to system (3.2) by integrating the system (3 15),
subject to the initial conditions
Let us define £:W.,
i
:=
—
^
V/,j_x
W^as
follows:
(4.4)
[^,z7r.
By definition of our (k+ Dst order constraints (that
is,
(3.20) with
and the Fredholrn Alternative Theorem, v/e can express z
in the
/?
= k),
system
(3.15) as a function of x and r. More explicitly, because certain consistency
conditions are satisfied at each
(x,v.,z)
constraints (3.20) hold at each {x;iz) e
Theorem can be invoked
system (3.15)
Thus,
we
enn,
that
is,
the (k+1)st order
the Fredholrn Alternative
to prove the existence of a solution to the innplicit
at each (x,ii^2) e
n.
can rewrite the implicit system (3.15) as the follov-/ing
system
subiect to the initial conditions
exDl'Mii'
Vi ill ±jj
•'
- ••'•0'i---'0- ±.---ij-
such that
rr,.
11..,
11;,
,\
•aPij
h
is
n
-^
'i
defined as
2,k
,[^:]
[J^'^(xxi)
The results
initial value
P c n,
of the following existence and
problem
which
is
(4.5) will be
deduced for
uniqueness theorem for the
all
(;<,
z)
in a region
defined as follows;
such that
\a/
- fv.
.
W^
Where
:=
a
>
v.+f»1
UlU-ZoiUbl,
0, b
0, W^- c W^,
>
•
and
W^ c w^^x
w^.
!f
h is of
class
CHP)
then h is bounded there. Let
;
/V;=
and
sup
llhIL
i
^-f.
::•
I
t-
= rnirua, ib///)).
Theorem
14
3).
4.2:
Suppose the
hypothes):-: of Theorprn
v^lue Drcbiem (4.5) is such that
He
4
i
Suppose fne
j? x.st.isiied
(r-. Hd)
m
initial
the LiDSch'tz
:7, rn\'r\
constant
L
sup
:=
II
t(xx:)
!4Qi
II,.
(x.j:)6i>
where
tCxiZj
;=
r
-jj
I.I
for all (y..t'V «
"i"hen,
',X,V.,Z,ii
J
i.
V
'.t
7
i
>*
there exists a unique solution
$.:= ^(X),
of (4.5),
-k
X
s (Xq, Xy + e^l
(4.10)
sucn that ^(xq) = Jq
'^l^'^. ^-^e
unique solution to the nonlinear
sustern (3.2) is the first n components of ^: that
**:=**(x):=[*i(x),
Proof of Theorem
..
.,
*n(.x)F.D
4.2:
Invoke the Picard-Lindelof Theorem.
is,
the solution to (3 2) is
(4.11)
Termination
4.2
Rank-Deficient Differential
of Process:
Equations Case
Next
(,-
ncnunear systems
will discLiss
cur recursive process tarrni nates
vv'h^ch
Rt
we
R2
[:
;
r<2
'^''''
k
^-
'''^
initial conditions for tne
•'''''
problem
smoothness properties hold on
existence of
a
-'^^^ ^-"'^^
wun
'•"
(3.2),
we
we
Jt, then
of the
'''
P;
j,
form
<
n,
i,Z-2)
and [Z.]2) for
and
^dd'ti jn to fiavlrg consistert
also have R2
1,
<
n.
and certain
will only le atle to prove
solution to (3.2). This particular situation is summarized
in
two theorems.
the following
Theorenrj 4.3;
Let
c
Vv'.,
[xQ.XfmJ snd
respectively.
rfklr..
j_^
r =
of
..
^
c
'1
-
be neignnorhoods of Xq and Vq,
a Zn s R^,
such that
n
^
i''4
'^
k ;= ^^[J^''^(:<o.&.iO''J < ^-
(xo,yo.Zj-j).
R""'
Suppose there exists
•'-•••O'ZiO'iJD'
R2
'fi,^
^^^ ** =
Suppose the function
well-defined function
£t'<l,
L
'^'x
^
'^^'i
^
'^'r
'^
^'"'"'^
"-
D be a neighbornood
defined by (3.1), and the corresponding
defined by (4.2). exist on
n and are of class
run).
