# On Perturbations of Generators ofC0-Semigroups

Authors: Martin Adler; Miriam Bombieri; Klaus-Jochen Engel
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We present a perturbation result for generators ofC0-semigroups which can be considered as an operator theoretic version of the Weiss-Staffans perturbation theorem for abstract linear systems. The results are illustrated by applications to the Desch-Schappacher and the Miyadera-Voigt perturbation theorems and to unbounded perturbations of the boundary conditions of a generator.


---

## Body

## 1. Introduction

In his classic [1]“Perturbation Theory for Linear Operators”, Kato addresses, among others, the following general problem.Given (unbounded) operators
A
and
P
on a Banach space
X
, how should one define their “sum”
A
+
P
and which properties of
A
are preserved under the perturbation by
P
?In the present paper we study this problem in the context of operator semigroups. Given the generatorA of a C0-semigroup on X, for which operators P is the (in a suitable way defined) sum A+P again a generator?Numerous results are known in this field (see, e.g., [2, Sections III.1–3 and related notes]), but no unifying and general theory is yet available.Our aim is to go a step towards a more systematic perturbation theory for such generators. To this end we choose the following setting. For the generatorA with domain D(A)⊂X consider perturbations
(1)P:DP⊂X⟶X-1A,
where X-1A is the extrapolated space associated with A (see [2, Section II.5.a]). The sum is then defined as AP≔(A-1+P)|X; that is,
(2)APx≔A-1x+Pxforx∈DAP≔z∈DP:A-1z+Pz∈X.
For which P remains AP a generator on X? The bounded perturbation theorem ([2, Section III.1]), the Desch-Schappacher ([2, Section III.3.a]), and the Miyadera-Voigt theorems ([2, Section III.3.c]) give some well-known answers in these cases.It seems that the Weiss-Staffans theorem on the well-posedness of perturbed linear systems (cf. [3, Theorems 6.1 and 7.2] and [4, Sections 7.1 and 7.4]) is a general result in this direction. In the present paper we formulate and prove this result in a purely operator theoretic way avoiding, in particular, notions like abstract linear systems and Lebesgue- or Yosida-extensions.More precisely (here we use the notation of Weiss, cf. [3]), the classical Weiss-Staffans theorem starts from an abstract linear system, that is, a quadruple (T,Φ,Ψ,F) of operator families verifying a set of functional equations (for the precise definition see [3, Definition 5.1]). It then shows that for an admissible feedback operator K (cf. [3, Definition 3.5]) there exists a unique corresponding closed-loop system (TK,ΦK,ΨK,FK). Moreover, it relates the generating operators (A,B,C,D) and (AK,BK,CK,DK) of these two systems. Since here T and TK are C0-semigroups with generators A and AK, respectively, this result implicitly contains a perturbation theorem for generators of C0-semigroups.However, to apply this theorem to a perturbed operatorAP as appearing in (2) one first has to construct an abstract linear system with appropriate generating operators and a suitable admissible feedback operator incorporating the unperturbed generator A and the perturbation P. This makes it quite cumbersome to formulate and to apply the Weiss-Staffans theorem as a perturbation result for generators.For this reason we start directly from a triple(A,B,C) of operators and then give conditions in terms of the semigroup generated by A and the operators B and C implying that AP for P=BC generates a C0-semigroup. Even though in our approach it is not necessary, it is nevertheless helpful to interpret the perturbed generator as the state operator of a control system with feedback in order to give some motivation for the various definitions of “admissibility.” For this reason in the sequel we use some common terminology from control theory.More precisely, choose two Banach spacesX and U calledstate- andobservation-/control space, respectively. (We assume that the observation and control spaces coincide. This is no restriction of generality and somewhat simplifies the presentation.) On these spaces consider the following operators:(i)
A:D(A)⊂X→X, called thestate operator (of the unperturbed system);(ii)
B∈L(U,X-1A), called thecontrol operator;(iii)
C∈L(Z,U), called theobservation operator,where A is the generator of a C0-semigroup (T(t))t≥0 on X. Moreover, D(C)=Z is a Banach space such that
(3)X1A↪cZ↪cX,
where “↪c” denotes a continuous linear injection and X1A is the domain D(A) equipped with the graph norm. Then consider the linear control system
(4)ΣA,B,Cx˙t=Axt+But,t≥0,yt=Cxt,t≥0,x0=x0.
The solution of Σ(A,B,C) is formally given by the variation of parameters formula
(5)xt=Ttx0+∫0tT-1t-sBusds.
Closing this system by putting u(t)=y(t), one formally obtains the perturbed abstract Cauchy problem
(6)x˙t=A-1+BCxt,t≥0,x0=x0,
which is well-posed in X if and only if AP for P≔BC∈L(Z,X-1A) is a generator on X (cf. [2, Section II.6]).Before elaborating this idea, we give a short summary of this paper.Section2 is dedicated to the notions ofadmissibility for control, observation, feedback, and pairs of operators. In Section 3 we state and prove the main results, that is, Theorems 10 and 14. In Section 4 we show how the Desch-Schappacher and Miyadera-Voigt theorems easily follow from Theorem 14 and give an application to the perturbation of the boundary condition of a generator in the spirit of Greiner [5].


