On logarithmic norms for differential algebraic equations

On logarithmic norms for di erential algebraic equations

Renate Winkler

Humboldt-University Berlin

Unter den Linden 6, D-10099 Berlin, Germany

Abstract

Logarithmic matrix norms are well known in the theory of ordinary di erential equations (ODEs) where they supply estimates for error growth and the growth of the solutions. In this paper we present a natural generalization of logarithmic norms which makes it applicable to di erential-algebraic equations (DAEs) and yield sense-full estimates for the growth of the solutions of the DAE. As there are various possibilities we show how they correspond to each other.

1 Introduction

We want to present und discuss possibilities for generalizing the concept of logarithmic norms to make it applicable to DAEs. Logarithmic norms for matrices were introduced in 1958 by Dahlquist and Lozinskij 1],5]. For a square matrix A²L(IR^m) the logarithmic matrix norm, or for short the logarithmic norm, is dened by

A] = lim_h

!0⁺

kI+hA^k;1

h (1.1)

where^kkdenotes a matrix norm induced by a vector norm^jj. In general, the logarithmic matrix norm depends on the used vector norm ^jj. If ^jj2 denotes the Euklidian vector norm, E is a nonsingular matrix, and ^jjE denotes the vector norm with ^jx^jE = ^jEx^j2

then we have for the induced matrix norm ^kA^kE = ^kEAE^;1k2 and ,analogously, for the corresponding logarithmic matrix norm

EA] = 2EAE^;1]:

] can take values in IRand is not a usual matrix norm. It can be written as A] = lim_h

!0⁺ ln^ke^hAk

h ^(1.2)

1

and, if ^jj is an inner product norm^jj2 =<>, also as A] = max_z

6= 0 < Azz >

jz^j2 : (1.3)

To illustrate the use of logarithmic matrix norms in the theory of ODEs we follow the lines of 2] and quote the following theorem of Dahlquist 1] :

Theorem 1.1

^{Let x and x} be solutions of

x⁰(t) =f(x(t)t) t²0T] (1.4)

with values x(t) x(t) lying in some neighborhood Mt := ^fz ² IR^m : ^jz^;h(t)^j (t)^g, where h is some continuous auxiliary function. Let be a piecewise continuous scalar function satisfying

f_x⁰(zt)](t) ⁸t²0T]⁸z ²Mt: Then it holds

jx(t2)^;x(t2)^j exp(

Z t²

t¹ (s)ds)^jx(t1)^;x(t1)^j for all t1 t2 satisfying 0t1 t2 T .

If f_x⁰(zt)] 0⁸t ² 0T]⁸z ² Mt the dierential equation (1.4) is called dissipative.

The freedom of choosing an arbitrary vector norm can be exploited to select a vector norm for which (t) is as small as possible. For a detailed representation we refer to 2].

As a conclusion of the theorem 1.1 we obtain for time-dependent linear ODEs the following

Corollary 1.2

^Consider

x⁰(t) =A(t)x(t) +q(t) (1.5)

with continuous coecients A() and continuous q(). Let denote L(t) =

Z t

0 A()]d:

Then, for a solution x() of (1.5) it holds

jx(t)^je^L(t)jx(0)^j+e^{L(t)Z t}

0 e^;L(s)jq(s)^jds ⁸t 0: 2

Aiming at similar estimates for the growth of the solutions of DAEs we describe a gener- alization of the concept of logarithmic norms which was presented by Marz 6]. Here we take care of the fact that the solutions of DAEs proceed in lower dimensional subspaces.

Our way is to use the correspondence of the solutions of DAEs to the solutions of corre- sponding inherent regular ODEs in invariant sub-manifolds . Linearization then lead to time-dependent linear ODEs with time-dependent invariant subspaces.

This leads us to a concept of logarithmic matrix norms with respect to subspaces which we will describe in chapter 2.It gives useful estimates for the solution of ODEs with invariant solution manifolds. We already mentioned that the logarithmic matrix norm generally depends on the chosen vector norm. But we found the freedom to use dierent vector norms (corresponding to the use of constant transformations of the solutions) to restric- tive to obtain optimal results and to compare results obtained by dierent approaches . We therefore investigate how the estimates are inuenced by time-dependent transforma- tions.

Based on this we are able to derive estimates for the solutions of DAEs. They are pre- sented in chapter 3. We start with linear index-1-DAEs, explain the concept of inherent regular ODEs and show that dierent choices for the inherent regular ODE lead to sys- tems that represent time-dependent transformations of each other. We show how these results are connected to those obtained by the concept of logarithmic norms for matrix pencils by Higueras/Garcia-Celayeta 4] and deal nally with nonlinear index-1 DAEs, where the state-space-form (see 3]) yields a nonlinear inherent regular ODE.