Without loss of generality, we will assume locally that
2,3
can be uniquely expressed
components
of Tq.
terms
More explicitly,
z^Q^^irir components
and ^-r
in
of Xq, '^,
components
of
and the remaining n-r
let us relabel the
of z^) can be uniquely
r
components
expressed
^ R""^ (remaining n-r components of Zo).D
in
terms
of
z.
of Xq,
Then
'Jfj,
Proof of Theorem 4.3
Let u? relabel the
system
i3 12)
with
= k
/?
as follovv's:
1
-7^
iuch that
^'^^'(xq, abj, 2p-r,o,
is of
rank
Then
uy
t
c
f^M.K, jt 5,-..
Thus
if
^
and ^tf-'I/a?
h e implicit F u n c 1 1 o n
5- n, and a unique
al; (x,
=
-
evaluated at
R2n-r+i
fundi on
5t 5i-r) ^
(v^^
r,f
;^:'J
iJ
value problem (3
2).
we
j|^^ 5i-r,o).
V,
k n o vv
t
h at
there
;i^is
Theorem
of class
R",
i
t
s a
c
R''
of
we can
obtain an
R-n+i of consistent initial conditions for the initial
Now we
will obtain conditions for the existence of a
order constraints ((3.20) with
the implicit
ex s
CVD
4.3 is satisfied
subject to any point on the surface
^
r,)
(4.14)
inteyratmo the system
r.
^7
= k) hold,
-
that
system
(3.15).
such that g
Suppose we can choose
is in the null
space of
is,
the
the Fredholrn
Alternative Theorem can be invoked to prove the existence of
n—
^
such that
Because certain consistency conditions are satisfied
3.:
,-,,
and a neighborhooa v
local solution to the initial value problemi (3.2) by
(k + l)st
jp. ^.,.
^
Furthermore,
the hypothesis of
c
T h e o rem,
-^
SL ^-r)) =
iZ^x,
entire surface r
(3. i5),
;:<,-,,
r.
neighuurhood U
for
^,q)
a
J^'^ on
a
solution to
smooth function
n.
and the initial
conditions on r are consistent for the following explicit system:
u
-
;k,-,
^
(4 15;
f
2 W
:v.v,zjj
cv^^K.^ZJ ^g
where
g:
n^
Rr',
z: \l, -^ R-^.
and
h:
The
n^
R2n.
local results of the foiiowing existence
problem (4.15) will be deduced for
defined bu (4.7), (4.3), and
Theorem
(x,^) in a region
D
--
initial value
n, which
is
(4.9).
4.4:
Suppose the hypothesis
value problem (4.15)
constant
all
theorem for the
is
of
theorem
such that
he
4.3 is satisfied.
{c^. Lip)
\k\
D
,
Suppose the
initial
with the Lipschitz
(4.9).
Then, there exists a unique solution
$1 ;=
$a(x), X e [Xo, Xo+£^]
of (4.15), such that $a(xo) = Tq.
unique once
a
(4.
The notation
^i denotes
smooth g has been chosen for which the
1
1
•
that the solution is
initial conditions
h\'\i
CGrri-i starit.
AiS.o a i-jluticin to thn
comDonents
of $i: that, is, a
$»
:=
$*(x)
:=
[$a,
(;;),
,
.
nonlinear syst.jm (3 2)
nonunique i.olutinn
,
$9„(x)r.G
Proof of Theorem 4.4
invoke the Picard-Lindelof Theorem.
QiVc-':
'hp
t'r::t
to (3,2) is
'
(4
4.3 Termination of Process: Algebraic
Now
move on
Systems Case
thai vvp have discussed our primary cateqonzations,
to our
iHcondary categorizations-
It
'vve
wish
to
can be point vvise proven, that
the Jacotian rriatrix J^>(x,v^i can be transformed, via Gaussian elimination
with pivotinq, to
1
3
1
a block
matrix
of the
form
(
.k
2.k.
,k
{^:ll>
'"'\'lLii.'
such that
n^ Rm(k]-rxn
J^M, J2M
and
3'>'
has rank
r
exi silence, and in
on
VV..
x
The
'tl-^.
criteria, v/hich gives us conditions for the
some instances,
the uniqueness of a local solution to
rectangular nonlinear systems of differential/algebraic equations of the
form
(3.2), for
Theorem
Let W><
c
which (4 16) holds, are stated
in the
following theorem.
4.5:
[xo,Xf„] and
W^ c
R" be neighborhoods of Xq and
respectively. Suppose there exists a Zq e Wj
^^'"dO'Zo^ e
n
:=
f:f'^'(xo,i!o,2o) = 0,
W^
X
Wy
X
Wj
C
c
j/o,
R", such that
D.
