
==== Front
J Evol Equ
J Evol Equ
Journal of Evolution Equations
1424-3199
1424-3202
Springer International Publishing Cham

1003
10.1007/s00028-024-01003-3
Article
On the Weierstraß form of infinite-dimensional differential algebraic equations
http://orcid.org/0009-0007-1214-2929
Erbay Mehmet erbay@uni-wuppertal.de

1
Jacob Birgit 1
Morris Kirsten 2
1 https://ror.org/00613ak93 grid.7787.f 0000 0001 2364 5811 IMACM, University of Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
2 https://ror.org/01aff2v68 grid.46078.3d 0000 0000 8644 1405 Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1 Canada
2 9 2024
2 9 2024
2024
24 4 737 8 2024
© The Author(s) 2024
2024
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The solvability for infinite-dimensional differential algebraic equations possessing a resolvent index and a Weierstraß form is studied. In particular, the concept of integrated semigroups is used to determine a subset on which solutions exist and are unique. This information is later used for a important class of systems, namely, port-Hamiltonian differential algebraic equations.

http://dx.doi.org/10.13039/501100000038 Natural Sciences and Engineering Research Council of Canada Bergische Universität Wuppertal (3089)Open Access funding enabled and organized by Projekt DEAL.

issue-copyright-statement© Springer Nature Switzerland AG 2024
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pmcIntroduction

Linear differential algebraic equations (DAEs), sometimes also called descriptor systems or implicit differential equations, arise in various fields such as physics, engineering and economics. They can be written as1 ddtEx(t)=Ax(t),t≥0.

Here, for complex Hilbert spaces X and Z, E:X→Z is a bounded operator (denoted by E∈L(X,Z)) and A:D(A)⊆X→Z is a closed and densely defined operator. We will often write (E, A) to refer to (1).

In comparison with ordinary differential equations, DAEs include both algebraic and differential constraints. Because of this, in general E has a non-trivial kernel. In finite dimensions it is always possible to decouple the system into an algebraic and a differential part by transforming (1) into a so-called Weierstraß form2 ddtI00Nx1(t)x2(t)=J00Ix1(t)x2(t),t≥0,

where J is in Jordan form and N is nilpotent, as long as (λE-A)-1∈L(Z,X) for some λ∈C, [11, Ch. 2.1]. In this context, the nilpotency degree of N is called the nilpotency index of (E, A) [2, 11]. For finite-dimensional spaces, this definition of index coincides with other common definitions of the index. In this context, investigation of the different index definitions can be found in [5, 22, 23] for the resolvent index, [7, 19] for the radiality index, [2] for a general comparison of different known indices in infinite dimensions and [11, 12] for the finite-dimensional case.

However, in infinite dimensions the existence of a Weierstraß form is not guaranteed, nor is there a general procedure for calculating it. In view of this difficulty, we will specify a condition under which such a form always exists, namely the existence of the radiality index. One of the main uses of the Weierstraß form is to analyse the well-posedness of the system, since the DAE is divided into an algebraic part and an ODE part, see (2). For the ODE part it is possible to use known solution methods. The solvability of infinite-dimensional DAEs has been intensively studied; see for example, [5, 7, 13, 14, 17, 18, 20–23]. In [20] the splitting of X into kerE and ranE∗ and the restriction of the DAE to the factorized space X/kerE is studied, in [7, 19] and [21] the splitting of the space, given that the radiality index exists, and the growth of the pseudo-resolvents (λE-A)-1E and E(λE-A)-1 is analysed, in [5] a more general observation of the pseudo-resolvent growth is provided and a useful dissipativity condition for solving the DAE is presented, in [22] and [23] with the use of Wong-sequences and the resolvent index a super-set of the solution space is determined, in [18] a sufficient condition in terms of Hille-Yosida-type resolvent estimates is offered, in [14] with the help of integrated semigroups the well-posedness is investigated, and in [13], the solvability of a class of port-Hamiltonian DAEs is discussed.

As in [14], we will study the well-posedness of (1) using integrated semigroups. There is a strong connection between integrated semigroups and the growth rate of the resolvent (λE-A)-1, for λ∈ρ(E,A)≠∅, whereρ(E,A):={λ∈C|(λE-A)-1∈L(X,Z)}.

A densely defined linear operator A generates an integrated semigroup if there exists an n∈N, such that3 (λI-A)-1≤Cλn-1,λ∈CRe≥ω,

for some C,ω>0 [15, Thm. 4.8]. For n=0 this matches the condition of a sectorial operator A, which is used for the generation of a analytic semigroup. Notice that (3) coincides with the complex resolvent index for E=I. Based on this, we build on existing solution methods for the abstract Cauchy problem ddtx(t)=Ax(t), x(0)=x0 on the subspace D(An)=ran((λI-A)-1)n [10, Sect. 1.3] and extend these studies to the differential algebraic Eq. (1) with x0∈ran((λE-A)-1E)n. For further details on integrated semigroups we refer to [1, 15].

We subsequently focus on a special class of systems, namely, port-Hamiltonian differential algebraic equations (pH-DAEs). These systems provide a framework for modelling and analysing energy-dissipating physical systems, including electrical circuits, mechanical systems, fluid dynamic, thermal systems or robotics and control systems (just to name a few). There are different ways to define and approach pH-DAEs, such as using relations [4] or Dirac structures [24]. Here, we use an operator formulation, as in [13]. To be more specific, we consider4 ddtEx(t)=AQx(t),t≥0

with E,Q∈L(X,Z), Q invertible, A:D(A)⊆Z→Z closed, densely defined and dissipative and E∗Q≥0 non-negative, i.e. ⟨E∗Qx,x⟩≥0 for all x∈X. The Hamiltonian of the pH-DAE (4) is given by ⟨E∗Qx,x⟩. Additionally to this, we will assume that ranE is closed. This guarantees the dissipation of the energy5 ddt⟨E∗Qx(t),x(t)⟩≤0,

along all classical solutions of (4).

The paper is organized as follows. In Sect. 2 a set of initial conditions for which (1) has a solution is characterized. In Sect. 3 the radiality index is defined and it is shown that if ranE is closed, existence of this index implies existence of a Weierstraß form. Several new results relating the radiality index and the degree of nilpotency are also obtained. Furthermore, provided that the complex resolvent index exists, and the Weierstraß form exists with densely defined A1, then A1 generates an integrated semigroup.

In Sect. 4 port-Hamiltonian DAEs are formally described and their dissipativity is proven. Solvability of port-Hamiltonian DAEs is studied in Sect. 5. With the aid of the results earlier in the paper, it is shown that pH-DAEs have complex resolvent index at most 3. This extends a result that is known in finite dimensions to the infinite-dimensional setting. Existence of solutions to pH-DAEs is then obtained, along with generation of an integrated semigroup.

