76. Adjoint Pairs of Differential Algebraic Equations and Their Lyapunov Exponents

76. Adjoint Pairs of Differential Algebraic Equations and Their Lyapunov Exponents tài liệu, giáo án, bài giảng , luận v...

DOI 10.1007/s10884-015-9474-6

Adjoint Pairs of Differential-Algebraic Equations

and Their Lyapunov Exponents

Vu Hoang Linh 1 · Roswitha März 2

Received: 12 January 2015 / Revised: 7 July 2015

© Springer Science+Business Media New York 2015

Abstract This paper is devoted to the analysis of adjoint pairs of regular

differential-algebraic equations with arbitrarily high tractability index We consider both standard formDAEs and DAEs with properly involved derivative We introduce the notion of factorization-adjoint pairs and show their common structure including index and characteristic values

We precisely describe the relations between the so-called inherent explicit regular ODE(IERODE) and the essential underlying ODEs (EUODEs) of a regular DAE We prove thatamong the EUODEs of an adjoint pair of regular DAEs there are always those which areadjoint to each other Moreover, we extend the Lyapunov exponent theory to DAEs witharbitrarily high index and establish the general class of DAEs being regular in Lyapunov’ssense The Perron identity which is well known in the ODE theory does not hold in generalfor adjoint pairs of Lyapunov regular DAEs We establish criteria for the Perron identity to

be valid Examples are also given for illustrating the new results

Keywords Differential-algebraic equation· Tractability index · Adjoint equation ·Essential underlying ODE· Lyapunov exponent · Lyapunov regularity · Perron identity

Mathematics Subject Classification 34A09· 34D08

Dedicated to the memory of Katalin Balla (1947–2005).

List of symbols and abbreviations

K Set of real numbersRand set of complex numbersC

L (Ks ,Kn ) Set ofK-valued n × s—matrices and linear operators fromKstoKn

C ( I , X) Space of continuous functions mappingI into X

C1( I , X) Space of continuously differentiable functions mappingI into X

(·, ·) Scalar product in function spaces

| · | Vector and matrix norms

 ·  Norms on function spaces, operator norms

χ u ( f ) The upper Lyapunov characteristic exponent of f

χ l ( f ) The lower Lyapunov characteristic exponent of f

DAE Differential-algebraic equation

ODE Ordinary differential equation

IVP Initial value problem

IERODE Inherent explicit regular ODE

EUODE Essential underlying ODE

1 Introduction

In the classical theory of explicit ordinary differential equations (ODEs) the adjoint equation

is introduced as equation satisfied by the adjoint inverses of the fundamental solution matrices,e.g [2,13,16,17] If

which is valid for each arbitrary pair of solutions of Eqs (1) and (2) Thereby the interval

I⊆Ris arbitrary We are most interested in an infinite one.1Particularly the Lagrange

iden-1 In contrast, when looking for adjoint operators of the operator representing the given ODE, one supposes a compact interval and, additionally, boundary conditions.

tity accounts for the benefit of adjoint ODEs, for instance, when investigating asymptotics,boundary value problems, and also optimal control problems.

The nature of differential-algebraic equations (DAEs) is much more complicated Exceptfor the less interesting case of index-0 DAEs, all fundamental solution matrices of regularDAEs are everywhere singular matrix functions such that one is coerced into finding appro-priate generalized inverses It was Katalin Balla who initiated to clarify the relevant structure

of regular index-1 and index-2 DAEs and who made profound contribution to this topic[4 10]

In the present paper, first we continue the investigations and take up the intentions ofKatalin Balla concerning adjoint pairs of DAE, now for regular DAEs with arbitrarily hightractability index We adress both standard form DAEs

E(t)x(t) + F(t)x(t) = 0, t ∈ I , (4)and DAEs with properly involved derivative

A(t)(Dx)(t) + B(t)x(t) = 0, t ∈ I (5)Together with the DAEs (4) and (5) we consider the equations

− (E∗y )(t) + F(t)∗y (t) = 0, t ∈ I , (6)and

− D∗(t)(A∗y )(t) + B(t)∗y (t) = 0, t ∈ I , (7)

later on justified as their adjoint counterparts.