## 2. Admissibility

Being only interested in the generator property ofA+P for some perturbation P, we can in the sequel assume without loss of generality that the growth bound ω0(A)<0 and hence
(7)0∈ρA.
Taking C=0 in the system Σ(A,B,C) and considering the initial value x0=0 it is natural to ask that for every control function u∈Lp([0,t0],U) one obtains a state x(t0)∈X for some/all t0>0. Hence formula (5) is leading to the following definition (cf. [6, Definition 4.1], see also [7]).Definition 1.
The control operatorB∈L(U,X-1A) is called p-admissible for some 1≤p<+∞ if there exists t0>0 such that
(8)∫0t0T-1t0-sBusds∈X,∀u∈Lp0,t0,U.Note that (8) becomes less restrictive for growing p∈[1,+∞).Remark 2.
The range condition (8) in the previous definition means that the operator Bt0:Lp([0,t0],U)→X-1A given by
(9)Bt0u≔∫0t0T-1t0-sBusds,u∈Lp0,t0,U
has range rg(Bt0)⊆X. Since obviously Bt0∈L(Lp([0,t0],U),X-1A), the closed graph theorem implies that for admissible B thecontrollability mapBt0 belongs to L(Lp([0,t0],U),X). On the other hand, using integration by parts, it follows that for every u∈W1,p([0,t0],U)(10)∫0t0T-1t0-sBusds=A-1-1T-1t0Bu0-But0∫0t0T-1t0-sBu′sds+∫0t0T-1t0-sBu′sds∈X.
Since W1,p([0,t0],U) is dense in Lp([0,t0],U), this shows that the range condition (8) is equivalent to the existence of some M≥0 such that
(11)∫0t0T-1t0-sBusdsX≤M·up,∀u∈W1,p0,t0,U.Next, considerΣ(A,B,C) with B=0. Then it is reasonable to ask that every initial value x0∈D(A) gives rise to an observation y(·)=CT(·)x0∈Lp([0,t0],U) for some/all t0>0 which also depends continuously on x0. This yields the following definition (cf. [8, Definition 6.1], see also [7]).Definition 3.
The observation operatorC∈L(Z,U) is called p-admissible for some 1≤p<+∞ if there exist t0>0 and M≥0 such that
(12)∫0t0CTsxUpds≤M·xXp,∀x∈DA.Note that (12) becomes more restrictive for growing p∈[1,+∞).Remark 4.
The norm condition (12) in the previous definition combined with the denseness of D(A)⊂X implies that there exists anobservability mapCt0∈L(X,Lp([0,t0],U)) satisfying Ct0≤M such that
(13)Ct0xs=CTsx,∀x∈DA,s∈0,t0.Finally, consider the systemΣ(A,B,C) with (possibly nonzero) p-admissible control and observation operators B and C. The following compatibility condition is needed to proceed (cf. [9, Section II.A]). For more information and several related conditions see [10, Theorem 5.8] and [4, Definition 5.1.1]. Recall that Z=D(C).Definition 5.
The triple(A,B,C) (or the system Σ(A,B,C)) is calledcompatible if for some λ∈ρ(A) we have
(14)rgRλ,A-1B⊂Z.If the inclusion (14) holds for some λ∈ρ(A), then it holds for all λ∈ρ(A) by the resolvent identity. Moreover, the closed graph theorem implies that the operator
(15)CRλ,A-1B∈LU,∀λ∈ρA.
Consider now a compatible control system Σ(A,B,C) with initial value x0=0. Then the input-output map of Σ(A,B,C) which maps a control u(·) to the corresponding observation y(·) by (5) is formally given by
(16)u·⟼y·=C∫0·T-1·-sBusds.
Of course, the right-hand side does not in general make sense for arbitrary u∈Lp([0,t0],U) since the integral might give values ∉Z=D(C). However, if
(17)u∈W02,p0,t0,U≔u∈W2,p0,t0,U:u0=u′0=0,
then integrating by parts twice and using (14) one obtains
(18)∫0rT-1r-sBusds=-A-1-1-Bur+A-1-1Bu′r∫0rTr-sA-1-1Bu′′sds-∫0rTr-sA-1-1Bu′′sds∈Z.
At this point it is reasonable to ask that the input-output map is continuous. This gives rise to the following definition.Definition 6.
The pair(B,C)∈L(U,X-1A)×L(Z,U) (or the system Σ(A,B,C)) is calledjointlyp-admissible for some 1≤p<+∞ if B is a p-admissible control operator and C is a p-admissible observation operator and there exist t0>0 and M≥0 such that
(19)∫0t0C∫0rT-1r-sBusdsUpdr≤M·upp,∀u∈W02,p0,t0,U.Remark 7.
IfΣ(A,B,C) is jointly p-admissible, then there exists a boundedinput-output map(20)Ft0∈LLp0,t0,U,suchthatFt0u·=C∫0·T-1·-sBusds,∀u∈W02,p0,t0,U.
We need a further definition.Definition 8.
An operatorF∈L(U) is called a p-admissible feedback operator for some 1≤p<+∞ if there exists t0>0 such that Id-FFt0∈L(Lp([0,t0],U)) is invertible.Note thatF=Id∈L(U) is admissible if Ft0<1. For further reference we summarize some of the previous notions in a single notation.Definition 9.
LetA be the generator of a C0-semigroup on a Banach space X, B∈L(U,X-1A) and C∈L(Z,U) for a Banach space Z satisfying X1A↪cZ↪cX. Then P≔BC∈L(Z,X-1A) is called aWeiss-Staffans perturbation forA if for some 1≤p<∞ the following hold:(i)
(A,B,C) is a compatible triple;(ii)
B is a p-admissible control operator;(iii)
C is a p-admissible observation operator;(iv)
(B,C) is a p-admissible pair;(v)
Id∈L(U) is a p-admissible feedback operator.