3

2 Logarithmic matrix norms with respect to sub- spaces

Let us consider the linear homogeneous time-dependent ODE

x⁰(t) = W(t)x(t) +q(t) t²IR (2.1)

x(t0) ² U(t0) (2.2)

whereW is a continuous matrix function andU(t) is a subspace ofIR^mdepending contin- uously on t in which all solutions of (2.1,2.2) proceed. i.e. U(t) is an invariant subspace of the ODE (2.1).

We are now interested in estimates for the growth of solutions in the invariant subspace U(t). Due to this invariant subspace the estimates obtained here may dier substantially from estimates concerning all solutions of the ODE (1.1).

We start with a denition of an induced matrix norm with respect to some subspace.

Denition 2.1

For a given vector norm^jjand a subspaceU IR^mwe dene the induced matrix norm with respect to the subspace U by

kA^kU := max_z

2U z⁶=0

jAz^j

jz^{j 8}A²L(IR^m):

Based on this we dene a logarithmic matrix norm with respect to a subspace.

Denition 2.2

For a given subspace U IR^m we dene the logarithmic matrix norm with respect to the subspace U by

A]^U := lim_h

!0⁺

kI +hA^{kU ;}1

h ⁸A ²L(IR^m):

Analogously to the usual theory of logarithmic matrix norm one easily justies the fol- lowing properties :

Let ^jj2 denote the Euklidian vector norm, E be a nonsingular matrix, and ^jjE

denote the vector norm with^jx^jE =^jEx^j2. Then, it holds

kA^k_UE =^kEAE^;1kEU2 :

4

Analogously, for the corresponding logarithmic matrix norm it holds

UEA] = ^EU2 EAE^;1] : (2.3)

U] can be written as

UA] = lim_h

!0⁺ ln^ke^hAkU

h : (2.4)

If^jj is an inner product norm ^jj2 =<>, then it holds

UA] = max_z

2U z⁶= 0< Azz >

jz^j2 : (2.5)

In the same analogous way estimates for the solutions of (2.1) in the invariant subspace U(t) are obtained: Let x() and x() be solutions of (2.1) in the invariant subspaceU(t).

Denote the dierence of the two solutions by v(t) := x(t)^;x(t). Considering m(t) :=^jv(t)^j=^jx(t)^;x(t)^j

and taking into account that v(t) = (x(t)^;x(t))²U(t) one obtains liminf_h

!0⁺ m(t+h)^;m(t)

h ^U(t)W(t)]m(t) and, hence, with L^U(t) = ^R₀^{t U()}W()]d

jv(t)^j e^LU(t)jv(0)^{j 8}t0:

It holds analogies to the theorem 1.1 and the corollary 1.2 with the only dierence that the solutions must ly in the invariant subspace U(t) and the logarithmic norm is taken with respect to this subspace.

Now, we ask , how do time-dependent transformations of variables inuence the estimates?

Let E() be a smooth matrix function and let E(t) be nonsingular for all t. Aiming to estimate the dierence of the transformed solutions ~v(t) =E(t)v(t) = E(t)x(t)^;E(t)x(t) we look for a dierential equation which is solved by them. Here we nd

~v⁰(t) = (Ev)⁰(t) = E⁰(t)v(t) +E(t)v⁰(t) =E⁰(t)v(t) +E(t)W(t)v(t)

= (E⁰E^;1+EWE^;1)(t)~v(t)

=: ~W(t)v~(t) (2.6)

5

for the dierence of the solutions ,and, analogously,

x~⁰(t) = (Ex)⁰(t) = (E⁰E^;1+EWE^;1)(t)~x(t) + (Eq)(t)

=: ~W(t)x~(t) + ~q(t) (2.7) for the solutions itself. Further, we note that ~x(t) v~(t)²(EU)(t) . Now, we obtain the estimate

j~v(t)^j exp(

Z t

0 (EU)(s)(E⁰E^;1+EWE^;1)(s)]ds)^jv~(0)^j

=: e^{~LEU(t) j}v~(0)^j =: ~c(t)^j~v(0)^j : (2.8) One may use the trivial estimate

j~v(t)^{j k}E(t)^kjv(t)^j

kE(t)^ke^LU(t)jv(0)^j

kE(t)^ke^LU(t)kE^;1(0)^kjv~(0)^j

=: c(t)^j~v(0)^j or, more accurate,

j~v(t)^{j k}E(t)^kU(t)e^LU(t)kE^;1(0)^kEU(0)jv~(0)^j

=: c^U(t)^jv~(0)^j: (2.9)

But, since there are used estimates of the form ^jAx^{j jj}A^kjx^j or ^jAx^{j jj}A^{kU j}x^j for x²U one may loose information.

