Stability of periodic solutions of index-2 differential algebraic systems

Stability of periodic solutions of index-2 dierential algebraic systems

Rene Lamour, Roswitha Marz and Renate Winkler Humboldt-University Berlin

Unter den Linden 6, D-10099 Berlin, Germany

Abstract

This paper deals with periodic index-2 dierential algebraic equations and the question whether a periodic solution is stable in the sense of Lyapunov. As the main result, a stability criterion is proved.This criterion is formulated in terms of the original data so that it may be used in practical computations.

Introduction

This paper deals with periodic index-2 dierential algebraic equations (DAEs) of the form A(x t)x⁰ +b(x t) = 0

and the question whether a periodic solution is stable in the sense of Lyapunov. As the main result, a stability criterion is proved. It sounds as nice as the well-known original model for regular ordinary dierential equations (ODEs).

This criterion is formulated in terms of the original data so that it may be used in prac- tical computations, too.

In view of various applications we try to do with smoothness conditions as low as possible.

The notion of stability to be used should reect the geometrical meaning of Lyapunov stability properly. In the case of index-2 DAEs we have to consider also the so-called hidden constraints. However, in practice, we cannot proceed on the assumption that the state manifold and its tangent bundle are explicitly available. This is why we use special projectors to catch the neighbouring solutions on that manifold properly in order to com- pared with the given solution (e. g. Marz 9]).

We follow the lines of the standard ODE theory that combines linearization and Lyapunov reduction. Hence, what we have to do in essence is

- to clarify what Lyapunov reduction means for index-2 DAEs and to construct the respective transformations and

- to make sure that linearization works as expected.

1

The paper is organized as follows. Fundamentals on linear continuous coecient index-2 DAEs and on linear transformations of them are given in Section 1 and 2. In Section 3, we construct special regular periodic matrix functions that transform a given periodic index-2 DAE into a constant coecient Kronecker normal form. By this we prove a kind of Floquet-Theorem and a Lyapunov-reduction for index-2 DAEs (Theorems 3.1 and 3.2).

Section 4 concerns nonlinear DAEs. There, the main result of the present paper, the sta- bility criterion for periodic solutions, is given by Theorem 4.2. In Section 5, we discuss an application to multibody systems. Finally, we show the practical use by checking the stability of an oscillator circuit numerically.

With the present paper we continue and complete, for the time being, our attempts to generalize standard stability results known for regular ODEs to low-index DAEs.

In Lamour, Marz and Winkler 12] a respective reduction theorem and stability criterion were obtained for index-1 DAEs. The Perron-Theorem for index-2 DAEs proved in Marz 8] provides an appropriate theoretical background for Theorem 4.2 of the present paper.

In this context, it should be pointed out once more that index-2 DAEs are much more complex than those having index 1, mainly in the particular case of non-autonomous equations.

The authors are of the opinion that the stability results obtained are sucient, for the moment, for non-stationary solutions of DAEs. As far as the stability of stationary solu- tions of easier autonomous DAEs is concerned, this problem has been under consideration for a longer time, (e.g. Griepentrog and Marz 2]).

It should be mentioned that there are nice results in a more general geometric context (e. g.

Reich 13]),which provides a good theoretical insight into the case of smooth systems.

1 Linear continuous coecient equations

Consider the linear equation

A(t)x⁰(t) +B(t)x(t) =q(t) t ²J IR (1.1) with continuous coecients. Introduce the basic subspaces

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

S(t) := ^fz ²IR^m:B(t)z²imA(t)^gIR^m

and assume N(t) to be nontrivial as well as to vary smoothly with t, i.e., to be spanned by continuously dierentiable basis functions n1 ::: nm^;r ²C¹(J IR^m). Then, A(t) has constant rank r.

The smoothness of N(t) is equivalent (see e.g. Griepentrog and Marz (1989)) to the existence of a projector functionQ²C¹(J L(IR^m)) such that

Q(t)² =Q(t) imQ(t) =N(t) t²J:

Further, let P(t) :=I^;Q(t).

2

The nullspace N(t) determines what kind of functions we should accept for solutions of (1.1). Namely, the trivial identity A(t)Q(t) = 0 implies

A(t)x⁰(t) =A(t)P(t)x⁰(t) =A(t)(Px)⁰(t)^;A(t)P⁰(t)x(t)

and, therefore, we use Ax⁰ as an abbreviation of A(Px)^0;AP⁰x in the following. Thus, (1.1) may be rewritten as

A(t)(Px)⁰(t) + (B(t)^;A(t)P⁰(t))x(t) =q(t) (1.2) which shows the function space

C_1N(J IR^m) :=^fy ²C(J IR^m) :Py²C¹(J IR^m)^g

to become the appropriate one for (1.1). The realization of both the expression Ax⁰ and the space C_1N is independent of the special choice of the projector function.

Hence, we should ask for C_1N-solutions, but not necessarily for C¹-solutions.

Obviously, S(t) is the subspace in which the homogeneous equation solution proceeds.