## 3. The Weiss-Staffans Perturbation Theorem

In this section we state and prove the main results of this paper. These results can be considered as purely operator theoretic versions of perturbation theorems for abstract linear systems due to Weiss [3, Theorems 6.1 and 7.2 (1994)] in the Hilbert space case and Staffans [4, Theorems 7.1.2 and 7.4.5 (2005)] for Banach spaces. In particular, our approach avoids the use of the notions of abstract linear systems and Lebesgue extensions which are not needed if one is only interested in generators. For related results see also [11] and [12, Theorems 4.2 and 4.3].Theorem 10.
Assume thatP=BC∈L(Z,X-1A) is a Weiss-Staffans perturbation for A. This means that there exist 1≤p<+∞, t0>0 and M≥0 such that
(21)irgRλ,A-1B⊂Z,forsomeλ∈ρA,ii∫0t0T-1t0-sBusds∈X,∀u∈Lp0,t0,U,iii∫0t0CTsxUpds≤M·xXp,∀x∈DA,iv∫0t0C∫0rT-1r-sBusdsUpdr≤M·uppmmmmmmmmmmmmmmi∀u∈W02,p0,t0,U,v1∈ρFt0,whereFt0∈LLp0,t0,U
is  given  by  (20). Then
(22)iiinABC≔A-1+BCX,DABC≔x∈Z:A-1+BCx∈X,
generates a C0-semigroup (S(t))t≥0 on the Banach space X. Moreover, the perturbed semigroup (S(t))t≥0 verifies the variation of parameters formula
(23)Stx=Ttx+∫0tT-1t-s·BC·Ssxds∀t≥0,x∈DABC.For the proof we extend the controllability-, observability-, and input-output maps introduced in Remarks2, 4, and 7 on R+ as follows. Recall that by assumption ω0(A)<0.Lemma 11.
Let(A,B,C) be compatible and (B,C) jointly p-admissible for some 1≤p<+∞. Then there exist(i)
a strongly continuous, uniformly bounded family(Bt)t≥0⊂L(Lp([0,+∞),U),X);(ii)
a bounded operatorC∞∈L(X,Lp([0,+∞),U));(iii)
a bounded operatorF∞∈L(Lp([0,+∞),U)),such that
(24)Btu=∫0tT-1t0-sBusds,∀u∈Lp0,+∞,U,C∞xs=CTsx,∀x∈DA,s∈0,+∞,F∞u·=C∫0·T-1·-sBusds∀u∈W02,p0,+∞,U.Proof.
The assertion for(Bt)t≥0 was proved in [13, Corollary 3.16]. The assertion for C∞ was shown in [13, Lemma 3.9]. Finally, the assertion for F∞ follows from [13, Remark 3.23].Forμ≥0 we indicate in the sequel the controllability-, observability-, and input-output maps associated with the triple (A-μ,B,C) with the superscript “μ”, for example,
(25)F∞μu·=C∫0·e-μ·-sT-1·-sBusds,∀u∈W02,p0,+∞,U.
Lemma 12 gives a condition such that the invertibility of I-Ft0 (see condition (v) of Theorem 10) implies the one of I-F∞μ for μ sufficiently large.Lemma 12.
Let the assumptions of Theorem10 be satisfied. If for μ≥0(26)Tt0+Bt01-Ft0-1Ct0<eμt0
holds, then 1∈ρ(F∞μ).Proof.
Inspired by [14, (2.6)] and the proof of [15, Proposition 2.1] consider for n∈N the surjective isometry (denote by vT the transposed vector of a vector v)(27)J:Lp0,nt0,U⟶∏k=1nLp0,t0,U,u⟼u1,…,unT,
where uk:[0,t0]→U, uk(s)≔u((k-1)t0+s), and (u1,…,un)Tpp≔∑k=1n‍ukp.
ThenFnt0 is isometrically isomorphic to the matrix(28)JFnt0J-1=Ft000⋯⋯0Ct0Tt00Bt0Ft00⋱⋮Ct0Tt01Bt0Ct0Bt0⋱⋱⋱⋮⋮⋱⋱⋱00⋮⋱Ct0Bt0Ft00Ct0Tt0n-2Bt0⋯⋯Ct0Tt0Bt0Ct0Bt0Ft0.Since by assumption 1-Ft0 is invertible, 1-Fnt0 is invertible as well and(29)J1-Fnt0-1J-1=G00⋯⋯0GCt0Tt0+Bt0GCt00Bt0GG0⋱⋮GCt0Tt0+Bt0GCt01Bt0GGCt0Bt0G⋱⋱⋱⋮⋮⋱⋱⋱00⋮⋱GCt0Bt0GG0GCt0Tt0+Bt0GCt0n-2Bt0G⋯⋯GCt0Tt0+Bt0GCt0Bt0GGCt0Bt0GG,where we put G≔(1-Ft0)-1. By Lemma A.1 applied to J(1-Fnt0)-1J-1 one obtains the estimate
(30)1-Fnt0-1≤G+GCt0·Bt0G·∑l=1n-1Tt0+Bt0GCt0l-1.
This shows that (1-Fnt0)-1 remains bounded as n→+∞ if (26) holds for μ=0.
If the estimate (26) only holds for some μ>0, consider the triple (A-μ,B,C). Let Mɛμ∈L(Lp([0,t0],U)) be the multiplication operator defined by
(31)Mɛμus≔eμs·us,u∈Lp0,t0,U.
Then Mɛμ is invertible with inverse Mɛ-μ and a simple computation shows that
(32)Bt0μ=e-μt0Bt0Mɛμ,Ct0μ=Mɛμ-1Ct0,Ft0μ=Mɛμ-1Ft0Mɛμ.
By similarity this implies that 1∈ρ(Ft0μ). Hence, repeating the above reasoning for (A-μ,B,C) one obtains from (30) that (1-Fnt0μ)-1 remains bounded as n→+∞ if
(33)e-μt0Tt0+Bt0μ1-Ft0μ-1Ct0μ<1.
Since by (32) one has
(34)e-μt0Tt0+Bt0μ1-Ft0μ-1Ct0μ=e-μt0Tt0+Bt01-Ft0-1Ct0,
the estimates (33) and (26) are equivalent. Summing up this shows that (26) implies that
(35)K≔sup⁡n∈N1-Fnt0μ-1<+∞.
Using this fact we finally show that 1∈ρ(F∞μ). Observe first that (1-F∞μ)u=0 for some u∈Lp([0,+∞),U) implies that (1-Fnt0μ)(u|[0,nt0])=0 for every n∈N. Since (1-Fnt0μ) is injective for every n∈N, this gives that u=0; that is, 1-F∞μ is injective.
To show surjectivity fix somev∈Lp([0,+∞),U) and define for n∈N(36)un≔1-Fnt0μ-1v0,nt0∈Lp0,nt0,U;
that is, un is the unique solution in Lp([0,nt0],U) of the equation
(37)1-Fnt0μu=v0,nt0.
However, for m≥n one has (Fmt0μum)|[0,nt0]=Fnt0μ(um|[0,nt0]); hence also um|[0,nt0]∈Lp([0,nt0],U) solves (37). This implies that
(38)um0,nt0=un.
Thus one can define
(39)us≔lim⁡n→+∞uns,s∈0,+∞.
Since by (35) it follows that un≤K·v for all n∈N, Fatou’s lemma implies that u∈Lp([0,+∞),U). Moreover, by construction
(40)1-F∞μu0,nt0=1-Fnt0μun=v0,nt0∀n∈N,
which implies (1-F∞μ)u=v. Since v∈Lp([0,t0],U) was arbitrary, this shows that 1-F∞μ is surjective. Hence 1-F∞μ is bijective and therefore 1∈ρ(F∞μ) as claimed.Next we show that the invertibility ofId-F∞μ implies for sufficiently large λ the invertibility of the“transfer function”Id-CR(λ,A-1)B of the system Σ(A,B,C) with feedback u(t)=y(t). In the following the Laplace transform of a function u is denoted by
(41)Luλ≔u^λ≔∫0+∞e-λrurdr.Lemma 13.
Assume that1∈ρ(F∞μ) for some μ≥0. Then 1∈ρ(CR(λ,A-1)B) for all λ∈C satisfying Reλ>μ and
(42)LId-F∞μ-1uλ=Id-CRλ,A-1B-1·u^λ∀u∈Lp0,+∞,U.Proof.
Assume first thatμ=0. Then it is well known that F∞=F∞μ is shift invariant (cf. [16]); that is, F∞ commutes with the right shift. Then also G≔Id-F∞∈L(Lp([0,+∞),U)) is shift invariant and by [16, Theorem  2.3] and [13, Lemma 3.19] one obtains for u∈Lp([0,+∞),U)(43)Gu^λ=Id-CRλ,A-1B·u^λ,Reλ>0.
Let R≔G-1∈L(Lp([0,+∞),U)). Then clearly the right shift also commutes with R; that is, this operator is shift invariant as well. Hence again by [16, Theorem  2.3] there exists R(λ)∈L(U) such that
