On the criterion of asymptotical stability for index-1 tractable DAEs

On the criterion of asymptotical stability for index-1 tractable DAEs

Tatyana Shtykel

Abstract

This paper considers the index-1 tractable dierential-algebraic equation. The Lyapunov stability of the trivial solution is discussed. As a criterion of the asympto- tical stability we propose a numerical parameter (^{A B}) characterizing the property of the index-1 matrix pencil^{fA Bg}to have all nite eigenvalues within the negative complex half-plane. An algorithm for computing this parameter is described.

Introduction

The implicit systems of the di erential-algebraic equations (DAEs) f(x⁰(t) x(t) t) = 0

with the nontrivial nullspace of the Jacobian f_y⁰(y x t) and its numerical solution have been discussed for many years 1-3] already. These equations often arise in various applications, e.g. in the simulation of electronic circuits, in control problems, in modelling constrained mechanical systems. Therefore the stability analysis is of great interest both from theoretical as well as from the practical point of view. The stability of DAEs was studied in 2, 4, 5].

In this paper we propose an approach to study the asymptotical stability of the trivial solution to the linear system of di erential equations

Ax⁰(t) +Bx(t) = 0 (1)

with constant matrix coecientsAandB. This problem is well investigated in the case of the nonsingular matrixA, when (1) turns into an explicit system of ordinary di erential equations (ODEs)

x⁰(t) =Mx(t) (2) with the matrix M = ^;A^;1B. According to the classical Lyapunov stability theory the trivial solution of (2) is asymptotically stable if and only if the eigenvalues ofM lie within the negative complex half-plane (see, e.g. 6]).

If the matrixAis singular, then the investigation of the spectrum of the matrix pencil

fA B^gis necessary. In 2] the trivial solution of the DAE (1) is shown to be asymptotically stable if all nite eigenvalues of the pencil ^fA B^g have negative real parts. However, checking this condition in practice involves some diculties due to the instability of the eigenvalues with respect to perturbations of the initial data.

1

As a matter of fact, as follows from the modern theory of Unsymmetric Eigenvalue Problem (see, e.g. 7]), it is more useful to deal not with individual eigenvalues but with some parameters that characterize their position with respect to a certain subset of the complex plane. For example, studying the asymptotical behaviour of solutions of equation (2) as a characteristic in 6] it was suggested a so-called dichotomy parameter

(M) = 2^kM^kkH^k , where the matrix H is the solution of the Lyapunov equation HM +MH =^;I:

In this paper we derive an analogous criterion of the asymptotical stability for the index-1 tractable DAEs. In Section 1 we recall the fundamental facts for the index-1 tractable DAE and some properties of the matrix pencils. Section 2 contains the in- vestigation of the asymptotical stability of the trivial solution of (1) similar to the one of ODEs in 6]. In Section 3 we introduce a numerical parameter (A B) and show that this parameter can be used as a quantitative characteristic of the "quality" of the asymptotical stability. In Section 4 we derive a system of matrix equations generalizing the classical Lyapunov equation and permitting to work up an algorithm for computing (A B). This algorithm is discussed in Section 5. We also indicate methods of reducing the pencil ^fA B^g with index 1 to the Kronecker canonical form and of computing the projectors onto the subspaces of^fA B^gassociated with the nite and innite eigenvalues, respectively.

The proposed criterion of numerical checking whether all nite eigenvalues of the matrix pencil ^fA B^g belong to the negative complex half-plane can also be used by the investigation of the asymptotical stability of the stationary solution of the index- 1 nonlinear autonomous equation 5] as well as of the nonautonomous equation with constant linear part and small nonlinearity 8].

1 The index-1 tractable DAE

Consider the di erential-algebraic equation

Ax⁰(t) +Bx(t) = 0 (3)

with constant matrix coecients A and B of order m.

De nition 1

2] The equation (3) is called index-1 tractable if the matrix pencil ^fA B^g is regular with index 1.