Existence of solutions on a subspace

Throughout this article, let X and Z be Hilbert spaces, and, if not mentioned otherwise, A:D(A)⊆X→Z is a closed and densely defined operator and E∈L(X,Z). In this section, we prove that if that the complex resolvent index exists, solutions exist on a subspace. We first recall the definition of the complex resolvent index, see [2, 22].

Definition 2.1

The (complex) resolvent index of (1) is the smallest number p=pres(E,A)∈N (p=pc,res(E,A)), such that there exists a ω∈R, C>0 with (ω,∞)⊆ρ(E,A) (CRe>ω⊆ρ(E,A)) and6 ‖(λE-A)-1‖≤C|λ|p-1,λ∈(ω,∞)(λ∈CRe>ω).

In the following we determine a set of initial conditions for which the differential algebraic Eq. (1) has a classical solution.

In this context, we say that x:R≥0→X is a classical solution of (1) for an initial condition x0∈X, if x(·) is continuous on R≥0, Ex(·) is continuously differentiable on R>0, x(t)∈D(A) for all t≥0, x(0)=x0 and (1) holds, and we say that x:R≥0→X is a mild solution, if Ex(·) is continuous, and for almost all t≥0, it holds ∫0tx(s)ds∈D(A) withEx(t)-Ex0=A∫0tx(s)ds.

The following theorem is a generalization of the solution theory of the abstract Cauchy problem ddtx(t)=Ax(t), x(0)=x0 and extends the proofs of [10, p. 34] and [16, Ch. 4, Thm. 1.2] to the DAE case with the help of pseudo-resolvents.

Theorem 2.2

Assume that (E, A) has a complex resolvent index. Then, for every x0∈ran((μE-A)-1E)pc,res(E,A)+2 there exists a unique classical solution x:R≥0→X of (1).

Proof

Existence: Let p=pc,res(E,A)+2 and, for the sake of simplicity, define R(μ):=((μE-A)-1E) for μ∈ρ(E,A). By assumption, the complex resolvent index exists. Thus, there are ω,C>0, such that7 R(λ)≤Cλpc,res(E,A)-1E,λ∈CRe≥ω.

Let x0∈ranR(μ)p. Thus, there exist a z0∈X and a μ∈CRe>ω such that x0=(-1)p-1R(μ)pz0 and therefore8 R(λ)(λ-μ)p≤C~λp-3λ-μp,

for a C~>0 and for all λ≠μ. Thus, the integral of eλtR(λ)z0(λ-μ)p along the imaginary axis at Reλ=ω exists and, especially, the function9 x(t):=(-1)p2πi∫ω-i∞ω+i∞eλtR(λ)z0(λ-μ)pdλ,t≥0,

is continuously differentiable. Since E is bounded, the function Ex(t) becomes differentiable as well and10 ddtEx(t)=(-1)p2πi∫ω-i∞ω+i∞λeλtER(λ)z0(λ-μ)pdλ

11 =(-1)p2πi∫ω-i∞ω+i∞eλtAR(λ)z0(λ-μ)pdλ+(-1)p2πi∫ω-i∞ω+i∞eλtEz0(λ-μ)pdλ⏟=0=Ax(t).

Here, we used that Ez0(λ-μ)p converges uniformly to 0 for λ→∞ and Jordan’s lemma to show that the second integral in (11) vanishes. Furthermore,x(0)=(-1)p2πi∫ω-i∞ω+i∞R(λ)z0(λ-μ)pdλ.

Using again Jordan’s lemma and, additionally, the residue theorem, we derivex(0)=(-1)p-1(p-1)!limλ→μdp-1dλp-1R(λ)z0=(-1)p-1R(μ)pz0=x0.

Here, we used ddλR(λ)nz0=-nR(λ)n+1z0 for all n∈N and λ∈ρ(E,A) ([19, Lem. 2.4.1]). Thus, x is a classical solution of (1).

Uniqueness: We assume that x is a solution with x(0)=0 and prove that x(t)=0 for all t≥0. Since x is a solution of (1), we deduceddtR(λ)x(t)=(λE-A)-1Ax(t)=λR(λ)x(t)-x(t),

for an λ∈ρ(E,A). The solution of this equation can be written as12 R(λ)x(t)=eλtx(0)-∫0teλ(t-τ)x(τ)dτ=-∫0teλ(t-τ)x(τ)dτ.

Let σ>0. Inequality (7) then implies13 limReλ→∞e-σλR(λ)=0.

By (12), (13) and the fact that a solution x is bounded on [0, t] we concludelimReλ→∞‖∫0t-σeλ(t-σ-τ)x(τ)dτ‖=limReλ→∞e-σReλ∫0t-σeλ(t-τ)x(τ)dτ≤limReλ→∞e-σReλ‖∫0teλ(t-τ)x(τ)dτ⏟=R(λ)x(t)‖+e-σReλ∫t-σteλ(t-τ)x(τ)dτ⏟→0=0.

Thus,limReλ→∞∫0t-σeλ(t-σ-τ)x(τ)dτ=0

and by [16, Ch. 4, Lem. 1.1] x(τ)=0 for all 0≤τ≤t-σ. Since σ and t were arbitrary, we deduce x(τ)=0 for all τ≥0. □

Remark 2.3

Of course it is also possible to formulate the initial condition x(0)=x0 of (1) as ddtEx(t)=Ax(t),t≥0,Ex(0)=x0.

Accordingly, one must assume x0∈E(ran((μE-A)-1E)pc,res(E,A)+2) in Theorem 2.2.

The term p=pc,res(E,A)+2 in the proof of Theorem 2.2 is chosen in such a way that R(λ)(λ-μ)p falls quickly enough, such that the integral in (9) not only exists; the function x(t) also becomes continuously differentiable. This means that the function x in (9) is only continuous, if x0∈ran((μE-A)-1E)pc,res(E,A)+1. However, in this case x is a mild solution of (1), as one can compute Ex(t)-Ex(0)=-12πi∫ω-i∞ω+i∞(eλt-1)⏟=∫0tλeλsdsER(λ)z0(λ-μ)pdλ=∫0t(-12πi∫ω-i∞ω+i∞λeλsER(λ)z0(λ-μ)pdλ)ds=∫0t(-12πi∫ω-i∞ω+i∞eλsAR(λ)z0(λ-μ)pdλ-12πi∫ω-i∞ω+i∞eλsEz0(λ-μ)pdλ⏟=0)ds=∫0tAx(s)ds=A∫0tx(s)ds.