(44)Ru^λ=Rλ·u^λ,Reλ>0,u∈Lp0,+∞,U.
Summing up one obtains for all u∈Lp([0,+∞),U)(45)u^λ=RGu^λ=Rλ·Gu^λ=Rλ·Id-CRλ,A-1B·u^λ=GRu^λ=Id-CRλ,A-1B·Ru^λ=Id-CRλ,A-1B·Rλ·u^λ.
Taking u(s)=e-sv for some v∈U, this implies
(46)11+λ·v=Rλ·Id-CRλ,A-1B·11+λ·v=Id-CRλ,A-1B·Rλ·11+λ·v,mmmmmmmmmmmmlmmmReλ>0.
Hence R(λ)=(Id-CR(λ,A-1)B)-1.
Ifμ>0, then by the same reasoning applied to F∞μ one obtains that
(47)1∈ρCRλ,A-1-μB=ρCRλ+μ,A-1B,∀Reλ>0.
Clearly this implies our claim in case μ>0 and the proof is complete.We are now well prepared to prove the main result of this section.Proof of Theorem10.
The idea of the proof is to define an operator family(S(t))t≥0⊂L(X) and then to verify that it is a C0-semigroup with generator ABC.
To this end, assume that the condition (26) in Lemma 12 holds for μ=0. Then Id-F∞ is invertible and one can define for t≥0(48)St≔Tt+BtId-F∞-1C∞∈LX.
Since (T(t))t≥0 and (Bt)t≥0 are both strongly continuous and uniformly bounded, the same holds for (S(t))t≥0. We proceed and compute the Laplace transform of S(·)x:[0,+∞)→X for x∈X. Since
(49)S·x=T·x+T-1·B*1-F∞-1C∞x,
the convolution theorem for the Laplace transform (or [13, Lemma 3.12]) and Lemma 13 imply for every x∈X and Reλ>0(50)LS·xλ=Rλ,Ax+Rλ,A-1B·L1-F∞-1C∞xλ=Rλ,Ax+Rλ,A-1B·Id-CRλ,A-1B-1·CRλ,Ax=:Qλx.
We now show that Q(λ)=R(λ,ABC). First note that by the compatibility condition (14) one has
(51)rgQλ⊂DA+Z=Z=DC.
Moreover,
(52)λ-A-1-BC·Qλ=iId-BCRλ,A+B·Id·Id-CRλ,A-1B-1·CRλ,A-B·CRλ,A-1B·Id-CRλ,A-1B-1CRλ,A=Id.
This implies that Q(λ) is a right inverse and rg(Q(λ))⊂D(ABC). To show that it is also a left inverse take x∈D(ABC)⊂Z=D(C). Then we obtain
(53)Qλ·λ-A-1-BCx=x-Rλ,A-1BCx+Rλ,A-1·BId-CRλ,A-1B-1·Id·Cx-Rλ,A-1·BId-CRλ,A-1B-1·CRλ,A-1B·Cx=x,
and hence it follows that Q(λ)=R(λ,ABC) as claimed. Summing up we showed that (S(t))t≥0⊂L(X) is a strongly continuous family with Laplace transform R(λ,ABC). By [17, Theorem 3.1.7] this implies that (S(t))t≥0 is a C0-semigroup with generator ABC.
To verify the variation of parameters formula (23) one first notes that by Lemma 13 and the explicit representation of R(λ,ABC) in (50) one has for all x∈D(ABC) and Reλ>μ=0 that
(54)L1-F∞-1C∞·xλ=LCS·xλ.
By the uniqueness of the Laplace transform this implies that
(55)1-F∞-1C∞·x=CS·x,
and the assertion follows from the definition of (S(t))t≥0 in (49).
Now assume that (26) only holds for some μ>0. Then repeating the same reasoning for the triple (A-μ,B,C) one concludes as before that (A-μ)BC=((A-μ)-1+BC)|X=ABC-μ is a generator. Clearly this implies that ABC  generates a strongly continuous semigroup (S(t))t≥0. Moreover, one obtains that the pair of rescaled semigroups (e-μtT(t))t≥0 and (e-μtS(t))t≥0 verify the variation of parameters formula (23) which implies that this formula holds for the pair (T(t))t≥0 and (S(t))t≥0 as well.As already remarked in the introduction, with increasingp∈[1,+∞) the p-admissibility of the control and observation operator becomes weaker and stronger, respectively.Assuming that the input-output map mapsLα to Lβ for some (the main cases we have in mind are α<p=β<+∞ and 1<α=p<β) 1≤α≤p≤β<+∞ satisfying α<β, one can drop the invertibility condition 1∈ρ(Ft0) in Theorem 10 (and sometimes even the compatibility condition (14), cf. Remark 15).Theorem 14.
Assume that conditions (i)–(iii) in Theorem10 are satisfied. Moreover, suppose there exist 1≤α≤p≤β<∞ with α<β, p>1, and M≥0 such that
(56)iv′∫0t0C∫0rT-1r-sBusdsUβdr≤M·uαβ∀u∈W02,α0,t0,U.
Then ABC given by (22) generates a C0-semigroup (S(t))t≥0 on the Banach space X verifying the variation of parameters formula (23).Proof.
By Theorem10 it suffices to show that 1∈ρ(Ft1) for some t1>0. By assumption the operator
(57)iiFt~:W02,α0,t,U⊂Lα0,t,U⟶Lβ0,t,U,Ft~u≔C∫0·T-1·-sBusds
has a bounded extension Ft∈L(Lα([0,t],U),Lβ([0,t],U)) for every t∈(0,t0]. We distinguish 2 cases and use in both of them Jensen’s inequality as follows:
(58)ur≤t1/r-1/s·us,
for 1≤r<s≤+∞ and u∈Ls([0,t],U)⊂Lr([0,t],U).
Case
α
<
p
. Then Ft belongs to L(Lα([0,t],U),Lp([0,t],U)) with norm (denote the norm of a bounded linear operator F:Lr→Ls by Frs)  Ftαp≤Ft0αp. This implies, by (58) for r=α and s=p, that
(59)Ftup≤Ftαp·uα≤t1/α-1/p·Ft0αp·up.
Case
p
<
β
. In this case, Ft∈L(Lp([0,t],U),Lβ([0,t],U)) with norm Ftpβ≤Ft0pβ. This implies, by (58) for r=p and s=β, that
(60)Ftup≤t1/p-1/β·Ftuβ≤t1/p-1/β·Ft0pβ·up.
Hence in both cases, considering Ft≔Ft|Lp([0,t],U)∈L(Lp([0,t],U)), one concludes that there exists t1>0 such that Ft1<1 which implies 1∈ρ(Ft1).Remark 15.
Assume as in Theorem14 that 1≤α≤p≤β≤+∞ with α<β and 1<p<+∞. If there exist t0>0 and a dense subspace D⊂Lα([0,t0],U) such that for every u∈D(i)
∫0r‍T-1(r-s)Bu(s)ds∈Z for almost all 0<r≤t0;(ii)
the map[0,t0]∋r↦C∫0r‍T-1(r-s)Bu(s)ds is in Lβ([0,t0],U);(iii)
there existsM≥0 such that
(61)∫0t0C∫0rT-1r-sBusdsUβdr≤M·uαβ,∀u∈D,then also the compatibility condition (14) is satisfied.
To verify this assertion defineFt~:D⊂Lα([0,t],U)→Lβ([0,t],U) as in (57) with W02,α([0,t],U) replaced by the space D. By assumption, Ft~ has a unique bounded extension Ft:Lα([0,t],U)→Lβ([0,t],U). As above take Ft≔Ft|Lp([0,t],U)∈L(Lp([0,t],U)). Then, by Hölder’s inequality (or (58) for r=1 and s=β), one obtains for every v∈U (define (f⊗x)(s)≔f(s)·x for all s∈[0,t0] where f:[0,t0]→C is a scalar function; moreover, by 1 denote the constant one function on the interval [0,t0])
(62)1t∫0tFt1⊗vsds≤1t∫0tFt1⊗vsds=1t·Ft1⊗v1≤1t·t1-1/β·Ft1⊗vβ≤t-1/β·Ftαβ·1⊗vα≤t1/α-1/β·Ft0αβ·vU⟶0,ast⟶0+.
By [10, Theorem 5.8] in the Hilbert space case or [4, Theorems 5.6.4 and 5.6.5] in the general case this convergence implies the compatibility condition (14).