The attempt [19] to treat DAEs as operator equations in appropriate function spacesprovides the adjoint operators as a byproduct when looking for the biadjoint operators rep-resenting the closures, e.g.,[19, Theorem 1] In this context, Eq (6) is already justified asadjoint equation associated with (4); and (7) is justified as adjoint of (5), see also [26] Incontrast, here we do not make use of functional-analytic arguments, but we try to argue fromappropriate aspects of the theory of differential equations

For smooth coefficients E and F, the Eq (6) is introduced as dual descriptor system

in [11]; and the original DAE and its dual are shown to be solvable at the same time Incase of merely continuous coefficients, the DAE (6) and a generalized Lagrange identityare applied in [5,28] to describe solution manifolds of boundary problems for the DAE(4) with index 1 In [6] the notion adjoint DAE is used and justified for the index-1 case by rigorous solvability investigations In particular, it is shown that, if X (t) denotes the maximal

fundamental solution matrix of the DAE (4) normalized at t0∈I, then

and the further characteristic values For an important class of self-adjoint DAEs, the inherent

explicit regular ODE (IERODE) is proved to be Hamiltonian in [10] Supposing so-called

completely decoupling projectors to define the generalized inverses D−and A∗ −, it is proved

in [7] that

Y (t, t0) := A(t)∗ −D(t)∗ −X(t, t0)− ∗D(t0)∗A(t0)∗

is the fundamental solution matrix of the adjoint DAE (7) normalized at the same point t0.These investigations are continued for DAEs with index≤2 in [4,9] Conditions ensuringthe inherent regular ODEs of the adjoint pair to be adjoint each to other are given Moreover,

also adjoint pairs of essential underlying ODEs (EUODE) are studied.

The standard form DAE (9) with smooth coefficients together with the standard form DAE

− E(t)∗y(t) + (F(t)∗− E(t)∗)y(t) = 0, t ∈ I , (8)resulting from (6) are revisited in [20], there called formally adjoint pair It is shown that the

DAEs (4) and (8) share in the differentiation index and the size of the differential part.The second purpose of the paper is to characterize the stability of regular DAEs by usingthe Lyapunov exponent theory, which is well known for ODEs, see [1,17,25] Recently,Lyapunov exponents and their properties have been extended to index-1 DAEs given ineither the standard form or the strangeness-free form, see [14,15,22–24] Now we aim toextend the Lyapunov exponent theory to regular DAEs of arbitrarily high index In particular,

we invesigate the relation between the sets of the Lyapunov exponents for adjoint pairs ofregular DAEs which is known as the Perron identity

The present paper is organized as follows: Sect.2describes the general assumptions andthe Lagrange identity for DAEs Section3collects required material concerning transfor-mations and refactorizations The basic structure of regular DAEs is exposed in Sect.4 Inparticular, we discuss how the IERODEs and the EUODEs are related to each other Thesepreliminaries are followed by Sect.5which gives a definition of adjoint DAEs and providesresults concerning the joint basic structure of adjoint pairs It is also shown that an adjointDAE pair possesses EUODEs adjoint each to other Finally, Sect.6presents new insightsconcerning the stability analysis of regular DAEs by investigating their Lyapunov exponents

The list of symbols and abbreviations is given at the end We drop the argument t if ever

possible without causing confusion

2 Basics and Lagrange Identity

We investigate standard form DAEs

E(t)x(t) + F(t)x(t) = 0, t ∈ I , (9)and DAEs with properly involved derivative

A (t)(Dx)(t) + B(t)x(t) = 0, t ∈ I (10)The intervalI⊆Ris arbitrary, possible infinite The coefficients are supposed to be contin-uous, that is,

E, F ∈ C ( I , L (Km ,Km )),

A∈C ( I , L (Kn ,Km )), D ∈ C ( I , L (Km ,Kn )), B ∈ C ( I , L (Km ,Km )),

withK=RandK=Cin the real and complex versions, respectively Additionally, out the paper we assume the time-varying subspaces

through-ker E (t), ker A(t), and im D(t), t ∈ I ,

to beC1- subspaces When dealing with a DAE of the form (10), we presume the transversalitycondition

to be valid, which means that the derivative is properly involved and the DAE shows actually

a properly stated leading term The decomposition (11) determines the so-called border