Tractability with index 1 is characterized by the fact that, for any projector Q onto kerA, the matrixG=A+BQ is nonsingular and

R

m = kerAS

where the subspace S := ^fz ^{2 Rm} : Bz ² imA^g 2, Theorem A13]. The projector Q onto the nullspace ofA along S is called the canonical projector.

2

Let Q be the the canonical projector onto kerA, then the relation Q = QG^;1B is valid 2, Lemma A14]. DenoteP =I^;Q. We multiply equation (3) byG^;1. Taking into account the easily checked equalities G^;1A =P and G^;1B =PG^;1BP +Q, we obtain the equation

(Px)⁰(t) +PG^;1BPx(t) +Qx(t) = 0:

Further, multiplying this equation by P and Q, respectively, we obtain the system (Px)⁰(t) +PG^;1BPx(t) = 0

Qx(t) = 0:

Therefore, the solutions of (3) can be written as x(t) =Px(t) +Qx(t) =Px(t).

De nition 2

A matrix-valued function^G(t)^G(t A B)²C¹ is called the Green matrix of equation (3) if it satises the initial value problem

dtd^G(t) = M^G(t) t >0

G(0) = P with M =^;PG^;1B.

Remark 1. The Green matrix^G(t A B) is a generalization of the classical notion of the fundamental matrix of solution for linear ODEs (see, e.g. 6]).

It is easy to verify that the Green matrix can be represented as^G(t) =Pe^tM. Unique- ness of ^G(t) immediately follows from the theory of ordinary di erential equations. Then the general solution of equation (3) is of the form

x(t) =^G(t)x⁰ =Pe^tMx⁰ wherex⁰ is an arbitrary vector.

Thus, we have proved the following

Theorem 1

2]. Let^fA B^gbe a regular pencil with index 1,Q be the canonical projector onto kerA along S and P =I^;Q. Then the initial value problem

Ax⁰(t) +Bx(t) = 0 P(x(0)^;x⁰) = 0 for all x^{0 2Rm} has a unique solution x(t) given by

x(t) =Pe^tMx⁰ with the matrix M =^;P(A+BQ)^;1B.

Now we recall some facts for the matrix pencil with index 1 which will be used in the sequel.

3

Let ^fA B^g be a regular matrix pencil with index 1, r = rankA. Then there are nonsingular matrices W and T such that

A=W I_{0 0}^{r 0}

T^;1 and B =W ^;B¹ 0 0 Im^;r

T^;1 (4) whereIrdenotes the unit (rr)-matrix. The representation (4) is the Kronecker canonical form of the pencil ^fA B^g 9].

It is easy to check that

Q=T ^{0 0}₀ I

T^;1 (5)

is the canonical projector onto kerA along the subspaceS. Indeed, the relations AQ= 0 and Q=Q(A+BQ)^;1B are valid. Then

P =I^;Q=T I_{0 0}⁰

T^;1 (6)

and the matrixM can be written in the form

M =^;P(A+BQ)^;1B =T B_{0 0}^{1 0}

T^;1: (7)

De nition 3

A complex value ⁶=¹ is said to be a nite eigenvalue of the matrix pencil

fA B^g if det(A+B) = 0. If is an eigenvalue, then there is a vector x⁶= 0 such that Ax=^;Bx. The vector x is called an eigenvector of the pencil.

De nition 4

The matrix pencil ^fA B^g has the innite eigenvalue = ¹ if there is a vectorx⁶= 0 such that Ax = 0. The vectorx is called an eigenvector of the pencil ^fA B^g corresponding to the eigenvalue =¹.

By the representation (4) we have

A+B =W I^;B¹ 0

0 I

T^;1: (8)

It follows from (7) and (8) that the matrixM has zero eigenvalue with multiplicitym^;r and the remaining eigenvalues of M are exactly the r nite eigenvalues of the matrix pencil^fA B^g. Moreover, the matrices Q and P given by (5) and (6) are the projectors onto the invariant subspaces of the pencil ^fA B^g associated with the innite and nite eigenvalues, respectively 5].

