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On Differential-Algebraic Equations in Infinite Dimensions

We consider a class of differential-algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.


Introduction
Let H be a Hilbert space and let M 0 , M 1 be bounded linear operators in H. In this article we are concerned with the implicit initial value problem of finding u : R ≥0 → H such that M 0 u ′ (t) + M 1 u(t) = 0, t > 0, u(0+) = u 0 ( * ) for some given initial datum u 0 ∈ H, where ( * ) has to be interpreted in a particular sense. This problem is commonly known as a "differential-algebraic equation" and has attracted a lot of interest in recent years. Most prominently, for finite-dimensional H, differential-algebraic equations have applications to control and electrical circuit theory, see e.g. [2]. For so-called regular systems and a finite-dimensional state space H, the discussion of ( * ) can be simplified by applying a generalised Jordan normal form, see e.g. [1,2,5]. A similar strategy can be used for particular infinite-dimensional cases, see [13]. We are unaware of any other treatment of differential-algebraic equations in infinite spatial dimensions. Thus, for the convenience of the reader, we present this work as much as self-contained as possible.
Our main motivation for discussing the infinite-dimensional case comes from the viewpoint of so-called evolutionary equations, a certain class of partial differential-algebraic equations introduced in [10]. The focus of [10] was to derive a particular Hilbert space setting such that a large class of linear equations in mathematical physics can be dealt with in a unified framework. A particular case is the equation for some (unbounded, skew-selfadjoint) operator A in H. In order to cover a large class of problems and to allow for M 0 = 0, as well, the basic setting in [10] uses homogeneous initial conditions. This, on the other hand, lead to the question of what possible initial values can be assumed for (1) in order to have a solution u.
In the habilitation thesis [18] this question has been addressed. In this article we shall carry out a more detailed analysis for the case of when A = 0. For this reason, we shall furthermore focus on the case of regular systems with index 0. Moreover, we will assume that R(M 0 ) ⊆ H is closed. Under the assumptions mentioned, we will derive a space of admissible initial values u 0 such that ( * ) admits a continuous differentiable ("strong") solution. Moreover, we will discuss the regularity of solutions for ( * ), if u 0 is not admissible. It turns out that the correct notion of a solution can be phrased as a "mild" solution, that is, u satisfies ( * ) only in an once integrated sense. Having obtained well-posedness of ( * ), we will address the asymptotic behaviour of both strong and mild solutions. We will formulate a spectral characterisation of both mild and strong exponential dichotomy as well as mild and strong exponential stability and will recover the well-known finite-dimensional stability results, see e.g. [3,2,15]. We emphasise that in our approach we do not use the generalised Jordan normal form or Weierstrass normal form, simply because these strategies are not applicable under the assumptions stated. A more thorough analysis of the case of non zero index has been initiated in [19] and will be addressed in future work. We shall comment on the organisation of this text. In the next section, we gather some material on evolutionary equations and establish the time-derivative as a suitable continuously invertible operator in a weighted vector-valued L 2 -space. In Section 3 we introduce our central object of study and define regular linear operator pencils (of index 0). We also provide a characterisation of regularity given the closedness of the range of M 0 . The characterisation of admissible initial values u 0 such that ( * ) has a continuous differentiable solution will be given in Section 4. Mild solutions, that is, where ( * ) holds in an integrated sense, only, will be considered in Section 5. It turns out that the problem of finding continuously differentiable solutions to ( * ) can be written as an abstract Cauchy problem leading to a norm-continuous semigroup as a fundamental solution. The spectrum of the generator of this semigroup is shown to coincide with the spectrum of the operator pencil in Section 6. In Section 7 we address the asymptotics (exponential dichotomy) of both the derived Cauchy problem as well as for mild solutions. Note that the difficult part is to define an appropriate notion for the case of mild solutions as they are a priori only locally integrable.

Preliminaries
Following [10,4,9], we introduce the temporal derivative as a normal operator in an exponentially weighted L 2 -space. One can view the exponential weight as an L 2 -variant of the well-known Morgenstern norm that is used for proving the Picard-Lindelöf theorem for arbitrary large Lipschitz constant, see [7] or [11,Section 4] for a more recent reference.