In [22, Thm. 5.7] the set of initial values 14 U:={x0∈X|∃x:R≥0→Xmild solution}

was analysed in more detail. To be more precise, it was shown that E-1A generates a C0-semigroup on ran((μE-A)-1E)pc,res(E,A)+2¯ if and only if U is closed and for each x0∈U the mild solution is unique. The assumption that U is closed is a bit restrictive and it also requires information about U.

In [5, Sect. 8] the existence of solutions on the subspace ranR(λ)k¯, for R(λ):=(λE-A)-1E, has been studied under slightly stronger conditions compared to the ones in Theorem 2.2. More specifically, under the assumption that there exists k∈N with k≥1, ω∈R and M>0, such that (ω,∞)⊆ρ(E,A) and R(λ)x≤Mλ-ωx,∀λ>ω,x∈ranR(ω)k-1

the following operator AR was defined through its graph graphAR={(R(λ)x,x+λR(λ)x)|x∈ranR(λ)k¯}.

Then, under the assumption that AR generates a C0-semigroup, it was shown that for every x0∈ranR(λ)k¯ there exists a unique mild solution of (1) and for every x0∈R(λ)ranR(λ)k¯ there exists a classical solution of (1). As it was remarked by the authors, the generation of the C0-semigroup can be achieved if λR(λ)x≤Mx∀λ∈C≥ω,x∈ranR(ω)k

holds, which is a slightly stronger assumption compared to the complex resolvent index. In the latter case, we only have λR(λ)R(μ)kx≤Mx∀λ∈C≥ω,x∈X,

for k=pc,res(E,A).

Sufficient condition for a Weierstraß form

In this section we start with the analysis of a sufficient condition for the existence of a Weierstraß form. To do that, we recall the definition of the radiality index [2] or, more generally, of the radiality [19].

We call two differential-algebraic equations ddtEx=Ax and ddtE~x~=A~x~ defined on Hilbert spaces X, Z and X~, Z~ equivalent, denoted by (E,A)∼(E~,A~), if there are two bounded isomorphisms P:X→X~, Q:Z→Z~, such that E=Q-1E~P and A=Q-1A~P. Based on that, we define the notion of a Weierstraß form as follows.

Definition 3.1

The system (1) has a Weierstraß form, if there exists a Hilbert space Y=Y1⊕Y2, such that15 (E,A)∼IY100N,A100IY2,

where N:Y2→Y2 is a nilpotent operator, A1:D(A1)⊆Y1→Y1 is a linear operator, and IYi indicates the identity operator on the associated subspace Yi, i=1,2. Furthermore, the nilpotency degree of N is known as the nilpotency index pnilp(E,A) [2, Sect. 2].

Definition 3.2

The system (1) has a (complex) radiality index, if there exists a p=prad(E,A)∈N (p=pc, rad(E,A)∈N) and ω,C>0, such that (ω,∞)⊆ρ(E,A) (CRe>ω⊆ρ(E,A)) and 16 (λ0E-A)-1E·⋯·(λpE-A)-1En≤C∏k=0p|λk-ω|n,E(λ0E-A)-1·⋯·E(λpE-A)-1n≤C∏k=0p|λk-ω|n,

holds for all λ0,…,λp∈(ω,∞) (λ0,…,λp∈CRe>ω) and n=1.

The system (1) is p-radial, if it has radiality index p and (16) holds for all n∈N.

Theorem 3.3

Let E∈L(X,Z) with closed range and A:D(A)⊆X→Z be a closed and densely defined operator. If (E, A) has a radiality index, then it also has a Weierstraß form, which is unique up to isomorphisms.

Proof

Let p=prad(E,A) denote the radiality index. For arbitrary λ0,…,λp∈ρ(E,A) defineX1=ran(λ0E-A)-1E·⋯·(λpE-A)-1E¯‖·‖X,X2=ker(λ0E-A)-1E·⋯·(λpE-A)-1E,Z1=ranE(λ0E-A)-1·⋯·E(λpE-A)-1¯‖·‖Z,Z2=kerE(λ0E-A)-1·⋯·E(λpE-A)-1.

By [19, Sects. 2.1, 2.2 and 2.5] these spaces are independent of the choice of λi, i∈{0,…,p},E2:=E|X2∈L(X2,Z2),A2:=A|D(A2):D(A2)⊆X2→Z2,D(A2):=D(A)∩X2

is boundedly invertible operator with A2-1∈L(Z2,X2), such that A2-1E2∈L(X2) and E2A2-1∈L(Z2) are nilpotent with degree less or equal p+1. Furthermore, there are two projections, namelyP:X→X,x↦limλ→∞(λ(λE-A)-1E)p+1x

onto X1 with kerP=X2, ranP=X1, Px1=x1 for all x1∈X1 andR:Z→Z,z↦limλ→∞(λE(λE-A)-1)p+1z

onto Z1 with kerR=Z2, ranR=Z1, Rz1=z1 for all z1∈Z1, such that X=X1⊕X2 and Z=Z1⊕Z2. Furthermore, (λE-A)-1Ex∈D(A) for all x∈D(A) and thus, since A is closed, Px∈D(A). Hence,17 APx=Alimλ→∞(λ(λE-A)-1E)p+1x=limλ→∞A(λ(λE-A)-1E)p+1x=limλ→∞(Eλ(λE-A)-1)p+1Ax=RAx.

Since E is bounded we derive18 EPx=Elimλ→∞(λ(λE-A)-1E)p+1x=limλ→∞E(λ(λE-A)-1E)p+1x=limλ→∞(Eλ(λE-A)-1)p+1Ex=REx

for all x∈X. Now, we need to show thatE1:=E|X1∈L(X1,Z1)

andAi:=A|D(Ai):D(Ai)⊆Xi→Zi

is closed and densely defined, whereby D(Ai):=D(A)∩Zi, i∈{1,2}. This was already proved for p=0 in [7, Prop. 2.5] and under stronger radiality assumptions in [19, Cor. 2.5.2, Thm. 2.5.3]. Let x∈X1. Since X=X1⊕X2, x∈X1 if and only if Px=x. By (18) we concludeEx=EPx=REx∈Z1

and, since E is bounded, the same yields for E1. Analogously, using (17) we have Ax=APx=RAx∈Z1 for all x∈D(A1). Now, let x∈X1. Thus, there exists a sequence (xn)n⊆D(A) with xn→x for n→∞. Since Pxn∈D(A1) and Pxn→Px=x for n→∞, one has D(A1)¯=X1. Similarly, using I-P one can show D(A2)¯=X2. Since A is closed and Ai(D(Ai))⊆Xi, Ai is closed as well, i∈{1,2}.

