Constant-coefficient differential-algebraic operators and the Kronecker form

Linear Algebra and its Applications 552 (2018) 29–41

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Constant-coefficient differential-algebraic operators and the Kronecker form Marc Puche, Timo Reis ∗ , Felix L. Schwenninger Universität Hamburg, Fachbereich Mathematik, Bundesstraße 55, 20146 Hamburg, Germany

a r t i c l e

i n f o

Article history: Received 25 October 2017 Accepted 3 April 2018 Available online 12 April 2018 Submitted by P. Semrl MSC: 34A09 65L80

a b s t r a c t We consider constant-coefficient differential-algebraic equations from an operator theoretic point of view. We show that the Kronecker form allows to determine the nullspace and range of the corresponding differential-algebraic operators. This yields simple matrix-theoretic characterizations of features like closed range and Fredholmness. © 2018 Elsevier Inc. All rights reserved.

Keywords: Differential-algebraic equation Matrix pencil Kronecker form Differential operator

1. Introduction The purpose of this article is to highlight properties of linear constant-coefficient differential-algebraic operators T(E,A) :

x →

d dt Ex

− Ax

* Corresponding author. E-mail addresses: [email protected] (M. Puche), [email protected] (T. Reis), [email protected] (F.L. Schwenninger). https://doi.org/10.1016/j.laa.2018.04.005 0024-3795/© 2018 Elsevier Inc. All rights reserved.

(1)

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M. Puche et al. / Linear Algebra and its Applications 552 (2018) 29–41

on Lp (0, T ; Kn ), the space of p-integrable Kn -valued functions on (0, T ), with E, A ∈ Kk×n , where K denotes the field of real or complex numbers. We show that many properties of such operators can be inferred from the algebraic theory of matrix pencils sE−A, i.e., first order matrix polynomials. In particular, we will see that important characteristics of operators (1) such as range and nullspace can be determined directly from the pencil’s Kronecker form. The latter refers to a canonical form under the group action of multiplication from the left and right with constant invertible matrices, see Gantmacher’s book [4]. This form allows for a blockdiagonal decomposition of (1) into certain prototypes of differential-algebraic operators, which can easily be analyzed. Naturally, (1) corresponds to the linear, time-invariant differential-algebraic equation (DAE) d dt Ex

= Ax + f,

(2)

with suitable initial conditions. DAEs appear in a wide range of fields, from electric circuit theory across multi-body systems in mechanics to economics, and their theory is subject of several textbooks [3,7,9]. One could be tempted to view DAEs only as a slight generalization of ordinary differential equations, but their intrinsic algebraic properties lead to very different behaviour — in theory as well as in numerical analysis, see e.g., [11]. An operator-theoretic viewpoint to DAEs is not entirely new, see [11,6]. In fact, März [11] recently already gave an overview on functional-analytic aspects of a class of differential-algebraic operators, but also stressed that “... an adequate sophisticated functional-analytic characterization of DAEs has not been accomplished yet”.1 d Her approach deals with time-varying DAEs of the form dt E(t)x = A(t)x + f that are, loosely speaking, neither under- nor overdetermined. With our contribution we focus on the constant-coefficient case, but allowing for under- or overdetermined systems (which refers to the so-called “regularity” of the matrix pencil sE − A, see Definition 1). We will show that the stronger assumptions on the coefficients enables a rather elegant interplay between the theory of matrix pencils and the functional analytic properties of the operator (1). In particular, this approach gives rise to matrix-theoretic interpretations on certain properties of the DAE operator in (1), among them are simple characterizations of surjectivity, closed range, dense range, injectivity and Fredholmness. With the present note, we do not pretend to accomplish the above-mentioned complete functional-analytic characterization of DAEs, but rather clarify on fundamental properties of the pencil sE − A in the view of operator theory. This may also be seen as a first step towards the ultimate goal. Ideally, by doing so, well-known features of DAEs will appear in a different light. DAE theory has always been at the intersection of different fields, primarily linear algebra, differential equations and consequently, e.g., numerical analysis and control theory. However, it is remarkable that, even in terms of linear algebra, the understanding of DAEs is still subject to current research as the recent article [2] shows. There, it is shown that for linear relations the Kronecker form 1

See R. März [11], p. 165.