Definition. Let ρ ∈ R. We define the Hilbert space L 2,ρ (R; H) of (equivalence classes of) H-valued functions as follows equipped with the usual inner product Moreover, we define the operator ∂ 0,ρ : Note that every continuous function f : R → H with support bounded below that satisfies the exponential growth condition |f (t)| H ≤ M e ωt for some ω ∈ R, M ≥ 0 and all t ∈ R belongs to L 2,ρ (R; H) for all ρ > ω. We shall need this observation later on. Some more remarks are in order. (b) The domain H 1 ρ (R; H) is an exponentially weighted variant of the Sobolev space H 1 (R; H) and becomes a Hilbert space with respect to the graph inner product of ∂ 0,ρ , that is, It turns out that the operator ∂ 0,ρ is a normal operator, whose spectral representation is given in terms of the so-called Fourier-Laplace transform, which is defined as follows.
We conclude the section with the following variant of the Sobolev-embedding theorem.
In the next section, we recall the notion of regularity for differential-algebraic equations, see e.g. [1,6]. Note that we shall deviate slightly from the usual notion of regularity in as much as we restrict our consideration to index 0, only. For differential-algebraic equations with nontrivial index, we refer to the study initiated in [19].

Regular linear operator pencils
Throughout this section, let M 0 , M 1 ∈ L(H).
Definition. We consider the function We call M regular (of index 0), if there exists ν ∈ R such that: (a) C Re>ν ⊆ ̺(M) and Moreover, we set  for each x ∈ R(M 0 ) and y ∈ N (M 0 ). Then, the linear pencil associated with (M 0 , M 1 ) is regular. Note that in the context of partial differential equations the mentioned positive definiteness conditions happen to be satisfied in many applications, see [10,Section 3.3.3].
We note that in the latter example, the strict accretivity of M 0 on R(M 0 ) in particular implies that R(M 0 ) is closed. In this case, we can characterise the regularity of the pencil by the invertibility of a particular operator. For the next statement, we introduce for a closed subspace S ⊆ H the canonical embedding It is easy to see that ι * S : H → S is surjective and acts as the orthogonal projection onto S, see also [12,Lemma 3.2].
Proposition 3.2. Assume that R(M 0 ) is closed and denote by M the linear pencil associated with (M 0 , M 1 ). Then the following statements are equivalent: In the latter case, σ(M) = σ − M −1 0 M 1 is compact and for each z ∈ C we have that as well as Proof. (i) ⇒ (ii): Assume that M is regular. By the closed graph theorem, it suffices to prove that the operator ι * Then, (u n ) n is bounded and thus, by passing to a suitable subsequence (not relabelled), we can assume that it is weakly convergent. We denote its weak limit by u. Then i.e., u ∈ N (M 0 ). Moreover, since for sufficiently large n ∈ N. As the right-hand side tends to 0 as n tends to infinity, we infer ι N (M 0 ) u = 0 and thus, u = 0.
(ii) ⇒ (i): For z ∈ C we have that Since M 0 is bijective, it is boundedly invertible by the closed graph theorem, and hence,

A first initial value problem
Throughout, let M 0 , M 1 ∈ L(H) such that R(M 0 ) is closed. Moreover, we assume that the linear operator pencil M associated with (M 0 , M 1 ) is regular. In this section, we are concerned with the following initial value problem where u 0 ∈ H is a given initial value and u : R ≥0 → H is to be determined. We recall that (IVP1) is a differential-algebraic equation. The aim of this section is to find a characterisation of possible initial values u 0 such that the solution of (IVP1) is continuously differentiable on R >0 . We present a necessary condition for u 0 in the next statement. The main result of this section will be to identify this necessary condition also as sufficient, see Corollary 4.5 below.
Proof. Since we have due to the closedness of R(M 0 ), i.e. u 0 ∈ IV. For the converse of Lemma 4.1, we need a couple of preparations.
is bijective and hence, continuously invertible.