What is left to show is that E1 is boundedly invertible. Since E1 is bounded, kerE1⊆kerE⊆X2 and X=X1⊕X2, it follows that E1 is injective. What remains to show is the surjectivity of E, because then the assertion follows from the closed graph theorem. Since the space Z1 does not depend on the choice of the λi, we can choose a λ∈ρ(E,A), such that Z1=ran(E(λE-A)-1)p+2¯‖·‖Z [19, Lem. 2.2.8]. Let y∈ran(E(λE-A)-1)p+2. Then there exists a z∈Z withy=(E(λE-A)-1)p+2z=E((λE-A)-1E)p+1(λE-A)-1z

and thusran(E(λE-A)-1)p+2⊆E(ran((λE-A)-1E)p+1)⊆E(X1).

Since ranE is closed, the same yields for E(X1)=ranE1=Z1∩ranE. Thus, the closure of ran(E(λE-A)-1)p+2 is still a subset of E(X1). Since ranE1⊆Z1=ran(E(λE-A)-1)p+2¯‖·‖Z one obtains the surjectivity of E. DefineP~=PIX-P∈L(X,X1×X2),R~=RIZ-R∈L(Z,Z1×Z2).

Then,P~-1=IX1IX2∈L(X1×X2,X),R~-1=IZ1IZ2∈L(Z1×Z2,Z).

Thus, 19a (E,A)∼(R~EP~-1,R~AP~-1)

19b ∼E100E2,A100A2

19c ∼IZ100E2A2-1,A1E1-100IZ2.

The uniqueness of the Weierstraß form follows from the uniqueness of the nilpotency index [2, Prop. 6.2], which exists since the nilpotency index of (E, A) is at most prad(E,A)+1 [2, Prop. 6.3]. □

Remark 3.4

With (19c) one can rewrite the DAE ddtEx=Ax as follows20 ddtIZ100E2A2-1z1(t)z2(t)=A1E1-100IZ2z1(t)z2(t),

on Z1×Z2.

Further, one can also show that(E,A)∼IX100A2-1E2,E1-1A100IX2

and rewrite the DAE Eddtx=Ax in the following way21 IX100A2-1E2ddtx1(t)x2(t)=E1-1A100IX2x1(t)x2(t),

on X1×X2.

In the following, we will recall and generalize an example from [2, Ex. 6.5], which shows that the radiality index can be greater than 0.

Example 3.5

Let W be a Hilbert space and (A0,D(A0)) generate a C0-semigroup on W. For b∈D(A0) and c∈D(A0∗) with ⟨b,c⟩≠0 define the operators Bu=bu, where u∈C and Cz=⟨c,z⟩ for any z∈W. We define the following DAE on Z=W×C by22 ddtI000⏟=:Ex(t)=A0BC0⏟=:Ax(t),t≥0.

In [2] it was already proven that the resolvent index is at most 2 and for specific A0, B and C it was also shown that the radiality index is 1. We want to generalize the last statement a bit. To do this, we look again at the left- and right-E resolvents.

We want to show that the radiality index exists by using the Weierstraß form. Define Qbz=z-⟨c,z⟩⟨c,b⟩b. Then, Qbb=0, ranQb=kerC and W=W1⊕W2:=kerC⊕span(b). For more concise notation, we additionally define C~=1⟨c,b⟩C, B~=1⟨c,b⟩B and K:=C~A0. Note that Kz:=C~A0z=1⟨c,b⟩⟨c,A0z⟩=1⟨c,b⟩⟨A0∗c,z⟩ holds for all z∈W and therefore K is a bounded operator on W. Thus, we can define the isomorphisms U:W×C→W1×W2×C and V:W1×W2×C→W×C viaU:=Qb-QbA0B~0B~C~-KB~,V:=IW1IW20-K01,

with inversesU-1:=IW1A0B0C0,V-1:=Qb0IW-Qb0KQb1.

These mappings applied to (22) we deriveE~:=UIW000V=IW100N,A~:=UA0BC0V=QbA000IW2×C,

with N=00C~0. Since A0 is a generator of a C0-semigroup, QbA0z=A0z-⟨c,A0z⟩⟨c,b⟩b=A0z-BKz, z∈W1 and BK is bounded, QbA0 also generates a C0-semigroup. Moreover, (E~,A~) has a Weierstraß form as in (15) and together with the following Theorem 3.6 b) (E~,A~) is 1-radial.

Further examples of DAEs with existing radiality index (such as the linearized Navier–Stokes equation) can be found in [3]. Next, the radiality index for DAEs in Weierstraß form gets investigated.

Theorem 3.6

Assume that (E,A)=[IY100N],[A100IY2], where A1 and N are defined as in (15). Then (E, A) has radiality index prad(E,A)=p∈N if and only if (IY1,A1) has radiality index prad(E,A)=p∈N and N has nilpotency degree smaller or equal to p+1.

If A1 generates a C0-semigroup and N has nilpotency degree smaller or equal to p+1, then (E, A) is p-radial and, especially, prad(E,A)=p.

Proof

Let N has nilpotency degree k+1∈N and λ∈ρ(E,A). Then, λ∈ρ(IY1,A1). Conversely, if λ∈ρ(IY1,A1), then (λN-IY2)-1=-∑l=0k(λN)l. Thus,23 ρ(E,A)=ρ(IY1,A1)=ρ(A1).

In particular, for λ∈ρ(E,A) we obtain24 (λE-A)-1E=(λIY1-A1)-100(λN-IY2)-1N=E(λE-A)-1

and∏l=0k(λlE-A)-1E=[∏l=0k(λlIY1-A1)-1000]

for λ0,…,λp∈ρ(E,A). In particular, we have25 ∏l=0k(λlIY1-A1)-1=∏l=0k(λlE-A)-1E.

Thus, Part a) follows directly from (23) and (25). To show Part b) assume that A1 generates a C0-semigroup and N has a nilpotency degree smaller or equal to p+1. Using the theorem of Hille-Yosida there exists ω,C>0 such that (ω,∞)⊆ρ(IY1,A1)=ρ(A1) and ‖((λIY1-A1)-1)n‖≤C(λ-ω)n for all λ≥ω, n∈N. Together with (24) and (25) we have∏l=0p(λlE-A)-1En=∏l=0pE(λlE-A)-1n≤Cp+1∏l=0p+1λl-ωn

for all λ0,…,λp+1≥ω. Thus, (E, A) is p-radial. □

A direct consequence of Theorem 3.6 is the equality of different index terms introduced in [2]. Here, we will denote the differentiation index, the chain index and the perturbation index by pdiff(E,A), pchain(E,A) and ppert(E,A).