2 Asymtotical stability

In this section we derive the necessary and sucient condition for the asymptotical stabil- ity of the trivial solution of the index-1 tractable equation (3). The following denitions (see, e.g. 2]) describe the meaning of the Lyapunov stability for the linear di erential- algebraic equation.

4

De nition 5

The trivial solutionx(t)0 of (3) is stable in the sense of Lyapunov if, for a certain projector along the maximal invariant subspace of the matrix pencil ^fA B^g associated with the innite eigenvalue, the initial value problem

Ax⁰(t) +Bx(t) = 0

(x(0)^;x⁰) = 0 (9)

for all x^{0 2Rm} has a solution x(t x⁰) dened on 0 ¹). Moreover, for each " >0 there exists a = (") > 0 such that ^kx(t x⁰)^k < " for all t 0 and for all x^{0 2 Rm} with

kx^0k< .

De nition 6

The trivial solution x(t)0 of (3) is asymptotically stable in the sense of Lyapunov if it is stable and if there is a ⁰ >0 such that for allx^{0 2Rm} with^kx^0k< ⁰ the solution x(t x⁰)^!0 for t^!1.

Remark 2. The Lyapunov stability, as it was noted in 4], does not depend on the special choice of the projector , the only relevant characteristic feature is its nullspace, which is fully determined by the matrix pencil^fA B^g.

Remark 3. For the ODE these denitions coincide with the classical notions of stability and asymptotical stability with =I.

The following theorem is already known and was shown in 2]. We prove it in another way using the representation of the solution by means of the matrix exponential.

Theorem 2

. The trivial solutionx(t)0 of equation (3) is asymptotically stable if and only if all nite eigenvalues of the pencil ^fA B^g have negative real parts.

Proof. Assume that all nite eigenvalues of the pencil ^fA B^g have negative real parts. Then, by (8), all eigenvalues of the matrix B¹ belong to the negative complex half-plane, i.e., Rej(B¹) ^; < 0. In this case we have the following estimation for t0

ke^tB1k (r)^kB^1k

r^;1

e^{;t =2} (10)

where(r) is a constant that depends on r only 6].

Further, taking into account that Pe^tM =T I_{0 0}⁰

e^tB1 0

0 I

T^;1 =T e^tB_{0 0}^{1 0}

T^;1 =Pe^tMP (11) we can estimate

kPe^{tMk k}T^kkT^;1kke^tB1k (r)^kT^kkT^;1kkB^1k

r^;1

e^{;t =2}: (12) By Theorem 1 the initial value problem (9) with = P has the unique solution x(t) =Pe^tMx⁰. For each " >0 we assign

= "^r;1

(r)^kT^kkT^;1kkB^1kr;1: 5

Then, for all t0 and x^{0 2Rm} with ^kPx^0k< , we obtain

kx(t)^k=^kPe^tMx^0k< e^{;t =2}" "

i.e., the trivial solution of (3) is stable. Moreover, x(t)^!0 fort ^!1. This means that the solution x(t)0 of equation (3) is asymptotically stable with =P.

Now, let the matrix pencil ^fA B^ghas a nite eigenvalue with nonegative real part, andz ²im be the corresponding eigenvector. Thenx(t) =e^tz is a solution of equation (3). In this case the trivial solution is not asymptotically stable since

kx(t)^k=^je^tjkz^{k k}z^k6= 0: The theorem is proved.

3 The criterion of asymptotical stability

Let all nite eigenvalues of the pencil^fA B^gwith index 1 have negative real parts. Given the matricesM andP having the structures described above. We consider the Lyapunov equation

XM +MX =^;PFP: (13) for the unknown matrix X. The matrix F is supposed to be hermitian and positive denite (F =F >0).