Proof. To show that the operator is one-to-one, let x ∈ IV ∩ N (M 0 ). Hence, there is y ∈ H such that M 1 x = M 0 y and consequently, Thus, for n large enough, we get For showing that it is onto, let w ∈ R(M 0 ). For n ∈ N large enough, we set We claim that u n ∈ IV. Indeed, using that Since nw ∈ R(M 0 ), we derive the claim. Moreover, we find y ∈ H such that w = M 0 y and we compute for all n ∈ N which shows that (u n ) n∈N is a bounded sequence in IV ⊆ H and hence, by passing to a suitable subsequence, we may assume without loss of generality that u n ⇀ u for some u ∈ IV. We obtain which proves the assertion.
yields the desired isomorphism.
Corollary 4.5. Let u 0 ∈ IV. Then there exists a unique solution u : R ≥0 → H of (IVP1), which is given by Proof. We address uniqueness first: If u : R ≥0 → H is a solution of (IVP1), then and by Proposition 4.3 for t > 0. Hence, For the existence part consider Then clearly u(0) = u 0 and for each t > 0. Hence, u solves (IVP1).

A second initial value problem
In this section, we aim at providing a different perspective to the initial value problem stated in (IVP1). In the previous section, we have addressed finding solutions u in the "strong sense", that is, we were looking for continuously differentiable solutions. The differentiability, in turn, restricted the class of admissible initial values. Here, we study the solvability of the differential-algebraic equation in a "weak" or "mild" sense. Weakening the solution concepts, we will be able to solve the differential-algebraic equation for all initial data from H. For this, we provide a different solution representation of the solution to (IVP1). Any function defined on R ≥0 is considered to be a function on R by extension by 0. The support of a function f will be denoted by spt f .
Proposition 5.1. Let u 0 ∈ IV and u : R ≥0 → H the solution of (IVP1). Then there is Proof. Since by Corollary 4.5 for t ≥ 0, we infer that |u(t)| ≤ M e ρ 1 t for some M ≥ 1, ρ 1 ∈ R and all t ≥ 0. Hence, using that spt u ⊆ R ≥0 , we obtain u ∈ ρ>ρ 1 L 2,ρ (R; H). We define ρ 0 := max{0, ρ 1 , s 0 (M)} and claim that As we have seen in Proposition 5.1, the solution u of (IVP1) is given by the relation for some ρ > 0 large enough and u 0 ∈ IV. Thus, we obtain the following consequence of Corollary 4.5 and Proposition 5.1: Then for ρ > 0 large enough we get Note that the right-hand side of the equation in Corollary 5.2 also makes sense for u 0 ∈ H. Indeed, if we just require u 0 ∈ H, then u given as above solves a weaker variant of the initial value problem (IVP1): We introduce the respective notion first: Note that, if u is a solution to (IVP2), then it follows from Lebesgue's dominated convergence theorem that M 0 u : R ≥0 → H is continuous and that the initial datum is attained.
. Then the following are equivalent i.e., u is a mild solution of (IVP1).
Note that, by Theorem 5.4, we deduce that if u is a mild solution of (IVP1), then u ∈ L 2,ρ (R; H). Moreover, we obtain that which yields another representation for the solution u. Then f (i · +κ) ∈ L 2 (R; H) for each µ ≤ κ ≤ ρ and Proof. The first claim follows from the boundedness of t → (it + κ)f (it + κ) for each µ ≤ κ ≤ ρ. For showing the second claim, let (R n ) n∈N be a sequence in R >0 such that R n → ∞ and for almost every t ∈ R as n → ∞. For n ∈ N, t ∈ R we compute by the Cauchy integral theorem. Hence, we can estimate Together with (3), this yields the assertion.
Remark 5.7. The latter lemma is a special case of the famous Paley-Wiener Theorem (see [8] or [14, 19.2 Theorem]), characterising the L 2 functions supported on the positive real line by their Laplace transform lying in the Hardy space H 2 (C Re>0 ; H). However, to keep this article self-contained and since we do not need this deeper result here, we restrict ourselves to the easier case outlined in Lemma 5.6.