Corollary 3.7

Assume that (E, A) has a Weierstraß form and nilpotency index pnilp(E,A), i.e. (E,A)∼[IY100N],[A100IY2]. If A1 generates a C0-semigroup on Y1, thenprad(E,A)+1=presd(E,A)=pnilp(E,A)=pdiff(E,A)=pchain(E,A)=ppert(E,A).

Proof

This follows from Theorem 3.6 and [2]. □

Next, we want to investigate the connection between integrated semigroups and linear differential algebraic equations.

Definition 3.8

A linear operator A on a Banach space X is the generator of an (n-1)-times integrated semigroup (S(t))t≥0, if there exists an n∈N, M,ω>0 and a strongly continuous family of operator (S(t))t≥0 in L(X) with S(t)≤Meωt for all t≥0, (ω,∞)∈ρ(A) and(λIX-A)-1x=λn-1∫0∞e-λtS(t)xdt

for x∈X.

It should be clear that for n=1 this coincides with the more known notion of a C0-semigroup. The importance of integrated semigroups becomes clear when one is interested in the well-posedness of the Cauchy problem26 ddtx(t)=Ax(t),t≥0,x(0)=x0,

for an x0∈X. Because then for all x0∈D(An) the unique solution of (26) is given byx(t)=S(t)An-1x0+∑k=0n-21k!tkAkx0

[15, eq. (4.1)]. Furthermore, for all x0∈X the function t↦S(t)x is a solution of the n-times integrated Cauchy problem and, thus, a mild solution of (26).

With this information we can illustrate the relation between the complex resolvent of a DAE and the generator of an integrated semigroup, given that (E, A) has a Weierstraß form.

Theorem 3.9

Assume that (E, A) has a Weierstraß form as in (15). If A1 is densely defined and (E, A) has a complex resolvent index pc, res(E,A), then A1 generates an (at least) pc, res(E,A)+2-times integrated semigroup.

Proof

Assume that (E,A)∼[IY100N],[A100IY2] has a complex resolvent index p=pc,res(E,A). Then, by (23) ρ(E,A)=ρ(A1) and27 ‖(λIY1-A1)-1‖≤C~‖(λE-A)-1‖≤C‖λ‖p-1,λ≥ω,

for given C,C~>0 and ω>0. Thus, (IY1,A1) has a complex resolvent index and by [15, Cor. 4.9] A1 generates an (at least) p+2-times integrated semigroup. □

Corollary 3.10

Assume that (E, A) has a complex radiality index. Then A1E1-1 and E1-1A1 from (20) and (21) generate integrated semigroups.

Proof

In [2, Prop. 5.4] it was shown that the existence of the radiality index already implies the existence of the resolvent index. This implication extends naturally to the complex resolvent and complex radiality index. Therefore, if (E, A) has a complex radiality index, then by Theorem 3.3(E,A)∼[IZ100E2A2-1],[A1E1-100IZ2]∼[IX100A2-1E2],[E1-1A100IX2] and by Theorem 3.9A1E1-1 and E1-1A1 generate integrated semigroups. □

Infinite-dimensional port-Hamiltonian DAEs

Let X,Z be Hilbert spaces, E,Q∈L(X,Z) and A:D(A)⊆Z→Z be a closed and densely defined operator. Then, the following differential-algebraic equation28 ddtEx(t)=AQx(t),t≥0,Ex(0)=z0,

for a z0∈Z is called a port-Hamiltonian differential-algebraic equation (short pH-DAE), if A is dissipative, ranE is closed, Q is invertible and29 E∗Q=Q∗E≥0

holds. In this context, we call30 H(x):=⟨Ex,Qx⟩Z,x∈X

the Hamiltonian of (E, AQ). Note that here T:X→X is called non-negative, denoted by T≥0, if ⟨Tx,x⟩X≥0 for all x∈X.

These systems provide a framework for modelling and analysing energy-dissipating physical systems, including electrical circuits, mechanical systems, fluid dynamic, thermal systems or robotics and control systems. In this section we want to examine a few properties of pH-DAEs and show that the derivative of the Hamiltonian dissipates on all classical solutions of (28).

Remark 4.1

One can define a pH-DAE similarly for the case 31 Eddtx(t)=AQx(t),t≥0,x(0)=x0,

for x0∈X. Hence, x a classical solution of (31), if x(t) is continuously differentiable on R≥0, x(t)∈D(AQ) for all t≥0 and (31) holds.

It can be observed that (29) implies 32 EQ-1=Q-∗E∗≥0.

In fact, using (29) we derive EQ-1=Q-∗Q∗EQ-1=Q-∗E∗, with Q-∗=(Q-1)∗=(Q∗)-1. Applying the positivity of EQ∗ we have ⟨Q-∗E∗z,z⟩Z=⟨Q-∗E∗QQ-1z,z⟩Z=⟨E∗Q(Q-1z),(Q-1z)⟩X≥0,z∈Z.

Since Q is invertible it is possible to assume Z=X and Q=IZ, such that E becomes non-negative and self-adjoint. This results from defining z(t):=Qx(t) in (28), such that ddtEx(t)=AQx(t) is equivalent to ddtEQ-1z(t)=Az(t) and EQ-1 being non-negative and self-adjoint as seen in (32). Similarly, Eddtx(t)=AQx(t) is equivalent to EddtQ-1z(t)=Az(t).

By c) (28) is equivalent to ddtEQ-1z(t)=Az(t). Since ranE is closed, the same holds for ranEQ-1 and one can split up the space into Z=ranEQ-1⊕kerEQ-1=:Z1⊕Z2. Thus, ddtEQ-1z(t)=Az(t) is equivalent to ddtE1000z1(t)z2(t)=A1A2z1(t)z2(t),t≥0,

whereby Ai:Z→Zi and E1:=EQranEQ-1-1. Thus, the representation for port-Hamiltonian differential algebraic equations chosen here corresponds with the notion of an abstract Hamiltonian differential-algebraic equation defined in [13, Assumption 7]. Furthermore, it should be mentioned that there are more ways to define and approach port-Hamiltonian systems, such as using relations [4] or Dirac structures [24].

Next, we want to show that the derivative of the Hamiltonian dissipates on solutions, i.e. for a solution of (28), ddtH(x(t))≤0 for all t≥0.

Proposition 4.2

Let B∈L(Z) be a non-negative and self-adjoint operator on Z and assume that ranB is closed. Then there exists a c>0 and T∈L(Z) with T≥cIZ andBTB=B.

Furthermore, ⟨·,·⟩T:=⟨T·,·⟩ induces a norm on Z equivalent to ‖·‖Z.