## 4. Applications

We now give some applications of our abstract results. First we show that Theorem14 can be considered as a simultaneous generalization of the Desch-Schappacher and the Miyadera-Voigt perturbation theorems. Moreover, we generalize a result of Greiner concerning the perturbation of the boundary conditions of a generator.


### 4.1. The Desch-Schappacher Perturbation Theorem

The following result was proved in [18, Theorem 5, Proposition 8]; see also [2, Corollary III.3.4] and [19, Corollary 5.5.1].Theorem 16 (see [18]).
Assume that forB∈L(X,X-1A) there exist 1≤p<+∞, t0>0, and M≥0 such that
(63)∫0t0T-1t0-sBusds∈X,∀u∈Lp0,t0,X.
Then (AB,D(AB)) given by
(64)iiiABx≔A-1+Bx,DAB≔x∈X:A-1+Bx∈X,
is the generator of a C0-semigroup (S(t))t≥0 on X.Remark that one could consider the condition (63) also for p=∞ or u∈C([0,t0],U). However, in this case one needs an additional norm estimate to ensure that condition (v) in Theorem 10 is satisfied (cf. [2, Corollary III.3.3.]). Moreover, note that in a certain sense the Desch-Schappacher theorem depends only on the range but not on the “size” of the perturbation B. In particular, if B satisfies the assumption of Theorem 16, then also BF satisfies it for every F∈L(X).Proof of Theorem16.
LetU=Z=X and C=Id. Then by assumption B∈L(X,X-1A) is a p-admissible control operator and conditions (i)–(iii) in Theorem 10 are clearly satisfied. We will prove that (ii) implies condition (iv′) from Theorem 14. To this end we first verify that the function
(65)0,t0∋r⟼vr≔∫0rT-1r-sBusds∈X
is continuous for every u∈Lp([0,t0],X). For such u define ut:[0,t0]→U by
(66)uts≔0if0≤s≤t0-tus-t0+tift0-t<s≤t0;
that is, ut is just the right translation of u by t0-t. Then ut∈Lp([0,t0],X) and using Remark 2 one obtains from v(r)=Bt0ur that for r0,r1∈[0,t0](67)vr0-vr1=Bt0ur0-ur1≤Bt0·ur0-ur1p⟶0,asr1⟶r0,
where the last step follows from the strong continuity of the nilpotent right translation semigroup on Lp([0,t0],X). Next define the operator
(68)Ft0:Lp0,t0,X⟶C0,t0,X,Ft0ur≔∫0rT-1r-sBusds,iiimmiiiir∈0,t0.iiimmiiii
The operator Ft0 is well-defined. Moreover, the estimate
(69)Ft0ur≤Bt0·urp≤Bt0·up∀u∈Lp0,t0,X,r∈0,t0,
shows that Ft0∈L(Lp([0,t0],X),C([0,t0],X))⊂L(Lp([0,t0],X),Lβ([0,t0],X)) for all β≥1. Choosing β>p this implies condition (iv′) and hence the proof is complete.Remark 17.
The proofs of Theorems14 and 16 imply the following: if B∈L(U,X-1A) is a p-admissible control operator for some 1≤p<+∞ then for every bounded C∈L(X,U) the triple (A,B,C) is compatible and jointly p-admissible. Moreover, in this case every F∈L(U) is a p-admissible feedback operator for the system Σ(A,B,C).


### 4.2. The Miyadera-Voigt Perturbation Theorem

As another application we consider the following version of the Miyadera-Voigt perturbation theorem (cf. [20, 21], see also [2, Corollary III.3.16] and [19, Theorem 5.4.2]).Theorem 18 (see [20, 21]).
Assume that forC∈L(X1A,X) there exist 1<p<+∞, t0>0, and M≥0 such that
(70)∫0t0CTsxXpds≤M·xXp,∀x∈DA.
Then (AC,D(AC)) given by
(71)ACx≔A+Cx,DAC≔DA,
is the generator of a C0-semigroup on X.Observe that one could consider condition (70) also for p=1. However, in this case one needs M<1 to ensure that condition (v) in Theorem 10 is satisfied (cf. [2, Corollary III.3.16]). Moreover, note that in a certain sense the Miyadera-Voigt Theorem 18 (for p>1) depends only on the domain but not on the “size” of the perturbation C. In particular, if C satisfies the assumption of Theorem 18, then also FC satisfies it for every F∈L(X).Proof of Theorem18.
LetU=X, Z=X1A, and B=Id. Then, by assumption, C∈L(Z,X) is a p-admissible observation operator and conditions (i)–(iii) in Theorem 10 are clearly satisfied. We will show that condition (iii) implies condition (iv′) from Theorem 14. To this end fix 0≤γ<δ≤t0 and x∈D(A). Then for u=1[γ,δ]⊗x one obtains
(72)C∫0rTr-susds=CA-1∫γr1γ,δs·Tr-sAxds=∫γr1γ,δs·CTr-sxds.Using (72), condition (iii), the triangle and Hölder’s inequality for f∈L1(a,b) and p≥1(73)∫0t0C∫0rTr-susdsXpdr≤∫γt0∫γr1γ,δs·CTr-sxXdspdr=∫γδ∫γrCTr-sxXdspdr+∫δt0∫γδCTr-sxXdspdr≤∫γδr-γp-1∫γrCTr-sxXpdsdr+∫δt0∫γδδ-γp-1CTr-sxXpdsdr≤∫γδr-γp-1M·xXpdr+∫γδδ-γp-1∫δt0CTr-sxXpdrds≤Mp·δ-γp·xXp+∫γδδ-γp-1M·xXpds=M1+1p·δ-γp·xXp=:Kp·δ-γp·xXp.
Let now u=∑k=1n‍1[γk,δk]⊗xk∈L1([0,t0],X) be a step function where the intervals [γk,δk]⊂[0,t0] are pairwise disjoint and xk∈D(A) for k=1,…,n. Then from (73) one obtains
(74)∫0t0C∫0rTr-susdsXpdr1/p≤K·∑k=1nδk-γk·xkX=K·u1.
Since the step functions having values in D(A) are dense in L1([0,t0],X), this implies condition (iv′) for α=1 and β=p. This completes the proof.Remark 19.
The proofs of Theorems14 and 18 imply the following: if C∈L(Z,U) is a p-admissible observation operator for some 1≤p<+∞ then for every bounded B∈L(U,X) the triple (A,B,C) is compatible and jointly p-admissible. Moreover, if p>1 then in addition every F∈L(U) is a p-admissible feedback operator for the system Σ(A,B,C).