Corollary 5.8. Let ρ 0 ∈ R, F : C Re≥ρ 0 → L(H) bounded, continuous and analytic in C Re>ρ 0 . Let g ∈ L 2,ρ 0 (R; H) with spt g ⊆ R ≥a for some a ∈ R. Then Proof. Let (φ n ) n∈N be a sequence in C ∞ c (R; H) such that spt φ n ⊆ R ≥a for all n ∈ N and φ n → g in L 2,ρ 0 (R; H) as n → ∞. By continuity and hence, it suffices to show that for ψ ∈ C ∞ c (R ≥a ; H). Next, we shall argue that we may assume without loss of generality a = 0. Indeed, let ψ ∈ C ∞ c (R ≥a ; H) and consider the function ψ := ψ(· + a) ∈ C ∞ c (R ≥0 ; H). Then we have Thus, we need to check spt L * ρ 0 F (i · +ρ 0 )L ρ 0 ψ ⊆ R ≥0 . Hence, proving the case a = 0 indeed implies the general statement. So, consider the function f : C Re≥ρ 0 → H, given by where ψ(z) = 1 √ 2πR e −zs ψ(s) ds = (L Re z ψ) (Im z).
Then clearly, f is continuous and analytic in C Re>ρ 0 and Having these prerequisites at hand, we are now in the position to prove the main result of this section.
Since ξ was arbitrary, we conclude the first step and deduce that Moreover, we compute for each t ∈ R and hence, with (4) and (6) we get By using (ii) and (5), we further obtain Since ϕ ∈ C ∞ c (R; H) was chosen arbitrarily, we obtain (iii).
Then, f is continuous and analytic in C Re>ρ and satisfies Thus, by Lemma 5.6 we infer that is supported on R ≥0 . Moreover, we compute for each t ∈ R. Hence, since For showing the remaining claim, let ϕ ∈ C ∞ c (R >0 ). From (7) we read off with the help of Proposition 2.2 ∂ 0,ρ M 0 (u − χ R >0 u 0 ) + M 1 u = 0 and consequently, In the following, we shall address the asymptotic properties of the solution u, if considered as a solution to (IVP1) or to (IVP2). For this, we associate a spectrum to (IVP1) and (IVP2). The next section proves equality of the spectra to be introduced.
In this section, we will assume throughout that M is a regular linear operator pencil associated with (M 0 , M 1 ) in some Hilbert space with R(M 0 ) ⊆ H closed. We consider the following two spectra We note that the first spectrum is strongly related to the asymptotics of the initial value problem treated in Section 4, while the second spectrum determines the asymptotic behaviour of the initial value problem studied in Section 5 (for details see Section 7). Before we address the spectra just introduced, we elaborate a bit more on the connection of IV and N (M 0 ) ⊥ , see also Remark 4.4. We reformulate IV first: Then u ∈ IV if, and only if, Proof. With the unitary operators U 0 and U 1 as introduced in Proposition 3.2, we have Thus, for u ∈ H with (x, y) := U 0 u we obtain where in the last equivalence we have used the invertibility of ι * Since, M is regular and R(M 0 ) ⊆ H is closed, we obtain the assertion by Proposition 3.2. Indeed, an application of Proposition 3.2 yields that ι * Proof. We shall prove (a) first. Using Lemma 6.1 we observe which shows the asserted equivalence. Moreover, if N (M 0 ) ⊥ ⊆ IV, we infer where we used that by Proposition 4.3 ι * R(M 0 ) M 0 ι IV is an isomorphism. As For the proof of (b), we observe that by Lemma 6.1 Proof. Recall from Proposition 3.2 that By definition, we have For z ∈ C we show z M 0 + M 1 is continuously invertible if, and only if, For this, let u ∈ R(M 0 ) and v ∈ IV be such that The latter is the same as saying By Lemma 6.1, we deduce that the latter can be written as Hence, for u ∈ R(M 0 ) and v ∈ IV the equation implies On the other hand, following the argument in reverse direction, we get that if x ∈ N (M 0 ) ⊥ solves (10), then solves (9). Note that v ∈ IV, by Lemma 6.1. The assertion follows.