Recall the condition

S(t)N(t) =IR^m t²J (1.3)

to characterize the class of

index-

1 DAEs (Griepentrog and Marz (1986)). (1.3) implies the matrix

A1(t) :=A(t) + (B(t)^;A(t)P⁰(t))Q(t) (1.4) to be nonsingular for all t ² J. Multiplying (1.2) by PA^;1¹ and QA^;1¹ we decouple this equation into the system

(Px)^0;P⁰Px+PA^;1¹BPx = PA^;1¹q

Qx+QA^;1¹BPx = QA^;1¹q ^(1.5)

which immediately provides a solution expression. We have

x=Px+Qx= (I^;QA^;₁¹B)y+QA^;₁¹q=: can(1)y+QA^;₁¹q wherey solves the regular linear ODE

y^0;P⁰y+PA^;1¹By=PA^;1¹q

and starts aty(to)²imP(to) for someto ²J.

Since can(1)(t) = (I^;(QA^;₁¹BP)(t))P(t) represents the canonical projector onto N(t) alongS(t), we knowS(t) to be lled by the homogeneous equation solution. On the other hand, nontrivial parts QA^;1¹q of the inhomogeneity cause the solution to bulge from the subspaceS(t), and to cover the wholeIR^m. Of course, such eects do not occur in regular ODEs.

For higher index DAEs, in particular for those having index 2, the condition (1.3) gets 3

lost. Consequently, dierent subspaces are relevant for those equations. In contrary to the above index-1 case, now a certain subspace of S(t) is only lled by the homogeneous equation solution.

Introduce the two additional subspaces N1(t) := kerA1(t)

S1(t) := ^fz ²IR^m :B(t)P(t)z²imA1(t)^g:

Denition:

The DAE (1.1) is said to be index-2 tractable if the conditions dim(N(t)^\S(t)) = const>0

N1(t)S1(t) = IR^m t²J ^(1.6)

are valid.

Remarks:

1) It holds that N1 = (I^;PA⁺(B^;AP⁰)Q)(N ^\S), and, consequently, N1(t) has the same dimension asN(t)^\S(t). Therefore, (1.6) impliesA1(t) to have constant rank.

2) (1.6) implies both the matrices G2(t) :=A1(t) +B(t)P(t)Q1(t) and

A2(t) := A1(t) + (B(t)^;A1(t)(PP1)⁰(t))P(t)Q1(t)

=G2(t)(I^;P1(t)(PP1)⁰(t)PQ1(t))

to become nonsingular, but A1(t) to be singular now. Thereby, Q1(t) denotes the projector onto N1(t) along S1(t), P1(t) :=I^;Q1(t). By construction, Q1 is contin- uous. In the followingQ1 is assumed to be C_1N.

3) With B1 := (B^;A1(PP1)⁰)P the subspace S1(t) rewrites S1(t) =^fz ²IR^m :B1(t)z ²imA1(t)^g:

We obtain the identities

Q1 =Q1A^;₂¹B1 =Q1A^;₂¹BP =Q1G^;₂¹BP Q1Q= 0: (1.7) 4) Each DAE (1.1) having Kronecker index{2 is index{2 tractable (Marz 6]).

The index-2 conditions (1.6) imply the decompositions IR^m = N(t)P(t)S1(t)P(t)N1(t)

which are relevant now instead of (1.3), which was true in the index-1 case.

4

Let us introduce further projectors T, which projects pointwisely onto S(t)^\ N(t) = imQ(t)Q1(t) andU :=I^;T. Taking this into account, we decompose the DAE solution x²C_1N(J IR^m) into

x=TQx+PP1x+ (PQ1+UQ)x=:w+u+v: (1.8) Multiplying (1.2) by A^;₂¹ forms (1.2) into

P1P(Px)⁰+A^;₂¹BPP1(I+P1(PP1)⁰PQ1)Q1+Q=A^;₂¹q: (1.9) Multiplying (1.9) by PP1, TQ and (PQ1 +UQ), respectively, and carrying out a few technical computations, we decouple the index-2 DAE into the system

u^{0 ;}(PP1)⁰u+PP1A^;₂¹Bu = PP1A^;₂¹q (1.10) QQ1(PP1)⁰u^;QQ1(Pv)⁰+TQP1A^;₂¹Bu+w = TQP1A^;₂¹q (1.11) UQA^;2¹Bu+v = (PQ1+UQ)A^;2¹q: (1.12) Looking at system (1.10)-(1.12) we know the index{2 DAE (1.1) to become solvable if PQ1A^;₂¹q belongs toC¹.

Remarks:

1) We ask for C_1N solutions again. Any higher regularity of solutions, say C¹, needs additional smoothness of the coecients, projectors and sources involved. Again, the decoupled system provides some help to state right conditions. In particular, for C¹ solutions at least Q1A^;₂¹q²C², QP1A^;₂¹q ²C¹ have to be valid additionally.

2) The inherent regular ODE (1.10) is aected by the complete coecient matrix PP1A^;2¹B ^;(PP1)⁰, but not only by the rst term PP1A^;2¹B. If (PP1)(t) varies quickly, the second term (PP1)⁰ may be the dominant one. This should be taken into account when considering the asymptotic behaviour.