(4 19)
and
r
Fs
= R^^. -
Suppose the lunct'iun
(,
correspondinq vvell-deiinHi] lunction
are of class
F^
F-'-^l,
defined ty (31), jnd tne
defined Dy (43), exist cm M. and
Pin).
The system
(3.12).
with
/•
= k,
can De transformed to the foiiOvvinq
decoupled sustem:
,*-''[t''](y
z)
\!
A
-
Ofi
..I
1
m[k]-r.
,^
l'
Sr-
'..'
7
'l
sucn tnat
(xxZJ' ^
/rr.[ki-r-
Then
^
^
-* Rm[kl-r
at each
(x,;£.,z)
s M-,
we
can solve the first r equations of the
nonlinear system (4 20) for r components of v
in
terms
of x
D
Proof of Theorem 4.5:
Suppose
open set
in
('<,an-r..Slr)
=
vve relabel the
components
R"»+V Suppose
we extend
^
^hat
l!'(x,an-r,Ur),
and
IS, let
of
%
Let
U
:=
W^
x V/^,_r x
w^
be an
the definition of ;^ for arbitrary
us consider a function of the form
(4.21)
rank
is 01
r.
Then by the implicit
neighborhood
of Mn-r
Vv'i^-
I'vv'x-x
'i''
F...n::t;.:n
Vv'i^-r]'-
''"'i*-,
Jf'"^
d
Tneorem
{'vVv;:-. Vv'i^-f',
^-inl
vve knovv
of ixn,
mat
there e^^isti a
iirr-r,oX. ^''"^
^i.h f..,nC.tiOn 0;-. [Wv-
:<
3
Vf|^-rJ "^
neiQhborhood
'-'•''-''
''''^'V'
U223)
ir(".ar-r,.^'^'<..yn-r)) = CL
for
all (:i,\ir,-/}
A
I
s
.
^-^^
e (W.^ X WV-r).
b e c a u s e vv e c a n o n y u n q u e y o b t a n r c o rn p o n e n t s o f y
i
i
and the remaining n-r components of
i
i
v..
i
n
e fn'> j o f
the remaining m[kl-r equations of
(4.20) are functionally dependent on the first r equations of (4.20)
the function J2r algo saUsf i£« these. ei^uation$.' the.t
f''^^'\}<.Xi^.r.^iXAin-r>.2) = ClD
t
15,
Thus,
we &lso> have,
(422b)
5
Illustration of Nonlinear Constraint Theory (Rigid
Pendulunrj Problem)
Suppose
a
(yr,t).ij2(ty'.
pendulum swings
in a
plane with rectanqular coordinates
and corresponding velocities (y3(t),y4(t))
L:= (U3^ + U4'^''/2 lor a unit rnass,
while the external force
unit gravitational force in the negative U2 direction
algebraic equation is c
:=
The Lagrangian
y^^ + y2^ -1 =
if
is
1=
[0.
-Ir for
the rnass is a unit distance from
Also, the authors in [2] interpret the term -2y5(t) to be the
tension
the pendulum rod
represented by
a
Therefore, the rigid pendulum problem can be
decoupled system of differential/algebraic equations of
the form (3.2). That
is.
yry3
y2-y4
Fi(s<,i(x)..5i'(x))
F2(x,;i(x),ii'(x))
FCx^CxjJiCx))
:=
F3(x,li(x),ii'(x))
y3*2yiy5
F4(x,v(??),v (?<:))
F5(xi(x)y(x))
y4*2y2y52
y i*y
The first order constraint Tor system (5.0
F6:=-2v5(i)(y,y',
Forsake
is:
+ y2y'2) = 0.
of simplicity
we
a
The constraint or
the pivot.
in
is
will let 275^') = -1.
2
2-
LL.
(5 1)
41.J
EHC
3 u ': 9
ciF^,/3Z = [y,
is in the
1^2
row space
i
1
1;
'J
J
of
u
and
{dFfc/ayUF^/azjr
IS not in the
row space
iy,
y
Olyi
y2
0]
of
-1
'3,0-
n
2y5
n
_2y,
2yi
'!
I.
•il5
2y2
n
F^ IS a first order constraint of type B, for all
Now we
can formf.^'^
(xxz) s n.