Inequality (12) yields the convergence of the integral in the following form H =

Z

1

0

e^tM PFPe^tMdt+QFQ: (14) It is easy to verify that the matrix H is hermitian, positive denite and satises equation (13). Indeed, H =H and

HM +MH =

Z

1

0

e^tM PFPe^tMMdt+QFQM+MQFQ+ +

Z

1

0

Me^tM PFPe^tMdt=

Z

1

0

dtd⁽e^tM PFPe^tM)dt=^;PFP:

Here we used the property MQ = QM = 0. Further, by means of the Cholesky decomposition the matrix F can be represented as the product F =LLwith detL⁶= 0.

Then we have for each vector z (Hz z) =

Z

1

0

(e^tM PFPe^tMz z)dt+ (QFQz z) =

Z

1

0

kLe^tMPz^k2dt+^kLQz^k2: Since the matrices L and e^tM are regular and the vectors Pz and Qz cannot vanish simultaneously for z ⁶= 0, we conclude that (Hz z) > 0, i.e., the matrix H is positive denite.

6

In (14) we assign F :=G^;G^;1 = (A+BQ)^;(A+BQ)^;1 and dene (A B) = 2^kA^kkB^kkH^k:

Here ^; denotes the composition of inversion and conjugate transforms. If the matrix pencil ^fA B^g has index 1 and all its nite eigenvalues belong to the negative complex half-plane, then (A B) < ¹, obviously. We set (A B) = ¹ if the index of ^fA B^g is greater than 1 or the pencil ^fA B^g has at least one eigenvalue with nonnegative real part.

It is interesting that the parameter (A B) can be used in pointwise estimates of solution of equation (3). More precisely, similar to 6], 7] the following estimation

kx(t)^k (G)^p (A B)e^{;tkAkkBk=(kGk2 (AB))k}Px(0)^k (15) can be proved, where (G) =^kG^kkG^;1k is the condition number of the matrixG.

Indeed, if (A B) =¹, then there is nothing to prove. Let (A B) <¹ and H be the solution of the matrix equation (13) with F = G^;G^;1. We note that the matrix H can be rewritten in terms of the Green matrix^G(t) =Pe^tMas follows

H =

Z

1

0 G

(t)F^G(t)dt+QFQ:

Let us consider the matrix-valued function fort >0 Y(t) =

Z

1

t ^G

(s)F^G(s)ds:

Using the obvious properties of the Green matrix

G(t) =^G(t)P =P^G(t)

G(t)Q=Q^G(t) = 0

G(t+s) =^G(t)^G(s) =^G(s)^G(t) we have

Y(t) =

Z

1

t ^G

(s)F^G(s)ds =^G(t)

Z

1

0 G

(s)F^G(s)ds

G(t) =^G(t)H^G(t): After di erentiating the matrixY(t) taking into account the inequality

(Hz z) ^kH^kkF^;1k(Fz z) we obtain, for an arbitrary vector z, the estimation

dtd⁽Y(t)z z) = ^G(t)(MH+HM)^G(t)z z=^;(^G(t)F^G(t)z z)

;

(^G(t)H^G(t)z z)

kH^kkF^{;1k =; (}Y(t)z z)

kH^kkF^;1k 7

which implies

dtd

e^t=(kHkkF;1k)(Y(t)z z) 0: Consequently,

(^G(t)H^G(t)z z) = (Y(t)z z) e^{;t=(kHkkF;1k)}(Y(0)z z) =

= e^{;t=(kHkkF;1k)}(PHPz z): ⁽¹⁶⁾ Moreover, using the inequality^ke^tMPz^ke^;jtjkMkkPz^k (see, e.g. 6]) we have

(HPz Pz) =

Z

1

0

(Fe^tMPz e^tMPz)dt ^kPz^k2

kF^;1k

Z

1

0

e^;2tkMkdt = ^kPz^k2

2^kM^kkF^;1k: (17) Further, combining (16) and (17) for z =x(t) =^G(t)x(0) we obtain

kx(t)^k2 = ^kG(t)x(0)^k2 2^kM^kkF^;1k(H^G(t)x(0) ^G(t)x(0)) 2^kM^kkF^;1ke^{;t=(kHkkF;1k)}(HPx(0) Px(0)) 2^kM^kkF^;1kkH^ke^{;t=(kHkkF;1k)k}Px(0)^k2:

Finally, taking into account that the relations M = ^;PG^;1B, P = G^;1A and F = G^;G^;1 imply the inequality

kM^kkF^{;1k k}A^kkB^kkG^;1k2kG^k2 =^kA^kkB^{k 2}(G) to be valid, we obtain the required estimation

kx(t)^k (G)^p (A B)e^{;tkAkkBk=(kGk2 (AB))k}Px(0)^k:

Remark 4. Note that the estimation (15) is a generalization of the classical one for explicit ODEs 6].

It follows from (15), in particular, that by (A B) < ¹ the trivial solution of the index-1 tractable equation (3) is asymptotically stable. On the other hand, if the trivial solution of (3) is asymptotically stable, then all eigenvalues of the pencil ^fA B^g with index 1 have negative real parts. In this case the matrixH dened in (14) has the nite norm, i.e., (A B)<¹.

Thus, the parameter (A B) can be used as a criterion of the asymptotical stability of the trivial solution of the index-1 tractable equation (3).

Concluding this section, we derive the integral representations for the matrix H and the projector P in terms of A and B

P = 1

Z

1

;1

(iA+B)^;1Ad (18)

H _{= 12}

Z

1

;1

(A+BQ)(iA+B)^;PFP(iA+B)^;1(A+BQ)d+QFQ:

8

The rst equation follows from the decompositions (4) and (8). In this formula we mean the principal value (in the sence of Caushy) of the integral. In order to prove the second equation we transfom the matrixH into

H =

Z

1

0

e^tM PFPe^tMdt+QFQ=

= T^{;nZ 1}

0

e^tB1 0 0 0

TFT e^tB_{0 0}^{1 0}

dt+ 0 00 I

TFT ^{0 0}₀ I

oT^;1: If the matrix TFT has the form

TFT = T¹¹ T¹² T²¹ T²²

(19) where the matrices T¹¹ and T²² are, obviously, hermitian and positive denite, then

H =T^;

0

@ Z

1

0

e^tB1T¹¹e^tB1dt 0

0 T²²

1

AT^;1 =T^; H¹ 0 0 T²²

T^;1: (20) Note that the matrix

H¹ =

Z

1

0

e^tB1T¹¹e^tB1dt satises the Lyapunov equation

XB¹+B¹X =^;T¹¹ (21) with unknown matrixX. By the Lyapunov theorem this equation is uniquely solvable for any matrixT¹¹ if all eigenvalues of the matrixB¹ have negative real parts (see, e.g. 6]).

Moreover, in 7] it was shown that the solution of (21) can be represented as X _{= 12}

Z

1

;1

(iI ^;B¹)^;T¹¹(iI^;B¹)^;1d:

In view of the uniqueness of the solution we conclude that H¹ _{= 12}

Z

1

;1

(iI ^;B¹)^;T¹¹(iI^;B¹)^;1d: (22) Consequently,

H=T^;

0

@

21

Z

1

;1

(iI ^;B¹)^;T¹¹(iI ^;B¹)^;1d 0

0 T²²

1

AT^;1 =

=T^;

1 2

Z

1

;1

(iI ^;B¹)^; 0

0 0

TFT ⁽iI ^;B¹)^;1 0

0 0

d

T^;1+QFQ:

Finally, taking into account that

P(iA+B)^;1(A+BQ) =T ⁽iI^;B¹)^;1 0

0 0

T^;1 we obtain the required integral representation for the matrixH.

9

4 Matrix equations

Consider the problem of numerical checking whether all nite eigenvalues of the pencil

fA B^g belong to the negative complex half-plane. In 6], 7] this problem is completely investigated for the case of A=I. For the general case in 7] it is proposed to reduce the problem of linear dichotomy of the matrix pencil^fA B^gto the problem dichotomy with respect to the unit circle of the pencil^fA+B A^;B^g. However, it is easy to see that for the index-1 pencil^fA B^gthe number = 1 is an eigenvalue of the pencil^fA+B A^;B^g. In this section we investigate this independently, essentialy using ideas from 7].

