On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems

This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates a C 0 -semigroup. Next, a necessary and sufficient condition for L-admissibility (p ∈ [1,∞[) of the system’s control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).


Introduction
The purpose of this paper is to analyze conditions for the inversion formula and the   -admissibility for control operators for the solution of the following integrodifferential equation: ẋ () =  () + ∫  0  ( − )  ()  +  () ,  ≥ 0,  (0) =  0 ∈ , (1) which has a "big" intersection with the class of scalar Volterra integral equations.Here we assume that  is a closed linear densely defined operator in a Banach space , and the kernel  belonging to  1 loc (R + ) is real-valued and of at most exponential growth. is a (possibly unbounded) linear operator on another Banach space  and the control function  ∈   loc (R + ; ).
It is further assumed that the uncontrolled system, that is, is well-posed, which is equivalent to the existence of a unique family of bounded linear operators (()) ≥0 on  called the resolvent or solution family for (2), where we write (()) ≥0 ⊂ L()) (see Section 2) and that (()) ≥0 is exponentially bounded.
Many authors have studied this class of Volterra integral equations by the classical approach and for different reasons, using the Laplace transform (see, e.g., Da Prato and Iannelli [1,2], Grimmer and Pritchard [3], Lunardi [4], and Prüss [5,6]).Maximal-  regularity results are due to Clément and Da Prato [7] and Prüss [6].In addition to the classical approach, there is a semigroup approach which was used in, for example, Miller [8], Chen and Grimmer [9,10], Desch and Grimmer [11], Desch and Schappacher [12], Di Blasio et al. [13], 2 International Journal of Differential Equations Nagel and Sinestrari [14], and Engel and Nagel [15].It was the main objection against the semigroup approach for many years that it is not possible to obtain regularity of the solutions.This is not true, as it was proved recently in [16].
From the point of view of complex inversion formula, early Hille and Philllips have proved in [17, p. 349] the validity of the complex inversion formula of the Laplace transform for  0 -semigroups (i.e., (2) with () = 0) on the domain ().In 1995, Yao has proved [18], in Hilbert spaces, the validity of the complex inversion formula of the Laplace transform for  0 -semigroups on .In 1999, Driouich and El-Mennaoui have proved [19] (see also [20,Proposition 3.12.2])that this inversion result remains true on UMD spaces.These results have been extended to strongly continuous cosine families by Cioranescu and Keyantuo in [21] and to strongly continuous resolvent families by Cioranescu and Lizama in [22].Recently, Haase has improved some results in [23] based on Fourier analysis and left as an open problem the corresponding result for convoluted semigroups, which has been solved affirmatively very recently in [24].
Several authors have been investigating the Cauchy problem from the point of view of admissibility of control operators (i.e., (2) with () = 0) in the past and the present [25][26][27][28][29][30][31][32] et al.But the first studies on  2 -admissibility of control operator for Volterra integral scalar systems began with the paper of Jung [33].The idea of treating  2 -admissibility for Volterra integral equations has been exploited in the past years by several authors, for example, [34][35][36].In [33], the notion of finite-time  2 -admissibility for Volterra integral scalar system is linked with finite-time admissibility of the well-studied semigroups' (i.e., () = 0) case for completely positive kernel.Likewise, in [34] infinite-time admissibility for a Volterra scalar system is linked with infinite-time admissibility for semigroups (i.e., () = 0) for a large class of kernels and the result subsumes that of [33].Other results are related to the case where the generator of the underlying semigroup has a Riesz basis of eigenvectors in [36].In [35], the authors have given necessary and sufficient conditions for finite-time  2admissibility of linear Volterra integrodifferential systems (2) when the underlying semigroup is equivalent to a contraction semigroup, which generalizes an analogous result known to hold for the standard Cauchy problem and it subsumes the result in [33].Recently, the authors in [37] have introduced the notion of Favard spaces with respect to resolvent families and have established a relationship between  admissibilities to these Favard spaces.This extends the results obtained for the semigroups' case in [32].Furthermore, it was proved in [37] that, for scalar Volterra integral systems with a creep kernel, finite-and infinite-time  1 -admissibility are equivalent to exponentially stable resolvent family, and if the state space  is reflexive then finite-time and uniformly finite-time  1 -admissibility are equivalent, extending wellknown results for semigroups.
We proceed as follows.In Section 2, we review some wellknown properties of resolvent families for scalar Volterra integral equations and their properties.Section 3 contains the definition of the UMD space and recalls some results on the complex inversion formula for wide classes of families of bounded linear operators on UMD spaces and prove the analogue of [17,Theorem 11.6.2]which is applied in Sections 4 and 5. Our hypotheses on the kernel differ from those considered by [22,23] and can contain a class of completely positive functions (see [35,Example 4.5]).In Section 4, we are concerned with a class of scalar integrodifferential Volterra equations.First we embed this class in a larger Cauchy system, a technique originating in Engel and Nagel [15,VI.7], in order to prove some results concerning the validity of the complex inversion formula.In Section 5, we go back to the study of the admissibility of control operators for Volterra integrodifferential equations (1) in the same spirit of semigroups and we get a new criterion to judge  admissibility ( ∈ [1, ∞[) of control operators in terms of UMD property of its underlying control space.If we set () = 0 then we recover the result in [38] for the semigroups.Note that this paper involves in particular a nonscalar kernel of the form "() = ()" and so a natural question is whether the situation extends when () is a nonscalar kernel.In a forthcoming work, we will consider a class of nonscalar kernels.