Remark:

For scalar equations it holds c(t) = ~c(t) .

Proof:

^{Let be m = 1 , E(t) = ((t)) with(t)6= 0 ,W(t) = ( (t})) . Then one computes c(t) = ^j(t)^jexp(

Z t

0 ()d)^j(0)^;1j = (t) (0) exp(

Z t

0 ()d) and

c~(t) = exp(

Z t

0 (⁰()^;1() +() ()^;1())d)

= exp(ln()]^t0)exp(

Z t

0 ()d) = (t) (0)exp(

Z t

0 ()d) = c(t) q.e.d.

Remark:

For vector-valued equations ( m > 1 ) we don't have c(t) = ~c(t) in general.

This can already be seen for constant coecients W and constant scalings E , where we have

c(t) =^kE^kkE^;1ke^W]t and ~c(t) = e^EWE;1]t : 6

3 Logarithmic norms for index-1-DAEs

3.1 Linear index-1-DAEs

Consider the linear equation

A(t)x⁰(t) +B(t)x(t) =q(t) t²J IR (3.1) with continuous coecients. Introduce the basic subspaces

N(t) := kerA(t)IR^m

S(t) := ^fz ²IR^m :B(t)z ²imA(t)^gIR^m

and assume N(t) to be nontrivial as well as to vary smoothly with t Then, A(t) has constant rankr. The null-spaceN(t) determines what kind of functions we should accept for solutions of (2.1). To distinguish solution components which appear dierentiated (dierential components) and those components which do not (algebraic components) let us introduce a projector functionQ(t) onto N(t), i.e.

Q(t)² =Q(t) imQ(t) =N(t) t²J:

Q shall be chosen such that it is as smooth as N . Further, let P(t) := I^;Q(t) denote the complementary projector. Then we may split a solution x(t) and represent it as a sum of dierential and algebraic components in the form

x(t) = (Px)(t) + (Qx)(t) :

Now, the trivial identity AQ= 0 implies Ax⁰ =APx⁰ =A(Px)^0;AP⁰x

and, therefore, we use Ax⁰ as an abbreviation of A(Px)^0;AP⁰x in the following. Thus, (3.1) may be rewritten as

A(Px)⁰+ (B^;AP⁰)x=q (3.2)

which shows the function space

C_1N(JIR^m) :=^fx²C(JIR^m) :Px²C¹(JIR^m)^g 7

to become the appropriate one for (3.1). The realization of both the expression Ax⁰ and the spaceC_1N is independent of the special choice of the projector function (see e.g. 3]) . Obviously, S(t) is the subspace in which the homogeneous equation solution proceeds. It is determined by the constraints and may be only continuous. Recall the condition

S(t) N(t) =IR^m t²J (3.3)

to characterize the class of index-1 DAEs (3]). Let us denote the special projector onto N(t) which projects along S(t) by

Qcan(t):

As the subspace S(t) also the projector Qcan(t) may be only continuous. Aiming to decouple the DAE (3.1) into some inherent dynamical part and an assignment for the derivative-free parts of the solution we will make use of these projectors.

The index-1 condition (3.3) implies the matrices

G(t) := (A+BQ)(t) and Gcan(t) := (A+BQcan)(t) (3.4) to be nonsingular for allt²J. Exploiting the relations (see 3])

G^;1A = P (3.5)

G^;1B = Q+G^;1BP (3.6)

QG^;1B = Qcan (3.7)

we see that multiplying byG^;1 orG^;_can¹ leads to a natural scaling of the dae. Multiplying (3.2) by PcanG^;_can¹ and QcanG^;_can¹ we decouple this equation into the system

Pcanx⁰ +PcanG^;_can¹BPcanx = PcanG^;_can¹q (3.8)

Qcanx = QcanG^;_can¹q: (3.9)

Here, Pcanx⁰ is used as an abbreviation for Pcan(Px)^{0 ;}PcanP⁰x. Since both projector functionsP and Pcan project along the same subspace N it holds

PPcan = P and PcanP = Pcan: (3.10)

Thus, multiplying the equation (3.8) by P leads to the equivalent equation Px⁰ +PG^;_can¹BPcanPx = PG^;_can¹q:

8

Using (3.9) we now compute

Px⁰ = (Px)^0;P⁰x = (Px)^0;P⁰(Pcanx+Qcanx)

= (Px)^0;P⁰(PcanPx+QcanG^;_can¹q)

which leads us to an regular dierential equation in u:=Px, an so-called inherent regular dierential equation:

u⁰+ (^;P⁰Pcan+PG^;_can¹BPcan)u = (P +P⁰Qcan)G^;_can¹q: (3.11) Ifu solves (3.11) one easily computes that Qusolves the linear homogeneous ODE

(Qu)^0;Q⁰Qu = 0

Thus, we conclude from u(t0) ² imP(t0) or, equivalently, (Qu)(t0) = 0 that for all t it holds u(t) ² imP(t), i.e. imP(t) is an invariant subspace for (3.11). Is u a solution of (3.11) with u(t0)²imP(t0) then

x := Pcanu+QcanG^;_can¹q (3.12)

is a solution of (3.1). Vice versa, each solution of (3.1) can be represented in the form (3.12) with u(t0) :=P(t0)x(t0) .