41
y, " y?
y;-y4
—
*
1
i-.i
J
r
y4*-y2y5-
'
Fa
yi * y2
yiy, ^ y2y2
Differentiating the nonlinear system (5.2) with respect to
the substitution z
J
1
- ll.
yields
a
system
of the
form
x,
(3.i5), for
and utilizing
a:
:.uch that
1
J
J1.1
,0
43
For the complptt' demonst nation of the application of our constraint
theory
in
order to prove the existence and uniqueness of the solution to this
problem, see
s y s t e rr\ o f
t
[4].
he
:
Successive differentiations with respect to
o rrr)
(31 5 )
dF'^/i?:
•iFiLi/a;
3Fi ]/di
3F12/.1;
,
f
or
x, etc.
yield a
1
J"
IJ
u
y't
•-yi
y 3
"-y
1
j-2y5y i-4yiy^5
-2y5y
45
_7
V
'
J
- ,^'4
,
.^y
which are such that
-,
u u -
^y, ^y^ u U
(3 15) holds, for
//
=
i
oj,
3 and
1
:.
i
^
7
Thu:,
wh
.:.!:it.a;ri
tne-
following fourth order constraints;
F,3
:=
[:i3(1)]W3l = Fy,
F,4:=[V3(2)]Ta[3 =
'15.=
F,6
iV^'-'j'al-^
= ly3W]W3
Fe..
= 2Ffl,
=
F 12'
F,7:=iV3^5)]T^r3 = 2F8.
F,o;=iv.C6)]TcJ3
F,o,
and
Thus we see that we have seven nontnvial fourth order constraints, which
are
all of
type
C.
Therefore, our recursive process, applied to the system of differential/
algebraic equations (5.1), representing the rigid pendulum problem, has
termnnated with
-3 3 "
we
1^2
all
type C fourth order constraints, R2
3 = 2. ^'or all {)<;i,2) in a
can invoke Theorems
solution to (5.1).
4.1
neighborhood
n of
3 = n r 5,
and
a point (xq^Vo.Zj].)
Thus
and 4.2 to prove the existence of a unique local
47
6.
Extension of Local Results to Semi-Local Results
The Picard-L'ridelnf Theorem 'or tne
approximations)
is
'"oetnod of
successive
used to investigate conditions under whicn an
initial
valued ordinary differential eouation has a solution and that that solution
unique,
in
the previous sections of this article,
we extended
that idea and
optained necessary and also sufficient conditions for the existence, and
some instances
is
m
the uniaueness of a local solution to nonlinear systems of
differential/aiqebraic equations.
Now
local results to serni-local results.
r/e will
We
local results to semi-local results for
will
discuss the extension of our
eschew the extension
systems
of our
of differential/algebraic
equations with discontinuities and/or turning points, because these systems
require that certain problem-dependent nnatching conditions be supplied.
We
utilized our constraint theory to prove the existence, and in
some
cases, the uniqueness of linear and nonlinear systems. of differential/
algebraic equations via transforming them to explicit systems of
differential equations of the form
subject to the initial conditions
such that
X e W,,
'//here
anu
Zo..
c
'/-/^
[xn;i^,rii,
Wv c
and Vv^ z
Rr',
are ne^qhtiorhooGi of
R"^
:-rope:t;vely.
.v,.
ij-y.
'
Recall that our existence and uniqueness results tor the initial va'iue
problem
defined
deduced for
(6.1) vvere
in (4.6), (4.7),
we proved
and
all (x.j;) in a
(4.8).
^
^
on
D
(:<o.-i^o^
M
<
CO
"'"'^6
,
authors
c
1)
[;<;o..:<finl
prove that
in [1]
D c n,
where
D
is
Via invoking the Picard-Lindelof Theorem,
that the initial value problem (6
exists on a finite interval [xq/x)
region
had a solution $,
Sf""^
r/hich
passes through the point
if ||hl[is
bounded by
then the following limit exists:
$(x-0)=
lim
(6.2)
*(x).
x-»x-o
Suppose that the point
(x,
$(x
- 0)) is in
D.
If
Yis
the function defined
by
Y(x)
= *(x),
Y(x)
=
then
*(x
to [xo,x).
is
- 0),
known
Yis
(6.3b)
X = X,
T is a solution to the
The function
(6 3a)
X e [xo,x)
called a
The authors
in [1]
initial value
problem
(6.1) of class
continuation or extension
C^ on
of the
[x,j,xi].
solution
discuss other extensions of the solution $.
that there exists at
most one solution through
^
if it
UQ
^I'x - 0);, thyn one can E-pAak of
IX,
[:<Q,
c
xv']
[:<o,XfinJ. ^^''f
some
on Iaq,x) exists on
say
^can
•^''
^-
-^
''"'
/.vr
continuation
general,
h IS
m
(5. 1)
bounded on
i7
- 0) =
exists.