Review on Resolvent Families
In this subsection, we collect some elementary facts about scalar Volterra integral equations and resolvent families.These topics have been covered in detail in [6].We refer to these works for reference to the literature and further results.
Let (, ‖ ⋅ ‖  ) be a Banach space; let  be a linear closed densely defined operator in ;  ∈  1 loc (R + ) is a scalar kernel.We consider the linear Volterra integral equation We denote by [()] the domain of  equipped with the graph norm.
We define the convolution product of the scalar function  with a vector-valued function  by Obviously, every strong solution of (3) is a mild solution.Conditions under which mild solutions are strong solutions are studied in [6].
For instance, when () = 1, then () corresponds to a  0semigroup and when () = , then () corresponds to cosine operator function.In particular, when () =  −1 /Γ() with 0 <  ≤ 2, they are the -times resolvent families studied by [39] and correspond to the solution families for fractional evolution equations, that is, evolution equations where the integer derivative with respect to time is replaced by a derivative of fractional order.
The existence of a resolvent family allows one to find the solution for (3).Several properties of resolvent families have been discussed in [6,40].
The following well-known result [6, Proposition 1.1] establishes the relation between well-posedness and existence of a resolvent family.
Note that, contrary to the case of  0 -semigroup, the resolvent for (3) needs not to be exponentially bounded; a counterexample can be found in [6,41].However, there are checkable conditions guaranteeing that (3) possesses an exponentially bounded resolvent operator.
We will use the Laplace transform at times.Suppose  : R + →  is measurable and there exist  > 0 and  ∈ R, such that ‖()‖ ≤   for almost  ≥ 0.Then, the Laplace transform exists for all  ∈ C with Re  > .