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31

can be seen as the analogue of the Jordan-normal form for matrices. The work [2] serves as a good example for how different view-points on obviously connected things — DAEs, matrix pencils, linear relations — can contribute to a better understanding of the whole picture. With the following, we want to add the keyword “differential-algebraic operator” to that list. Besides we want to point out that this operator-theoretic approach is a “toy-case” of the situation where sE − A is an operator pencil, which refers to partial differential-algebraic equations. As classical semigroup theory is no longer the “universal” tool for such equations — in contrast to the abstract ODE setting — solution theory can, in principle, be established in the spirit presented here; by directly investigating the solution operator. Of course, then crucial technical difficulties enter the picture. In this context we refer to recent progress [13,14] which centers around the theory of evolutionary equations, see [12] — a framework that seems promising for a general DAE theory in infinite dimensions. In the following, D(T ) and im(T ) will denote domain and range of a, possibly unbounded, linear operator T : D(T ) ⊂ X → Y on Banach spaces X, Y . We use ·, ·X  ,X for the duality brackets of the Banach space X, where we omit the subscripts, if clear from context. K[s] stands for the ring of polynomials with coefficients in K. Furthermore, Gln (K) is the set of invertible matrices of size n × n and In , denote the identity matrix in Kn×n . The conjugate transpose of an element A ∈ Kn×m will be denoted by A∗ ∈ Km×n . By 0n,m , n, m ∈ N ∪ {0} we refer to the zero matrix of size n × m. Note that here we particularly allow for n, m being zero, cf. p. 34. The symbols (α), |α| respectively stand for the length and the absolute value of a multi-index α. For an interval J ⊂ R, W m,p (J; Kn ) ⊂ Lp (J; Kn ) denotes the usual Sobolev space of m-times (weakly) differentiable functions. 2. The DAE operator Definition 1. For E, A ∈ Kk×n , the expression sE − A ∈ K[s]k×n is called a matrix pencil. Moreover, rankK[s] (sE − A) stands for the rank of the K[s]-module spanned by the columns of sE − A. A pencil sE − A is regular, if n = k and rankK[s] (sE − A) = n. Note that rankK[s] (sE−A) coincides with the generic rank of the mapping λ → λE−A. Definition 2. Let sE − A ∈ K[s]k×n , T > 0 and p ∈ [1, ∞]. The DAE operator T(E,A) on Lp is defined by T(E,A) : D(T(E,A) ) ⊂ Lp ([0, T ]; Kn ) → Lp ([0, T ]; Kk ),

x →

d

dt E

 − A x,

(3a)

with domain   D(T(E,A) ) = x ∈ Lp ([0, T ]; Kn ) | Ex ∈ W 1,p ([0, T ]; Kk ), (Ex)(0) = 0 .

(3b)

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Remark 1. For p ∈ [1, ∞) it holds: (a) The operator T(E,A) is closed, which is linked to generalized Sobolev spaces of the form WE1,p := {x ∈ Lp | Ex ∈ W 1,p }, see also [6].  (b) As D(T(E,A) ) is dense, the dual operator T(E,A) is well-defined [5, Thm. II.2.6]. Let us formulate our main result which relates algebraic characteristics of the pencil sE − A to properties of the DAE operator. Theorem 1. Let p ∈ [1, ∞) and consider the DAE operator in Lp associated to the pencil sE − A ∈ K[s]k×n . The following statements hold: (i) (ii) (iii) (iv) (v)

ker(T(E,A) ) = {0} if, and only if, rankK[s] (sE − A) = n. im(T(E,A) ) ⊂ Lp ([0, T ]; Kk ) is dense if, and only if, rankK[s] (sE − A) = k. im(T(E,A) ) ⊂ Lp ([0, T ]; Kk ) is closed if, and only if, imA ⊂ imE + A ker E. sE − A is regular if, and only if, T(E,A) has trivial nullspace and dense range. The following statements are equivalent: a) T(E,A) is Fredholm (that is, both dim ker T(E,A) and codim imT(E,A) are finite); b) T(E,A) has a bounded inverse; c) imA ⊂ imE + A ker E and sE − A is regular.

The dual of T(E,A) is again a DAE operator as we show next. Similar results have been achieved in [6,10,11] for regular DAEs with varying coefficients. Further note that, the adjoints of a certain class of DAEs arising in optimal control has been considered in [8]. Proposition 1. Let sE − A ∈ K[s]k×n , T > 0, p ∈ [1, ∞) and q such that Then the dual operator of T(E,A) is given by   T(E,A) : D(T(E,A) ) ⊂ Lq ([0, T ]; Kk ) → Lq ([0, T ]; Kn ),

1 p

+

1 q

= 1.

 d T(E,A) z = − dt E ∗ z − A∗ z, (4a)

and    D(T(E,A) ) = z ∈ Lq ([0, T ]; Kk ) | E ∗ z ∈ W 1,q ([0, T ]; Kn ), (E ∗ z)(T ) = 0 .

