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Abstract

Impasse points can occur in the analysis of Differential-Algebraic-Equations (DAE's) that, for example, describe the dynamic behavior of electrical networks. Such points are usually characterized by the condition that solutions of the DAE in question cannot be continued beyond them. However, it turns out that several classes of impasse points, each of which represents different behavior of the DAE, do exist. In this paper, relations between these classes and between impasse points and pseudolimit points are examined for the first time. Sufficient conditions as well as necessary conditions for a point to be an impasse are given. Known results similar to ours are restricted to the class of forward and backward impasse points and concern continuously differentiable solutions only. We extend and generalize them; some become special cases of theorems in this paper. Other results are disproved by our counterexamples.
122
IEEE
TRANSACTIONS ON CIRCUITS AND SYSTEMS-[ FUNDAMENTAL THEORY AND APPLICATIONS,
VOL
43,
NO
2,
FEBRUARY
1996
Differential-Algebraic-Equations (DAE’s) that, for example,
points are usually characterized by the condition that solutions
of the DAE in question cannot be continued beyond them.
describe the dynamic behavior of electrical networks. Such
However, it turns out that several classes of impasse points,
each of which represents different behavior
of
the DAE, do
V,
-
ifferential-A gebraic
Equations
a
GD
VD
Gunther
ReiBig,
Student Member,
IEEE
firit time. Sufficient conditions a.3- Well as nec=ary conditions
for
a
point
to
be
an
impasse
are
given.
Known
to ours are restricted to the class
of
forward and backward
~i~, 1.
exist.
(b)
VCR
of
branch D. (c) Illustration
of
the solutions
of
system
(1)
(a) Simple network, the global equations
of
state
of
which do not
impasse points and concern continuously differentiable solutions
only. We extend and generalize them; some become special
cases of theorems in this paper. Other results are disproved
by our counterexamples.
I. INTRODUCTION
MPASSE POINTS can occur in the analysis of Differential-
Algebraic-Equations (DAE’s). In particular, one has to deal
with such points
if
the DAE in question describes the dynamic
behavior of an electrical network the global equations
of
state
of which do not exist.
As an example, consider the simple normalized circuit
shown in Fig.
1.
L
is an inductor with inductance 1 and
D
is a
tunnel diode with voltage-current-relation’ (VCR)
Go,
which
is
shown in Fig.
1
as well. The analysis of this network leads
to the following system
I,(t)
=
-vD(t)
(vD(t),iD(t))
E
GD.
(1)
The point is that a tunnel diode
is
not current-controlled.
Therefore, the analysis
of
(1) cannot be reduced to the analysis
of some ordinary differential equation (ODE).
We now observe the following:
i)
For any
p
E
GD
\
{A,
B}
there is a local solution of
(1)
passing through
p.
ii)
There is
no
solution of (1) passing through
A,
nor
through
B.
iii) There are noncontinuable solutions approaching
A
(resp.
B)
in backward (resp. forward) time direction,
“reaching”
A
(resp.
B)
at some finite time.
Because of these phenomena,
A
(resp.
B)
is
called backward
(resp. forward) impasse point in
[l],
singular source (resp.
sink) in
[2],
accessible (resp. inaccessible) impasse point in
[3],
and simply impasse point in [4].
Among other things, the presence of a forward impasse point
implies that the circuit model is defective [4]. The correspond-
ing “real-world circuit” exhibits a well-defined behavior at any
time, while the solutions of the circuit model are bounded,
but do not exist on
R+.
Therefore, in order to predict the
behavior of those “real-world circuits,” which correspond to
such defective circuit models, a number
of
techniques has been
developed:
Van der Pol was one of the first who knew
of
the importance
of parasitic reactances; concerned with the examination
of
an
oscillator, he discovered that, “in order to explain the reason
for the maintenance of oscillation.
.
.,
we found it necessary to
take into account the inductance
L
of
the wires. . .”
[5],
[6].
Closely related to augmenting the circuit model with par-
asitic reactances (see also [7]-[ll] and [4]) is the theory
of
singular perturbations, which was used to study the phenom-
enon of
Jump
behavior in [12]. Further, it allows to handle
discontinuous solutions of so-called gradient systems, a special
class of semiexplicit
DAE’s
[13].
Among the research that has been done on impasse points,
we have to mention here the work of Chua and Deng [l],
of
Venkatasubramanian et al. [14], [2], as well as
of
Rabier and
Rheinboldt
[3].
In
[l],
[14], and [2], semiexplicit DAE’s of
form
with
f:
R”
x
R”
t
R“,
g:
IW”
x
R”
+
Rk,
and
m
=
k
have
been analyzed.
consider the mapping
Manuscript received February 11, 1994, revised September 9, 1994 This
The
author
is
with the Institut fur Grundlagen der Elektotechnik und
paper
was
recommended by Associate Editor
M M.
Green
In [I,
pm
I] the authors Qscovered that
It
1s useful
to
Elektronik, Technische Universitat Dresden
Publisher Item Identifier
S
1057-7122(96)01358-X
Other terms such as
"constitutive
relation” and “driving-point characteris-
tic” are also
in
use
h:
(A,
v)
H
g(z0
+
f(X0,
Yo),
Y)
1057-7122/96$05
00
0
1996 IEEE
REIBIG: DIFFERENTIAL-ALGEBRAIC EQUATIONS
123
in order to find out whether or not the point
(ZO,
yo)
E
g-l(O)
is an impasse. Their main result is, roughly speaking, that
(20,
yo)
is an impasse point of
(2)
iff
(0,
yo)
is a limit point
of
h-l(O).
This and another criterion of
[
11
are wrong, which
we will show in Sections
IV.8,
IV.9,
111.4,
and 111.11.
Numerical simulation phenomena near singular points, i.e.,
points
(zo,
yo)
E
g-'(O)
with
&g(xo,
yo)
being not bijective
have also been investigated. In
[l,
Part
111,
the Liapunov-
Schmidt procedure is used to identify limit points of
h.
It
is further shown that, for generic
f
and
9,
any singular point
is a limit point.
In [14] the behavior of
(2)
on certain subsets of
g-l(O)
is
examined. The Singular Dynamics Hierarchy Theorem
[
141,
[2] includes a criterion for the presence of impasse points.
Under conditions on the derivatives of order less than or equal
to
2
(resp.
3)
of
g
with respect to
y,
local phase portraits
of (2) near singular points are given, called by the authors
codimension
1
(respectively,
2)
dynamics. Also, codimension
3
dynamics has been partially dealt with and stability properties
have been considered. The treatment of system
(2)
in [14] and
[2] is mainly based on singular transformations.
Rabier and Rheinboldt dealt with the more general class
of quasilinear DAEs [3], but we will report here their results
only in view of DAE
(2)
with
IC
=
m.
Similar to [14] and
[2], derivatives of order less than or equal to
2
of
g
with
respect to
y
are used in order to give sufficient conditions for
the presence of impasse points. In doing
so,
the authors make
heavy use of theorems on the dynamics of implicit differential
equations near singular points.