projector function R∈C1( I , L (Kn ,Kn )) by

ker R (t) = ker A(t), im R(t) = im D(t), t ∈ I (12)Since both involved subspaces areC1-subspaces, the projector function R is actually contin-

uously differentiable

Since ker E is a C1-subspace, owing to [26, Theorem 3.1], we find a proper

factor-ization E =: AD, A ∈ C ( I , L (Kn ,Km )), D ∈ C1( I , L (Km ,Kn )) such that ker E(t) =

ker D (t), t ∈ Iand the condition (11) is valid For instance, one can use A = E, D = P, with a projector function P along ker E as applied already in [18] Then we can rewrite thestandard form DAE (9) as

A (t)(Dx)(t) + (F(t) − A(t)D(t))x(t) = 0, t ∈ I , (13)which is a DAE with properly stated leading term

Conversely, each DAE (10) with a continuously differentiable coefficient D and C1-solutionscan be written also as standard DAE

A(t)D(t)x(t) + (B(t) + A(t)D(t))x(t) = 0, t ∈ I (14)

We emphasize that the standard form DAE and the DAE with properly stated leading termshare most their structural properties However, thoughC1-solutions are supposed for standardform DAEs, the DAE (10) naturally admits continuous functions x showing a continuously differentiable part Dx.2

Together with the DAEs (9) and (10) we consider the equations

− (E∗y)(t) + F(t)∗y(t) = 0, t ∈ I , (15)and

− D∗(t)(A∗y )(t) + B(t)∗y (t) = 0, t ∈ I , (16)

later on justified as their adjoint counterparts.

The DAE (16) has a properly stated leading term at the same time as (10), with the associated

border projector function R∗.

The DAE (15) is obviously out of the scope of a standard form DAE, but, supposing

addi-tionally that E and y are continuously differentiable, one can turn to the standard form DAE

− E∗(t)y(t) + (F(t)∗− E(t)∗)y(t) = 0, t ∈ I (17)

On the other hand, applying the proper factorization E∗= [AD]∗= D∗A∗, Eq (17) leads

to

− D(t)∗(A∗y )(t) + (F(t)∗− D(t)∗A (t)∗)y(t) = 0, t ∈ I , (18)which is the precise counterpart of (13)

For any solution pair x ∈C1

D(t)x(t), A(t)∗y(t) = constant, t ∈ I , (19)

as well as the generalized Lagrange identity for the pair (9) and (15),

x(t), E(t)∗y(t) = D(t)x(t), A(t)∗y(t) = constant, t ∈ I (20)The last identity (20) is valid for all solutions x ∈C1

D ( I ,Km ) (including all x ∈ C1( I ,Km ))

and y∈C1

A∗( I ,Km ) of the DAEs (13) and (15), respectively If E is continuously

differen-tiable, then (20) makes sense for all solutions x∈C1( I ,Km ) and y ∈ C1( I ,Km ) of (9) and(17) at least

3 Transformations and Refactorizations

Let pointwise nonsingular matrix functions

In summary the following relations are valid:

Next we turn to DAEs with properly stated leading term

and consider multiplications from left and coordinate transformations x = K ˜x by pointwise

nonsingular matrix functions

L∈C ( I , L (Km ,Km )), K ∈ C ( I , L (Km ,Km )).

Additionally we allow refactorizations of the leading term A D = (AH)(H−D) by H with

H ∈C1( I , L (Ks ,Kn )), H− ∈C1( I , L (Kn ,Ks )), n, s ≥ r := rank D,

In particular, one can apply refactorizations with n = s and nonsingular H.

The resulting DAE (cf [21, Sect 2.3])

has the coefficients

˜A := L AH, ˜D := H−D K , ˜B := L BK − L AH(H−R)D K

It inherits the properly stated leading term from (25), and its border projector function is

− ˜D∗( ˜A∗˜y)+ ˜B∗˜y = K∗p. (29)

In summary the following relations are valid:

A(Dx)+ Bx = q ad j oi nt

←−−−→ −D∗(A∗y)+ B∗y = p

⇓ L, K, H ⇑ L−1, K−1, H− ⇓ K∗, L∗, H− ∗ ⇑ K∗ −1, L∗ −1, H∗

˜A( ˜D ˜x)+ ˜B ˜x = Lq ad j oi nt

←−−−→ − ˜D∗( ˜A∗˜y)+ ˜B∗˜y = K∗p

The following observation will play its role for Definition1below

Let the matrix function H ∈ C1( I , L (Ks ,Kn )) describe a refactorization of the leading

term in the DAE (25) and let H− ∗induce a refactorization in (28) A refactorization does

not change neither the DAE solutions nor the relevant function spaces housing the DAEsolutions In particular, we have

A∗( I ,Km ) of the homogenous versions of

the DAEs (25) and (28), respectively, it holds that

H(t)−D(t)x(t), H(t)∗A(t)∗y(t)

= R(t)H(t)H(t)−D(t)x(t), A(t)∗y(t) = D(t)x(t), A(t)∗y(t), t ∈ I ,

and hence, next to (19), also

H(t)−D(t)x(t), H(t)∗A(t)∗y(t) = constant, t ∈ I (30)Thereby, the constant is the same as in (19)

4 The Basic Structure of a Regular DAE

In the context of the projector based analysis of DAEs, the basic structure of a regular DAE isdetermined by its tractability indexμ and the characteristic values r0≤ · · · r μ−1 < r μ = m.

We refer to [21] for general relations with other index notions

The DAE with properly stated leading term

as described in Sect.2has continuous coefficients A , D, B If necessary, the coefficients

are supposed to be smooth enough for regularity and the existence of complete decouplings,e.g., [21, Sect 2.4.3] We apply the regularity notion given in [21, Definition 2.25], which issupported by several constant-rank requirements yielding the tractability indexμ ∈Nand

the characteristic values

r0≤ · · · ≤ r μ−1 < r μ = m,

of a regular DAE Regularity is formally determined by means of admissible projector

func-tions

P0, , P μ−1∈C ( I , L (Km ,Km ))

associated with the construction of admissible matrix functions sequences starting from

G0:= AD and ending up with a nonsingular G μ, see [21, Definition 2.6]

The tractability index generalizes the Kronecker index of a regular matrix pencil, and, in

case of such a matrix pencil, the characteristic values r i provide a complete description ofthe formal structure of the corresponding Weierstraß–Kronecker form

We use the further denotations

with sufficiently smooth coefficients E , F, as follows: the standard form DAE (33) is regularwith tractability indexμ and characteristic values r0 ≤ · · · ≤ r μ−1 < r μ = m, if any (equivalently: each) proper factorization of the leading coefficient E = AD yields a regular

DAE of type (31),

A(Dx)+ (F − AD)x = q, (34)being regular with these characteristics, e.g., [21, Sect 2.7] Similarly, the equation

with sufficiently smooth coefficients E , F, is called regular DAE with tractability index μ

and characteristic values r0 ≤ · · · ≤ r μ−1 < r μ = m, if any (equivalently: each) proper factorization E = AD yields a regular DAE of type (31),

− D∗(A∗y )+ (F∗− D∗A∗)y = q, (36)being regular with these characteristics

The sequence of projector functions P0, , P μ−1serves as tool for the decoupling of the

DAE itself and the decomposition of the solution x into their characteristic parts, see [21,

Sect 2.4] In particular, the component u = DΠ μ−1 x satisfies the so-called IERODE

u−D Π μ−1 D−

u + DΠ μ−1 G−1

μ B Π μ−1 D−u = DΠ μ−1 G−1

The componentsv0= Q0x, v1= Π0Q1x , , v μ−1 = Π μ−2 Q μ−1 x satisfy the triangular

subsystem involving several differentiations

The subspace im D Π μ−1 is an invariant subspace for the IERODE The components

v0, v1, , v μ−1 remain within their subspaces im Q0, imΠ μ−2 Q1, , im Π0Q μ−1,respectively The structural decoupling is associated with the decomposition

speaks about a fine decoupling, if H1= · · · =H μ−1 = 0, and about a complete decoupling,

ifH0= 0, additionally A complete decoupling is given, exactly if the coefficientKvanishesidentically

If the DAE is regular and the original data are sufficiently smooth, then fine and completedecouplings exist and can be constructed, see [21, Sect 2.4.3] Below, we suppose at least afine decoupling