(4b)

Proof. Define   D := z ∈ Lq ([0, T ]; Kk ) | E ∗ z ∈ W 1,q ([0, T ]; Kn ), (E ∗ z)(T ) = 0 .   First we show that D(T(E,A) ) ⊂ D. Let z ∈ D(T(E,A) ) ⊂ Lq ([0, T ]; Kk ). For all ϕ ∈ ∞ n C (R; K ) with supp ϕ ⊂ (0, T ) it follows that ϕ ∈ D(T(E,A) ) and

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 d     d (T(E,A) + A∗ )z, ϕ = z, T(E,A) ϕ + Aϕ = z, dt Eϕ = E ∗ z, dt ϕ . Hence, by definition of the weak derivative, E ∗ z ∈ W 1,q ([0, T ]; Kn ) with ∗ d dt E z

 = −(T(E,A) + A∗ )z.

(5)

 To see that any z ∈ D(T(E,A) ) satisfies the boundary condition (E ∗ z)(T ) = 0, take ∞ n x ∈ C ([0, T ]; K ) with x(0) = 0. Since x ∈ D(T(E,A) ), a similar calculation and integration by parts lead to  T(E,A) z, x = (E ∗ z)(T )∗ x(T ) +



  d − dt E ∗ − A∗ z, x .

By (5), we conclude that (E ∗ z)(T )∗ x(T ) = 0 and thus, (E ∗ z)(T ) = 0.  To show that D ⊂ D(T(E,A) ), let z ∈ D and x ∈ C ∞ ([0, T ]; Kn ) with x(0) = 0. Again by integration by parts, z, T(E,A) x =



  d − dt E ∗ − A∗ z, x .

In order to conclude the proof, we claim that {x ∈ C ∞ ([0, T ]; Kn ) | x(0) = 0} is a dense subspace of D(T(E,A) ) equipped with the graph norm induced by T(E,A) . Then the assertion follows from continuity of ·, ·. To show the density, consider a subspace S ⊂ Kn such that Kn = ker E ⊕ S. By the decomposition x = x1 + x2 , with x1 ∈ ker E and x2 ∈ S, we obtain that 1,p ψ : D(T(E,A) ) → Lp ([0, T ]; ker E) × W(0) ([0, T ]; ES),

x → ψ(x) = (x1 , Ex2 )

1,p is a topological isomorphism, where W(0) ([0, T ]; ES) denotes the closed subspace of functions in W 1,p ([0, T ]; ES) that vanish at t = 0. Now the density follows by well-known properties of Sobolev spaces. 2

Remark 2 (Dual DAE operator). Consider the reflection operator R : L1loc (J; Kn ) → L1loc (−J; Kn ),

f (·) → f (−·),

and, for a ∈ R, the shift operator τa : L1loc (J; Kn ) → L1loc (a + J; Kn ),

f (·) → f (· − a).

Then Proposition 1 states that for p ∈ [1, ∞) it holds that  T(E,A) = τ−T RTE ∗ ,A∗ RτT .

(6)

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For n ∈ N let (Xk )nk=1 and (Yk )nk=1 be K-vector spaces and let Ak : Xk → Yk be linear mappings for k ∈ {1, ..., n}. We recall the diagonal operator n

diag (A1 , . . . , An ) :

n

× Xk → k=1 × Yk , (x1, . . . , xn) → (A1x1, . . . , Anxn). k=1

Further note that, for p ∈ N, the matrix 0p,0 ∈ Kp×0 stands for the unique linear mapping from {0} to Kp , whereas the matrix 0p,0 ∈ Kp×0 stands for the unique mapping from Kp to {0}. Hence the diagonal operator composed of A ∈ Km×n with 00,p (0p,0 ) is the matrix formed by adding p zero columns (rows) to A, that is A ∈ K(m+p)×n . diag (A, 0p,0 ) = 0p,n 

diag (A, 00,p ) = [A, 0m,p ] ∈ K

m×(n+p)

,

For m ∈ N we introduce the matrices ⎡0 ⎤ 0 1 1 0   ⎢ 1 ... ⎥ m×m .. .. .. .. ⎥ Nm = ⎢ , Km = , Lm = K(m−1)×m . . . . . ⎣ .. . . . . ⎦ ∈ K 0 1 1 0 . . . 0 ···

1 0

Moreover, for multi-indices α ∈ N(α) , β ∈ N(β) , γ ∈ N(γ) , we set Nα = diag (Nα1 , . . . , Nα(α) ),