Using the decomposition (20) for the matrix H we obtain the following relations PH = T^; H¹ 0

0 0

T^;1 =HP =PHP QH = T^{; 0 0}₀ T²²

T^;1 =HQ=QHQ=QFQ:

Then H =PHP +QHQ: (23)

On the other hand, we have

B(A+BQ)^;PHP(A+BQ)^;1A = T^{; ;}B¹H¹ 0

0 0

T^;1 B(A+BQ)^;QHQ(A+BQ)^;1A = 0

A(A+BQ)^;PHP(A+BQ)^;1B = T^{; ;}H¹B¹ 0

0 0

T^;1 A(A+BQ)^;QHQ(A+BQ)^;1B = 0:

Therefore, from (23) we obtain

A(A+BQ)^;H(A+BQ)^;1B +B(A+BQ)^;H(A+BQ)^;1A=

=T^{; ;}H¹B^1;B¹H¹ 0

0 0

T^;1: (24)

Since the matrix H¹ given by (22) satises the Lyapunov equation (21) and T^; T¹¹ 0

0 0

T^;1 =T^; I 0 0 0

T¹¹ T¹² T²¹ T²²

I 0 0 0

T^;1 =PFP we can rewrite (24) as

A(A+BQ)^;H(A+BQ)^;1B+B(A+BQ)^;H(A+BQ)^;1A=PFP: (25) Denote

Z := (A+BQ)^;H(A+BQ)^;1 =

= 12

Z

1

;1

(iA+B)^;PFP(iA+B)^;1d+ (A+BQ)^;QFQ(A+BQ)^;1: 10

By (4) and (20) we have

Z =W^; H¹ 0 0 T²²

W^;1: Then (25) implies

AZB+BZA=PFP:

Moreover, the matrixZ satises the relations

(BQ)Z(A+BQ) = (A+BQ)ZBQ and (BQ)ZBQ=QFQ which immediately follow from (23).

Theorem 3

^{. Let f}A B^g be a regular matrix pencil with index 1 andF be a hermitian, positive denite matrix. Assume that there exist the matrix Q and the hermitian, positive denite matrix Z (Z =Z >0) which satisfy the matrix equations

AZB+BZA= (I^;Q)F(I ^;Q) (26) AQ= 0 Q=Q(A+BQ)^;1B (27) (BQ)Z(A+BQ) = (A+BQ)ZBQ (28)

(BQ)ZBQ=QFQ: (29)

Then all nite eigenvalues of the pencil ^fA B^g have negative real parts and I^;Q is the projector onto the subspace corresponding to the nite eigenvalues of ^fA B^g.

Proof. Assume that the matrix pencil ^fA B^g is reduced to the Kronecker canonical form (4). Let the matrices

Q=T Q¹¹ Q¹² Q²¹ Q²²

T^;1 and Z =W^; Z¹¹ Z¹² Z²¹ Z²²

W^;1

satisfy the equations (26)-(29). The equality AQ = 0 implies Q¹¹ = Q¹² = 0. Since the matrixA+BQ is nonsingular,Q²² is so, too. Further, from the second relation of (27) it follows that

0 0

Q²¹ Q²²

= 0 0

Q²¹ Q²²

I 0

;Q^;122Q²¹ Q^;122

B¹ 0 0 I

= 0 00 I

i.e. Q²¹= 0 and Q²²=I. Thus, we see that the matrix I^;Q=T I_{0 0}⁰

T^;1

is the projector onto the subspace corresponding to the nite eigenvalues of^fA B^g. Denote

R :=BQ(A+BQ)^;1 =W ^{0 0}₀ I

W^;1: 11

Equality (28) impliesRZ =ZR. Then

RZ(I ^;R) = RZ^;RZR= 0 (I^;R)ZR= (RZ(I^;R)) = 0: Further, we obtain

Z = (R+(I^;R))Z(R+(I^;R)) =RZR+(I^;R)Z(I^;R) =W^; Z¹¹ 0 0 Z²²

W^;1 where the matrices Z¹¹ and Z²² are hermitian and positive denite.