Next we turn shortly to the homogeneous equation. Forq = 0 the system (1.10) { (1.12) yieldsv = 0 and

x = (I+QQ1(PP1)^{0 ;}QP1A^;₂¹B)u

= (I+ (QQ1(PP1)^0;QP1A^;₂¹B)PP1)u =: ku:

The matrix k(t) is nonsingular. This denes the canonical projector for the index{2 case can(2) :=kPP1

which projects on the solution space. Clearly, not the whole spaceS(t) is lled by solutions of the homogeneous equation, as in the index{1 case, but a proper subspace ofS(t) only.

The fundamental matrixX(t) as a matrix solution of the homogeneous equation with the initial values

(PP1)(t0)(X(t0)^;I) = 0 has the structure

X(t) = can(2)(t)Y(t)(PP1)(t0)

whereY(t) represents the ordinary fundamental matrix of the ODE (1.10).

In the following we simply use can for can(2). 5

2 General linear transformations

We have characterized the index{2 condition by (1.6). Do linear nonsingular transfor- mations x(t) = F(t)x(t) of the unknown function keep this condition invariant? It is adequate to choose F ²C_1N.

The coecients of (1.1) are transformed by

A=AF B =BF +AF⁰: (2.1)

In this context AF⁰ is used as an abbreviation of A((PF)^0;FP⁰) (see 12]).

The spaces N and S are transformed into N =F^;1N and S =F^;1S

hence,

N ^\S=F^;1(N^\S):

The nullspace N(t) varies smoothly with t if N(t) does so ( 12], Lemma 2.1 ). Let Q denote a C¹ projector function onto ker A, but A1 S1 etc. the respective matrices and subspaces formed by A B.

Lemma 2.1

^:

A1 = A1F(I^;F^;1QFP) S1 = F^;1S1 and

N1 = (I^;F^;1QFP)F^;1N1

S1 = (I^;F^;1QFP)F^;1S1:

Proof:

It holds that PFQ = 0 and PF^;1Q = 0 because AQ = 0(= APFQ) and AQ= 0(= APF ^;1Q). The transformed chain matrix A1 is

A1 = A+ B0Q = AF + (BF +A^f(PF)^0;P⁰F^g;AFP⁰) Q (2.2)

F Q=QF Q= AF +BQFQ^;AP⁰QFQ (2.3)

= (A+ (B ^;AP⁰)Q)(PF +FQ) (2.4)

= A1F(F^;1PF + Q) =A1F(F^;1PFP+ Q) (2.5)

= A1F(I^;F^;1QFP) (2.6)

with nonsingular (I^;F^;1QFP).

This shows that im A1 = imA1 and N1 = (I^;F^;1QFP)F^;1N1. Further

S1 := ^fz: BPz²im A1^g (2.7)

= ^fz: (BF +AF⁰) Pz²im A1^g (2.8)

= ^fz:B(P +Q)FPz²im A1^g (2.9)

= ^fz:BPFPF ^;1Fz+ ((B^;AP⁰)Q+AP⁰Q)FPz²im A1^g (2.10)

= ^fz:BPFz²im A1^g (2.11)

6

i.e., S1 =F^;1S1. Finally, it holds that

(I ^;F^;1QFP)F^;1S1 = ^fz:BPF(I^;F^;1QFP)z ²im A1^g

= ^fz:BPFz²im A1^g

= F^;1S1 = S1:

Theorem 2.2

The tractability index 2 is invariant under transformations F ² C_1N and it holds that PQ1 ²C¹ i PQ1 ²C¹:

Proof:

The relations of Lemma 2.1 lead to N1^\S1 = (I^;F^;1QFP)F^;1(N1^\S1). Because of the non-singularity of I ^;F^;1QFP, the relations N1 ^\S1 = ^f0^g and N1 ^\S1 = ^f0^g are equivalent. Taking into account that N ^\S= F^;1(N ^\S), we know the invariance of index{2 tractability. The transformed projector Q1 is given by

Q1 = (I+F^;1QFP)F^;1Q1F(I ^;F^;1QFP) therefore PQ1 = PF^;1Q1F = PF^{| {z;1}}

2C¹ PQ1

|{z}

2C^{1 |{z}}PF

2C¹

2C¹.

Denition.

Two linear DAEs given onIRare said to be kinematically equivalent if there are nonsingular matrix functions F ² C_1N, E ² C which transform the coecients by (2.1), and if sup_t

2IR^jF(t)^j<¹, sup_t

2IR^jF(t)^;1j<¹.

3 Linear periodic index{2 DAEs

Let us turn to linear homogeneous DAEs with periodic coecients

A(t)x⁰(t) +B(t)x(t) = 0 (3.1)

whereA B ²C(IR L(IR^m)), A(t) =A(t+), B(t) =B(t+) for all t²IR.

Note that the spacesN(t) andS(t) are -periodic since the coecientsA(t) andB(t) are so.Does a smooth periodic basis of N(t) exist ?