Supposed that the canonical projector functionPcan is bounded the stability behavior of the solutions of the original DAE is determined by the solution properties of the inherent regular ODE (3.11) in the invariant subspace imP(t).

In general the supposition that Pcan is bounded may be rather restrictive and should be replaced by the condition that Pcan behaves moderately.

Remark 3.1

Starting the decoupling of the DAE (3.1) by multiplying byPG^;1 andQG^;1 is just another technique to obtain the same inherent regular equation (3.11) .

Proof:

(3.1) is equivalent to

Px⁰+PG^;1BPx = PG^;1q (3.13)

Qx+QcanPx = QG^;1q: (3.14)

9

Using (3.14) we obtain

Px⁰ = (Px)^0;P⁰x= (Px)^0;P⁰(Px+Qx)

= (Px)^0;P⁰(Px^;QcanPx+QG^;1q)

= (Px)^0;P⁰(Pcanx+QG^;1q) :

This leads to the following regular dierential equation inu:=Px:

u⁰+ (^;P⁰Pcan+PG^;1BP)u = (P +P⁰Q)G^;1q: (3.15) (3.15) coincides with (3.11). To show this we will check now, that the coecients coincide.

First, we look at the right-hand sides. Here we see:

(P +P⁰Q)G^;1 = (P +P⁰Qcan)G^;_can¹ since

(P +P⁰Q)G^;1Gcan = (P +P⁰Q)G^;1(A+BQcan) = (P +P⁰Q)(G^;1A+G^;1BQcan)

= (P +P⁰Q)(P + (Q+G^;1BP)Qcan) = (P +P⁰Q)(P +Qcan)

= P +P⁰Qcan :

Next, we look at the coecient matrices. Here the rst term ^;P⁰Pcan is obviously the same in both matrices. We will see that also the second terms coincide. First, we note that PG^;1BP =PG^;1BPcan . This equality follows since

QQcan=Qcan , hence G^;1BQcan =G^;1BQQcan =QQcan =Qcan and ,hence PG^;1BP ^;PG^;1BPcan =PG^;1BQcan^;PG^;1BQ=PQcan^;PQ= 0 : Then, we compute

PG^;1BP ^;PG^;_can¹BPcan = PG^;1BPcan^;PG^;_can¹BPcan

= G^;1(AG^;1BPcan^;AG^;_can¹BPcan)

= G^;1((G^;BQ)G^;1BPcan^;(Gcan^;BQcan)G^;_can¹BPcan)

= G^;1(BPcan^;B QG^;1B

| {z }

Qcan

Pcan^;BPcan+B QcanG^;_can¹B

| {z }

Qcan

Pcan = 0 : Summarizing we have

PG^;1BP =PG^;1BPcan =PG^;_can¹BPcan : (3.16) q.e.d.

10

As P is not uniquely determined also the inherent regular dierential equation is not unique. Let ~P be some other smooth projector function along N. Then we have with

u~⁰+ (^;P~⁰Pcan+ ~PG^;_can¹BPcan)~u = ( ~P + ~P⁰Qcan)G^;_can¹q (3.17) another inherent regular dierential equation in ~u := ~Px with im ~P as an invariant subspace. Again we obtain by

x := Pcanu~+QcanG^;_can¹q (3.18)

a solution of (3.2) if ~u is a solution of (3.17) with ~u(t0)²im ~P(t0).

If the subspace S(t) depends smoothly on t also the canonical Projector Pcan =I^;Qcan

may be chosen as a smooth projector function alongN . We then obtain

u⁰_can+ (^;P_can⁰ Pcan+PcanG^;_can¹BPcan)ucan = (Pcan+P_can⁰ Qcan)G^;_can¹q (3.19) as inherent regular dierential equation in ucan := Pcanx with S(t) = imPcan(t) as an invariant subspace. Solutions of (3.2) are composed by

x := ucan+QcanG^;_can¹q (3.20)

if ucan is a solution of (3.19) with ucan(t0)²S(t0) = imPcan(t0).