From the relation (26) it follows that I 0

0 0

TFT I_{0 0}⁰

= I 0

0 0

Z¹¹ 0 0 Z²²

;B¹ 0

0 I

+ + ^;B¹ 0

0 I

Z¹¹ 0 0 Z²²

I 0 0 0

= ^;Z¹¹B^1;B¹Z¹¹ 0

0 0

= T¹¹ 0 0 0

in addition there exists the hermitian matrixZ¹¹ that satises the Lyapunov equation Z¹¹B¹ +B¹Z¹¹=^;T¹¹: (30) In this case by the Lyapunov theorem all eigenvalues of the matrix B¹ have negative real parts 6], i.e., all nite eigenvalues of the pencil ^fA B^g belong to the negative complex half-plane. The theorem is proved.

The converse is also true.

Theorem 4

. If all nite eigenvalues of the matrix pencil ^fA B^g with index 1 have neg- ative real parts, the system of matrix equations (26)-(29) with hermitian, positive denite matrix F for the unknown matrices Qand Z has an unique solution such that the matrix Z is hermitian and positive denite. Moreover, the solution can be represented as follows

Q = I^;1

Z

1

;1

(iA+B)^;1Ad (31)

Z _{= 12}

Z

1

;1

(iA+B)^;(I^;Q)F(I^;Q)(iA+B)^;1d+ + (A+BQ)^;QFQ(A+BQ)^;1:

Proof. Equations (27) imply that Q is the canonical projector onto N along S. Then the representation (31) for Qfollows immediately from (18).

The proof of Theorem 3 has shown that any solutionZ of (26)-(29) is of the following

form Z =W^; Z¹¹ 0

0 Z²²

W^;1

where the matrix Z¹¹ satises the Lyapunov equation (30). As mentioned above, the equation (30) has exactly one solution, which is given by

Z¹¹_{= 12}

Z

1

;1

(iI ^;B¹)^;T¹¹(iI^;B¹)^;1d:

The matrixZ²² can be computed from equation (29). It is uniquely dened and given by Z²² =T²². The theorem is proved.

12

5 Computing the parameter

^{(A B)}

In this section we propose a method for reducing the regular index-1 pencil^fA B^gto the Kronecker canonical form (4) and discuss the computational aspects of the criterion of the asymptotical stability (A B).

Let A=U ⁰_{0 0}

V (32)

be the singular value decomposition of A, where U and V are unitary matrices, is a regular (rr)-matrix,r= rankA. Then

Q¹ =V ^{0 0}₀ Im^;r

V

is the orthogonal projector onto N = kerA. We can use it to determine the canonical projectorQ=Q¹(A+BQ¹)^;1B. If the matrixB has the form

B =U B¹¹ B¹² B²¹ B²²

V then the matrixG=A+BQ¹ is given by

G=U B¹² 0 B²²

V:

The blockB²² is nonsingular, since Gis supposed to be nonsingular. Then the projectors Qand P =I^;Q can be represented as

Q=V B^22;10B^{21 0}I

V and P =V I 0

;B^22;1B²¹ 0

V: Further, for the matrixM =^;P(A+BQ)^;1B we obtain

M =V ^;;1(B^11;B¹²B^22;1B²¹) 0 B^22;1B^21;1(B^11;B¹²B^22;1B²¹) 0

V =V M¹ 0

;B^22;1B²¹M¹ 0

V whereM¹ =^;;1(B^11;B¹²B^22;1B²¹).