Obviously, if B is invertible, then T=B-1.

Proof

Since ranB is closed and B is self-adjoint, Z has an orthogonal decomposition into Z=ranB⊕kerB. Let PranB and PkerB be the projections of Z onto ranB and kerB. Define B~:ranB→ranB, B~(z)=Bz. This is a bounded, positive self-adjoint operator on the Hilbert space ranB, which is bijective (note that kerB~=kerB∩ranB={0}). Hence, since B~-1 is bounded, B~ is strictly positive with⟨B~z1,z1⟩≥c‖z1‖2,z1∈ranB

for all z1∈ranB, for some c>0.

Let ιranB:ranB→Z be the embedding and defineT:ιranBB~-1ιranB∗+PkerB.

This operator satisfies BTB=B and is bounded. Let z=z1+z2∈Z=ranB⊕kerB. Then⟨Tz,z⟩=⟨B~z1+z2,z1+z2⟩=⟨B~z1,z1⟩+⟨z2,z2⟩≥c‖z1‖2+‖z2‖2.

Thus, T is strictly positive. The self-adjointness of T follows from that of B, completing the proof. □

Corollary 4.3

Let E,Q∈L(X,Z), whereby E has a closed range, Q is invertible and (29) holds. Then, there exists a c>0 and T∈L(Z) with T≥cIZ and33 E∗TE=E∗Q=Q∗E.

Furthermore, ⟨·,·⟩T:=⟨T·,·⟩ induces a norm on Z equivalent to ‖·‖Z.

Proof

This follows from Remark 4.1, Proposition 4.2 with B=EQ-1 and the fact that Q is invertible. □

Theorem 4.4

Let (E, AQ) be a pH-DAE with Hamiltonian H. Then, for all classical solutions x:R≥0→Xddt⟨Ex(t),Qx(t)⟩≤0,t≥0.

Proof

By Corollary 4.3 we know that T≥cIZ, such that (33) holds. Let x:R≥0→X be a classical solution of (28). Thenddt⟨Ex(t),Qx(t)⟩Z=ddt⟨x(t),E∗Q⏟=E∗TEx(t)⟩Z=2ReEx(t),ddtTEx(t)Z=2ReQx(t),ddtEx(t)Z=2ReQx(t),AQx(t)Z≤0,t≥0,

where we used the continuity of E, Q and T in the second and third equation. Note that we used ⟨Ex(t)=TEx(t)⟩=⟨x(t),E∗TEx(t)⟩=⟨x(t),Q∗Ex(t)⟩=⟨Qx(t),Ex(t)⟩ in the third equality. □

Remark 4.5

From the proof of Theorem 4.4 it becomes clear that for all classical solutions x of (28) the DAE is already dissipating, i.e. one hasRe⟨Ex(t),AQx(t)⟩T≤0,t≥0.

It is possible to show the same for the adjoint system and, if the set containing all trajectories of the solutions of (28) is closed, one can apply [7, Thm. 3.7] to gain a Weierstraß form on a subset. To be more precise, we know that ranE∗ is closed and Q-1 is invertible. Thus, there also exists a S∈L(X) with S≥c~IX, for a c~>0, withESE∗=EQ-1=Q-∗E∗

and ⟨·,·⟩S-1=⟨S-1·,·⟩ induces a norm on X equivalent to ‖·‖X. Define XS-1:=(X,⟨·,·⟩S-1) and ZT:=(Z,⟨·,·⟩T). Let z∈{z∈Z|Tz∈D(A∗)} and x∈D(AQ). Then,⟨AQx,z⟩T=⟨x,SQ∗A∗Tz⟩S-1,.

withIn fact, by simple reformulations it is easy to see thatD((AQ)S-1,T∗)={z∈Z|∃x∗∈X,∀x∈D(AQ):⟨z,AQx⟩T=⟨x∗,x⟩S-1},

where (AQ)S-1,T∗ denotes the adjoint of AQ:D(AQ)⊆XS-1→ZT. Consider now the adjoint system of (28)34 ddtE∗z(t)=Q∗A∗z(t),t≥0,E∗z(0)=x0∈X.

Then, by choosing z(t)=Tz~(t), this is equivalent to35 ddt(E)S-1,T∗z~(t)=(AQ)S-1,T∗z~(t),t≥0,(E)S-1,T∗z~(0)=x~0∈X.

Let z be a classical solution of (35). Assuming that A∗ is dissipative, one hasRe⟨(E)S-1,T∗z(t),(AQ)S-1,T∗z(T)⟩S-1=Re⟨E∗Tz(t),SQ∗A∗Tz(T)⟩X=Re⟨E∗Tz(t),ddtSE∗Tz(T)⟩X=Re⟨Tz(t),ddtESE∗Tz(T)⟩X=Re⟨Q-1Tz(t),S-1ddtSE∗Tz(T)⟩X=Re⟨Q-1Tz(t),S-1SQ∗A∗Tz(T)⟩X=Re⟨Tz(t),A∗Tz(T)⟩X≤0,t≥0.

Thus, if U:={u∈Z|∃x classical solution of (28) ∃t≥0:x(t)=u}⊆Z and V:={v∈Z|∃z classical solution of (35) ∃t≥0:z(t)=v}⊆Z are both closed sets, one can apply [7, Thm. 3.7] to (E|U,(AQ)|U) on the spaces XS-1 and ZT.

Example 4.6

Consider longitudinal vibrations in a viscoelastic nanorod. Let l be the length of the nanorod, N(x, t) be the resultant force of axial stress, w(x, t) be the displacement of the nanorod in x direction, C be the elastic modulus, D be the cross sectional area, μ be a non-local parameter, ρ the mass density, τd the viscous damping and a2, b2 be the stiffness and damping coefficients of the light viscoelastic layer and consider the system introduced in [9]. Consider 36a ∂N(x,t)∂x=a2w(x,t)+b2∂w(x,t)∂t+ρD∂2w(x,t)∂t2,

36b N(x,t)-μ∂2N(x,t)∂x2=CD∂w(x,t)∂x+τd∂2w(x,t)∂x∂t,

with boundary conditions37 ∂w∂t(0,t)=∂w∂t(l,t)=0.

In [6] the associated port-Hamiltonian system is given through38

and state39 z(x,t)=w(x,t)ρD∂w(x,t)∂tμρD∂2w(x,t)∂x∂t∂w(x,t)∂xN(x,t).

Here, E and Q are bounded operators living on X=Z=L2((0,l);R5) and A:D(A)⊆X→X withD(A):={(z1,…,z5)T∈X|z2,z5∈H1(0,l),z2(0)=z2(l)=0}.