It should be added at this point, that the coefficients of the IERODE depend on the specialchoice of admissible projector functions However, they are uniquely determined in the scope

of fine decouplings

The so-called canonical projector function Π can of a regular DAE (see [21, Definition2.37]) is actually a generalization of the spectral projector onto the finite eigenspace alongthe infinite eigenspace of a regular matrix pencil (cf [21, Sect 1.4])

By means of fine decoupling projector functions P0, , P μ−1 , the canonical projector

function of the DAE ([21, Definition 2.37]) can be represented as

Π can = (I − H0)Π μ−1

It follows that

We emphasize thatΠ canitself is independent of the choice of projector functions Therefore,

also D Π μ−1does not depend of the construction

One can find fine decoupling projector functions P0, , P μ−1 with arbitrarily fixed start

projector function P0 along ker D This allows to prescribe the generalized inverse D−in

(32)

In contrast, complete decoupling projector functions P0, , P μ−1yield the representation

Π can = Π μ−1, which is very useful in theory, but less comfortable in practice when dealing

with D−(cf (32)).

If the DAE is regular, then the IVP

with U (·, t0) being the classical (nonsingular) fundamental solution matrix of the IERODE

(37) from a fine decoupling, U (t0, t0) = I Furthermore,

X(t, t0)−:= Π can (t0)D(t0)−U (t, t0)−1D(t)Π can (t), t ∈ I , (43)turns out to be the appropriate generalized inverse concerning the flow, see [21, Sect 2.6]

Example 1 The system comprising the m = m1+ m2equations

x

1+ B11x1+ B12x2= q1,

B21x1= q2,

is said to be a DAE in Hessenberg form of size 2, supposed the product B21B12is nonsingular

Then this DAE is regular with tractability index 2, and with characteristic values r0 = r1=

m1, r2= m1+ m2 Denoting

Ω := B12B−

12, B12− := (B21B12)−1B21, (44)

we obtain the projector functionΩ onto im B12along ker B21 SupposeΩ to be continuously

differentiable The canonical projector function of this DAE reads (e.g., [21, p 107])

Choosing a matrix functionΓ d ∈C1( I , L (Km1 ,Kd )), d := m1− m2, whose columns form

a basis of im(I − Ω) = ker B21, and determining the generalized inverseΓ d−so that

Γ−

d Γ d = I − Ω, Γ d Γ−

d = I d ,

we may condense the IERODE to the so-called essential underlying ODE (cf [3,9])

As exposed in Sect.3, the multiplication from left by L, the coordinate transformation

x = K ˜x, and the refactorization by H transforms the DAE (31) into the DAE

with coefficients

˜A := L AH, ˜D := H−D K , ˜B := L BK − L AH(H−R )D K

Theorem 1 The tractability index μ and the characteristic values r0 ≤ · · · ≤ r μ−1 < r μ

of any regular DAE with properly stated leading term persist under multiplications from left by continuous nonsingular matrix functions, continuous coordinate transformations and refactorizations with continuously differentiable matrix functions.

Proof ForK =R, the assertion is a direct consequence of [21, Theorems 2.18, 2.21] Theproof of the complex case can be carried out in the same way 

In the context of standard form DAEs one supposes continuously differentiable solutionsand, correspondingly, continuously differentiable coordinate transformations

Corollary 1 The tractability index μ and the characteristic values r0 ≤ · · · ≤ r μ−1 < r μ

of any regular standard form DAE (33) persist under multiplications from left by continuous

nonsingular matrix functions and continuously differentiable coordinate transformations.

Proof The tractability index as well as the characteristic values of a standard form DAE are

defined via the form (34) Thereby, the special choice of the factorization does not matter

Supposing K and x to be continuously differentiable in

with ˜ D− := K−1D−H Furthermore, the border projector function and the canonical

projector function of the transformed DAE (46) can be described by ˜ R = H−R H and

˜

Π can = K−1Π can K

Proof The real case is verified in [21, Sect 2.3] The proof of the complex case can be carried

The next theorem says that each regular DAE can be transformed into a form with pled fast and slow parts, similar to the Weierstraß–Kronecker form of a regular matrix pencil.This provides the main tool to be used in the next section for proving properties of adjointDAE pairs.

decou-Theorem 2 Each regular DAE (31) with tractability index μ and characteristic values r0≤

· · · ≤ r μ−1 < r μ = m can be transformed by pointwise nonsingular matrix functions

L∈C ( I , L (Km ,Km )), K ∈ C ( I , L (Km ,Km )), and a refactorizations of the leading term by H with

Recall that the IERODE lives inKn , n ≥ d, and it is unique in the scope of fine decouplings.