Let us consider the nonsingular matrix

T =V I 0

;B^22;1B²¹ I

: (33)

Then the matrix M is of the following form

M =T I 0

B^22;1B²¹ I

M¹ 0

;B^22;1B²¹M¹ 0

I 0

;B^;122B²¹ I

T^;1 =T M_{0 0}^{1 0}

T^;1 13

and for the canonical projectors we obtain the representations Q=T I 0

B^22;1B²¹ I

0 0

B^22;1B²¹ I

I 0

;B^22;1B²¹ I

T^;1 =T ^{0 0}₀ I

T^;1 and

P =T I 0

B^;122B²¹ I

I 0

;B^22;1B²¹ 0

I 0

;B^22;1B²¹ I

T^;1 =T I_{0 0}⁰

T^;1: Let us now consider the nonsingular matrix

W =U B¹² 0 B²²

: (34)

Then the matricesA and B can be rewritten as A=W ^{;1 ;;1}B¹²B^22;1

0 B^22;1

0

0 0

I 0

;B^22;1B²¹ I

T^;1 =W I_{0 0}⁰

T^;1 and

B = W ^{;1 ;;1}B¹²B^22;1 0 B^;122

B¹¹ B¹² B²¹ B²²

I 0

;B^22;1B²¹ I

T^;1 =

= W ^;1(B^11;B¹²B^22;1B²¹) 0

0 I

T^;1:

Thus, we obtain the representation of the matrix pencil ^fA B^g in Kronecker canonical form (4) with nonsingular matricesT and W in the form (33) and (34), respectively, and B¹ =^;;1(B^11;B¹²B^22;1B²¹) =M¹.

By the decomposition (20) the computation of the matrix H is reduced to calculating the matricesT¹¹, T²² and solving the Lyapunov equation

H¹B¹ +B¹H¹ =^;T¹¹: (35) It follows from (19) that

T¹¹ T¹² T²¹ T²²

=TFT = (TW^;1T)TW^;1T:

For the solving of equation (35) we can use the quickly convergent procedure described in detail in 7]. First, one has to compute the axiliary matrices

L = e^B1 =I+B¹+²

2!B¹²+:::

C =

Z

0

e^B1T¹¹e^B1 =D⁰+ ²

2!D¹+³

3!D²+:::

14

whereD⁰ =T¹¹,Dj =Dj^;1B¹+B¹Dj^;1 and is chosen to be not too large, for example = 1=2^kB^1k. Then it is easy to verify that the

H¹ =C+^X1

j⁼¹(L)^jCL^j is the solution of equation (35).

Acknowledgement:

The author would like to thank R.Marz and V.I.Kostin for valuable advice and useful discussions.

References

1] Boyarintcev Yu.E. Regulyarnye i singulyarnye sistemy lineinyh uravnenii. { Novosi- birsk: Nauka, 1980.

2] Griepentrog E., Marz R. Dierential-algebraic equation and their numerical treat- ment. { Teubner-Texte Math.88, Leipzig, 1986.

3] Hairer E., Wanner G. Solving ordinary dierential equations II. Sti and dierential- algebraic problems. { Springer Verlag, Berlin, Heidelberg, New York, 1991.

4] Marz R. Criteria for the trivial solution of dierential algebraic equations with small nonlinearities to be asymtotically stable. { Preprint Nr. 97-13, Humboldt-Univer.

Berlin, 1997.

5] Tischendorf C. On the stability of solutions of autonomous index-1 tractable and quasilinear index-2 tractable DAEs. { Circuits Systems Signal Process, Vol. 13, N 2-3, 1994, Pp. 139-154.

6] Godunov S.K. Obyknovennye dierencialnye uravneniya s postoyannymi koecien- tami. { Novosibirsk: Izd-vo Novosib. un-ta, T.1 Kraevye zadachi, 1994.

7] Godunov S.K. Sovremennye aspekty lineinoi algebry. { Novosibirsk: Nauchnaya kniga, 1997.

8] Lamour R., Marz R., Winkler R How Floquet-theory applies to dierential-algebraic equations. { Preprint Nr. 96-15, Humboldt-Univer. Berlin, 1996.

9] Gantmacher F.R. Matrizentheorie. { Springer Verlag, Berlin, Heidelberg, New York, 1986.

15