Kβ = diag (Kβ1 , . . . , Kβ(β) ),

Lγ = diag (Lγ1 , . . . , Kγ(γ) ). Next we present Kronecker’s famous result on matrix pencils. Theorem 2 (Quasi-Kronecker form [4, Chap. XII]). Let sE − A ∈ K[s]k×n be a matrix pencil. Then there exist W ∈ Glk (K), V ∈ Gln (K) and A1 ∈ Kn1 ×n1 as well as multi-indices α ∈ N(α) , β ∈ N(β) , γ ∈ N(γ) , such that ⎡

sIn1 − A1 0 ⎢ W (sE − A)V = ⎣ 0 0

0 sNα − I|α| 0 0

0 0 sKβ − Lβ 0

⎤ 0 0 ⎥ ⎦. 0 ∗ ∗ sKγ − Lγ

(7)

The multi-indices α, β, γ are uniquely determined by sE − A. Further, the matrix A1 is unique up to similarity. The right hand side of (7) is called quasi-Kronecker form of the pencil sE − A. Remark 3. a) If, additionally, the matrix A1 in (7) is in Jordan canonical form, then (7) is called Kronecker form. Note that, in contrast to the quasi-Kronecker form, the Kronecker form of a real matrix pencil is not necessarily real.

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b) The numbers (β) and (γ) respectively express the column and row rank deficiency of sE − A ∈ K[s]. That is, (β) = n − r (γ) = k − r for r = rankK[s] (sE − A). c) In the particular case (β) = (γ) = 0, that is, the pencil sE − A is regular, then the right-hand side of (7) is called quasi-Weierstraß form and similarly, if A1 is in Jordan canonical form the quasi suffix is dropped. Theorem 2 allows to block-diagonalize DAE operators. Namely, for W ∈ Glk (K), V ∈ Gln (K) leading to quasi-Kronecker form (7), we have  TW EV,W AV = diag

(α)

(β)

(γ)

i=1

i=1

i=1



TIn1 ,A1 , diag (TNαi ,Iαi ), diag (TKβi ,Lβi ), diag (TKγ∗

i

,L∗ γ

i

) .

(8)

This block diagonalization gives rise to the fact that D(T(E,A) ), ker(T(E,A) ), im(T(E,A) ) can be expressed by Cartesian products, i.e.,   D(T(E,A) ) =V −1 D(TIn1 ,A1 ) × D(TNα ,Iα ) × D(TKβ ,Lβ ) × D(TKγ∗ ,L∗γ ) ,   ker(T(E,A) ) =V −1 ker(TIn1 ,A1 ) × ker(TNα ,Iα ) × ker(TKβ ,Lβ ) × ker(TKγ∗ ,L∗γ ) ,   im(T(E,A) ) =W −1 im(TIn1 ,A1 ) × im(TNα ,Iα ) × im(TKβ ,Lβ ) × im(TKγ∗ ,L∗γ ) with, further (α)

D(TNα ,Iα ) =

× i=1

(β)

D(TNαi ,Iαi ),

D(TKβ ,Iβ ) =

D(TK × i=1

βi ,Lβi

)

(γ)

D(TKγ∗ ,L∗γ ) =

D(TK × i=1

γi ,Lγi

),

and analogous representations of the nullspaces and ranges of TNα,Iα , TKβ ,Lβ and TKγ∗ ,L∗γ as Cartesian products. The previous findings yield that the specification of nullspace and range of a DAE operator leads to the determination of nullspace and range of the prototypes TIm ,A , ∗ ,L∗ . TNm ,Im , TKm ,Lm and TKm m Proposition 2. Let m ∈ N and A ∈ Km×m . Then (i) ker(TIm ,A ) = {0}; (ii) ker(TNm ,Im ) = {0};

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M. Puche et al. / Linear Algebra and its Applications 552 (2018) 29–41

⎧⎛ (m−1) ⎞ ⎫ ⎨ y ⎬ (iii) ker(TKm ,Lm ) = ⎝ .. ⎠ | y ∈ W m−1,p ([0, T ]; K), y(0) = . . . = y (m−2) (0) = 0 ; . ⎩ ⎭ y