Its coefficients are expressed in terms of the original DAE In contrast, the EUODE has

minimal size d, but it is accessible by suitable transformations only An EUODE can be seen

as condensed IERODE However, depending on the transformations used, there is a variety

of EUODEs We describe the condensing in detail, which is actually part of the proof ofTheorem2in [21]

The IERODE (37) resulting from a fine decoupling is independent of the construction.

We emphasize this fact by rewriting the IERODE as

u− (DΠ can D−)u + DΠ can G−1

μ B Π can D−u = DΠ can G−1

Since the projector function D Π can D− is continuously differentiable and has rank d, so

is (DΠ can D−)∗, and im(DΠ can D−)∗ is spanned by d continuously differentiable basis

functions This means that there is a matrix function

Γ d ∈C1( I , L (Kn ,Kd )), im Γ d∗= im (DΠ can D−)∗, ker Γ d∗= {0}.

Then we determine the generalizedΓ d− ∈C1( I , L (Kd ,Kn )) by

Γ d Γ d−Γ d = Γ d , Γ d−Γ d Γ d− = Γ d−, Γ d−Γ d = DΠ can D−, Γ d Γ d−= I d (50)Lettingη = Γ d u for the solutions u = DΠ can D−u = Γ d−Γ d u of the IERODE leads to an

EUODE (48) with (cf [21, Sect 2.8])

Finally in this section, we emphasize again that, though the IERODE is unique, the EUODE

is not, since it depends on the choice of the basis functions of im(DΠ can D−)∗to construct

Γ d

5 The Common Structure of Factorization-Adjoint Pairs

A standard form DAE and its adjoint are known to be regular with tractability indexμ ≤ 2

at the same time; the pair also shares the characteristic values r0 < r1 = m, resp r0 ≤

r1 < r2 = m, and it has the common dynamical degree of freedom d = r0 forμ = 1,

d = r0+ r1− m for μ = 2, see [6,27]

The same is shown for DAEs with properly stated leading term, see [7]

In the present section we verify and specify correspondent general properties of regular DAEswith arbitrary index Being about to do this we take a second look to Hessenberg size 2 DAEs

Example 2 (Continuation of Example1) The Hessenberg form DAE from Example1

The associated IERODEs are

u− (I − Ω)u + (I − Ω)B11(I − Ω)u = 0 (51)and

− v+ (I − Ω∗)v + (I − Ω∗)B∗

11(I − Ω∗)v = 0. (52)Observe that the IERODEs (51) and (52) form a classical adjoint pair, if and only if theprojector functionΩ is constant, or, equivalently, if the associated subspaces im B12 and

ker B21do not vary with t.

One has(DΠ can D−)∗= I −Ω∗= A∗Π ∗ can A∗ −, however, the canonical projector function

Π ∗ candoes not equalΠ∗

canunless very strong additional restrictions

On the other hand, choosing a matrix functionΓ d ∈C1( I , L (Km1,Kd )), d := m1− m2,whose columns form a basis of im(I − Ω) = ker B21, and determining the generalizedinverseΓ d−so that

We emphasize again that the EUODEs depend on the specially chosenΓ d

Correspondent properties as in Example2are verified for all regular DAEs with tractabilityindexμ ≤ 2 in [7,9] Moreover, supposing completely decoupling projector functions to

define also D−and A∗ −, it is proved in [7] that the relation

Y (t, t0) = A(t)∗ −D(t)− ∗X(t, t0)− ∗D(t0)∗A(t0)∗

concerning the fundamental solution matrices of the DAE and the adjoint DAE normalized

at the same point t0is valid, and in particular

resulting from a refactorization of the leading term by H have exactly the same solutions.

In particular, each fundamental solution matrix of (53) represents at the same time a damental solution of the DAE (54) Therefore, having in mind wanted appropriate relations

fun-of fundamental solution matrices, we regard this matter by extending the notion fun-of adjoint