### 4.3. Perturbing the Boundary Conditions of a Generator

In this section we show how Theorem10 can be used to generalize significantly the approach by Greiner in [5] to perturbations of boundary conditions of a generator. To explain the general setup we consider the following:(i)
two Banach spaces (in this section denote the elements ofX by f instead of x)   X and ∂X, the latter called “boundary space”;(ii)
a closed, densely defined “maximal” operator (“maximal” concerns the size of the domain, e.g., a differential operator without boundary conditions)Am:D(Am)⊆X→X;(iii)
the Banach space[D(Am)]≔(D(Am),·Am) where fAm≔f+Amf is the graph norm;(iv)
two “boundary” operatorsL,Φ∈L([D(Am)],∂X).Then define two restrictions A,AΦ⊂Am by
(75)iiDA≔f∈DAm:Lf=0=ker⁡L,DAΦ≔f∈DAm:Lf=Φf.
In many applications X, ∂X, and D(Am) are function spaces and L is a “trace-type” operator which restricts a function in D(Am) to (a part of) the boundary of its domain. Hence one can consider AΦ with boundary condition Lf=Φf as a perturbation of the operator A with abstract “Dirichlet type” boundary condition Lf=0.In order to treat this setup within our framework we make the following assumptions:(i)
the operatorA generates a strongly continuous semigroup (T(t))t≥0 on X;(ii)
the boundary operatorL:D(Am)→∂X is surjective.Under these assumptions the following lemma, shown by Greiner [5, Lemma 1.2], is the key to write AΦ as a Weiss-Staffans perturbation of A.Lemma 20.
Let the above assumptions (i) and (ii) be satisfied. Then for eachλ∈ρ(A) the operator L|ker⁡(λ-Am) is invertible and Dλ≔(Lker⁡(λ-Am))-1:∂X→ker⁡(λ-Am)⊆X is bounded.Using this so-calledDirichlet operatorDλ one obtains the following representation of AΦ where for simplicity we assume that A is invertible.Lemma 21.
If0∈ρ(A), then
(76)AΦ=A-1-A-1D0·ΦX;
that is, AΦ=ABC for  U≔∂X, Z≔[D(Am)], and
(77)B≔-A-1D0∈LU,X-1A,C≔Φ∈LZ,U.Proof.
Denote the operator on the right-hand side of (76) by A~Φ. Then
(78)f∈DA~Φ⟺f-D0Φf∈DA⟺Lf=LD0Φf=Φf⟺f∈DAΦ.
Moreover, for f∈D(AΦ) we have
(79)A~Φf=Af-D0Φf=Amf-D0Φf=Amf=AΦf,
as claimed.We mention that in [5, Theorem  2.1] the operator Φ∈L(X,U) is bounded and the assumptions imply that A-1D0 is a 1-admissible control operator. Hence in this case AΦ is a generator by the Desch-Schappacher theorem.By using Theorem10 one can now deal also with unbounded Φ.Corollary 22.
Assume that for some1≤p<+∞ the pair (A-1D0,Φ) is jointly p-admissible and that Id∈L(∂X) is a p-admissible feedback operator for A. Then AΦ is the generator of a C0-semigroup on X.Proof.
One only has to show the compatibility condition (14). This, however, immediately follows from
(80)rgRλ,A-1B=rgId-λRλ,AD0⊂ker⁡Am+DA⊆DAm=Z.Remark 23.
We note that in [22, Theorem 4.1] the authors study a similar problem in the context of regular linear systems.As a simple but typical example for the previous corollary consider the spaceX≔Lp[0,1] and the first derivative Am≔d/ds with domain D(Am)≔W1,p[0,1] (c.f. [5, Example 1.1.(c)]). As boundary space choose ∂X=C, as boundary operator the point evaluation L=δ1 and as perturbation some Φ∈W1,p0,1′. This gives rise to the differential operators A,AΦ⊂d/ds with domains
(81)DA≔f∈W1,p0,1:f1=0,DAΦ≔f∈W1,p0,1:f1=Φf.
Then clearly the assumptions (i) and (ii) made above are satisfied; in particular A generates the nilpotent left-shift semigroup given by
(82)Ttfs=fs+tifs+t≤1,0else.
However, AΦ is not always a generator. For example if Φ=δ1, then AΦ=Am and σ(AΦ)=C; hence AΦ is not a generator. Thus one needs an additional assumption on Φ.Definition 24.
A bounded linear functionalΦ:C[0,1]→C haslittle mass in r=1 if there exist q<1 and δ>0 such that
(83)Φf≤q·f∞,
for every f∈C[0,1] satisfying supp⁡f⊂[1-δ,1].Note thatW1,p[0,1]↪cC[0,1] and hence C0,1′⊂DAm′. Now the following holds.Corollary 25.
IfΦ∈C0,1′ has little mass in r=1, then for all 1≤p<+∞ the operator AΦ is the generator of a strongly continuous semigroup on Lp[0,1].Proof.
By Corollary22 it suffices to show that for the triple (A,A-1D0,Φ) the conditions (ii)–(v) of Theorem 10 are satisfied. To this end, note that 0∈ρ(A) and that the Dirichlet operator D0:C→Lp[0,1] is given by D0α=α·1 where 1(s)=1 for all s∈[0,1].
(ii) By Remark2 it suffices to verify estimate (11) where we may assume that u∈W01,p[0,t0] for some 0<t0≤1. Using integration by parts and [23, Theorem  4.2] we conclude that (for a function g defined on an interval denote in the sequel by g~ its extension to R by the value 0)
(84)∫0t0T-1t0-sBusds=-∫0t0T-1t0-sA-1D0usds=D0ut0-∫0t0Tt0-sD0u′sds=ut0·1-∫0t0Tt0-s1·u′sds=ut0·1-∫max⁡⁡0,·+t0-1t0u′sds=umax⁡0,·+t0-1=u~·+t0-1.
This implies Bt0uX=Bt0up≤up for all u∈W01,p[0,t0] which shows (ii).
(iii) By the Riesz-Markov representation theorem there exists a regular complex Borel measureμ on [0,1] such that
(85)Φf=∫01frdμr,∀f∈C0,1.
Using Fubini’s theorem and Hölder’s inequality one obtains for 0<t0≤1 and f∈D(A)(86)∫0t0CTsfpds=∫0t0Φf~·+spds≤∫0t0∫01f~r+sdμrpds≤∫0t0μ0,1p-1·∫01f~r+spdμrds=μp-1·∫01∫0t0f~r+spdsdμr≤μp·fpp,
where μ≔μ[0,1] (which coincides with Φ∞). This proves (iii).
(iv) From (84) one obtains for 0<t0≤1 and u∈W01,p[0,t0] by similar arguments as in (iii)
(87)∫0t0C∫0rT-1r-sBusdspdr=∫0t0Φu~·+r-1pdr=∫0t0∫1-r1us+r-1dμspdr≤∫0t0μ1-r,1p-1·∫1-r1us+r-1pdμsdr≤μ1-t0,1p-1·∫1-t01∫1-s1us+r-1pdrdμs≤μ1-t0,1p·upp.
This shows (iv).
(v) Since by assumptionΦ has little mass in r=1, it follows that μ[1-t0,1]<1 for sufficiently small t0>0. Hence from estimate (87) and the denseness of W01,p[0,t0] in Lp[0,t0] it follows that Ft0≤μ[1-t0,1]<1 for 0<t0≤1 sufficiently small. This implies 1∈ρ(Ft0) as claimed.Remarks. (i) Corollary 25 could be easily generalized (with essentially the same proof) to the first derivative on Lp([0,1],Cn). One could even go further and prove a similar result on Lp([0,1],E) for a (possibly infinite dimensional) Banach space E provided the boundary operator Φ has a representation as a Riemann-Stieltjes integral as in (85). See also [22, Example  5.1].(ii) In most cases the admissibility of the identity as a feedback operator is verified by showing thatFt0<1 for sufficiently small t0>0. Choosing Φ=αδ1, by (84), one obtains Ft0=αId for all t0>0; hence 1∈ρ(Ft0) if and only if α≠1. This provides an example where our perturbation theorem is applicable even if Ft0>1 for all t0>0. Note that for α=1 one obtains AΦ=Am; hence in this case AΦ cannot be a generator.