Aiming at estimates for the solutions of (3.1)we use the composition of the solutions (3.12) and apply the concept of logarithmic matrix norms for ODEs with invariant subspaces to the inherent regular ODE (3.11). This leads us to the following theorem:

Theorem 3.2

^Let x and x be both solutions of (3.1). Let denote L(t) =

Z t 0

imP()^;M()]d where M(t) = (^;P⁰Pcan+PG^;_can¹BPcan)(t)

is the coecient matrix in the inherent regular ODE . Further, let qP denote the right- hand side of the inherent regular ODE (3.11) : qP = (P +P⁰Qcan)G^;_can¹q : Then it holds for all t0 :

jx(t)^{j k}Pcan(t)^kimP(t)e^L(t)j(Px)(0)^j+e^L(t)

Z t

0 e^;L(s)jqP(s)^jds

+^j(QcanG^;_can¹q)(t)^j (3.21)

and

j(x^;x)(t)^jkPcan(t)^kimP(t)e^L(t)j(P(x^;x))(0)^j: (3.22) 11

Remark:

Under the condition thatS(t) is smooth we may choose P :=Pcan which leads directly toPcanx=ucan (see (3.20)). In relation to this we see that

kPcan^kS =^kPcan^kimPcan = max_z

2imPcanz⁶=0

jPcanz^j

jz^{j = maxw:Pcanw6=0}

jPcanPcanw^j

jPcanw^{j = 1} : With Lcan(t) =^R₀^{t S();}Mcan()]d where Mcan =^;P_can⁰ Pcan+PcanG^;_can¹BPcan

we obtain as a special case of (3.21) and (3.22)

jx(t)^j e^Lcan(t)j(Pcanx)(0)^j+e^{Lcan(t)Z t}

0 e^;Lcan(s)jqPcan(s)^jds+^j(QcanG^;_can¹q)(t)^j

j(x^;x)(t)^j e^Lcan(t)j(Pcan(x^;x))(0)^j: (3.23) Now, the question arises, how do these estimates depend on the chosen projector P ? First, note that only the rst term Pcanx = PcanPx = Pcanu of (3.12) causes estimates that may depend on the special choice of the projector P. Let us assume that Pand P~ are both smooth projector functions along N. Choosing u := Px resp. ~u := ~Px as dierential components of the solution x we obtain (3.11) resp. (3.17) as the inherent regular ODE. Now, we will show that these underlying inherent regular ODEs represent transformations of each other.

Theorem 3.3

^Let Pand ~P are both smooth projector functions along N. Let denote Q=I^;P and ~Q=I^;P^~ the corresponding complementary projectors. Then ,E = ~P+Q is a smooth matrix function and E(t) is nonsingular for all t . It is E^;1 = P + ~Q and EP = ~P . It holds: If u is a solution of (3.11) with u(t)²imP(t) then ~u:=Eu is a solution of (3.17) with u~(t)²im ~P .

Proof:

^{Let u} be a solution of (3.11) with u(t) ²imP(t) and ~u :=E u. Then it holds u~(t)²E(t)imP(t) = im ~P(t) and

u~⁰ = (Eu)⁰ = E⁰u+Eu⁰ =E⁰u+E(^;Mu+ (P +P⁰Qcan)G^;_can¹q)

= (E⁰E^;1;EME^;1)(Eu) +E(P +P⁰Qcan)G^;can¹q : First, we deal with the inhomogeneous term. It is:

E(P +P⁰Qcan) = ( ~P +Q)(P +P⁰Qcan) = ~P + ~^|{z}PP⁰

~P^0;~P⁰PQcan+QP⁰

|{z}

;Q⁰PQcan

= ~P + ~P⁰Qcan

12

sincePQcan = 0 . Next, we deal with the coecient matrix. SinceEu²im ~P foru²imP we are interested in the values of this coecient matrix applied to elements of im ~P only.

Here we compute:

(E⁰E^;1;EME^;1) ~P = (E^0;EM)(P + ~Q) ~P =E⁰P ^;EMP

= ( ~P⁰ +Q⁰)P ^;( ~P +Q)(^;P⁰Pcan+PG^;_can¹BPcan)P

= ~P⁰P +Q⁰P + ~PP^|{z}0

~P^0;~P⁰PPcan+QP⁰

|{z}

;Q⁰PPcan^;PG~ ^;_can¹BPcan

= ~P⁰Pcan^;PG~ ^;_can¹BPcan = ^;M~ =^;M~P :^~

Summarizing it follows that ~u solves (3.17). q.e.d.

Corollary 3.4

Under the suppositions of theorem 3.3 it holds

im ~P(t)^;M~(t)] = E(t)im P(t)(E⁰E^;1+E(^;M)E^;1)(t)] :

3.2 Relation to the logarithmic norm for matrix pencils

There is another idea to extend the usual concepts of induced matrix norm and logarithmic matrix norm by Higueras and Garcia-Celayeta 4] . They dene a logarithmic norm for matrix pencils and use this to study the growth of the solutions of linear time-dependent DAEs. We start quoting their denitions, some remarks and main theorems.