N(t) is supposed to be smooth. Consequently the orthoprojectorQ^?(t) :=I^;A⁺(t)A(t) depends continuously dierentiable on t. Obviously, it holds that Q^?(t) = Q^?(t+).

Given a basisn₀₁ ::: n_0m^;_r ²R^m of N(0), then the solutions of the initial value problems n⁰ =P^?0n n(0) =n0i i= 1 ::: m^;r

form a smooth basis of N(:). These functions are periodic, namely ni(t+) = exp(

Z t+

0 P^?0(s)ds)n0i

= exp(P^?(t+)^;P^?(0))n_0i

= exp(P^?(t)^;P^?(0))n_0i =ni(t): 7

Let us agree to choose periodic smooth projectorsQ P in the following. Then the matrices A1, etc. but also the subspaces N1 S1, are periodic, hence the projector Q1 is periodic, too. SincePQ1 is continuously dierentiable , we nd periodicC¹{functionsb1 ::: bthat span imPQ1.

In this section, we show how to transform a linear periodic index{2 DAE into a kinemati- cally equivalent one with constant coecients A and B. To construct such a transforma- tion we decompose IR^m using the projectors. Note that

Q1 = QQ1+PQ1 = (QQ1+I)PQ1

imQQ1 = N ^\S = imT = imTQ:

WithN = imQQ1imUQwe have the splittingIR^m= imPP1imPQ1imQQ1imUQ:

We span imPQ1 by -periodic functions b1(t):::b(t) ² C¹. With qi := (I +QQ1)bi ²

imQ1 we have a basis bi = Pqi for imPQ1 and ni = Qqi is a basis for imQQ1. With imPP1 =:

span

^fp1:::pr^;g pi ² C¹ and imUQ =:

span

^fn+1:::nm^;r^g we introduce the nonsingular matrix

V(t) := (p1 :::pr^; b1 :::b n1 :::n n+1 :::nm^;r): With the aid of V the projectors can be represented by P(t) =V

0

B

B

@

I I 0 0

1

C

C

AV^;1,PP1(t) =V

0

B

B

@

I 0 0 0

1

C

C

AV^;1 and PQ1(t) =V

0

B

B

@

0 I 0 0

1

C

C

AV^;1.

We are interested in a similar representation for the projector Q1, too. By

V

0

B

B

@

0 I I 0

0

1

C

C

A=

0

B

B

B

B

B

B

@

0 0 0

... ... ...

... b1+n1:::b+n ... ...

... ... ...

0 0 0

1

C

C

C

C

C

C

A

=

0

B

B

B

B

B

B

@

0 0 0

... ... ...

... q1:::q ... ...

... ... ...

0 0 0

1

C

C

C

C

C

C

A

=Q1V

we see Q1 =V

0

B

B

@

0 I I 0

0

1

C

C

AV^;1

We aim at constructing a transformation that transforms the time varying linear DAE into a constant one. Remember that, in the index{2 case, can=kPP1with a nonsingular

8

periodic k. The fundamental matrix given by AX⁰+BX = 0 (PP1)(0)(X(0)^;I) = 0 has the representation

X(t) = can(t)Y(t)(PP1)(0) (3.2)

= k(t)V(t)

0

B

B

@

I 0 0 0

1

C

C

AV^;1(t)Y(t)V(0)

0

B

B

@

I 0 0 0

1

C

C

A

| {z }

=:

0

B

B

B

B

B

B

@

Z(t) 0 0

0

1

C

C

C

C

C

C

A

V^;1(0) (3.3)

with Z(0) =I:

Also the so{called monodromy matrixX() is given by X() =k(0)V(0)

0

B

B

@

Z() 0 0

0

1

C

C

AV^;1(0):

From linear algebra (see e.g. 10]) it is known that every nonsingular matrix C ²L(IR^r) can be represented in the form

C=e^W with W ²L(IC^r) and C² =e^W with W ²L(IR^r): Now, let

Z() =e ^W0 W0 ²L(IC^r) (3.4)

and

Z(2) =Z()² =e^{2 W0} W0 ²L(IR^r): (3.5) We introduce the transformation

F(t) := k(t)V(t)

0

B

B

@

Z(t)e^;tW0 I I

I

1

C

C

A (3.6)

= X(t)V(0)

0

B

B

@

e^;tW0 0 0

0

1

C

C

A+k(t)V(t)

0

B

B

@

0 I I I

1

C

C

A: (3.7)

From (3.6) we see that F is nonsingular and not smooth, butPF ²C¹. 9

Theorem 3.1

The fundamental matrix X(t) of the DAE (3.1) can be written in the form X(t) =F(t)

0

B

B

@

e^tW0 0 0

0

1

C

C

AF(0)^;1

where F ²C_1N(IR L(IC^m)) is nonsingular and -periodic.