## 4.1. The Desch-Schappacher Perturbation Theorem

The following result was proved in [18, Theorem 5, Proposition 8]; see also [2, Corollary III.3.4] and [19, Corollary 5.5.1].Theorem 16 (see [18]).
Assume that forB∈L(X,X-1A) there exist 1≤p<+∞, t0>0, and M≥0 such that
(63)∫0t0T-1t0-sBusds∈X,∀u∈Lp0,t0,X.
Then (AB,D(AB)) given by
(64)iiiABx≔A-1+Bx,DAB≔x∈X:A-1+Bx∈X,
is the generator of a C0-semigroup (S(t))t≥0 on X.Remark that one could consider the condition (63) also for p=∞ or u∈C([0,t0],U). However, in this case one needs an additional norm estimate to ensure that condition (v) in Theorem 10 is satisfied (cf. [2, Corollary III.3.3.]). Moreover, note that in a certain sense the Desch-Schappacher theorem depends only on the range but not on the “size” of the perturbation B. In particular, if B satisfies the assumption of Theorem 16, then also BF satisfies it for every F∈L(X).Proof of Theorem16.
LetU=Z=X and C=Id. Then by assumption B∈L(X,X-1A) is a p-admissible control operator and conditions (i)–(iii) in Theorem 10 are clearly satisfied. We will prove that (ii) implies condition (iv′) from Theorem 14. To this end we first verify that the function
(65)0,t0∋r⟼vr≔∫0rT-1r-sBusds∈X
is continuous for every u∈Lp([0,t0],X). For such u define ut:[0,t0]→U by
(66)uts≔0if0≤s≤t0-tus-t0+tift0-t<s≤t0;
that is, ut is just the right translation of u by t0-t. Then ut∈Lp([0,t0],X) and using Remark 2 one obtains from v(r)=Bt0ur that for r0,r1∈[0,t0](67)vr0-vr1=Bt0ur0-ur1≤Bt0·ur0-ur1p⟶0,asr1⟶r0,
where the last step follows from the strong continuity of the nilpotent right translation semigroup on Lp([0,t0],X). Next define the operator
(68)Ft0:Lp0,t0,X⟶C0,t0,X,Ft0ur≔∫0rT-1r-sBusds,iiimmiiiir∈0,t0.iiimmiiii
The operator Ft0 is well-defined. Moreover, the estimate
(69)Ft0ur≤Bt0·urp≤Bt0·up∀u∈Lp0,t0,X,r∈0,t0,
shows that Ft0∈L(Lp([0,t0],X),C([0,t0],X))⊂L(Lp([0,t0],X),Lβ([0,t0],X)) for all β≥1. Choosing β>p this implies condition (iv′) and hence the proof is complete.Remark 17.
The proofs of Theorems14 and 16 imply the following: if B∈L(U,X-1A) is a p-admissible control operator for some 1≤p<+∞ then for every bounded C∈L(X,U) the triple (A,B,C) is compatible and jointly p-admissible. Moreover, in this case every F∈L(U) is a p-admissible feedback operator for the system Σ(A,B,C).


## 4.2. The Miyadera-Voigt Perturbation Theorem

As another application we consider the following version of the Miyadera-Voigt perturbation theorem (cf. [20, 21], see also [2, Corollary III.3.16] and [19, Theorem 5.4.2]).Theorem 18 (see [20, 21]).
Assume that forC∈L(X1A,X) there exist 1<p<+∞, t0>0, and M≥0 such that
(70)∫0t0CTsxXpds≤M·xXp,∀x∈DA.
Then (AC,D(AC)) given by
(71)ACx≔A+Cx,DAC≔DA,
is the generator of a C0-semigroup on X.Observe that one could consider condition (70) also for p=1. However, in this case one needs M<1 to ensure that condition (v) in Theorem 10 is satisfied (cf. [2, Corollary III.3.16]). Moreover, note that in a certain sense the Miyadera-Voigt Theorem 18 (for p>1) depends only on the domain but not on the “size” of the perturbation C. In particular, if C satisfies the assumption of Theorem 18, then also FC satisfies it for every F∈L(X).Proof of Theorem18.
LetU=X, Z=X1A, and B=Id. Then, by assumption, C∈L(Z,X) is a p-admissible observation operator and conditions (i)–(iii) in Theorem 10 are clearly satisfied. We will show that condition (iii) implies condition (iv′) from Theorem 14. To this end fix 0≤γ<δ≤t0 and x∈D(A). Then for u=1[γ,δ]⊗x one obtains
(72)C∫0rTr-susds=CA-1∫γr1γ,δs·Tr-sAxds=∫γr1γ,δs·CTr-sxds.Using (72), condition (iii), the triangle and Hölder’s inequality for f∈L1(a,b) and p≥1(73)∫0t0C∫0rTr-susdsXpdr≤∫γt0∫γr1γ,δs·CTr-sxXdspdr=∫γδ∫γrCTr-sxXdspdr+∫δt0∫γδCTr-sxXdspdr≤∫γδr-γp-1∫γrCTr-sxXpdsdr+∫δt0∫γδδ-γp-1CTr-sxXpdsdr≤∫γδr-γp-1M·xXpdr+∫γδδ-γp-1∫δt0CTr-sxXpdrds≤Mp·δ-γp·xXp+∫γδδ-γp-1M·xXpds=M1+1p·δ-γp·xXp=:Kp·δ-γp·xXp.
Let now u=∑k=1n‍1[γk,δk]⊗xk∈L1([0,t0],X) be a step function where the intervals [γk,δk]⊂[0,t0] are pairwise disjoint and xk∈D(A) for k=1,…,n. Then from (73) one obtains
(74)∫0t0C∫0rTr-susdsXpdr1/p≤K·∑k=1nδk-γk·xkX=K·u1.
Since the step functions having values in D(A) are dense in L1([0,t0],X), this implies condition (iv′) for α=1 and β=p. This completes the proof.Remark 19.
The proofs of Theorems14 and 18 imply the following: if C∈L(Z,U) is a p-admissible observation operator for some 1≤p<+∞ then for every bounded B∈L(U,X) the triple (A,B,C) is compatible and jointly p-admissible. Moreover, if p>1 then in addition every F∈L(U) is a p-admissible feedback operator for the system Σ(A,B,C).