In [6] the existence of solution was studied by reducing the system to a homogeneous port-Hamiltonian system (see [8, Ch. 7]). Here, we provide another approach by simply examining the dissipativity of the DAE as seen in Remark 4.5.

Since the various physical constants in Q are positive, it is easy to show that E∗Q is non-negative and self-adjoint and with⟨Az,z⟩X=∫0lz2(x)∂z5(x)∂x+z5(x)∂z2(x)∂x-b2z22(x)-(CDτd+μb2)z32(x)dx=-b2z2X2-(CDτd+μb2)z3X2≤0,z=(z1,…,z5)∈D(A),

A is dissipative. Thus, (E, AQ) fits into our definition of a port-Hamiltonian DAE.

Now, as in [6, p. 452] we impose the boundary conditions directly on the space and show that (38) has radiality index 0. DefineXt:={(z1,…,z5)T∈L2((0,l);R5)|z2,z5∈H1(0,l),z2(0)=z2(l)=0,μ∂z2(x)∂x=z3(x)}.

with the inner product ⟨·,·⟩X~t:=⟨Q·,·⟩Xt. Since Q is coercive this induces a norm equivalent to ‖·‖X. Xt is in fact closed [6, Lem. 4.1]. Define X~:=(X,⟨·,·⟩X~t). Then,40 Re⟨AQz,Ez⟩X~t≤Re∫0l1ρDz2(x)∂z5(x)∂x+z5(x)∂z2(x)∂xdx=0

holds for all z=(z1,…,z5)∈D(AQ). Furthermore, (AQ)∗=A∗Q in Xt and, through similar calculations, one derives Re⟨A∗Qz,Ez⟩Xt≤0 for all z∈D((AQ)∗). Then, by [7, Thm. 3.6, 3.7] (E, AQ) is 0-radial and admits a decomposition as in (20) on Xt=Xt1⊕Xt2 with E2(AQ)2-1=0 and (AQ)2-1E2=0. In particular, (AQ)1E1-1 generates a contraction semigroup on Xt1.

Note that in this example the operator T from above is already given by Q, since E∗QE=E∗Q=Q∗E.

By [7, Thm. 3.7] it becomes clear that if (E, AQ) is a pH-DAE with Re⟨AQx,Ex⟩≤0 and Re⟨(AQ)∗z,E∗z⟩≤0, then (E, AQ) is in fact 0-radial (and therefore has radiality index 0), admits a Weierstraß form as in (20) or (21) and (AQ)1E1-1 and E1-1(AQ)1 generate contraction semigroups on Z~1 and X~1. In this case, (AQ)1E1-1 and E1-1(AQ)1 also generate C0-semigroups on Z1 and X1.

Weierstraß form and solutions of port-Hamiltonian DAEs

It is possible to show that the (complex) resolvent index for pH-DAEs always exists and that it is bounded by 2 (3). This is an already widely known result in finite dimensions and was shown for the infinite-dimensional case in [2] for X=Z and Q=I.

Proposition 5.1

Let E, Q and A be defined as before, such that (E, AQ) defines a pH-DAE. Assume that there exists a ω>0 with (ω,∞)⊆ρ(E,AQ) (CRe>ω⊆ρ(E,AQ)). Then the (complex) resolvent index exists and is at most 2 (at most 3).

Proof

Using Remark 4.1 and [2, Thm. 3.3] one knows that (EQ-1,A) is a pH-DAE with an existing (complex) resolvent index, which is at most 2 (at most 3), and together with (λEQ-1-A)-1=Q(λE-AQ)-1 for all λ∈ρ(EQ-1,A)=ρ(E,AQ) the assertion follows. □

Proposition 5.2

Let E, Q and A be defined as before, such that (E, AQ) defines a pH-DAE and assume that there exists a ω>0 with CRe>ω⊆ρ(E,AQ). Then, for every x0∈ran((μE-AQ)-1E)pc,res(E,AQ)+2 there exists a unique solution x:R≥0→X of the following differential algebraic equationddtEx(t)=AQx(t),t≥0,x(0)=x0.

Proof

This is a direct consequence of Theorem 2.2 and Proposition 5.1. □

The next goal is to examine the well-posedness of a pH-DAE on the whole domain, given that the radiality index exists. Let (E, AQ) be such a pH-DAE. By Theorem 3.3 (28) is equivalent to41 ddtIZ1-100E2(AQ)2-1z1(t)z2(t)=(AQ)1E1-100IZ2z1(t)z2(t),t≥0

and (31) is equivalent to42 IX1-100(AQ)2-1E2ddtx1(t)x2(t)=E1-1(AQ)100IX2x1(t)x2(t),t≥0.

The main problem with a transformation like that is that (41) and (42) are not necessarily pH-DAEs anymore. Thus, the resulting question is under which circumstances (AQ)1E1-1 and E1-1(AQ)1 generate a C0-semigroup. A possible condition for that is AQ being strongly (E, p)-radial [19, Thm. 2.6.1].

Theorem 5.3

Let E, Q and A be defined as before, such that (E, AQ) defines a pH-DAE. Assume that the radiality index exists and that CRe≥ω⊆ρ(E,AQ) for a ω>0. Then, (AQ)1E1-1 and E1-1(AQ)1 generate an (at least) pc,res(E,AQ)+2-times integrated semigroup. If additionally Q∗(Z1)=X1, then E1-1(AQ)1 generates a C0-semigroup on X1.

Q(X1)=Z1, then (AQ)1E1-1 generates a C0-semigroup on Z1.

Proof

Let x∈X and λ∈ρ(E,AQ) with λ>0. Thus, (λE-AQ)-1∈L(Z,X). As seen in the proof of Theorem 3.3(AQ)1E1-1 and E1-1(AQ)1 are densely defined and by Proposition 5.1 the complex resolvent index pc,res(E,AQ) exists. Thus, by Theorem 3.9(IZ1,(AQ)1E1-1) and (IX1,E1-1(AQ)1) generate an (at least) pc, res(E,AQ)+2-times integrated semigroup.

In order to show a), assume that Q∗(Z1)=Q∗(E(X1))=X1 holds. Thus, Q∗E:X1→X1 is non-negative and self-adjoint. In fact, since Q∗E is self-adjoint the mappings Q∗E±iIX are surjective. Hence, for every z∈X1 there are x±=x±1+x±2∈X=X1⊕X2 with (Q∗E±iIX)x±=z. Together with Q∗E(X1)=X1 one has PQ∗Ex±=PQ∗Ex±1+PQ∗Ex±2=Q∗EPx±1=Q∗EPx± andz=Pz=PQ∗Ex±±iPx±=Q∗EPx±±iPx±=(Q∗E±iIX1)x±.