If
(x,
nqht
X
.
lim
^(x
be such that
$
if
c
(6.1) on an interval [xq;x)
We
is a
Ix^.x^-^n],
[>::,-,,
iv^nl,
then
we can
following theorem from
in the
[1],
he C^
in a
domain
£/
c n,
and
solution to the initial valued problem
then the limit
^(x)
- 0)) IS in />
$ may
then the solution
can repeatedly apply Theorem
system
c
6.1:
Let the system
of
*
be continued (extended), or has a continuation (extension).
Theorem
^(x
^to
a continuation of a solutlDn
interval containing [xq.x)
The previous remarks are summarized
suppose
if
oi
6.1 to
be extended to the
continue a local solution to a
semi-local solution until
of differential/algebraic equations to a
the criteria that allowed us to invoke the Picard-Lindelof Theorem no longer
hold: that is, until one or
more
of the following
occur
(i)
A discontinuity
of F is
encountered on
n.
(ii)
A turning point
of F is encountered on
n:
that
encountered such that the rank of IdF/dz} varies on
is. a
ti.
point is
50
A
(iii)
Discontinuities of
(iv)
and/or
[tiF^'^l/Ciz]
A point
(v)
[;iFl''<:j/.3v).
(vi)
ijiS'iontinuitu oi
and/or
lti£f'<l/3z]
another point
(:<2,iii..Z2)
The function
The situations
1
t
and of the Jacobian matrices
he hyp
vary on ft
n.
,v, ,2,
s
e
)
does not satisfy
a
exarnp'e. a t.^pe a
type B or type C constraint at
n.
n in (6.1
hes s
f
1
a
Lipschitz condition on H.
and (vn) are the miost
in (i), (11)
1
n becomies
)
fc.r
t
h e P c a rd - L n d e
i
1
1
f
common
T h e re rn
and natural ones
c re a k 3 d
remiaining situations are consequences of our constraint analysis,
particular, the situation described in (vi). which
of the previous situations,
Definition
5.
1
motivates us
if
the
Thus we can conclude that
number
if
may
vv n
.
The
in
be caused by one or
all
to define the followinq:
A nonlinear function
order turning points
we
['?'£.-^-/3iJ.
n.
of constraints ^rif variable on
a point (xi
r vv h c h
[^F'^^:'/.ix],
encountered such that the ranks of the Jacobian matrices
IS
constraint at
f
HncountHred on Jt
are encountered on
The numiber
(vii)
is
F'^-!
of the
form
(6
1
)
has
higher
of constraints are variable on
the situations
(i)
nZ]
through (vn) do net occur.
can utilize our constraint analysis, the Picard-Lindelbf theorem, etc
.
to
continue a local solution to a system of differential/algebraic equauons to
a semi-local (and
perhaps
a global) solution.
References
[1]
Coddington, E
A.
and Levmson,
Equations, McGraw-Hiil,
Gear, C,w and Petzold,
New
Dold,
A and Eckmann,
B,,
Theory of Ordinary Differential
York, (1955).
Singular implicit Ordinary Differential
L.R,,
EQuations and Constraints,
N,,
in
Sparse Matrix Techniaues edited Dy
.
Lecture Notes
in
Matnematics, Springer
Verlag, Copenhagen, (1976), pp. 120-127.
[3]
Hanson,
A.,
Regge,
T.
and Teitelboim, C
,
Constrained Hamiltontan
Systems, Academia Nazionale Dei Lincei, Rome. (1976).
[4]
Mack,
I
M, Block Implicit Mettiods for Solving Smooth and
Discontinuous Systems of Differential/Algetiraic Equations, Doctoral
Thesis, (1986).
[5]
Marsden,
J.E.,
Elementary Classical Analysis, W.H. Freeman and
Company, San Francisco,
(
1
974).
16]
Ortega, J.M. and Rheinboldt.
W C,
Iterative Solution of Nonlinear
Equations in Several Variables, Academic Press,
New
York, (1970).
17]
Strang,
G.,
anesr A/getrd snd its AppJicstwns,
Third Edit iorf,
Harcourt, Brace, and Jovanoich, (1988)
IS]
Sudarsahn,
E.C.S.
and Mukunda,
N.,
Perspective, John Wiley and Sons,
2876
V
126
Classical Dynamics,
Inc.,
New
a Modern
York, (1974).
v-y/-//
Date Due
Lib-26-67
,IT
c^ofiO
llBf"^f"f?,
00720616
3