In the next section we discuss the asymptotic properties of the solutions of both initial value problems (IVP1) and (IVP2).

Asymptotic Properties
In the previous section, we have shown that the spectra connected to strong and mild solutions coincide. Hence, one might argue that the asymptotic properties (i.e., exponential stability or instability) of solutions to (IVP1) and (IVP2) are the same. For this, we note here that for mild solutions it does not make sense to talk about pointwise properties since mild solutions are in a certain L 2 -space, only. We shall however introduce the correct concepts anticipating this problem. It will turn out that the exponential weight in the L 2 -spaces considered in this exposition can be used to define asymptotically stable parts. We assume throughout this section that M is a regular linear operator pencil associated with (M 0 , M 1 ) and that the range of M 0 is a closed subspace of H. Moreover, we consider the linear pencil N associated with (M 0 , −M 1 ). Since M is regular, we have by Proposition 3.2 that σ(M) is compact and that M(z) −1 is uniformly bounded outside a ball B(0, ρ) ⊆ C for some ρ > 0 large enough. Since we infer that also N is regular and N (z) −1 is uniformly bounded outside the same ball B(0, ρ). We first show that there is a strong connection between the solutions of the initial value problems induced by M and by N .
We recall from Theorem 5.4 that for each u 0 ∈ H the mild solution u of (IVP1) satisfies Moreover, we have that R ≥0 ∋ t → (M 0 u) (t) is a continuous function. Hence, pointevaluation of M 0 u is well-defined. This will be used in the next statement.
Proposition 7.1. Let N be the linear pencil associated with (M 0 , −M 1 ) and ρ > 0 large enough, such that sup z∈C\B(0,ρ) M(z) −1 < ∞. Moreover, let u 0 ∈ H and let u ∈ L 2,ρ (R; H) be given by Let T > 0 and define Proof. We consider the following function We note that v ∈ L 2,ρ (R; H) since u, w ∈ L 2,ρ (R; H) and spt u ⊆ R ≥0 by Theorem 5.4. We claim that If the latter is true, the assertion follows, since then spt v ⊆ R ≥0 by Theorem 5.4 and thus, To verify (11), we compute using that spt u, spt w ⊆ R ≥0 , and thus, Since u satisfies (2) by Theorem 5.4, we may compute where we have again used Theorem 5.4. Summarising, we obtain which proves (11).
We introduce the following two solution operators: Let N the linear pencil associated with (M 0 , −M 1 ) and let ρ > 0 such that sup z∈C\B(0,ρ) M(z) −1 < ∞. Then we define Using this notation, we realise that Proposition 7.1 states that for all u 0 ∈ H and T > 0 for all µ ≥ ρ, which in turn is equivalent to for every t ∈ R, µ ≥ ρ. By the identity theorem, the latter is equivalent to for all z ∈ C \ B(0, ρ).
Definition 7.4. We say that M admits a strong exponential dichotomy, if there exist closed subspaces S, T ⊆ IV with the following properties: S ∔ T = IV, S and T are invariant under M and there is ρ > 0, C ≥ 0 such that for all t ≥ 0 where u := S(M)u 0 . Moreover, we say that M is strongly exponentially stable, if it admits a strong exponential dichotomy with S = IV and T = {0}.
We now reformulate the exponential growth on the subspace T as an exponential decay for the pencil N . We note that the IV-spaces for M and N coincide, which is a consequence of (i) There exists C > 0 such that (ii) There exists C > 0 such that Proof. (i) ⇒ (ii): Let w 0 ∈ T and set w := S(N )w 0 . We first prove that w(t) ∈ T for each t ≥ 0. Indeed, since we infer that also Hence, T is invariant under N and thus w(t) ∈ T for each t ≥ 0. We fix t > 0 and set u := S(M) (w(t)) . Since w(t) ∈ T we obtain |u(s)| H ≥ Ce ρs |w(t)| H for each s ≥ 0. Using now Proposition 7.1 (and interchanging the roles of M and N ), we derive u = w(t − ·) on [0, t] and hence, in particular Thus, (12) gives |w(t)| H ≤ C −1 e −ρt |w 0 | H which shows the claim.