In this paper, we investigate autonomous as well as
nonautonomous DAE'
s.
Our
considerations include degenerate
DAE's, i.e., DAE's
(2)
with
IC
#
m,
as well as nondifferen-
tiable solutions. Contrary to other theories, e.g., [ll] and
[
151,
we do not apriori assume that
g-'(O)
is a manifold.
After basic concepts and definitions will have been ex-
plained, the existence of maximally continued solutions is
established in Section
11.
The types of impasse points which have been dealt with
in publications are forward and backward impasse points.
However, we will show in Section
Ill
that several classes of
impasse points, each
of
which represents different behavior of
the DAE, do exist; relations between these classes are given.
The relation between impasse points and pseudolimit points
is examined in Section IV, the main result of which is Theorem
IV.5. We state Corollary IV.10. which is the simplest of
various possible conlusions from Theorem IV.5. It provides a
criterion for a point to be an impasse by means of second-order
derivatives of
g.
The generality and usefulness of Theorem
IV.5. and Corollary IV.10. are demonstrated in several Exam-
ples.
11.
BASIC
CONCEPTS
AND
DEFINITIONS
We will define in this section the basic terms, which will
be frequently used in our discussion. Further, we give a result
*Unless
the
following abbreviations lead to misunderstandings, we use
CO
and
C*
instead.
with
respect
to the
considered space
TABLE
I
NOTATION
"if' in definitions, is defined
as
logical
or,
and, not, equal
set
of
natural numbers
1,2,
. . .
;
N
U
{O}
real numbers,
{z
E
Rlz
2
0},
{z
E
Rlz
5
0)
image
of
function
f,
domain off
f
restricted to the set
A
C
dom(f)
identity mapping
least upper, greatest lower bound
of
A
derivative of
f
at
2:
partial derivative with respect
to
z,
j-th derivative, j-th partial derivative
set
of
continuous,
i
times continuously
differentiable functions' on
E
If
j,
:
M
-+
X,
and
x
E
M,
we set
inner product
of
x
and
y
open, closed interval,
.
. .
closure3, linear hull
of
A
set
of
open
neighborhoods3
of
z
open &-ball3 with center
z,
orthogonal3
complement of
V
there exists, there exists exactly
one
Graph
of
f,
{(z,f(z))lz
E
dom(f))
(fl,
fiN.1
:=
(fl(Z),f2(2.)).
on the existence of maximally continued solutions under mild
conditions.
First, the classes
of
differential-algebraic equations which
we will deal with are given and the concepts
of
solutions
are explained. For illustrations of the following definitions see
Example
11.5,
for notation see Table
I.
ZZ.1.
Defznition: (DAE, ADAE) Let
n
E
N,
m,k
E
Z+,
X
C
R",Y
C
R"
be open sets and
T
R
open and
connected. Let
f
and
g
be functions. If
f:X
x
Y
x
T
4
R"
E
CO
and
g:X
x
Y
x
T
-+
IWk
E
CO,
the system
.(t)
=
f(z(t),
Y(t),
t)
0
=
g(z(t),
Y(t>,
t)
(3)
is called a Differential-Algebraic-Equation (DAE). If
f:
X
x
Y
--t
R"
E
CO
and
g:
X
x
Y
-+
Rk
E
CO,
the system
(4)
is called an autonomous Differential-Algebraic Equation
11.2. Dejnition: (Solution) Consider DAE
(3)
and ADAE
(4)
and let
2:
1
-+
X
E
CO
and
y:
1
-+
Y
E
Co.
For the sake
of simplicity of our notation, set
T
:=
W
in case of ADAE
(4). Then,
cp
:=
(z,
y)
is said to be a solution
:*
(ADAE)
!
0
i)
I
C_
T
is open and connected.
ii)
z
E
C1
iii)
~t~~(~(t)
=
f(x(t),
YO),
t)
A
0
=
dz(t),
~(t),t))
in
'JtEI(W
=
f(Z(t),U(t))
A
0
=
g(z(t),y(t))
in case
case of DAE
(3),
and
of
ADAE (4).
0
41n the case
m
=
IC
=
0,
we identify
(3)
and
(4)
with
ODE'S
i(t)
=
f(s(t),t)
and
j:
=
f(s),
respectively.
124
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS,
VOL.
43,
NO.
2,
FEBRUARY
1996
ib*:-t
VC
tl
(a)
(b)
(c)
Fig
2
Illustration
of
Example
I1
5:
(a) Simple circuit exhibiting a
G1-2-FIP.
(b)
Voltage-chxge-relation of the capacitor
C
(c) Voltage
vc(t)
(-
-
-)
and
charge
q~(t)
(-)
when starting from an initial charge
40
=
1
stored in the
capacitor at time
to
=
0
From
tl
=
(:)
3
on voltage
and
charge are zero.
The
solution
set
S
and the
C1-solution
set
Scl
are defined by
S
:=
{$I$
is
a
solution}
Se1
:=
s
n
C1.
Elements of
S~I
are called @-solutions. Let now be
(20,
yo,
to)
E
X
x
Y
x
T.
The solution set
S(zo,yo,to)
and the
C1
-solution set
Scl
r(zo
,yo
,to)
according
to
the initial condition
(xo,
yo,
to)
are defined as follows:
S(zo,yo,to)
:=
{$
E
Sl?lr(to)
(xo,Yo)}
SCl,(zo,Yo,to)
:=
S(eo,yo,to)
n
C1
Elements of
S(zo,Yo,to)
(resp.
SCI,(~~,~~,~~))
are
called
solu-
tions (resp.
C1-soZutions)
passing through
(XO,
90)
at
to.
0
11.3.
Definition: (State set) Consider DAE
(3)
and
ADAE
(4).
The state set P and the C1-state set
Pcl
are defined by
P
:
=
(2
E
x
x
Y
x
TIS,
#
0}
PCl
:
=
(2
E
x
x
Y
x
T(SCl,,
#
0}
p
:
=
{(Z,Y)
E
x
x
YIS(z,y,O)
#
0)
PCl
:
=
{(GY)
E
x
x
YISCl,(z,y,O)
#
01
in case of DAE
(3),
and by
in case of ADAE
(4).
0
ZZ.
4.
Definition: (Continuation of solutions) Consider DAE
(3)
and ADAE
(4)
and assume that
a
E
S.
p
is called
continuation
of
0:
:*
p
E
S
A
r(a)
C
r(p)
C1-continuation
of
a
:*
P
E
Scl
A
r(0:)
&
r(p)
U!
is called
noncontinuable to the right
:-
'd'pEs(l'(cu)
C
F(p)
==+
supdomp
=
supdoma)
non-C'-continuable
to
the
right
:tj
'dpE~,l
(r(a)
2
r(p)
==+
supdomp
=
supdoma)
noncontinuable
to
the
left
:*
V/pEs(rl(cy)
non-C1-continuable
to
the left
:*
YfiEsCl(r(a)
non-C'-continuable
:-
noncontinuable
:e
r(p)
==+
inf domp
=
inf doma)
r(p)
==+
infdomp
=
infdoma)
'dPEScl
(r(a)
c
r(P)
==+
a
=
P)
v/P€s(r(Qi)
c
r(P)
===+
a
=
P)
0
In
order to clarify the definitions stated above in general terms,
we
consider the following simple example.