## 4.3. Perturbing the Boundary Conditions of a Generator

In this section we show how Theorem10 can be used to generalize significantly the approach by Greiner in [5] to perturbations of boundary conditions of a generator. To explain the general setup we consider the following:(i)
two Banach spaces (in this section denote the elements ofX by f instead of x)   X and ∂X, the latter called “boundary space”;(ii)
a closed, densely defined “maximal” operator (“maximal” concerns the size of the domain, e.g., a differential operator without boundary conditions)Am:D(Am)⊆X→X;(iii)
the Banach space[D(Am)]≔(D(Am),·Am) where fAm≔f+Amf is the graph norm;(iv)
two “boundary” operatorsL,Φ∈L([D(Am)],∂X).Then define two restrictions A,AΦ⊂Am by
(75)iiDA≔f∈DAm:Lf=0=ker⁡L,DAΦ≔f∈DAm:Lf=Φf.
In many applications X, ∂X, and D(Am) are function spaces and L is a “trace-type” operator which restricts a function in D(Am) to (a part of) the boundary of its domain. Hence one can consider AΦ with boundary condition Lf=Φf as a perturbation of the operator A with abstract “Dirichlet type” boundary condition Lf=0.In order to treat this setup within our framework we make the following assumptions:(i)
the operatorA generates a strongly continuous semigroup (T(t))t≥0 on X;(ii)
the boundary operatorL:D(Am)→∂X is surjective.Under these assumptions the following lemma, shown by Greiner [5, Lemma 1.2], is the key to write AΦ as a Weiss-Staffans perturbation of A.Lemma 20.
Let the above assumptions (i) and (ii) be satisfied. Then for eachλ∈ρ(A) the operator L|ker⁡(λ-Am) is invertible and Dλ≔(Lker⁡(λ-Am))-1:∂X→ker⁡(λ-Am)⊆X is bounded.Using this so-calledDirichlet operatorDλ one obtains the following representation of AΦ where for simplicity we assume that A is invertible.Lemma 21.
If0∈ρ(A), then
(76)AΦ=A-1-A-1D0·ΦX;
that is, AΦ=ABC for  U≔∂X, Z≔[D(Am)], and
(77)B≔-A-1D0∈LU,X-1A,C≔Φ∈LZ,U.Proof.
Denote the operator on the right-hand side of (76) by A~Φ. Then
(78)f∈DA~Φ⟺f-D0Φf∈DA⟺Lf=LD0Φf=Φf⟺f∈DAΦ.
Moreover, for f∈D(AΦ) we have
(79)A~Φf=Af-D0Φf=Amf-D0Φf=Amf=AΦf,
as claimed.We mention that in [5, Theorem  2.1] the operator Φ∈L(X,U) is bounded and the assumptions imply that A-1D0 is a 1-admissible control operator. Hence in this case AΦ is a generator by the Desch-Schappacher theorem.By using Theorem10 one can now deal also with unbounded Φ.Corollary 22.
Assume that for some1≤p<+∞ the pair (A-1D0,Φ) is jointly p-admissible and that Id∈L(∂X) is a p-admissible feedback operator for A. Then AΦ is the generator of a C0-semigroup on X.Proof.
One only has to show the compatibility condition (14). This, however, immediately follows from
(80)rgRλ,A-1B=rgId-λRλ,AD0⊂ker⁡Am+DA⊆DAm=Z.Remark 23.
We note that in [22, Theorem 4.1] the authors study a similar problem in the context of regular linear systems.As a simple but typical example for the previous corollary consider the spaceX≔Lp[0,1] and the first derivative Am≔d/ds with domain D(Am)≔W1,p[0,1] (c.f. [5, Example 1.1.(c)]). As boundary space choose ∂X=C, as boundary operator the point evaluation L=δ1 and as perturbation some Φ∈W1,p0,1′. This gives rise to the differential operators A,AΦ⊂d/ds with domains
(81)DA≔f∈W1,p0,1:f1=0,DAΦ≔f∈W1,p0,1:f1=Φf.
Then clearly the assumptions (i) and (ii) made above are satisfied; in particular A generates the nilpotent left-shift semigroup given by
(82)Ttfs=fs+tifs+t≤1,0else.
However, AΦ is not always a generator. For example if Φ=δ1, then AΦ=Am and σ(AΦ)=C; hence AΦ is not a generator. Thus one needs an additional assumption on Φ.Definition 24.
A bounded linear functionalΦ:C[0,1]→C haslittle mass in r=1 if there exist q<1 and δ>0 such that
(83)Φf≤q·f∞,
for every f∈C[0,1] satisfying supp⁡f⊂[1-δ,1].Note thatW1,p[0,1]↪cC[0,1] and hence C0,1′⊂DAm′. Now the following holds.Corollary 25.
IfΦ∈C0,1′ has little mass in r=1, then for all 1≤p<+∞ the operator AΦ is the generator of a strongly continuous semigroup on Lp[0,1].Proof.
By Corollary22 it suffices to show that for the triple (A,A-1D0,Φ) the conditions (ii)–(v) of Theorem 10 are satisfied. To this end, note that 0∈ρ(A) and that the Dirichlet operator D0:C→Lp[0,1] is given by D0α=α·1 where 1(s)=1 for all s∈[0,1].
(ii) By Remark2 it suffices to verify estimate (11) where we may assume that u∈W01,p[0,t0] for some 0<t0≤1. Using integration by parts and [23, Theorem  4.2] we conclude that (for a function g defined on an interval denote in the sequel by g~ its extension to R by the value 0)
(84)∫0t0T-1t0-sBusds=-∫0t0T-1t0-sA-1D0usds=D0ut0-∫0t0Tt0-sD0u′sds=ut0·1-∫0t0Tt0-s1·u′sds=ut0·1-∫max⁡⁡0,·+t0-1t0u′sds=umax⁡0,·+t0-1=u~·+t0-1.
This implies Bt0uX=Bt0up≤up for all u∈W01,p[0,t0] which shows (ii).
(iii) By the Riesz-Markov representation theorem there exists a regular complex Borel measureμ on [0,1] such that
(85)Φf=∫01frdμr,∀f∈C0,1.
Using Fubini’s theorem and Hölder’s inequality one obtains for 0<t0≤1 and f∈D(A)(86)∫0t0CTsfpds=∫0t0Φf~·+spds≤∫0t0∫01f~r+sdμrpds≤∫0t0μ0,1p-1·∫01f~r+spdμrds=μp-1·∫01∫0t0f~r+spdsdμr≤μp·fpp,
where μ≔μ[0,1] (which coincides with Φ∞). This proves (iii).
(iv) From (84) one obtains for 0<t0≤1 and u∈W01,p[0,t0] by similar arguments as in (iii)
(87)∫0t0C∫0rT-1r-sBusdspdr=∫0t0Φu~·+r-1pdr=∫0t0∫1-r1us+r-1dμspdr≤∫0t0μ1-r,1p-1·∫1-r1us+r-1pdμsdr≤μ1-t0,1p-1·∫1-t01∫1-s1us+r-1pdrdμs≤μ1-t0,1p·upp.
This shows (iv).
(v) Since by assumptionΦ has little mass in r=1, it follows that μ[1-t0,1]<1 for sufficiently small t0>0. Hence from estimate (87) and the denseness of W01,p[0,t0] in Lp[0,t0] it follows that Ft0≤μ[1-t0,1]<1 for 0<t0≤1 sufficiently small. This implies 1∈ρ(Ft0) as claimed.Remarks. (i) Corollary 25 could be easily generalized (with essentially the same proof) to the first derivative on Lp([0,1],Cn). One could even go further and prove a similar result on Lp([0,1],E) for a (possibly infinite dimensional) Banach space E provided the boundary operator Φ has a representation as a Riemann-Stieltjes integral as in (85). See also [22, Example  5.1].(ii) In most cases the admissibility of the identity as a feedback operator is verified by showing thatFt0<1 for sufficiently small t0>0. Choosing Φ=αδ1, by (84), one obtains Ft0=αId for all t0>0; hence 1∈ρ(Ft0) if and only if α≠1. This provides an example where our perturbation theorem is applicable even if Ft0>1 for all t0>0. Note that for α=1 one obtains AΦ=Am; hence in this case AΦ cannot be a generator.



---

*Source: 213020-2014-10-30.xml*