Denition 3.5

^Let AB ²L(IR^m) , let V IR^m a subspace such that V ^\kerA=^f0^g . Then we call V an admissible subspace for ^kA^k and dene

kAB^kV = max_v

2V^jAv^j6=0

jBv^j

jAv^{j (3.24)}

and

VAB] = lim_h

!0⁺

kAA^;hB^{kV ;}1

h : (3.25)

Remarks:

ForA=I and V =IR^m it is ^IRmIB] = ^;B] .

If^jj is an inner product norm, then

VAB] = max_x

2V Ax⁶=0< Ax^;Bx >

< AxAx > : 13

IfE is nonsingular and ^jx^jE =^jEx^j2 then

VEAB] = ^V2EAEB] and ^VAB] = ^EVAE^;1BE^;1] :

It holds the following theorem concerning the growth of the solutions of the linear homo- geneous DAE

A(t)x⁰(t) +B(t)x(t) = 0 : (3.26)

Theorem 3.6

^Let x be a solution of (3.26) such that x(t) ² V(t) where V(t) IR^m is an admissible subspace for ^kA(t)^k. Then it holds

j(Ax)(t)^je^{R0tV( )A() B();A0()]d j}(Ax)(0)^{j 8}t0 : (3.27)

Corollary 3.7

Let (3.26) have index 1. Then the solutions of (3.26) lie in S(t) and S(t) is an admissible subspace for ^kA(t)^k. Hence, it holds the estimate (3.27) with V(t) :=S(t) :

j(Ax)(t)^j e^LA(t)j(Ax)(0)^{j 8}t0 (3.28) where LA(t) =

Z t

0 S()A()B()^;A⁰()]d :

We now want to relate these results to those of the previous chapter. Therefore, let x be a solution of the linear homogeneous DAE (3.26) and let (3.26) have index 1. Then it holds x =Pcanx = Pcanu , where u solves the homogeneous inherent regular ODE u⁰ =^;Mu u(0) :=P(0)x(0) ( see (3.9),(3.11 ) withq= 0 ). This yields the estimate

j(Px)(t)^j e^L(t)j(Px)(0)^{j 8}t 0 (3.29) where L(t) =

Z t

0 imP()^;M()]d :

The estimate (3.29) concerns Px, while (3.28) concerns Ax. Note that it holdsA=GP, Gis a smooth matrix function and G(t) is nonsingular for allt . We may now start from the inherent regular ODE (3.11) inu:=Pxand look for the corresponding ODE for the transformed variables ^u:=Gu=GPx=Ax . This results in

u^⁰ = (G⁰G^;1+G(^;M)G^;1)u := ^;Mu ^ u^(0) := (GPx)(0) = (Ax)(0) (3.30)

14

and gives estimates

j(Ax)(t)^j e^{^L(t) j}(Ax)(0)^{j 8}t0 (3.31)

where ^L(t) =

Z t

0 imA()(G⁰G^;1+G(^;M)G^;1)()]d :

Now, we show that this estimate coincides with the estimate (3.28) obtained by Higueras and Garcia-Celayeta 4] with the help of their logarithmic norm for matrix pencils:

Theorem 3.8

Let the linear time-dependent DAE (3.1) have index 1. Then it holds

S(t)A(t)B(t)^;A⁰(t)] = ^imA(t)(G⁰G^;1+G(^;M)G^;1)(t)] ⁸t and ,hence , LA(t) = ^L(t) ⁸t0 .

Proof:

By denition it is

SAB^;A⁰] = liminf_h

!0⁺

kAA^;h(B^;A⁰)^kS;1

h ^{where k}AC^kS= max_v

2S Av⁶=0

jCv^j

jAv^j and

imA^;M^] = liminf_h

!0⁺

kI+h(^;M^)^kimA;1

h ^{where k}W^kU = max_u

2U u⁶=0

jWu^j

ju^j It remains to show that ^kAA^;h(B^;A⁰)^kS = ^kI^;hM^{^kimA} ,i.e.

v²maxS Av⁶=0

j(A^;h(B^;A⁰))v^j

jAv^{j = maxu2imA u6=0j(}I^;hM^{^})u^j

ju^j : Since

u²immaxA u⁶=0

j(I^;hM^{^})u^j

ju^{j = maxv Av6=0j(}I^;hM^{^})Av^j

jAv^{j = maxv=Pcanv Av6=0j(}I^;hM^{^})Av^j

jAv^j

it remains to show that ^;MAv^ = (A^0;B)v ⁸v ²S ,i.e. ^;MAP^ can = (A^0;B)Pcan . But this can be seen by:

MAP^ can = ^MA = (^;G⁰G^;1+GMG^;1)A = ^;G⁰P +GMP

= ^;(A+BQ)⁰P + (A+BQ)(^;P⁰Pcan+PG^;_can¹BPcan)

= ^;(A+BQ)⁰P ^;(A+BQ)P⁰Pcan+AG^;_can¹BPcan

= ^;(A+BQ)⁰P ^;(A^{0 ;}(A+BQ)⁰P)Pcan+ (Gcan^;BQcan)G^;_can¹BPcan

= ^;A⁰PPcan+BPcan^;BQcanPcan = ^;(A^0;B)Pcan :

q.e.d.