11.5. Example: (Chua
[4])
Let
R
in Fig. 2(a) be a linear
resistor with value
l5
and let
C
be described by
(5)
qc
=
-vc
where
qc
denotes the charge stored in the capacitor
C
and
vc
denotes the voltage across
C
(see Fig.
2(b)).
The analysis
of
this
circuit leads, by eliminating
VR,
ic,
and
i~,
to
the
following system:
23
3
4c
=
-we
=
f(qc,w)
0
=
qc
-
-213
c
-
-
g(qc,w).
(6)
2
Set
V(x)
:=
sign(z)
q$lxI.
Obviously,
g(qc,vc)
=
0
iff
wc
=
V(qc).
It is now intuitively clear6 that, in order to
obtain all solutions of ADAE
(6)
on
R
x
(R
\
{O}),
it suffices
to analyze ODE
(7),
on
R
\
(0).
For any initial value
qo
E
R
\
(0)
and
for
any
initial time
to
E
R,
let
I(qo,to)
be defined by
We now observe that
is a solution of ODE
(7)
passing through
qo
at
to.
Clearly,
qc
E
C'
and
qc
is unique by applying standard theorems on
ODE'S.
It remains to find all solutions
of
ADAE
(6)
passing
through
(O,O),
which is obviously an equilibrium point of
(6):
For
any
(q0,wo)
E
g-l(O)
and for any
to,t
E
R,
set
Q(t,
QO,
to)
:=
sign(qo)(lqolg
-
($)i(t
-
to));
if
40
#
0
and
{
0
It is easy to check that, for any
to
E
J,
any
qo
E
R,
and
any
J
C
I(q0,
to)
open and connected, the function
{Q(-,
40,
to),
V(G?(.,
40,
to)))l~
is
a
solution
of
ADAE
(61,
since
&(.,q~,to)
E
C1.
Hence, to show that
s={(&(.,qo,to),V(Q(.,4o,to)))i~
I
qo
E
RAto
E
JA
J
C
I(q0,
to)
is
open and connected},@)
it suffices to prove that solutions of ADAE
(6)
cannot
"leave"
(0,O).
More precisely, we have to show that for
any
(qc,
UC)
E
S
the following holds
qC(t0)
=
0
*
vtCdomqc,t>toqC(t)
=
0.
5For the sake
of
simplicity, we deal with
normalized
circuits only.
6This situation
IS
covered by Lemma
111.6.
REIRIG: DIFFERENTIAL-ALGEBRAIC EQUATIONS
125
(a)
(b)
(C) (4
Fig.
3.
Illustrations according to definition
111.1.
(Dashed lines denote sets without border). (a)
p
is
a
1-IP. (b)
p
is a 2-IP.
(c)
p
is
a I-FIP.
(d)
p
is
a
C1-2-FIP.
This is straightforward to check. We also have
To
clarify the concept of continuability, note that,
in our
example,
'p
E
S
is noncontinuable iff
domcp
=
R.
Finally, Fig. 2(c) shows the charge
qc(t)
and the voltage
wc(t)
when starting from an initial charge
qo
=
1
stored in
U
As we saw in the above example, any solution of ADAE
(6)
could be continued to some noncontinuable solution of
(6).
The following lemma shows that this is always possible.
11.6.
Lemma:
(Maximally continued solutions) Consider
DAE
(3)
and ADAE
(4).
Then the following hold:
i) For any
a
E
S
there exists a noncontinuable continuation
ii) For any
a
E
Scl
there exists a non-C1-continuable
U
Pro08
This proof [16] is a straightforward application
of
Zorn's
Lemma which states that an ordered set contains
a
0
the capacitor at time
to
=
0.
of
a.
C1-continuation of
a.
maximal element if every chain has an upper bound.
111. IMPASSE
POINTS
In this section we introduce several classes of impasse
points, each of which represents different behavior of the DAE
in question. After giving appropriate circuit examples, we will
state some simple relations between these classes.
For illustration of the following definition, see Fig.
3.
ZZZ.1.
Definition:
(Impasse point) Consider DAE
(3)
and
ADAE
(4).
For simplicity, set
Q
:=
X
x
Y
x
T,
p
:=
(zo,
yo,
to)
E
Rn
x
R"
x
R
in case of DAE
(3)
and set
Q
:=
X
x
Y,
p
=
(z0,yo)
E
R"
x
R",
and
to
:=
0
in
case of ADAE
(4).
p
is called
impasse point
(respectively,
C1
-impasse point)
of
the first
kind
(1-IP, resp. C1-1-IP)
:-
p
E
g-l(O)
\
P
impasse point
(respectively,
@-impasse
point)
of
the
second
kind
(2-IP, resp. C1-2-IP)
:-
p
E
(Pn
Q)
\
P
forward impasse point
of
the first kind
(1-FIP)
:*
resp.
p
E
g-l(O)
\
Pcl
resp.
p
E
(Gn
Q)
\
Pc~)~
7Here,
endowed with the topology induced by
some
norm.
denotes the closure
of
P
in
R"
x
Rm
x
R
(respectively,
R"
x
R")
forward impasse point
of
the second kind
(2-FIP)
:e
p
E
P
A
3~~s($
is noncontinuable to the right
AsupdomQ
=to
A
lim $(t)
=
(z0,yo))
C1-fonvard impasse point
of
the second kind
(0-2-
FIP)
:-
t+to
p
E
PCI
A
3+1E~,1
($I
is non-C'-continuable to the right
AsupdomIC,
=
to
A
lim
$(t)
=
(20,~~))
t-to
backward impasse point
of
the first kind
(1-BIP)
:*
p
E
g-l(O)
\
P
A
3GEs(inf dom
IC,
=
to
A
lim $(t)
=
(20,
YO))
t-to
C1-backward impasse point
of
the first kind
:*
p
E
g-l(O)
\
Pcl
A
3+E~cl
(inf
dam$
=
to
A lim
$(t)
=
(20,
YO))
t+t0
backward impasse point
of
the second kind
(2-BIP)
:-
p
E
P
A
3+€~($
is noncontinuable to the left
A
inf dom
$
=
to
A lim
IC,(t)
=
(Q,
yo))
C1-backward impasse point
of
the second kind
(C1-2-
BIP)
:U
p
E
Pcl
A
3$€scl
(IC,
is non-C1-continuable to the left
A inf dom
$
=
to
A
lim
$(t)
=
(20,
yo))
In the sequel, the sets of impasse points of the first kind, of
the second kind, of forward impasse points of the first kind, of
the second kind,
of
backward impasse points of the first kind,
and of the second kind (respectively, the corresponding kinds
of
C1-impasse points) will be refered
to
by 11,
I,,
IF,^, IF,^,
IB,~, and
IB,Z
(resp.