The Complex Inversion Formula and UMD Spaces
In this section, we review some results on the complex inversion formula of the Laplace transform, in the strong sense, for wide classes of families of bounded linear operators on UMD spaces.There are several equivalent definitions of a UMD space, one of which involves the so-called unconditional Martingale differences, but we will use a different characterization, due to [42][43][44] involving the vector-valued Hilbert transform (see [45]) for more about UMD spaces.Let  ∈ ]1, ∞[, and define the operator H  on   (R; ) by A Banach space  is called a UMD space (or said to have the UMD-property) if for some (and hence all)  ∈]1, ∞[ (see [44,46]) H := lim ↘0 H   exists in   (R; ) and defines a bounded operator H on   (R; ).The operator H is called the Hilbert transform on   (R; ).Every UMD space is reflexive and its dual is also a UMD space.Typical examples of UMD spaces are   (Ω)-spaces, Sobolev spaces    (Ω), and Besov spaces   , for ,  ∈]1, ∞[ and their closed subspaces.
The natural question that comes in mind is the following: let () ≥0 be the (exponentially bounded) resolvent family International Journal of Differential Equations for (3) on a Banach space , under what conditions does the complex inversion formula lim holds true?For the  0 -semigroups' case (i.e., () = 1) the classical result [17, p. 349] is that one always has the strong convergence (12) if  ∈ ().Recently, the authors in [38,Proposition 4.8] have used the notion of the admissibility and proved in [38,Proposition 4.8] that this inversion takes place on larger spaces than ().In Hilbert setting, the inversion (12) has been generalized to all  ∈  in [18] using Plancherel's Theorem.In the paper [19] Driouich and El-Mennaoui have extended the result in [18] in the case where  has the UMD-property and it has been proved that the UMD property is essential by exhibiting an example for which the inverse Laplace transform does not always converge.This was subsequently generalized from semigroups to solution families for scalar type Volterra integral equations (3) by Cioranescu and Lizama in [22] under some regularity assumptions on the kernel ().In particular, it has been proved in [22,Proposition 2] that for  ∈  1 (R + ) the inversion for solution families for Volterra equations (3) holds on ().On UMD spaces, Haase in [23] has presented new and much shorter proofs of these results (under less strong assumptions on ()), eventually, even generalizing them.His approach uses some elementary Fourier analysis.Recently, this was generalized from resolvent families to convoluted semigroups in [24].This inversion problem will be studied for a class of integrodifferential Volterra equations in the next section.Note that the class of integrodifferential Volterra equations has a "big" intersection with the class of Volterra integral one (partial results had been obtained earlier).Let us signalize that early, and in Hilbert setting, the first result to our knowledge on the inversion formula on () for scalar Volterra integrodifferential equations (3) under the conditions that  ∈  1 (R + ) and both  and   are exponentially bounded is implicitly contained in [8].
for Re  > .

Integrodifferential Equation with Bounded Variation Kernels
The purpose here is to prove some complex inversion of Laplace transform for the resolvent families of the integrodifferential Volterra equations ( 2).Although our hypotheses on kernel and the approach differ from the one considered by [22,23], there is a big overlap in the fundamental results.
For this class, it is known that the integrodifferential equation ( 2) can be converted to an abstract Cauchy problem on a product space (see, e.g., [15,VI.7]).This technique has been widely used (see, e.g., [9,10,12,16,35]).In the following, we restate some notations and related results for the sake of convenience.This idea in the following will be applied to the case of the complex inversion formula for (2).Although it is a special case of the situation considered in the above section, it is worthwhile to deal with the partial inverse formula case first, which generalizes the result from semigroups to solution families for scalar Volterra equations (2) and constitutes an extension of a result in [8].Next, we prove the strong convergence of the complex inversion formula for this class on UMD spaces.
It is easy to see that ( 2) is equivalent to In what follows, we assume that  ∈  1, (R + ) and  generates a  0 -semigroup (()) ≥0 on .Recall that with  ∈  loc (R + ) (the space of functions locally of bounded variation) only, the operator  has to be a generator of  0semigroups to obtain the well-posedness of ( 2), but this condition is not much restrictive, since generation of  is a necessary and sufficient condition for the well-posedness of (21) (which is equivalent to ( 2)) (see [6,Corollary 1.4]).
By virtue of Proposition 5, the Laplace transform of (⋅) 0 is well defined and it is given by for all Re  > max( 0 (),  0 ()) and  0 ∈ .
The following lemma is quite useful.) ,
Using ) . ( Here is finally the result about strong convergence of the complex inversion formula. Proposition 8. Let (()) ≥0 be the exponentially bounded resolvent family for (2) on .Suppose that  is a UMD space.Then there exists ω ≥  0 (), such that for all  ∈  and  > ω one has in , uniformly in  from compact subsets of ]0, ∞[.
We see that the results in [22, Theorem 1], [23, Theorem 4.2], and Proposition 8 assert only strong convergence and uniformity in  from compact subsets of ]0, ∞[.As for the semigroups' case, it would be of interest to see whether, for scalar Volterra integral systems, this convergence holds true for  = 0 (it does not for  ∈ (), (see Proposition 6)).For the semigroups, which are the special cases of scalar Volterra integral equations, an affirmative answer was given in [38,Proposition 2.3].
Proof.The proof is more or less the same for the semigroups' case.Assume first that  ∈ ().Then for all 0 ̸ =  >  and using the resolvent identity we obtain Let C +  be the boundary of the half-disc C +  defined by Since  generates an analytic semigroup on , the set {(1/â(  ))((1/â(  )), )} is uniformly bounded in C +  (see [47]) and, hence, {  (  )} is uniformly bounded in C +  .It follows that ∫ + − () is uniformly bounded in C +  .Using once again the resolvent identity we obtain for all  ∈ ().By the fact that  generates an analytic semigroup and by virtue of Riemann-Lebesgue's theorem (see [20, Theorem 1.8.1.c]),the integral on the right-hand side of the above equality converges to zero as  → ∞.