Proof

: We will show thatF given by (3.6) realizes this representation, indeed. First, we look at the transformed spaces and projectors. The basis functions of the nullspace N are represented by ni =V(t)ei+r i = 1:::m^;r, where ei are the unit vectors. What is the transformed nullspace N =F^;1N . We consider

F^;1ni =

0

B

B

@

e^tW0Z^;1(t) I I

I

1

C

C

AV^;1(t)k^;1(t)ni

=

0

B

B

@

e^tW0Z^;1(t) I I

I

1

C

C

AV^;1(t)ni since k^;1ni =ni

=

0

B

B

@

e^tW0Z^;1(t) I I

I

1

C

C

Aei+r =ei+r: It follows that

N =

span

^fer+1 ::: em^g:

Therefore, in the transformed nullspace we can choose the projectors Q=

0

B

B

@

0 0 I I

1

C

C

A

10

and P =I^;Q. What about PQ1 ? PQ 1 = PPQ 1 = PF^;1PQ1F

= P

0

B

B

@

e^tW0Z^;1(t) I I

I

1

C

C

AV^;1(t)k^;1(t)PQ1k

| {z }

PQ¹ V

| {z }

0

B

B

B

B

B

B

@

0 I 0 0

1

C

C

C

C

C

C

A 0

B

B

@

Z(t)e^;tW0 I I

I

1

C

C

A

=

0

B

B

@

0 I 0 0

1

C

C

A

It follows that PP1 = P ^;PQ 1 =

0

B

B

@

I 0 0 0

1

C

C

A.

The general transformation rules for the coecients A and B are given by (2.1). Hence, by the special transformation (3.6) the coecients

A=AF B =BF +AF⁰

are well dened. As we have constant projectors P Q PQ 1 etc., the following relations become true

A^;₂¹A = P1P =I ^;Q^;Q1

A^;₂¹B = A^;₂¹BPP1+ Q1+ Q:

11

In particular, we have now

BPP1 = (BF +A^f(PF)^0;P⁰F^g) PP1

= (BXV(0)

0

B

B

@

e^;tW0 0 0

0

1

C

C

A+A^f(PF)⁰PP1^;P⁰XV(0)

0

B

B

@

e^;tW0 0 0

0

1

C

C

A g

= A^f(PF)⁰PP1^;(PX)⁰V(0)

0

B

B

@

e^;tW0 0 0

0

1

C

C

A g

= APXV(0)

0

B

B

@

e^;tW0(^;W0) 0 0

0

1

C

C

A

= AF

0

B

B

@

;W0

0 0 0

1

C

C

A= A

0

B

B

@

;W0

0 0 0

1

C

C

A

Using the structure of our transformed projectors in more detail yields A^= A^;₂¹A=

0

B

B

@

I 0

;I 0 0

1

C

C

A

and B^ = A^;2¹B = A^;2¹BPP1+ Q1+ Q:

Now it becomes clear that scaling by A^;2¹ leads to B^ = A^;₂¹B = A^;₂¹A

0

B

B

@

;W0

0 0 0

1

C

C

A+ Q1+ Q

= P1PPP1

0

B

B

@

;W0

0 0 0

1

C

C

A+ Q1 + Q=

0

B

B

@

;W0 0 II I

I

1

C

C

A:

Finally, we know that using the transformation given by (3.6) and then scaling by A^;1 we succeed in reducing the variable coecient DAE (3.1) to a DAE that has the constant coecients

A^=

0

B

B

@

I 0

;I 0 0

1

C

C

A

B^ =

0

B

B

@

;W0 0 II I

I

1

C

C

A

12

and the fundamental solution matrix X^(t) =

0

B

B

@

e^W0t 0 0

0

1

C

C

A:

Denition.

Two linear, homogeneous,-periodic DAEs are said to be (periodically) equiv- alent i the relation

A=EAF and B =E(BF +AF⁰) (3.8)

where F ²C_1N, E ²C are -periodic and nonsingular matrix functions, is true for their coecients.

Periodic equivalence means kinematic equivalence by periodic transformations.

Verifying Theorem 3.1 we have proved, in fact, the following generalization of Lyapunov's Reduction Theorem.

Theorem 3.2

(i) If two linear homogeneous-periodic index{2 DAEs are(periodically) equivalent, then their monodromy matrices are similar and, hence, their character- istic multipliers coincide.

(ii) If the monodromy matrices of two linear-periodic index{2 DAEs are similar, then the DAEs are (periodically) equivalent.

(iii) Each index{2 DAE with periodic coecients is (periodically) equivalent to a T- periodic complex (2-periodic real) linear system with constant coecients.

Remark:

^{Let (t) := X(t 0)V}(0), where we chooseV(t) with (t) =V(t)

I 0

V^;1(t) and D(t) := (t)e^;Wt with W :=

w0

0

:

Denote by X^; the reexive general inverse ofX with XX^; = can(t) and

X^;X = (0): It follows that

^; = can(t) ^; =

I 0

and

DD^;= can(t) D^;D=

I 0

13

, and remains a periodic function. The transformationF is given by F :=D+ (I ^;)V (=D+kV

0 I

) and its inverse by

F^;1 =D^;+V^;1(I^;)k^;1:

This representation of F seems to be the direct generalization of the ODE-case one and it is valid at least for the cases

index 0 : I

index 1 : P and index 2 : PP1:

4 Quasilinear periodic index-2 DAEs

We consider the quasilinear DAE

f(x⁰(t) x(t) t) := A(x(t) t)x⁰(t) +b(x(t) t) = 0 (4.1) where the coecients A and b are continuous, continuously dierentiable with respect to the variable x, and - periodical, i.e., A(x t) = A(x t +) b(x t) = b(x t+).