Icl,l,
Ic1,2,
Ic~,F,I,
Ic~,F,~,
ICI,EI,I~
Some of the terms defined above seem to be interesting,
but nevertheless fictitious, unobservable phenomena. In order
to show that the defined classes occur even in the analysis of
very simple circuits, we will give appropriate circuit examples.
t+to
t-to
and
ICl,B,2).
126
IEEE
TRANSACTIONS ON CIRCUITS AND SYSTEMS-I. FUNDAMENTAL THEORY AND APPLICATIONS,
VOL
43,
NO
2,
FEBRUARY
1996
(a)
(b)
(C)
(dl
Fig.
4.
Circuit from Example
111.4.
(c)
VCR
of
tunnel diode
D.
(d)
Illustration
of
the solutions
of
the circuit.
Illustration
of
impasse point definitions: (a) Circuit
from
Example
III.3.
The
voltage-charge-relation
of
the capacitor
C
is
given
in
(5)
and
2(b).
(b)
111.2.
Example: Consider the circuit
from
Example
II.5.
see Fig. 4(c). The analysis of this circuit leads to the ADAE
shown in Fig.-2(a) again. Its analysis led to ADAE
(6).
Recall
that
(0,O)
is an equilibrium point
of
ADAE
(6).
Further, note
that
define a C1-solution
of
ADAE
(6)
on
{T
E
[WIT
<
tl}
for
tl
=
(i);
(see Fig. 2(c)). It is now a simple task to check
that lim.;c(t)
=
-NI
and thus, this solution
is
non-C1-
continuable to the right. In view
of
Definition
III.1.
we see
that
(0,O)
E
IcI,~,~. More precisely,
tltl
0
=
v1
-
v2
=
g((w1,
v2),
U,)
(10)
We will restrict our considerations to the examination of
the behavior of ADAE (10) in some neighborhood of
p
:=
i) We first show that
p
$!
P:
Assume there was some
$J
:=
((v~,v~,)wD)
E
S
passing through
p
at
0.
Then,
for all
t
E
dam$,
((Vl,O,
v2,o),m,o)
:=
((32,32),4).
0
=
.;ll(t)
-
.;2(t)
=
-2S(vo(t))
+
Ul(t),
and hence, vl(t)
=
2Q(WD(t)). Since
9
has a local
minimum at
VD,O
=
4, and since
VD
E
CO,
there
is
some
E
>
0
that
VtE~-E,Epl(t)
2
32,
but this clearly
contradicts .;ll(O)
=
-Q(wD,o)
=
-16.
ii) We now show that there
is
some C1-solution
cp
of
ADAE
(10) converging to
p
for t
-+
sup
dom
cp.
Set
IC",F,2
=
((0,O))
and
IF,2
=
-JCI,F,l
=
0.
This example shows that C1-2-FIP's can really occur in the
analysis of very simple circuits, and also, that the distinction
0
111.3.
Exumple: We modify the circuit from Example
II.5
(Fig. 2(a)) and substitute the linear resistor
R
by a constant
current source maintaining a current
io
=
1.
The analysis
of the resulting network, shown in Fig. 4(a), leads to the
following ADAE
between
solutions
and C1-solutions is essential.
vl(t)
:=
v~(t)
:=
64e-'/' and
WD(~)
:=
@(~j.(t))
fort
<
ln(4), where
:=
(Q1{7ER,T>4))-1
E
C1.
This
mapping
Q,
obviously exists (see
Fig.
4(c)). We now
have
Gl(t)
=
-Q(wD(t))
and+2(t)
=
Q(VD(t))-Ul(t).
Thus,
cp
:=
(211,
~2),
WD)
is a C1-solution of ADAE
(10).
It is now easy to show that limt+~n(4) cp(t)
=
p,
and
hence, we get that
p
is a I-FIP as well as a Cl-l-FIP.
4.c
=
-1
=
f(4c,
WC)
0
=
qc
-
-U3
c
-
-
s(Qc,vc).
(9)
2
It is a simple task
to check that
By further investigation we obtain
(qC,
vC):
iw
+
iw
x
iw:
t
w
(-t,
-
sign(t)
6)
P=Pcl
={((W1,2)2),'UD)E[W2X[WI'Ul=W2AvD
${2,4}
A
211
=
2.
*(U,)}.
is
a
solution,
but
not
a
C1-solution
of
ADAE
(9).
Namely,
we observe that limt,o
.;c(t)
=
-00,
limf,o(qc, vc)(t)
=
(O,O),
and (qc,vc)l~\{o}
E
C1.
Thus,
(0,O)
is a Cl-l-FIP
as well as a C1-1-BIP, but neither a
I-FIP
nor a 1-BIP. This
example shows again that the distinction between solutions
and C1-solutions is essential.
111.4.
Example: Consider the circuit shown in Fig. 4(b). Let
R
be a linear resistor with value
1,
C1
and
C2
be linear
capacitors with value
1,
S
be a controlled current source
maintaining a current
io
with value
io
=
i2
+
01,
and let
D
be a tunnel diode, which is described by
Among other things, this demonstrates that LIP'S are
not
0
necessarily 1-FIP's or 1-BIP's, see Fig. 4(d).
In the examples considered above we saw that the classes
of
impasse points are in general different from each other. The
following can be shown.
ZZ1.5.
Lemma: Consider DAE
(3)
or ADAE (4). Then
i,
IBJ
U
IF,1
c
12
c
11
ii)
IC1,B,l
iii)
I1
IC1,l
iv)
IF,2
=
IB,~
=
0
rC1,,~
c
IC1,2
c
IC1,l
v)
(kC',F,2
U
IC',B,2)
n
b,l
=
0
i~
=
Q(wD)
:=
W%
-
9~;
+
24~0,
0
REIBIG: DIFFERENTIAL-ALGEBRAIC EQUATIONS
127
Proof:
The proof of i), ii), iii), and v) is straightforward
[16]. The idea behind the proof of iv) is as follows [16]. Let,
without loss
of generality,
p
=
(zo,yo,to)
E
IFJ,
cp:]to
-
&,to+&[
+
XxY
E
S,,$:]to-~,to[
+
XxY
E
Sforsome
E
>
0,
limt-,to
$(t)
=
(xo,yo),
and
4
be noncontinuable to
the right. Define
p
=
(&,,By)
by
p(t)
=
+(t)
for
t
<
to
and
p(t)
=
y(t)
otherwise, for
t
-
to
small enough. One can show
that
,&
fulfills the ODE ,&
=
f(,&,
py,
t). (This is simple
though not trivial.) Thus,
p
is a solution of the DAE in question
in contradiction to the assumption that
+
is noncontinuable to
In the case
m
=
k
=
0,
we identify DAE
(3)
and ADAE (4)
with ODE’s
i(t)
=
f(z(t),
t)
and
i
=
f(x),
respectively. As
is well known from the theory of ODE’s, we get
P
=
Pc1
=
X
x
T
and P
=
Pcl
=
X,
respectively, in that case. This
is an important difference between DAE’s and ODE’s. In the
following lemma, situations are examined where the DAE or
ADAE in question is locally equivalent to an ODE in some
sense:
111.6.