∗ ,L∗ ) = {0}. (iv) ker(TKm m

Proof. Denote the i-th component of x by xi . d (i) Let x ∈ ker(TIm ,A ). Then dt x = Ax with x(0) = 0, whence x = 0. d (ii) Let x ∈ ker(TNm ,Im ). Then x1 = 0 and xi = dt xi−1 for all i ∈ {2, . . . , m}. A successive insertion gives x1 = x2 = . . . = xm = 0, and thus x = 0. (iii) To show the inclusion “⊃”, let y ∈ W m−1,p ([0, T ]; K) with vanishing first m − 2 derivatives at zero. Consider x ∈ Lp ([0, T ]; Km ) with xi = y (m−i) for all i ∈ {1, . . . , m}. This directly gives that Km x ∈ W 1,p ([0, T ]; Km−1 ) with (Km x)(0) = 0 d and dt Km x = Lm x, that is x ∈ ker(TKm ,Lm ). d To prove “⊂”, consider x ∈ ker(TKm ,Lm ). Then dt xi = xi−1 for all i ∈ (m−i) {2, . . . , m}. Hence, for y := xm , we have xi = y for all i ∈ {1, . . . , m}, m−1,p which further leads to y ∈ W ([0, T ]; K). By (Km x)(0) = 0, we obtain that x1 (0) = . . . = xm−1 (0) = 0, whence y(0) = . . . = y (m−2) (0) = 0. d ∗ ,L∗ ), then x1 =, xi = (iv) Assume that x ∈ ker(TKm dt xi−1 for all i ∈ {2, . . . , m − 1}. m Again, a successive insertion leads to x = 0. 2

Corollary 1. Let sE − A ∈ K[s]k×n be a matrix pencil. With the notation of Theorem 2 it holds that   ker(T(E,A) ) = V −1

(β)

{0} × {0} ×

ker(TK × i=1

βi ,Lβi

) × {0}

where, for i ∈ {1, . . . , (β)}, the spaces ker(TKβi ,Lβi ) are given by Proposition 2 (iii). ∗ ,L∗ . We now characterize the ranges of TIm ,A , TNm ,Im , TKm ,Lm and TKm m

Proposition 3. For m ∈ N and A ∈ Km×m holds (i) im(TIm ,A ) = Lp ([0, T ]; Km ); ⎧⎛ ⎞ ⎨ f1 ··· (ii) im(TNm ,Im ) = ⎝ ... ⎠ ∈ Lp ([0, T ]; Km ) ⎩ fm y1 := f1 ∈ W 1,p ([0, T ]; K), d ∈ W 1,p ([0, T ]; K), y2 := dt f1 + f2 d d y3 := dt ( dt f1 + f2 ) + f3 ∈ W 1,p ([0, T ]; K), .. . d d d (· · · dt ( dt f1 + f2 ) ym−1 := dt +f3 ) + · · · ) + fm−2 ) + fm−1 ∈ W 1,p ([0, T ]; K),

⎫ y1 (0) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ y2 (0) = 0 ⎪ ⎪ ⎪ ⎪ y3 (0) = 0 ⎬ ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ym−1 (0) = 0 ⎭

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(iii) im(TKm ,Lm ) = Lp ([0, T ]; Km−1 ); ∗ ,L∗ ) = im(TN ,I T0 ∗ ) = (iv) im(TKm m m m×(m−1) ,Lm ⎧⎛ m⎞ ⎫ ⎨ f1 ⎬ ⎝ .. ⎠ ∈ im(TNm ,Im ) d (· · · d ( d f1 + f2 ) + f3 ) + · · · ) + fm−1 ) + fm = 0 . . dt dt dt ⎩ ⎭ fm Furthermore, for p ∈ [1, ∞), one has that im(TNm ,Im ) is dense in Lp ([0, T ]; Km ) and p m ∗ ,L∗ ) is not dense in L ([0, T ]; K im(TKm ). m Proof. (i) The result follows by the fact that for any f ∈ Lp ([0, T ]; Km ), the ordinary d differential equation dt x − Ax = f with initial value x(0) = 0 has a solution 1,p m x ∈ W ([0, T ]; K ). (ii) We first show the inclusion “⊃”: Consider f ∈ Lp ([0, T ]; Km ) such that the condition on the right hand side of (ii) holds and define yi ∈ W 1,p ([0, T ]; K), for i ∈ {1, . . . , m − 1}, accordingly. Let x ∈ Lp ([0, T ]; Km ) be defined by xi = −yi for i ∈ {1, . . . , m − 1} and d d d xm = − dt (· · · dt ( dt f1 + f2 ) + f3 ) + · · · ) + fm−1 ) − fm ∈ Lp ([0, T ]; K).