15

Finally, we consider the special case that the DAE (3.1) is already scaled such thatA =P is a projector function ( this includes semi-explicit DAEs whereA =P =

I 0

) . Here, (3.28) as well (3.29) supply estimates of Px. Corresponding to that we will show that then the constants in (3.28) and (3.29) coincide, i.e. the value of the logarithmic norm for the pencil (PB) with respect to the subspace S(t) coincides with the value of the logarithmic norm for the inherent regular equation with respect to the invariant subspace imP .

Theorem 3.9

Let the linear time-dependent DAE (3.1) have index 1 and let A be a smooth projector function, i.e. A=P . Then it holds

S(t)P(t)B(t)^;P⁰(t)] = ^imP(t);M(t)] ⁸t

Proof:

Following the steps in the proof of theorem3.8 it remains to show that

;MPcan = (P^0;B)Pcan . This can be seen by MPcan = ^;P⁰Pcan+PG^;_can¹BPcan

= ^;P⁰Pcan+ (Gcan^;BQcan)G^;_can¹BPcan

= ^;P⁰Pcan+BPcan = ^;(P^0;B)Pcan

q.e.d.

Remark:

Applying this result to an semi-explicit index-1 DAE Ax⁰+Bx:=

I 0

( x⁰₁ x⁰2 ) +

B11 B12

B21 B22

( x1

x2 ) = ( q1

q2 )²IR^m1 IR^m2 we obtain

N = ^fz :z1 = 0^g S = ^fz :B21z1+B22z2 = 0^g = ^fz :z2 =^;B22^;1B21z1^g M =

B11^;B12B22^;1B21 0

0 0

forP :=A=

I 0

while Pcan =

I 0

;B₂₂^;1B21 0

and, hence

SA^;B] = ^Rm1f0gm2;M] = ^;(B11^;B12B₂₂^;1B21)] :

16

3.3 Examples

3.3.1 Example 1

We consider the linear DAE ( 3.1 ) with the coecients A(t) =

1 0 t 0

B(t) =I :

The solutions of the homogeneous DAE can be represented in the form x(t) =

1 t

e^;tx1(0) :

For the subspaces N(t) and S(t) we obtain N(t) = kerA(t) = span^f

0 1

g and S(t) = imA(t) = span^f

1 t

g:

In this example A is itself a projector function onto imA = S such that A = Pcan and Gcan=A+BQcan =Pcan+Qcan =I .

Now, let x be a solution of the linear homogeneous DAE (3.26).We want to see what the presented estimates for the growth of solutions give here . As mentioned there is a variety of possibilities to split the solution in dierential and algebraic components. Choosing the constant orthogonal projector ~P =

1 0 0 0

along N would yield a splitting x(t) = ~Px(t) + ~Qx(t) =

x1(t) 0

+

0 x2(t)

: Choosing the canonical projectorPcan(t) =A(t) yield

x(t) =Pcan(t)x(t) +Qcan(t)x(t) =

x1(t) tx1(t)

+

0

; tx1(t) +x2(t)

:

First we note that because of (3.9) for the solution of the homogeneous DAE Qcanx= 0 is fullled such that

x = Pcanx+Qcanx = Pcanx :

Based on this dierent splittings there are dierent possibilities to estimate the growth of x=Pcanx :

17

Choosing an inherent regular ODE (3.19) inucan =Pcanx we obtain the coecient matrix

Mcan(t) = (^;P_can⁰ Pcan+PcanG^;_can¹BPcan)(t) = (^;A⁰A+AIA)(t)

= ^;

0 0 0

+

1 0 t 0

=

1 0

t^; 0

and compute

S(t)2 ^;Mcan(t)] = max_x

2S(t) x⁶=0<^;Mcan(t)xx >2

< xx >2 = <^;Mcan(t)

1 t

1 t

>

<

1 t

1 t

>

= <^;

1 t^;

1 t

>

1 + ( t)² = ^;1 + ²t 1 + ( t)² By theorem 3.9 we also have

S(t)2 A(t)B(t)^;A⁰(t)] = ^S(t)₂ Pcan(t)I^;P_can⁰ (t)] = ^S(t)₂ ^;Mcan(t)]