Lemma:
Consider DAE
(3)
and let
(zo,yo,to)
E
X
x
Y
x
T.
If there are some neighborhood
U
of
(x0,to)
and a mapping
h:
U
+
Y
E
Co
such that
h(zo,
to)
=
yo
and
g(z,h(x,t),t)
=
0
for all
(x,t)
E
U,
then
If
h
is even C1, then
{(z,h(x,t),t)l(x,t)
E
U>
C
Pel.
In case of ADAE (4) let
(20,
yo)
E
X
x
Y.
If there are
some neighborhood
U
of
xo
and a mapping
h:
U
-+
Y
E
Co
such that
h(xo)
=
yo
and
g(x,
h(x))
=
0
for all
x
E
U,
then
0
Proof:
We omit this proof since it is trivial [17], [16].
0
the right.
U
{(.,h(~,t),t)l(z,c
E
U>
c
p.
{(z,h(z))lz
E
U>
c
p.
If
h
is even
C1,
then
{(x,h(x))lz
E
U}
g
Pel.
111.
7.
Remark:
i)
ii)
iii)
Since
h
does not need to be unique, Lemma 111.6. is
slightly more general than the statements in
[l,
Lemma
1 and
21.
Using a Surjective Implicit Function Theorem [18], one
could easily find simple sufficient conditions for a point
p
to be in P or in
Pel,
as well as sufficient conditions for
local uniqueness of solutions of the DAE in question. In
case
of
ADAE (4) with
m
=
k
and
&g(p)
bijective, this
approach leads to some vector field on g-l(O), which
is
equivalent to the considered ADAE and ensures both,
local existence
and
uniqueness of solutions
[
191.
In Example 11.5. we have already intuitively used
arguments similar to those of Lemma 111.6.
0
We now derive a simple criterion for a point to be an impasse.
To motivate our approach, consider ADAE (4) and assume,
for a while, that
g-l(O)
is
a
smooth submanifold of
Rn
x
W”.
Then, for any C1-solution
cp
=
(3,
y)
passing through
p
at
to,
+(to)
has to lie in the tangent space of
g-l(O)
at
p.
So
we
have
+(to)
=
(f(p),
y(to>>
E
ker
Dg(p)
and, hence,
Dlg(p)f(P)
E
imD2dP). (1 1)
Conversely, if (11) does not hold,
p
is clearly a C1-1-IP. As
we will see, this conclusion is valid whether or not g-l(O) is
a manifold. This suggests the following definition:
111.8. Dejinition:
Consider DAE
(3)
and ADAE (4), let
p
E
g-l(O),
and assume that g is
C1
on some neighborhood
of
p.
Then the
tangential
property
is defined as follows:
0
111.9.
Example:
Consider the circuit from Example 111.3,
Fig. 4(a), again. Recall that
f(0,O)
=
-1,
g(qc,wc)
=
qc
-
$I&,
and hence
Thus, in this example, we have that
7(f,
g,
(0,O))
does not
hold. We further observe that
(0,O)
E
P and (0,O)
E
Icl.l,Fn
U
As we saw above, there is a solution passing through
(0,O)
at
0,
but this solution is not even Lipschitz continuous near
0.
This is not an accidental observation.
111.10.
Lemma:
Consider DAE
(3)
and ADAE (4), let
p
=
(zo,yo,to)
E
gf1(0) in case of DAE
(3)
and let
p
=
(50,
yo)
E
g-l(O)
and
to
E
W
in case of ADAE (4). Under the
assumption that
g
is
C1
in some neighborhood of
p
and that
T(f,
g,
p)
does not hold, the following is true:
IC1
,l
,
B
.
i)
P
E
b,1
lIY(t)-Yoll
=
W.
ii)
(Xt.,Y)
E
S(zo,yo,to)
-
It--to(
0
The proof can be found in Appendix
B.
111.ll
Remark:
In [l, Lemma
31
it is asserted thats
p
is
an impasse point
--7’
il(f,g,p).
This statement is easy to disprove. Consider the circuit from
Example 111.4, Fig. 4(b), again. Recall that
p
=
((32,32),
4)
is
a Cl-l-FIP as well as a I-FIP of ADAE (10). We now observe
that
f(p)
=
(rI,1$!i2)
=
(I::)
and thus,
Dlg(p)f(p)
=
(1,
-
1)
(1;:)
=
0.
Hence,
I(
f,
g,
p)
holds, in contradiction
to [l, Lemma
3
1.
0
Iv.
PSEUDOLIMIT POINTS
AND
IMPASSE
POINTS
In [l], Chua and Deng discovered that it
is useful to consider
the mapping
h:
(A,
Y)
g(x0
+
A
f(zo,
yo),
9)
(12)
in order to find out whether or not the point
p
=
(50,
yo)
E
gP1(0) is an impasse. Their main result is, roughly speaking,
that
p
is an impasse point of
(2)
iff
(0,
yo)
is
a
limit
point
[l]
of
h-l(O). This criterion is wrong, which we will see in IV.8
and IV.9. Nevertheless, in this section we follow the idea to
examine
h,
called by us a
cut mapping,
in order to identify
impasse points.
=
0
A
-7(f,
g,p)
can be shown to be equivalent to
p
E
S3,
with
Sj
as
in
[I].
128
IEEE TRANSACTIONS
ON
CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS,
VOL. 43,
NO.
2,
FEBRUARY
1996
(a)
(b)
(c)
(d)
(e)
(0
Fig.
5.
Illustration
of
examples: (a) Circuit from Example IV.4
and
IV.12.
(b)
VCR's
of
humel
diodes
D
of Examples IV.4, IV.6
(-
-
-)
and
Example
IV.12
(-),
(c)
gpl(0)
as
in Example IV.4. (d) (0,4) is
a
right pseudolimit point
in
both,
Example N.4.
(-
-
-)
and Example IV.12
(-).
(e) Illustration
of
Example IV.6.
(0
Function
T
from Example IV.6.
W.1.
Dejnition: (Pseudolimit point) Let M
C
R
x
R",
(Xo,yo)
E
M,
and set
TA:
R
x
R"
4
R:
(X,y)
H
A.
(X0,yo)
is called right pseudolimit point (PLP)
of
M
if
~YI,EC([O,i],M)(C(O1
=
(X0,Yo)
===+
3t,o.rrx(c(t))
I
XO).
~cEc([o,l],M)(c(o)
=
(Xo,
Yo)
===+
%om(c(t))
L
A,).
(XQ,
yo)
is
called
left
pseudolimit point
of
M
if
0
ZV.2.
Dejinition: (Cut mapping) Consider
ADAE
(4)
and
The mapping
h
from
(12),
defined on some neighborhood
of
(0,
yo),
is
called a
cut
mapping
of
(4)
at
p.
Consider DAE
(3) and let
p
=
(20,
yo,
to)
E
g-'(O). The mapping
let
p
=
(Z0,YO)
E
9-".
h:
(4
Y)
g(xo
+
.
f@)i
Y,
to
+
A)
7
defined
on
some neighborhood
of
(0,
yo),
is called a
cut
U
IV.3. Remark:
i) The property of a point to be a PLP
of
a cut mapping
is
independent of the choice of the neighborhood of
(0,
yo)
in Definition
IV.2.
ii) Let
h
be a cut mapping of ADAE
(4).