(9)

Then x ∈ D(TNm ,Im ) and, moreover, by construction of xi , we observe that 0 = d x1 +f1 and dt xi−1 = xi +fi for all i ∈ {2, . . . , m}. Thus, f = TNm ,Im x ∈ imTNm ,Im . To prove “⊂”, let x ∈ D(TNm ,Im ) and f = TNm ,Im x. Then the components of d y = −x and f fulfill y1 = f1 and yi = dt yi−1 +fi for all i ∈ {2, . . . , m}. A successive insertion gives the desired result. d (iii) In the following, we make use of the fact that the DAE dt Km x = Lm x + f can d be rewritten as an ordinary differential equation dt x ˜ = Nm−1 x ˜ + e1 x1 + f , where x ˜ ∈ W 1,p ([0, T ]; Km−1 ) is composed of the last m − 1 components of x: Let f ∈ d Lp ([0, T ]; Km−1 ). Let x ˜ ∈ W 1,p ([0, T ]; Km−1 ) with x ˜(0) = 0 be a solution of dt x ˜= 0 d p m Nm−1 x ˜ + f . Then x = x˜ ∈ L ([0, T ]; K ) fulfills x ∈ D(TKm ,Lm ) and dt Km x = Lm x + f , i.e., f ∈ im(TKm ,Lm ). ∗ (iv) The first identity holds since Km = Nm L∗m and L∗m = Im L∗m . Also note that x T0m×(m−1) ,L∗m x = 0 . To prove “⊃” of the second identity, let f ∈ im(TNm ,Im ) be such that the condition on the right hand side of (iv) holds. Define yi , for i ∈ {1, . . . , m − 1}, as on the right hand side of (ii). With x chosen as in the first part of the proof of (ii), we have that TNm ,Im x = f . In particular, xm , defined by (9), equals zero by the x assumption on f . Thus, f = TNm ,Im 0 . x To see “⊂”, let f ∈ im(TNm ,Im T0m×(m−1) ,L∗m ), that is, f = TNm ,Im ( 0 ) for some x ( 0 ) ∈ D(TNm ,Im ). Hence, f ∈ imTNm ,Im and the components of y = −x and f d fulfill y1 = f1 and yi = dt yi−1 + fi for all i ∈ {2, . . . , m − 1}. We further have

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d − dt ym−1 =

whence

d d d dt (· · · dt ( dt f1

d dt xm−1

= fm ,

+ f2 ) + f3 ) + · · · ) + fm−1 ) + fm = 0.

2

Remark 4. a) Let us take a closer view on those elements of f ∈ im(TNm ,Im ) which have the additional property that only the jth component is possibly nonzero, i.e., fi = 0 for all i ∈ {1, . . . , m} \{j}. Then (ii) yields that fj is m −j −1 times weakly differentiable. Indeed, it can be seen that for all j ∈ {1, . . . , m} holds {f ∈ im(TNm ,Im ) | fi = 0 for all i ∈ {1, . . . , m} \ {j}} ⎧⎛ ⎞ ⎫ .. ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ fj ∈ W m−j,p ([0, T ]; K) ⎪ ⎪ ⎨⎜ 0 ⎟ ⎬ ⎜ ⎟ (m−j−1) = ⎜ fj ⎟ with fj (0) = · · · = fj (0) = 0 . ⎪ ⎜0⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎝ ⎠ ∀j ∈ {1, . . . , m − 1} ⎪ ⎪ ⎪ ⎩ .. ⎭ . b) If the components of f ∈ im(TNm ,Im ) are sufficiently smooth, then the sums in the derivatives on the right hand side in (ii) can be expanded, such that we obtain

yj =

j $

(j−i)

fi

for j ∈ {1, . . . , m}.

i=1

Hence, we obtain that ⎧⎛ ⎞ ⎫ j ⎨ f1 ⎬ m $ (j−i) ⎝ .. ⎠ ∈ W m−j,p ([0, T ]; K) fj (0) = 0 ∀j ∈ {1, . . . , m − 1} . ⎩ ⎭ j=1 i=1 fm

×

⊂ im(TNm ,Im ). Note that this set inclusion is strict except for the cases m = 1 and m = 2: Namely, for m > 2, consider f ∈ Lp ([0, T ]; Km ) with f1 = · · · = fm−3 = 0, fm = 0 and fm−2 (t) =

% t

: t ∈ [0, T /2),

T /2 : t > T /2,

fm−1 (t) =

% −1 0

: t ∈ [0, T /2), : t > T /2.