International Journal of Differential Equations
Finally, combining Propositions 8 and 9 leads to the following corollary.
Corollary 10.Let  be the generator of an analytic  0semigroup in  and let (()) ≥0 be the resolvent family for (2).Suppose that  is a UMD space.Then, for all  ∈  (resp.,  ∈ ()) and  >  * , one has in  (resp., in ()), which is uniform on  for any compact interval of [0, ∞[.

Characterization of Admissibility
We now turn our attention back to the notion of admissibility.
In this section we present sufficient and necessary conditions for the   -admissibility of control operators for integrodifferential Volterra control systems (1) which has the following equivalent form: extending the result for the semigroups [38,Proposition 3.2].
Here we assume that the control operator  ∈ L(;  −1 ) where  −1 is the extrapolation space with respect to  (see, e.g., [15]) and  is another Banach space.It is further assumed that  ∈  1 loc (R + ) is exponentially bounded and that the uncontrolled system (i.e., (21)) admits an exponentially bounded resolvent family (()) ≥0 .
The mild solution of ( 47) is formally given by the variation of the constant formula which is actually the classical solution if  ∈ L(, ),  0 ∈ (), and  is sufficiently smooth.In general however,  is not a bounded operator from  into  and so an additional assumption on  will be needed to ensure that () ∈  for every  0 ∈  and every  ∈   (R + ; ) or   loc (R + ; ).In the same spirit of the semigroups' case, the following are the most natural definitions of the   -admissibility for resolvent families.
(ii)  is called a finite-time   -admissible operator for (()) ≥0 if there exist  0 > 0 and a constant ( 0 ) > 0, Note that the definition of (infinite-time)   -admissible control operator for (() ≥0 ) was introduced in [34] when  = 2 and implies the finite-time  2 -admissibility condition considered in [33].Our definitions of finite-and infinitetime   -admissible control operator for (() ≥0 ) correspond to that of the semigroups and also imply that of [33] when  = 2.It is well known that ( 1 ): finite-time   -admissibility and the uniform finite-time   -admissibility, which means that for all  > 0 there exists a constant () > 0, such that ‖ −1 * ()‖  ≤ ()‖‖   ([0,];) for all  ∈   ([0, ]; ), are equivalent for semigroups and ( 2 ): finite-time  admissibility and the infinite-time   -admissibility are equivalent for exponentially stable semigroups.We emphasize that, in [35], the authors have found an example (i.e., (2) with  generator of exponentially stable semigroup) for which finitetime  2 -admissibility and infinite-time  2 -admissibility are not equivalent, but, in their example, we can see that the associated resolvent family is not exponentially stable.Thus, a question that remains open to our knowledge is, whether for Volterra integral systems, these problems (i.e., ( 1 )-( 2 )) are still true for resolvent families.In [37, Corollary 5.4 and Proposition 5.6], partial answers were given to these problems when  = 1.
It has been observed in [36, for  = 2] (resp., [37, for is equivalent to the fact that there exists a constant  > 0, such that for all  ∈    (R + ; ) (the space of functions in   (R + ; ) with compact support).
Of course,   -admissibility of  guarantees that the operator B ∞ :    (R + ; ) → , given by possesses an extension to a linear bounded operator from   (R + ; ) to .We denote this extension again by B ∞ .