Then, any limit
point of
h-'(O)
in the sense
of
[l] is a PLP. The
converse is not true, not even if
h-'(O)
is a Coo-curve
[17,
Example 3.31. However, in examples of this paper,
for the sake
of
simplicity, only such PLP's occur which
are also limit points.
0
N.4.
Example: (Takens [13], Rabier, Rheinboldt [319 Con-
sider the circuit shown in Fig. 5(a).
Let
C,
L:
R
+
W+
\
(0)
E
Co.
The tunnel diode
D
be
described by the same relation as in Example
III.4,
namely,
ZD
=
@(vD)
:=
t&
-
921:
+
24w~,
see also Fig.
5(b).
The
analysis
of
this circuit leads to the ADAE
mapping
of
(3)
at
p.
See Fig 5(c) for illustration
of
g-'(O).
We now consider points
p
:=
((~/c,o,z~,o),v~,o)
with
ZD,~
=
16,
VD,O
=
4,
similar to
90ur example
is
the dual
of
the circuit dealt with
in
[13,
8
23 11 and
[3
I11
deffnes
a
cut mapping
h,,,,
of (13) at
p,
defined on, say,
1-1,1[
x
]3,5[.
Although it could be rigorously shown that
(0,4)
is
a
right PLP of
h;;,,(O),
we content ourselves with
examining Fig. 5(d): Obviously, not any curve with values
in
h;;,,(O)
passing through (0,4) can pass through a point
U
Our
main result on the relation between impasse points and
pseudo
liimit points is:
N.5.
Theorem: Consider DAE
(3)
or ADAE
(4)
and let
p
=
(zo,yo,to)
E
g-'(O)
in case
.of
DAE (3) and
p
=
(zo,~~)
E
g-'(O)
in case of ADAE
(4).
Further, let
h
be
a cut mapping of
(3)
or
(4)
at
p
and the following conditions
be satisfied.
(X,VD)
with
X
>
0.
i)
g
is
C1
on
some neighborhood
of
p.
ii)
p
does not have the tangential property
7(f,
g,p).
rankDag(p)
=
k
-
1.
(0,
yo)
is
a right (resp. left) PLP of
h-'(O).
P
E
11.
Consider DAE
(3)
and let
V
5
R"
x
R
be some
al-
gebraic complement of span{
(f
(p),
l)}.
Assume there
are some neighborhood
U
E
ZA((z0,
to)),
mappings
T:
U
-+
Y
E
Co
and
y:
V
n
(U
-
(20,
to))
+
R
E
C1,
that
T(z0,to)
yo
and
$0)
=
0.
Set
in case
p
is a right PLP and set
?i
E
domy
A
X
>
-y(u)}
in case
p
is a left PLP. Suppose that for all
(2,
t)
E
6
we have
g(z,T(z,t),t)
=
0."
Then
p
is a
1-FIP
(resp. 1-BIP).
'OIn the case of ADAE (4) let
V
C
Rn
be
an algebric complement
of
span{f(p)},
U
E
u(z0),
T.
U
+
Y
E
CO,
Y.
v
n
(U
-
z0)
+
w
E
c1,
REIBIG: DIFFERENTIAL-ALGEBRAIC EQUATIONS
129
13) If, in addition to the requirements of I2),
TI,
E
C1
0
While conditions in 12) are illustrated within the proof
of
Theorem IV.5 for the general situation, see Fig. 6(a), let us
demonstrate first how to check the requirements
of
12) in
practice.
ZV.6
Example: Consider the circuit from Example IV.4,
shown in Fig. S(a), again. Its analysis led to ADAE (13).
Although we obtained that
p
=
(
(vc,~,
16),
4)
is a right
PLP of
h;;,,(O)
(see (14)) as long as
VC,~
2
0,
we will
restrict ourselves to the case
VC,~
=
1.
First, we mention
that assumptions i) through iii) in Theorem IV.5. are clearly
fulfilled, since
Dlg(p)f(p)
=
-&
<
0
and
Dzg(p)
=
0.
In order to check I2), set
holds, then
p
is a C1-1-FIP (resp. C1-1-BIP).
U
:=
(R+
\
(0))
x
]15,17[,
v
:=
Iw
x
{O},
y(v)
:=
0.
In this situation we obtain
6
=
(R+
\
(0))
x
]16,17[, see Fig.
5(e). Set further
1
where
@
:=
(ql{TERlr>4))-
E
C1.
This mapping
@
obvi-
ously exists (see Fig. S(b),(f)). It is also clear that
T
E
Co.
Let now (v~,i~) E
E.
Then we have
io
>
16, and hence,
~(VC,ZO)
=
@(zD)
and
g((VC,iD),T(VC,ZD))
=
0.
BY 12)
and 13) we see that
p
is
a 1-FIP as well as a C1-1-FIP.
0
We are now going to prove Theorem 1V.S. The crucial trick
is to assume the existence
of
some
(z,y)
E
S(zo,yo,to)
and
then to apply an sophisticated version of the Implicit Function
Theorem (see Appendix A) to the mapping
G:
(s,t,z)
++
g(5o
+
S.
f(p),y(t+
to)
+
2,s
+to).
(15)
Note that, in general, G is
not
differentiable on some open
neighborhood of
(0,
0,O).
So,
to show differentiability of
G
at
(O,O,O)
requires some extra work; we put this part into a
separate lemma:
IV.7.
Lemma: Consider DAE (3) and assume that
g
is
C1
on some open neighborhood
U
of
p
=
(20,
yo,
to)
E
g-'(O).
Let(z,y)
E
S,,I:=dom(z,y),E
:=RxIW~(kerD2g(p))~,
W
C
E
an
open ball with center
(O,O,O),
so
that
t+to
E
I
and
(20
+
S.
f(p),
y(t
+
to)
+
2,
s
+to)
E
U
for all
(s,
t,
z)
E
W.
Define G for
(s,t,z)
E
W
as in (15). Then
i)
G
E
Go
and G(O,O,O)
=
0.
ii) G is differentiable at
(0,0,0).
iv)
D1,3G
exists as a partial derivative on
W,
is continuous,
iii) D2G(O?
0,O)t
=
-D1g(p)f(p)t
-
&g(p)t.
and D1,3G(O,
O,O)(s,
2)
=
Dlg(p)f(p)s
+
&~(P)s
+
D2dPb
0.
The proof can be found in Appendix B.
T(rc0)
=
yo,
and
y(0)
=
0
instead.
Also,
set
6
:=
{z
E
Ulz
=
sofX
f(p)+aAv
E
domyAA
E
RAX
<
y(u)}
incasepisarightPLPand
set
fi
=
{.E
E
Uls
=
zo
+A
f(p)+vAv
E
domy
AX
E
RAX
>
y(v)}
in case
p
is a left PLP Suppose further that
V,,,;g(z,
T(z))
=
0.