We suppose here, as in Chapter 2, that kerA(x t) =: N(t) is independent of x and smooth, and, additionally, that also imA(x t) is independent of x and smooth. This allows us, analogously to Chapter 2, to work with the corresponding smooth and periodic projectors. Let us denote

Q(t) a smooth, periodic projector onto N(t) P(t) := I^;Q(t)

R(t) a smooth, periodic projector onto imA(x t):

Then we have for the space tangential to the constraint manifold S(x t) : = ^fz ²IR^m:b⁰_x(x t)z ²imA(x t)^g

= ^fz ²IR^m: (I^;R(t))b⁰_x(x t)z = 0^g:

Now, let x? ²C_1N be the periodic solution of (4.1), whose stability we want to check. We linearize (4.1) in this solution and rewrite the nonlinear DAE (4.1) in the form

0 = f(x⁰(t) x(t) t)^;f(x⁰_?(t) x?(t) t)

= A(x?(t) t)(x⁰(t)^;x⁰_?(t)) +B(x⁰_?(t) x?(t) t)(x(t)^;x?(t)) +h(x⁰(t)^;x⁰_?(t) x(t)^;x?(t) t)

where

B(y x t) :=f_x⁰(y x t) =b⁰_x(x t) + A(x t)y]⁰_x: 14

Shifting the solution and writingx(t) for x(t)^;x?(t) andx⁰(t) for x⁰(t)^;x⁰_?(t) we obtain 0 =A(x?(t) t)

| {z }

=:A(t)

x⁰(t) +B(x⁰_?(t) x?(t) t)

| {z }

=:B(t)

x(t) +h(x⁰(t) x(t) t) (4.2) with

h(y x t) := f(x⁰_?(t) +y x?(t) +x t)^;A(t)y^;B(t)x

= A(x?(t) +x t)(x⁰_?(t) +y) +b(x?(t) +x t)^;A(t)y^;B(t)x (4.3) where we have to check the stability of the trivial solution x = 0. By construction the function hdescribes a small nonlinearity. It holds that

h(0 0 t) = A(x?(t) t)x⁰_?(t) +b(x?(t) t) = 0 h⁰_y(y x t) = A(x?(t) +x t)^;A(t)

h⁰_y(y x t)z ² imA(x t) = imA(0 t) for all z ²IR^m h⁰_y(y x t)z = 0 for all z ²N(t)

h(y x t) = h(P(t)y x t)

h⁰_x(y x t) = b⁰_x(x?(t) +x t) + A(x?(t) +x t)(x?(t) +y)]⁰_x^;B(t):

To prove that the trivial solution is stable under certain conditions we will work with linearizations. Firstly, we suppose that the linear part

A(t)x⁰(t) +B(t)x(t) = 0 (4.4)

is of index 2. This index-2 property of the linear part (4.4) does not automatically imply the index-2 property for neighbouring equations like (4.2), too. Additional structural conditions are necessary. Illustrating examples of this phenomenon are given in 7], for a more detailed discussion we refer to 14]. In our situation these structural conditions can be formulated in terms of that partc of the small nonlinearityh that corresponds to the derivative-free equations of (4.1).

Therefore, we consider

c(x t) := (I^;R(t))h(0 x t)

= (I^;R(t))b(x?(t) +x t)^;b⁰_x(x?(t) +x t)x] (4.5) where we stress that c depends only on parts of b, and suppose that at least one of the following structural conditions shall be true:

(S1) c(x t) =c(P(t)x t) or

(S2) c(x t) =c((P +UQ)(t)x t) ,where U(t) is a projector along S(0 t)^\N(t), or (S3) c(x t)^;c(P(t)x t)²imA1(t), or

(S4) S(x t)^\N(t) =S(0 t)^\N(t).

15

In case of index-2 Hessenberg systems or linear index-2 systems each of these conditions is fullled.

To prove the desired stability theorem we will transform the DAE (4.2) by means of a nonsingular F ²C_1N for the transformation of variables and a nonsingularE ²C for the scaling of the equations. In this way we obtain a transformed DAE

A^x⁰(t) + ^Bx(t) + ^h(x⁰(t) x(t) t) = 0 (4.6) where

x = F(t)x

A^(t) = E(t)A(t)F(t) B^(t) = E(t)(BF +AF⁰)(t)

^h(y x t ) = E(t)h(F(t)y+F⁰(t)x F(t)x t): For the small nonlinearity ^h we compute

^h⁰y(y x t)z = E(t)h⁰_y(F⁰(t)x+F(t)y F(t)x t)F(t)z

^h⁰y(y x t)z ² E(t)imA(t) = im ^A for all z ²IR^m

^h⁰y(y x t)z = 0 for z ²N =F(t)^;1N(t) and

h^(y x t ) = ^h( ^P(t)y x t ) for any projector ^P(t) along N:

(4.7) Further, we will see that each of the structural conditions (S1),(S2),(S3),(S4) for the original problem carries over to the transformed one. For the transformed equations we have

^c(x t) = (I^;R^{^})E(t)h(F⁰(t)x F(t)x t)

= E(t)(I^;E(t)^;1RE^ (t))h(0 F(t)x t)

= E(t)c(F(t)x t)

whereR(t) := E(t)^;1RE^ (t) is used as a special projector onto imA(t) , and it holds:

Lemma 4.1

For quasilinear DAEs (4.1) with only time-dependent, smooth spaceskerA(x t) and imA(x t) any of the structural conditions (S1),(S2),(S3),(S4) is invariant under a nonsingular transformation of variables F ²C_1N and a scaling of the equations E ²C.

Proof:

Suppose that one of the structural conditions (S1),(S2),(S3),(S4) is true. Then we have for the conditions:

(S1): For the special projector ^P(t) :=F(t)^;1P(t)F(t) along N we compute

^c( ^P(t)x t) = E(t)c(F(t) ^P(t)x t)

= E(t)c(P(t)F(t)x t)

= E(t)c(F(t)x t) = ^c(x t)

and, hence, it follows for any projector P along N that

^c( Px t ) = ^c( ^P(t) Px t ) = ^c( ^P(t)x t) = ^c(x t): 16

(S2): First, we mention that also condition (S2) is independent of the special choice of the projectorsQ(t) andU(t). To see this letQ(t) and Q(t) be projectors ontoN(t), and U(t) and U(t) be projectors along N(t)^\S(0 t). If (S2) is true for the projectors Qand U, then (S2) is also true for Q and U, since

(P +UQ)( P + UQ) = PP + PUQ + UQP + UQUQ

= P ^; P(I^;U) Q + UQP + UUQ^;UPUQ

= P + 0 + UQP + UQ^;U0

= P + UQP + UQQ

= P + UQ

and, hence,

c(( P + UQ)x t) =c((P +UQ)( P + UQ)x t) =c((P +UQ)x t) =c(x t): Now, considering ^c(( ^P + ^UQ^{^})x t), where ^P =F^;1PF, and ^U =F^;1UF with the dropped argumentt, we obtain

^c(( ^P + ^UQ^{^})x t) = Ec(F( ^P + ^UQ^{^})x t)

= Ec((PF +UQF)x t) = Ec((P +UQ)Fx t )

= Ec(Fx t ) = ^c(x t):

(S3): Like (S1) and (S2) also (S3) is independent of the special choice of the projectorP and we see that

^c(x t)^;^c( Px t ) = E(t)c(F(t)x t)^;c(F(t) Px t )]

= E(t)c(F(t)x t)^;c((F(t) PF(t)^;1)F(t)x t)]

2 E(t)imA1(t) = im ^A1: (S4): (S4) implies

S(y x t )^\N = S(0 0 t)^\N where S(y x t ) := ^fz: ^Bz+ ^h⁰_x(y x t )z ²im ^A^g: Namely, we have

A^ = EAF

B^+ ^h⁰_x = E(BF +AF⁰) +E(h⁰_xF +h⁰_yF⁰) hence

S(y x t ) = ^fz: B(t) +h⁰_x(F(t)y+F⁰(t)x F(t)x t)]F(t)z ²imA(t)^g

= ^fz: b⁰_x(x?(t) +F(t)x t)]F(t)z ²imA(t)^g

= F(t)^;1S(F(t)x t)

17

thus

S(y x t )^\N = F(t)^;1(S(F(t)x t)^\N(t))

= F(t)^;1(S(0 t)^\N(t))

= S(0 0 t)^\N :

q.e.d.

As in 12] we now follow the lines of the well-known Floquet-theory for ODEs and look for a transformation of the linear part (4.4) to a linear DAE with constant coecients rstly.

Therefore, we apply Theorem 3.2, which guarantees (4.4) to be periodically equivalent to a system with constant coecients. More precisely, there exists a special-periodic non- singularF ²C_1N for the transformation of variables and a special-periodic nonsingular E ²C for the scaling of the equations such that

A^=E(t)A(t)F(t) =

0

B

B

@

I 0

;I 0 0

1

C

C

A and ^B =E(t)(BF+AF⁰)(t) =

0

B

B

@

;W0 0 II I

I

1

C

C

A

with a constant matrixW0 ²L(IC^m;r;). The system

A^x⁰(t) + ^Bx(t) = 0 (4.8)

possesses the same characteristic multipliers as (4.4) since the monodromy matrices of the systems are similar.