(10)

Then fm−2 ∈ / W 2,p ([0, T ]; K), fm−1 ∈ / W 1,p ([0, T ]; K). However, in terms of (ii), we d have y1 = · · · = ym−3 = 0, ym−2 = fm−2 ∈ W 1,p ([0, T ]; K), ym−1 = dt fm−2 +fm−1 = 1,p 0 ∈ W ([0, T ]; K). Therefore, f ∈ im(TNm ,Im ) by Proposition 3. c) Taking a closer look at the proof of Proposition 3 (ii), we see that x and f with TNm ,Im x = f are related by

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39

d d d d x = − dt Nm (· · · dt Nm ( dt Nm ( dt Nm f + f ) + f ) · · · + f ) − f.

(11)

k If Nm f ∈ W k,p ([0, T ]; Km ) for all k ∈ {1, . . . , m − 1}, then (11) simplifies to the well-known relation (cf. [3,4,7,9])

x=−

m−1 $

i di dti Nm f.

(12)

i=0

Note that the solution formula (12) can be applied without the assumption N k f ∈ W k,p ([0, T ]; Km ), if the distributional derivative is considered instead of the weak derivative in Sobolev spaces. d) By a similar argumentation as in b), we see that ⎧ ⎫ m & (m−j) ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ fj = 0 and ⎪ ⎪ ⎨ f1 ⎬ m j=1 . m−i,p ⎝ . ⎠∈ W ([0, T ]; K) . j ⎪ ⎪ & (j−i) i=1 ⎪ ⎪ fm ⎪ fi (0) = 0 ∀j ∈ {1, . . . , m − 1} ⎪ ⎩ ⎭

×

i=1

⊂im(TKm ,Lm ) Again, this subset relation is strict in the case m > 2: For f ∈ Lp ([0, T ]; Km ) with f1 = · · · = fm−3 = 0, fm = 0 and fm−2 , fm−1 as in (10) holds f ∈ im(TKm ,Lm ) though fm−2 ∈ / W 2,p ([0, T ]; K) and fm−1 ∈ / W 1,p ([0, T ]; K). Corollary 2. Let sE − A ∈ K[s]k×n be a matrix pencil. With the notation of Theorem 2 holds im(T(E,A) )  = W −1

(α)

Lp ([0, T ]; Kn1 ) ×

im(TN × i=1

αi ,Iαi

) × Lp ([0, T ]; K|β|−(β) ) ×

(γ)

× im(TK

j=1

∗ ∗ γj ,Lγj

 ) ,

where, for i ∈ {1, . . . , (α)}, j ∈ {1, . . . , (γ)}, the spaces im(TNαi ,Iαi ), im(TKγ∗ ,L∗γ ) are i i given by the expressions in Proposition 3 (ii)&(iv). Remark 5. Assume that a square pencil sE − A ∈ K[s]n×n is given which is additionally regular, that is det(sE − A) is not the zero polynomial. Then Remark 3 b) implies that the Kronecker form reads  W (sE − A)V =

sIn1 − A1 0

0 . sNα − I|α|

(13)

Further, by Proposition 2, we obtain that the nullspace of T(E,A) is trivial. Further, we can infer from Proposition 3 that imT(E,A) is a dense subspace of Lp ([0, T ]; Kn ).

40

M. Puche et al. / Linear Algebra and its Applications 552 (2018) 29–41

−1 Consequently, T(E,A) : im(T(E,A) ) ⊂ Lp ([0, T ]; Kn ) → Lp ([0, T ]; Kn ) is a densely defined   operator with dense range. Consider f ∈ im(T(E,A) ) and partition ff12 = W f according to the block structure in (13). The variations of constants formula and the previous remark then lead to the well-known solution formula

⎛ '· ⎜0 −1 T(E,A) f =V ⎜ ⎝

exp(A1 (· − τ ))f1 (τ )dτ −

ν−1 & i=0

di i dti N f2

⎞ ⎟ ⎟, ⎠

(14)