Proofi
(of Theorem IV.5.) The proof
is
done for DAE
11)
Assume
(z,
y)
E
S,
contrary to the assertion and let
I,
E,
W,
and
G
as in Lemma
1V.7.
Since
D1,3G(O,0,0)
is bijective"(has rank
IC
because
of
ii) and iii)), the
Implicit Function Theorem A.l. is applicable and one
gets
(3)
and for
p
being a right PLP only:
In the following we denote the point
(s,
z) correspond-
ing to
t
by
(s(t),x(t)),
i.e.,
Then
(s,
z)
is continuous on some neighborhood of
0,
differentiable at
0,
and
i(0)
=
1.
Hence,
contrary to the hypothesis that
(0,
yo)
is a right PLP
of
12) For this part, see
also
Fig. 6(a). Set
F(z,t)
:=
f(z,
T(z,
t),
t). Clearly,
F
E
Co(U),
and hence, there
is some solution
cp:I
+
U
E
C1
of
k(t)
=
F(z(t),t)
passing through
20
at
to,
where
I
R
denotes an open
interval,
to
E
I.
Let now
P
be the projection operator
onto
V
along span((f(p),
1)).
Then
h-l(O).
for
X
E
C1
defined by
Setting
r(t)
:=
X(t)
-
T(P(~~~;~~~)) yields
r(to)
=
0
and
=
1. Thus, there is some
E
>
0
that
r(t)
<
0,
i.e., ~(t)
<
y(~(pjtl$o)),
for all
t
E
]to
-
€,to[.
Due to the definition of
6
we have
('6"))
E
f?
and
g(cp(t),
T(cp(t),t),t)
=
0.
Lemma 111.6. and continuity
of
T
imply
limt,to
$(t)
=
(zo,yo)
and
$:]to
-
E,
to[
-+
R"
x
Rm:
t
c-$
(cp(t),
T(cp(t),
t))
E
s.
13) trivial.
U
Before we investigate a special situation in which the con-
ditions in 12) may be easily checked, some remarks on the
hypotheses in Theorem 1V.S. are in order.
"Obviously,
D2,3,~
is also bijective at
(O,O,O)
but does not exist on some
open neighborhood of
(O,O,O).
So
we have to go indirectly.
130
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43,
NO.
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FEBRUARY
1996
(a)
(b)
(c)
Fig.
6.
(a) Illustration of the proof
of
Theorem
IV.5.
12). (b)
g-’(O)
as
in
Example IV.9.
9
is
a
solution
of
ADAE
16.
(c)
h-l(O)
as in
Example
IV.9.
(0,O)
is
a right
PLP
in this case.
IV.8.
Remark:
i) Condition ii) in Theorem IV.5. cannot be omitted
(Ex-
ample IV.9).
ii) Theorem IV.5. gives sufficient conditions for a point to
be an impasse. These are not necessary ones [17], 1201.
iii)
iv)
An impasse point that is detected by Theorem IV.5.
is not necessarily a “singular source” or
“sink“
of
[2] or an impasse point of [3]. In Examples IV.11
ad IV.12 we show that Theorem
IV.5.
applies in
situations where the results of [14] and [3] may not.
Note also that these results do not say anything about
the existence of nondifferentiable solutions through the
point in question.
In
[l,
Theorem
1.1
it is asserted that
(Q,
yo)
E
g-’(O)
is a 1-FIP (resp. 1-BIP) if and only if
(0,
yo)
is a right
(resp. left) limit point of
h-l(O),
where
h
denotes a
cut mapping of ADAE (4). This criterion is wrong:
Example IV.9. contradicts its “if ’-part. Moreover, its
17
“only if”-part is also wrong [17][16].
ZV.9.
Example: In this example, which is illustrated in Fig.
6(b) and (c), we will content ourselves with examining the
following ADAE
Xl
=
1
x.2
=
2x1
(16)
22
O=xz-(x1+y
)
((t,t2),0)
E
~c~,((o,o),o,o)~
and hence,
((O,O),O)
E
Po.
As
without giving a circuit example, but it
is
obvious how to
realize ADAE (16). Clearly,
p:]-l,l[
-+
R2
x
R:t
H
can easily be seen,
h:
R
x
R
+
R:
(A,
y)
H
-(X+y2)’
is a cut
mapping of (16), and
(0,O)
is a right limit point in the sense
of [l] as well as a right PLP of
hP1(0).
Theorem IV.5. is not
applicable, since
((
0, 0),
0)
does have the tangential property.
Note that this example contradicts the “if”-part of
[
1, Theorem
We now attempt to find sufficient conditions for a point to be
a 1-FIP or a 1-BIP, which possibly do not apply to the whole
class of DAE’s covered
by
Theorem IV.5, but, in return, are
really simple
to
check.
First, note that i) through iii) of Theorem IV.5. ensure that
Dg(p)
is
surjective, and hence, that
h-l(O)
is a manifold.
Moreover, the Implicit Function Theorem yields that, roughly
11.
0
speaking,
h-l(O)
can be described by
X
=
A(y)
for some
A
E
C’
whenever
g
E
C’.
Obviously,
(0,yo)
is a right (respectively, left) PLP
of
h-l(O)
if
A
has a maximum (respectively, minimum) at
yo.
It is
now
straightforward to express derivatives of
A
in terms
of derivatives of
g.
Thus, we obtain sufficient conditions for
(0,
yo)
to be a PLP, depending only on derivatives of
g
at
p.
For the sake
of
convenience, we content ourselves with
examining the simplest case, namely, we assume that
m
=
k
in
(3)
and
(4),
and consider nondegenerate extrema of
A
only.
Condition iv) in the following Corollary is there precisely
to
ensure that
D2A(y0)
is definite.
The
surprising fact is, that these simplest conditions for the
presence of an extremum of
A
at
yo,
namely,
DA(y0)
=
0
and
D2h(yo)
is definite,
do not only ensure that condition iv) in Theorem IV.5. is
fulfilled, but also guarantee that the conditions included in
its
12)
and
13)
parts hold!
W.10.
Corollary:
Consider DAE
(3)
or ADAE
(4)
with
m
=
k
and let
p
=
(zo,
yo,
to)
E
g-l(O)
in case of DAE
(3)
and
p
=
(z0,yo)
E
g-’(O)
in case
of
ADAE (4). Let
U
E
(imD2g(p))l
\
(0)
and
h
E
kerDzg(p)
\
(0)
and the
following conditions be satisfied:
i) g is
C2
on some neighborhood of
p.
ii)
p
does not have the tangential property
T(f,
g,p).
iii) dimker&g(p)
=
1.
iv)
D;g(P)hh
$
imD2dp).
Then exactly one
of
the following two situations is met:
(F)
n(h)
>
0.
Then
p
is
a 1-FIP
as
well as a C1-l-FIP.
(B)
a(h)
<
0.
Then
p
is
a 1-BIP as well as a C1-l-BIP.
Here,
a
is, in case
of
ADAE
(4), defined by
and in case of
DAE
(3)
by
0
The proof can be found in Appendix
B.