In the next step we apply the special transformation F and scaling E to the nonlinear system (4.2) and obtain :

A^x⁰(t) + ^Bx(t) + ^h(x⁰(t) x(t) t) = 0 (4.9) which is by construction a DAE with a small nonlinearity and a constant linear part, which is of index-2 even in Kronecker-like normal form. It has the following block structure:

x⁰1 ^; W0x1 + ^h1((x⁰1 x⁰2 0 0) (x1 x2 x3 x4) t) = 0 x2 + ^h2(0 (x1 x2 x3 x4) t) = 0

;x⁰2 + x2+ x3 + ^h3((x⁰1 x⁰2 0 0) (x1 x2 x3 x4) t) = 0 x4 + ^h4(0 (x1 x2 x3 x4) t) = 0

(4.10)

where ^h =

0

B

B

B

@

^h1

^h2

^h3

^h4

1

C

C

C

A

.

18

For this specially structured equation we can also have a closer look at the structural conditions mentioned before. In our case, with ^R =

0

B

B

@

I 0 I 0

1

C

C

A as a projector onto im ^A=IR^r;f0^gIR^f0^gm;r; we have

^c(x t) = (I^;R^{^})^h(0 x t) =

0

B

B

@

^ 0

h2(0 x t)

^ 0

h4(0 x t)

1

C

C

A:

Choosing P =

0

B

B

@

I I 0 0

1

C

C

A and UQ =

0

B

B

@

0 0 0 I

1

C

C

A and taking into account that imA^{^}1 =im

0

B

B

@

I 0

;I I I

1

C

C

A=IR^r;f0^gIRIR^m;r;, we see that the structural conditions for (4.9/4.10) mean the following:

(S1) ^h2 and ^h4 are independent of x3 and x4

(S2) ^h2 and ^h4 are independent of x3

(S3) ^h2 is independent of x3 and x4

(S4) N ^\S(x t) = ^fz :z1 = z2 = 0 ^h⁰_2x³z3 + ^h⁰_2x⁴z4 = 0 h^⁰4x³z3 + (I+ ^h⁰4x⁴)z4 = 0 ^g

= N ^\S(0 t) =^fz : z1 = z2 = z4 = 0^g.

Now, we will use a result of 8] to prove that under certain smoothness conditions the trivial solution of (4.9) is stable in the sense of Lyapunov if all eigenvalues of the mon- odromy matrix ^X lie in ^fz ² IC : ^jz^j<1^g or, equivalently, if the nite spectrum ( ^A B^{^}) is contained in the left sideIC^; of the complex plane. Using the transformationx=F(t)x we will derive the following main theorem:

Theorem 4.2

^{Let kerA(x t) and imA(x t}) be only time-dependent and smooth and let x? be a -periodic solution of (4.1), let the linearized equation (4.4) be of index-2 and let one of the structural conditions (S1), (S2), (S3), (S4) be true. Suppose that (4.1) is suciently smooth, which will be specied later in the proof, and suppose that all eigenvalues of the monodromy matrix X of (4.4) lie inside the complex unit circle, i.e., in ^fz ²IC :^jz^j<1^g. Then the periodic solution x? is stable in the sense of Lyapunov.

Proof:

We will prove that the trivial solution of (4.9) is stable in the sense of Lyapunov since then the assertion of Theorem 4.2 follows by the transformation of variables x =

19

F(t)x. We know that all eigenvalues of the monodromy matrix ^X of (4.8) lie inside the complex unit circle since the corresponding property for the original monodromy matrix X also applies to ^X. Now, we look for properties of the small nonlinearity ^h. From (4.7) we see that

im ^h⁰y(y x t ) im ^A , and ker ^A ker ^h⁰y(y x t):

Further, we know by Lemma 4.1 that the structural conditions (S1), (S2), (S3), (S4) carry over to the transformed problem.

Next, by construction we have that ^his continuous together with its partial Jacobians ^h⁰y,

^h⁰_x,

^h(0 0 t) = E(t)h(0 0 t) = 0 for t ²IR

and, to each small " >0, a (")>0 can be found such that^jx^j ("),^jy^j (") yield

^h⁰_y(y x t )^j " ^{j^}h⁰_x(y x t )^j "

uniformly for allt²IR.

To apply Theorem 3.1 of 8] we nally need that the part ^c additionally has continuous derivatives ^c⁰_t c^⁰⁰xt ^c⁰⁰xx and

^c⁰_t(0 t) = 0 for all t²IR

^c^{0 0}_xt (x t) " and ^c⁰⁰_xx(x t) for^jx^j (") t²IR where are constants.

These smoothness and smallness conditions for ^clead to smoothness assumptions for the corresponding derivative-free part of the original problem after a suitable scaling of the equations. We compute:

^c(x t) = (I^;R^{^})^h(0 x t)

= (I^;R^{^})Eh(0 Fx t )

= (I^;R^{^})E(I^;R)h(0 Fx t ) for any projector R onto imA

= (I^;R^{^})Ec(Fx t ) since

(I ^;R^{^})ERh(0 Fx t ) = (I^;R^{^}) ^|{z}EA

= ^AF^;1A⁺Rh(0 Fx t)

| {z }

2im ^A

= 0

and for the special choice ofR = E^;1RE^{^} or ^R = ERE^;1 we obtain

^c(x t) = E(t)c(F(t)x t):

20