where ν = max α, cf. [3,4,7,9]. In particular, T(E,A) has a bounded inverse, if, and only if, ν = 1 and sE − A is regular. Proof of Theorem 1. (i) From Remark 3 we have that rankK[s] (sE − A) = n if, and only if, for in the Kronecker form of sE − A holds (β) = 0. On the other hand, by Corollary 1, we see that the latter is equivalent to ker(T(E,A) ) = {0}. (ii) We will use the fact that imT(E,A) is dense in Lp ([0, T ]; Kk ), if, and only if, the  nullspace of the dual operator T(E,A) is trivial, see e.g. [5, Thm. II.3.7]. By (6), we  have ker T(E,A) = ker TE ∗ ,A∗ RτT , where R is the reflection operator and τT is the shift operator. Now using that RτT : Lp ([0, T ]; Kk ) → Lp ([0, T ]; Kk ) is bijective, we obtain that imT(E,A) is dense in Lp ([0, T ]; Kk ), if, and only if, TE ∗ ,A∗ has trivial nullspace. The latter assertion is, by (i), equivalent to rankK[s] (sE − A) = k. (iii) First note that imA ⊂ imE + A ker E, if, and only if, for any W ∈ Glk (K), V ∈ Gln (K) holds imW AV ⊂ imW EV + W AV ker W EV . Since, further, imTW EV,W AV = W · imT(E,A) , we obtain that imT(E,A) has closed range, if, and only if, imTW EV,W AV has closed range. As a consequence, we may assume without loss of generality that sE − A is in Kronecker form. We can infer from Corollary 2 that im(T(E,A) ) is closed if, and only if, αi = 1 for all i = 1, ..., (α) and γi = 1 for all i = 1, ..., (γ). On the other hand, taking a close look at the Kronecker form (7), we see that imA ⊂ imE + A ker E holds if and only if αi = 1 for all i = 1, ..., (α) and γi = 1 for all i = 1, ..., (γ). (iv) This follows by a combination of (i) and (ii). (v) The implication “b)⇒a)” is trivial. “c)⇒b)”: Assume that sE − A is regular and that imA ⊂ imE + A ker E: Then a combination of (iii) and (iv) implies that T(E,A) is bijective, whence it possesses a bounded inverse. “a)⇒c)”: Assume that T(E,A) is Fredholm. Since Fredholm operators have closed range [1, Lem. 4.38], we obtain from (iii) that imA ⊂ imE +A ker E. If rankK[s] sE − A < n, then a combination of Remark 3 b), Proposition 2 and Corollary 1 implies dim ker T(E,A) = ∞. On the other hand, if rankK[s] sE − A < k, then Remark 3 b) together with Proposition 3 (iv) and Corollary 2 yields codim imT(E,A) = ∞, which is again a contradiction. Hence, sE − A is regular. 2

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Acknowledgement We acknowledge support by the German Research Foundation DFG via the grant RE 2917/4-1 “Systems theory of partial differential-algebraic equations”. References [1] Y.A. Abramovich, C.D. Aliprantis, An Invitation to Operator Theory, Grad. Stud. Math., vol. 50, American Mathematical Society, Providence, RI, 2002. [2] T. Berger, C. Trunk, H. Winkler, Linear relations and the Kronecker canonical form, Linear Algebra Appl. 488 (2016) 13–44. [3] S.L. Campbell, Singular Systems of Differential Equations, Res. Notes Math., vol. 40, Pitman (Advanced Publishing Program), Boston, Mass.–London, 1980. [4] F.R. Gantmacher, The Theory of Matrices, vols. 1, 2, Chelsea Publishing Co., New York, 1959, translated by K.A. Hirsch. [5] S. Goldberg, Unbounded Linear Operators: Theory and Applications, McGraw-Hill Book Co., New York–Toronto, Ont.–London, 1966. [6] M. Hanke, Linear differential-algebraic equations in spaces of integrable functions, J. Differential Equations 79 (1) (1989) 14–30. [7] P. Kunkel, V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, EMS Textbk. Math., European Mathematical Society (EMS), Zürich, 2006. [8] P. Kunkel, V. Mehrmann, Formal adjoints of linear DAE operators and their role in optimal control, Electron. J. Linear Algebra 22 (2011) 672–693. [9] R. Lamour, R. März, C. Tischendorf, Differential-algebraic equations: a projector based analysis, in: Differential-Algebraic Equations Forum, Springer, Heidelberg, 2013. [10] V.H. Linh, R. März, Adjoint pairs of differential-algebraic equations and their Lyapunov exponents, J. Dynam. Differential Equations 29 (2) (2017) 655–684. [11] R. März, Differential-algebraic equations from a functional-analytic viewpoint: a survey, in: Surveys in Differential-Algebraic Equations. II, Differ.-Algebr. Equ. Forum, Springer, Cham, 2015, pp. 163–285. [12] R. Picard, A structural observation for linear material laws in classical mathematical physics, Math. Methods Appl. Sci. 32 (14) (2009) 1768–1803. [13] S. Trostorff, M. Waurick, On differential-algebraic equations in infinite dimensions, preprint, available at arXiv:1711.00675. [14] S. Trostorff, M. Waurick, On higher index differential-algebraic equations in infinite dimensions, preprint, available at arXiv:1710.08750.