Thus, Q∗E±iIX1:X1→X1 are surjective and consequently, as a closed operator, (Q∗E)|X1 is self-adjoint. Furthermore, since Q∗ and E1 are invertible and (Q∗E)|X1 is self-adjoint, (Q∗E)|X1 becomes invertible and ((Q∗E)|X1)-1 self-adjoint as well. Hence, using Lemma 4.2 for B=((Q∗E)|X1)-1 there exists a c>0, such that T=((Q∗E)|X1)≥cIX1 and ⟨·,·⟩X~1:=⟨(Q∗E)|X1·,·⟩ induces a norm equivalent to ‖·‖X on X1. Define X~1:=(X1,⟨·,·⟩X~1). Since A is dissipative one derivesRe⟨E1-1(AQ)1x,x⟩X~1=Re⟨AQx,Qx⟩X≤0,x∈D(E1-1(AQ)1),

which means that E1-1(AQ)1 is dissipative in X~1. Given that X=X1⊕X2, E1 is bijective and (λE1-(AQ)1) is surjective, (λIX1-E1-1(AQ)1)=E1-1(λE1-(AQ)1):D((AQ)1)⊆X1→X1 becomes surjective as well.

By the theorem of Lumer-Phillips [1, Thm.3.4.5] E1-1(AQ)1 generates a contraction semigroup on X~1. In this case, E1-1(AQ)1 also generates a C0-semigroup on X1.

To show b) assume that Q(X1)=Q(E-1(Z1))=Z1 holds. As seen before it is possible to show that QE-1 is non-negative, self-adjoint and has a non-negative and self-adjoint inverse, such that QE-1≥cIX for a c>0. Define ⟨·,·⟩Z~1:=⟨QE-1·,·⟩ and Z~1:=(Z1,⟨·,·⟩Z~1). Thus,Re⟨(AQ)1E1-1z,z⟩Z~1=Re⟨AQE1z,QE1z⟩≤0,z∈D((AQ)1E1-1)

since A is dissipative. Then, the rest of this proof follows the previous part. □

Remark 5.4

Let X=Z and Q=IX and assume that E commutes with A on D(A). ThusEx=E(λE-A)(λE-A)-1x=(λE-A)E(λE-A)-1x

or equivalently (λE-A)-1Ex=E(λE-A)-1x. Thus, the left- and right-E resolvent coincide and X1=Z1, X2=Z2 as well as the projections from Theorem 3.3P=R. In this case Q∗E(X1)=X1 and QE-1(Z1)=Z1 hold. Unfortunately, the assumption that E and AQ commute massively limits the choice of systems. Because if one considers the following type of systemddtI00N⏟=:Ex1(t)x2(t)=A1A2A3A4⏟=:Ax1(t)x2(t),t≥0,

and additionally assume that EA=AE hold, then A2=A2N, NA3=A3 and NA4=A4N.

Example 5.5

Consider the systemddtI000⏟=Ex1(t)x2(t)=A1A2A3A4⏟=Ax1(t)x2(t)

with existing radiality index and A being dissipative.

Since kerE⊆X2 and X=X1⊕X2, one has E1=IX1. Thus, one has E(X1)=X1=Z1=E1(Z1). But, in this case, it is even possible to say directly something about the generation of C0-semigroups without using Theorem 5.3. Because in such a case A1=A1E1-1=E1-1A1 is dissipative (as a restriction of a dissipative operator) and λIX1-A1 is surjective (since (ω,∞)⊆ρ(E,A)=ρ(IX1,A1)). Hence, using the Lumer-Phillips theorem one obtains the same outcome. This result is similar to [7, Sect. IV].

Example 5.6

Recall the system from [2, Ex. 3.4]. Define A=diag(A0,A1,A2,…) withA0=0-110,Ak=0k4+1-k4+1-2,k∈N,

and D(A):={x∈ℓ2|Ax∈ℓ2}. Then A can be extended to ℓ2, which will denoted by A-1. Define E∈L(ℓ2), B:R→D(A∗)′ and C:D(A)→R with E=diag(E0,E1,E2,…) and B=(B0,B1,B2,…)T=C∗, whereE0=1000,Ek=1001,k∈N,B0=01,Bk=0k54,k∈N.

Define then the systemddtE00000000⏟Ex1x2x3=A-1B0-C0I0-I0⏟Ax1x2x3

on X=ℓ2×R×R. Obviously, E is non-negative and self-adjoint and, by its construction, A with maximal domain is dissipative. Furthermore, for λ∈ρ(E,A)(λE-A)-1=(λE-A)-10(λE-A)-1B00I-C(λE-A)-1-IC(λE-A)-1B.

It was shown in [2, Ex. 3.4] that this system has real resolvent index 2 and complex resolvent index 3.

We now compute the radiality index of (E,A). First, calculate(λE-A)-1E=(λE-A)-1E00000-C(λE-A)-1E00,E(λE-A)-1=E(λE-A)-10E(λE-A)-1B000000.

It is needed to determine the growth rate of (λE-A)-1=diag(M0(λ),M1(λ), M2(λ),…) withM0(λ)=0-11λ,Mk(λ)=1λ(λ+2)+k4+1λ+2k4+1-k4+1λ,k∈N.

It is easy to see that M0(λ) has a linear growth and Mk(λ) is decreasing with rate 1λ. Hence, due to the structure of E, (λE-A)-1E and E(λE-A)-1 are bounded and(λE-A)-1E(μE-A)-1E=E(λE-A)-1E(μE-A)-1≤Cλμ,λ,μ>0

for a given C>0. Moreover, by simple calculations it is easy to show that BkTMk(λ) and Mk(λ)Bk are still decreasing with rate 1λ. Thus, for all λ,μ>0C(λE-A)-1E(μE-A)-1E≤Cλμ,E(λE-A)-1E(μE-A)-1B≤Cλμ.

Since(λE-A)-1E(μE-A)-1E=(λE-A)-1E(μE-A)-1E00000-C(λE-A)-1E(μE-A)-1E00,E(λE-A)-1E(μE-A)-1=E(λE-A)-1E(μE-A)-10E(λE-A)-1E(μE-A)-1B000000

the system (E,A) has radiality index 1.

Acknowledgements

The financial support of NSERC (Canada) for the research described in this paper is gratefully acknowledged. The authors would like to thank Hannes Gernandt and Timo Reis for valuable discussions.

Funding

Open Access funding enabled and organized by Projekt DEAL.

Data availability

No data were used for the research described in the article.

Declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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