= ^;1 + ²t 1 + ( t)² :

With Lcan(t) = ^R₀^{t S();}Mcan()]d = ^R₀^t(^;1 +₁₊₍₎^{2 2})d and (3.23) we have

jx(t)^je^Lcan(t)j(Pcanx)(0)^j = e^Lcan(t)jx(0)^j

Choosing the constant orthogonal projector ~P =

1 0 0 0

along N and an inherent regular ODE (3.17) in ~u= ~Pxwe obtain the coecient matrix

M~(t) = (^;P~⁰Pcan+ ~PG^;_can¹BPcan)(t) = ( ~PG^;_can¹BPcan)(t)

=

1 0 0 0

t1 00

=

1 0 0 0

and compute

im ~P

2 ^;M~] = _x max

2im ~P x⁶=0<^;Mxx >^~ 2

< xx >2 = <^;M^~

1 0

1 0

>

<

1 0

1 0

>

= ^;1 :

18

Because of ~L(t) = ^R₀^t(^;1)d = ^;t we obtain

jPx~ (t)^je^;tjPx^~ (0)^{j 8}t 0 and

jx(t)^j=^jPcan(t) ~Px(t)^{j k}Pcan(t)^{kim ~P}e^;tjPx^~ (0)^j

kPcan(t)^{kim ~P}e^;tkP^~kimPcan(0)jx(0)^j :

Because ofPcan(0) = ~P it is ^kP~^kimPcan(0) =^kP~^{kim ~P} = 1. Further, we compute:

kPcan(t)^{kim ~P} = max_x

2im ~P x⁶=0

Pcan(t)x^j

jx^{j =} Pcan(t)e1^j

je1^j =

j 1

t

j

1 =

p1 + ( t)² : Hence, it follows

jx(t)^j=^jPcan(t)x(t)^jp1 + ( t)²e^;tjPcan(0)x(0)^j=^p1 + ( t)²e^;tjx(0)^j : Note that we end with the same result if we apply the estimates based on the logarithmic norm for matrix pencils to the DAE scaled by ~G^;1 = (A+BQ^~)^;1 = (Pcan+ ~Q)^;1 . We then obtain coecients

A~= ~G^;1A= ~P =

1 0 0 0

and ~B = ~G^;1B = ~G^;1 = ~P +Qcan =

1 0

; t 1

: Since a scaling of the DAE does not change the solutions, the subspaces N = ~N and S = ~S do not change. Therefore, computing

S(t)2 ~AB^~(t)] = max_x

2S(t) x⁶=0<Ax^{~ ;}Bx >^~

<Ax^~ Ax >^{~ =}

<A^~

1 t

^;B^~

1 t

>

<A^~

1 t

A^~

1 t

>

= <

1 0

^;

1 0

; t 1

>

<

1 0

1 0

> ^=;1 we again obtain the estimate

j( ~Ax)(t)^j=^j( ~Px)(t)^j e^;tj( ~Px)(0)^{j 8}t0

j(Ax)(t)^j=^j( ~GAx^~ )(t)^{j k}G~(t)^{kim ~P} e^;tj( ~Px)(0)^j

= ^p1 + ( t)²e^;tjx(0)^j : 19

Since dimim ~P = dimimPcan(t) = 1 we meet the same nice property that all this estimates coincide like for scalar ODEs. Indeed, we compute

p1 + ( t)²e^;t =e^Lcan(t) since

Lcan(t) =

Z t

0 S()^;Mcan()]d =

Z t

0 (^;1 + ² 1 + ( )²)d

= ^;t+

Z t 0

2

1 + ( )²)d = ^;t+

Z t

0 (ln^p1 + ²)⁰d

= ^;t+ln^p1 + ²t :

3.4 Nonlinear index-1 DAEs

We consider nonlinear DAEs

f(x⁰(t)x(t)t) = 0 (3.32)

where f is dened on IR^{m DJ} with ^D IR^{m J} IR. f is supposed to be at least continuous and to be continuously dierentiable with respect to the rst and second argument. We will use the concept of tractability with index 1 introduced by Griepentrog/Marz 3]. Using the notations:

A(yxt) := f_y⁰(yxt) B(yxt) := f_x⁰(yxt)

S(yxt) := ^fz ²IR^m :B(yxt)z ²imA(yxt)^{g 8}(yxt)²IR^mDJ we dene

Denition 3.10

The DAE (3.32) has (tractability - ) index 1 in IR^mDJ i

kerA(yxt) does not depend on y and x, and N(t) := kerA(yxt)

is smooth, and

N(t)^\S(yxt) = ^f0^{g 8}(yxt)²IR^mDJ. 20