(ii) ⇒ (i): The claim can be shown by following the same argumentation as above.
Lemma 7.6. Let S ⊆ IV be a closed subspace. Then the following conditions are equivalent (i) S is invariant under M and there exists C > 0, ρ > 0 such that (ii) There exists ρ ′ > 0 such that for all u 0 ∈ S.
(ii) ⇒ (i): Let u 0 ∈ S. As u := S(M)u 0 is continuous, we infer (S(M)u 0 ) (t) ∈ S for every t ≥ 0. Let now Then we compute for each t ≥ 1 using the representation of u(t) from Corollary 5.2 Since S(M) : S → L 2,−ρ ′ (R; S) is closed, we infer its boundedness by the closed graph theorem. Hence, for some C ′ ≥ 0 By Corollary 5.2 again, we deduce for some C ′′ ≥ 0 Thus, the assertion follows with ρ = ρ ′ .
With the latter two lemmas at hand, we immediately obtain the following proposition.
where N denotes the pencil associated with (M 0 , −M 1 ).
Since this characterisation of a strong exponential dichotomy does not require any continuity conditions on the solutions, we can use this notion to define exponential dichotomies for our second initial value problem (IVP2). For this, observe that by Theorem 5.4, we obtain that the unique solution u solving (IVP2) for u 0 ∈ N (M 0 ) is the zero function. Thus, for the description of asymptotic behaviour, we can dispense with the space N (M 0 ).
Definition 7.8. We say that M admits a mild exponential dichotomy if there exist closed subspaces S, T ⊆ N (M 0 ) ⊥ satisfying the following properties: S ∔ T = N (M 0 ) ⊥ and there exists ρ > 0 such that We say that M is mildly exponentially stable, if it admits a mild exponential dichotomy with S = N (M 0 ) ⊥ and T = {0}.
The main goal is now to provide a characterisation for exponential dichotomy in terms of the spectrum σ IV (M) = σ(M). As a prerequisite, we start to study the simple case M 0 = 1; moreover, we shall characterise mild and strong exponential stability. Note that we recover the stability theorem in [2, Theorem 3-1.1] for the finite-dimensional case.  Proof. We show that (i) implies (ii), first. Since M is a bounded operator, its spectrum is compact. Let ρ 2 > ρ 1 with σ(−M ) ⊆ C Re<ρ 2 . Assume by contradiction that σ(−M ) ∩ C Re>ρ 1 = ∅. The mapping ̺(−M ) ∩ C Re>ρ 1 ∋ z → (z + M ) −1 is holomorphic. By assumption we find a convergent sequence (z n ) n in C Re>ρ 1 such that, for every n ∈ N, z n lies in the component of C for each ρ > max{s 0 ( M), 0}. By assumption, the mapping is analytic. For ρ ′ ∈]ρ 1 , ρ[, we in particular deduce that f : {z ∈ C ; ρ ′ ≤ Re z ≤ ρ} → H is continuous and analytic in the interior. Since for |z| > 2 M we have by the Neumann series, we infer sup ρ ′ ≤Re z≤ρ |zf (z)| < ∞ and thus, u = L * ρ ′ f (i · +ρ ′ ) ∈ L 2,ρ ′ (R; H) according to Lemma 5.5. Due to Corollary 4.5 the next result is a direct consequence of Lemma 7.9: Theorem 7.10. M admits a strong exponential dichotomy, if, and only if, The pencil M is exponentially stable, if, and only if, From Section 6, Theorem 7.10 and Theorem 7.12 we derive the following statement.
Corollary 7.13. M admits a strong exponential dichotomy if and only if it admits a mild exponential dichotomy. Moreover, M is strongly exponentially stable if and only if it is mildly exponentially stable.