International Journal of Differential Equations 9
Thanks to Proposition 5, the Laplace-transform of  −1 (⋅) is well defined and it is given by ∀ Re () > max ( 0 () ,  0 ()) . (53) In the sequel we use the following notations. with We may now formulate and prove the main result of this section by giving a necessary or/and sufficient condition for finite-(or infinite-) time   -admissibility of .The necessary condition here is essentially based on a geometric property of the underlying control space  that is the UMDproperty.The result encompasses Hilbert control spaces, but the proposition below yields the criterion's necessity.
Proof.Part (i).The proof of this is similar to that of [38,Proposition 3.2].Notice that, for any  ∈ () and constant input  0 ∈ , we have  2  2 (, ) 0 ∈ ().By Proposition 6, we have Thanks to [48,Lemma 1] the integral ∫  0   −1 ()() takes value in  due to the fact that the considered input () is a step function.Thus by a density argument, to prove that  is finite-time   -admissible (for time  0 ) for (()) ≥0 , it suffices to prove that for any step function  : [0,  0 ] →  with compact support that does not contain zero, the following uniform estimate         ∫ holds for some   0 > 0.
So, consider a step function  with compact support that does not contain zero.Then there exists  > 0 such that   ‖ ≤   0 for some   0 > 0 and for all  > 0, then  2  2 (, )  0  is uniformly bounded as  → ∞ (see [48,Corollary 3]) and we have for some ,  0 > 0 and for all  >  0 which completes the proof of (i).Part (ii): (⇒) Assume that  is   -admissible for (()) ≥0 .Let  ∈    (R + ; ), with Supp() = [, ].By Fubini's theorem we have The use of Fubini theorem in this chain of equalities is justified by the fact that the maps (, ) →  Here the new input    is given by and   denotes the Dirichlet kernel given by Following the lines of the proof of [23, Lemma 3.2], one establishes that for a UMD space  and  ∈]1, ∞[ we have Thus, making use of the fact that  is a UMD space, appealing (70), we obtain Then, using the   -admissibility of  and the fact that  0 () < 0, we obtain for  1 >  2 > 0, which implies that (  ()) ≥0 is a Cauchy sequence in .Thus (  ()) ≥0 converges in  as  → ∞, for all  ∈    (R + ; ) and that (  ) ≥0 is uniformly bounded in L(   (R + ; ), ) according to Banach-Steinhaus's theorem.Furthermore, using once again  −⋅    →  −.  =  in   (R + ; ), we obtain ∫ for some  > 0.
Thanks to (i) we deduce that  is finite-time  admissible (for time  0 ) for (()) ≥0 .By examining the proof of (i) we deduce that  is   -admissible for (()) ≥0 .This ends the proof.

Remark 13. (1)
The result of Theorem 12 may extend to systems of Volterra integral equations (3) provided that ∫  0 () takes value in () for all  ∈ .One may of course ask whether it is a severe restriction to consider only () = 1 + (1 * )().This will be a subject of a forthcoming work.