In case of
f
E
C1,
the hypotheses of the foregoing result
are
equivalent to
those
of
[3,
Theorem 6.1.1. Equivalence to
the assumptions in [14, Theorem 5, Item 21 has been shown in
[16]. Further, any impasse point that is detected by Corollary
IV.10. is a “singular source” or “sink’ in the sense of
[2]
and
an
impasse point in the sense of
[3].
It should be noticed that Corollary IV.10. is still sharper
than [14, Theorem 5, Item 21 and
[3,
Theorem 6.1.1 since
these results do not say anything about the existence of non-
C1-solutions through the point in question.
As we will see in Example IV.11., it is very easy
to
check
the conditions in Corollary IV.lO. An application in the case
f
$
C1
is also given there. However, Example IV.12. shows
that Corollary IV.10 does not apply to all circuits to which
Theorem IV.5 applies.
So,
we consider it
to
be only one,
REIRIG: DIFFERENTIAL-ALGEBRAIC EQUATIONS
131
namely, the simplest and most practicable conclusion from
Theorem IV.5.
Indeed, Theorem IV.5. admits to find various conditions
for a point to be an impasse, all stated in terms of higher
derivatives of
g
at
p.
This is beyond the scope of this paper
and we bring our considerations on impasse points to an end
by giving two more examples.
ZV.11.
Example:
Consider the circuit Erom Examples IV.4
and IV.6, shown in Fig. 5(a), again. Its analysis led to ADAE
(13). In IV.6 we laboriously showed that the requirements
of 12) in Theorem IV.5. are met, Let us now look how
Corollary IV.10 works here, when examinig the same point
p
=
((1,16),4) as in IV.6.
Recall that
g
E
C2
and, from IV.6., that ii) and iii) of
Corollary IV.10 are fulfilled and that D2g(p)
=
0.
We also
have
02g(p)
-O2q(4)
=
-6Wo
f
181,,=4
=
-6.
Hence,
ker
Dzg(p)
=
(im&g(p))l
=
R.
Choose
U
=
h
=
1.
Then, since Dlg(p)f(p)
=
-&
<
0,
iv) is fulfilled and
a(h)
>
0.
Thus, p.is a 1-FIP as well as a Cl-l-FIP of
ADAE
(13). Note that
f
does not need to be differentiable. Especially,
if we choose one of the reactances to have a piecewise linear
characteristic, Corollary IV. 10. remains applicable and our
conclusions remain true. The results in
[14]
and [3], however,
U
ZV.12.
Example:
Consider the circuit from the previous
Example, shown in Fig. 5(a), again. We now assume the tunnel
diode to be described by
do not apply in this case.
(-4
+
wD)4
(-1
+
4WD)
16
io
=
@(WO)
:=
16
+
1
see Fig. 5(b). This leads to ADAE (13) again, with
XQ
as
above. It is intuitively clear that this modified circuit exhibits a
behavior qualitatively equivalent to that of the original circuit.
Especially,
p
=
((1,16),
4)
should be a 1-FIP.
Indeed,
(0,4)
is a right PLP
of
an appropriate cut mapping
of
ADAE (13) at
p,
see Fig. 5(d). Further,
y
and
'Y
can be
constructed in complete analogy to Example 1V.6, see
also
Fig.
5(e) and
(f).
Thus, Theorem IV.5. applies, and
p
is a 1-FIP.
However, it is impossible to conclude this fact from Corollary
IV.10, [14, Theorem
5,
Item 21, or
[3,
Theorem 6.11 since
0
Dig(p)
=
-D2Q(4)
=
0.
V. CONCLUSION
The types of impasse points (IP's) which have been dealt
with in publications are forward and backward IP's. However,
we have shown that several classes of IP's, each of which
represents different behavior of the DAE in question, do exist;
relations between these classes have been given.
Further, motivated by geometry, we have introduced the
tun-
gentiul property,
which plays
an
important role in conditions
for the presence of several kinds of IP's.
In order to identify forward and backward IP's, we have
introduced the term
pseudo limit point
(PLP) and examined a
certain mapping, called by us a
cut mapping.
This approach
was suggested earlier, but has led to success for the first time
in Theorem IV.5 of
our
paper.
Although it has become clear that Theorem IV.5 admits to
find various conditions
for a point to be a forward or backward
IP, all stated in terms of higher derivatives of
g,
we have
contended ourselves with giving the simplest of these in terms
of second-order derivatives in Corollary IV. 10.
It turned out that the'hypotheses in Theorem IV.5 are
considerably weaker and those of Corollary IV.lOare still
slightly weaker than the assumptions in other publications.
On
the other hand, Theorem IV.5 as well as Corollary IV.10 yield
sharper results since they take into account nondifferentiable
solutions. Such results have not been known and also, cannot
be derived in a straightforward manner from known ones.
On the whole, our results are sharper than known ones
and remain applicable in situations where those from other
publications do not apply.
APPENDIX A
AN
IMPLICIT
FUNCTION
THEOREM
A.I.
Theorem:
E,
Q,
Z be Banach spaces over
R,
and
U
be an open neighborhood of
(zo,yo)
E
E
x
Q.
Further, let
F:
U
+
Z and the following conditions be satisfied.
i>
F(Xo,Yo)
=
0.
ii)
D2
F
exists as a partial derivative on
U
and
02
F
(50,
yo)
iii)
F
and
D2F
are continuous at
(20,
yo).
is bijective.
Then
4
3r,,0>ovzEB(zo,ro)3!yEB(yo,r)F(X,
9)
=
0.
In the following, the corresponding mapping
x
H
y
is
denoted by
Y.
c) If
F
is continuous then
so
is
Y.
e) If
F
is differentiable at
(zo,
yo),
then
Y
is differentiable
at
xo
and
DY(z0)
=
-D2F(zo,
yo)-'
o
D1F(zo,
yo).
0
(a) and (c) are proved in [18], (e) in [16].
B.
1
Proofs
Proof:
(of Lemma 111.10.) It clearly suffices to show ii)
for DAE (3). Assume there is some
L
E
R
and some sequence
(tn)nE~
that t,
E
domy
\
{to}
for all
n,
t,
--f
to,
and
vnENllAYYnll
5
Wtnl
(17)
contrary to the assertion, where
Ay
=
y(t)
-
yo,
Ay,
=
y(t,)
-
yo,
At
=
t
-
to,
and
At,
=
t,
-
to.
Choose
X
>
0
that
q:B(to,X)
x
Y
-+
Rk:
(t,~)
H
g(x(t),z,t)
is in
C1.
Obviously, q(t,y(t))
=
0
as long as
t
E
B(to,X)
and
@(to,
YO)(t,
2)
=
Dg(p)(t
f(P),
z,
t).
In view of the definition of the derivative one obtains12
'dE>o
3
NE
NV~>
N
IIDg(p)(At,.f(p),
AY,,
Atn)ll
I&ll(At,,
AYn)II
IEMlAtTZl
"For
simplicity,
we
denote
any
used
norm
by
11.11.
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