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International Journal of Differential Equations
Volume 2014 (2014), Article ID 948597, 13 pages
On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems
Department of Mathematics, Faculty of Sciences, Ibn Zohr University, BP 8106, 8000 Agadir, Morocco
Received 13 January 2014; Revised 13 April 2014; Accepted 26 April 2014; Published 1 June 2014
Academic Editor: J. M. A. M. van Neerven
Copyright © 2014 Ahmed Fadili and Hamid Bounit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 -semigroup. Next, a necessary and sufficient condition for -admissibility 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).
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: 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 is real-valued and of at most exponential growth. is a (possibly unbounded) linear operator on another Banach space and the control function .
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 on called the resolvent or solution family for (2), where we write ) (see Section 2) and that 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 , Lunardi , and Prüss [5, 6]). Maximal- regularity results are due to Clément and Da Prato  and Prüss . In addition to the classical approach, there is a semigroup approach which was used in, for example, Miller , Chen and Grimmer [9, 10], Desch and Grimmer , Desch and Schappacher , Di Blasio et al. , Nagel and Sinestrari , and Engel and Nagel . 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 .
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 -semigroups (i.e., (2) with ) on the domain . In 1995, Yao has proved , in Hilbert spaces, the validity of the complex inversion formula of the Laplace transform for -semigroups on . In 1999, Driouich and El-Mennaoui have proved  (see also [20, Proposition ]) that this inversion result remains true on UMD spaces. These results have been extended to strongly continuous cosine families by Cioranescu and Keyantuo in  and to strongly continuous resolvent families by Cioranescu and Lizama in . Recently, Haase has improved some results in  based on Fourier analysis and left as an open problem the corresponding result for convoluted semigroups, which has been solved affirmatively very recently in .
Several authors have been investigating the Cauchy problem from the point of view of admissibility of control operators (i.e., (2) with ) in the past and the present [25–32] et al. But the first studies on -admissibility of control operator for Volterra integral scalar systems began with the paper of Jung . The idea of treating -admissibility for Volterra integral equations has been exploited in the past years by several authors, for example, [34–36]. In , the notion of finite-time -admissibility for Volterra integral scalar system is linked with finite-time admissibility of the well-studied semigroups’ (i.e., ) case for completely positive kernel. Likewise, in  infinite-time admissibility for a Volterra scalar system is linked with infinite-time admissibility for semigroups (i.e., ) for a large class of kernels and the result subsumes that of . Other results are related to the case where the generator of the underlying semigroup has a Riesz basis of eigenvectors in . In , the authors have given necessary and sufficient conditions for finite-time -admissibility 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 . Recently, the authors in  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 . Furthermore, it was proved in  that, for scalar Volterra integral systems with a creep kernel, finite- and infinite-time -admissibility are equivalent to exponentially stable resolvent family, and if the state space is reflexive then finite-time and uniformly finite-time -admissibility are equivalent, extending well-known results for semigroups.
We proceed as follows. In Section 2, we review some well-known 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 ] 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 () of control operators in terms of UMD property of its underlying control space. If we set then we recover the result in  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.
2. 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 . 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 ; 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
Definition 2. Equation (3) is called well-posed if, for each , there is a unique strong solution on of and, for a sequence implies in , uniformly on compact intervals.
Definition 3. Let . A strongly continuous family is called resolvent family for (3), if the following three conditions are satisfied: (S1);(S2) commutes with , which means that , for all , and for all and ;(S3)for each and all the resolvent equations hold: Note that the resolvent for (3) is uniquely determined. The proofs of these results and further information on resolvent can be found in the monograph by Prüss . We also notice that the choice of the kernel classifies different families of strongly continuous solution operators in . For instance, when , then corresponds to a -semigroup and when , then corresponds to cosine operator function. In particular, when with , they are the -times resolvent families studied by  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 following well-known result [6, Proposition 1.1] establishes the relation between well-posedness and existence of a resolvent family.
A resolvent family is called exponentially bounded, if there exist and such that for all , and the pair is called type of . The growth bound of is . The resolvent family is called exponentially stable if .
Note that, contrary to the case of -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 is measurable and there exist and , such that for almost . Then, the Laplace transform exists for all with .
The following well-known generation result, stated in , is quite important in this paper. It establishes the relation between resolvent family and Laplace transform.
Proposition 5. Let be -exponentially bounded. Then, (3) admits a resolvent family of type if and only if the following conditions hold: (i) and (the resolvent set of ), for all ; (ii) called the resolvent associated with satisfies
Under these assumptions the Laplace transform of is well-defined and it is given by for all . Note that, for , Proposition 5 becomes the well-known Hille-Yosida theorem.
3. 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 [42–44] involving the vector-valued Hilbert transform (see ) for more about UMD spaces.
Let , and define the operator on by A Banach space is called a UMD space (or said to have the UMD-property) if for some (and hence all) (see [44, 46]) exists in and defines a bounded operator on . The operator is called the Hilbert transform on . Every UMD space is reflexive and its dual is also a UMD space. Typical examples of UMD spaces are -spaces, Sobolev spaces , and Besov spaces for and their closed subspaces.
The natural question that comes in mind is the following: let be the (exponentially bounded) resolvent family for (3) on a Banach space , under what conditions does the complex inversion formula holds true?
For the -semigroups’ case (i.e., ) 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  using Plancherel’s Theorem. In the paper  Driouich and El-Mennaoui have extended the result in  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  under some regularity assumptions on the kernel . In particular, it has been proved in [22, Proposition 2] that for the inversion for solution families for Volterra equations (3) holds on . On UMD spaces, Haase in  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 . 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 and both and are exponentially bounded is implicitly contained in .
We will have the occasion to use the following observation which is the exact generalization of the corresponding well-known case (i.e., the semigroups) proved in [17, Theorem ]. We sketch the proof for the reader’s convenience.
Proposition 6. Let be exponentially bounded and let be the exponentially bounded resolvent family for (3). Then, for each and , we have in , uniformly in from compact subsets of .
Proof. Let and then by the resolvent equation (S3), we have , and by virtue of Proposition 5 we obtain
Thanks to [23, Proposition 2.1], for all we have where the limit exists uniformly with respect to in .
On the other hand, we have It is well-known that the second limit is if but if (see, e.g., ).
Indeed, we known for all and that in , uniformly in from compact subsets of where is the resolvent of the semigroup .
In particular, for , the generator is and , thus we obtain in , uniformly in from compact subsets of .
Then according to the resolvent equation (S1) and (S3) in Definition 3, for all and we obtain
Hence we deduce that for all and , we have in , uniformly in from compact subsets of .
4. 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 . 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 and generates a -semigroup on . Recall that with (the space of functions locally of bounded variation) only, the operator has to be a generator of -semigroups 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]).
From now on, we denote by the resolvent family associated with (21). As in [15, VI.7, Part C], we first introduce the product space , which is a Banach space with the norm Next, we define on the operator matrices where (the left shift semigroup) and are defined on : Then forms a -semigroup on with the generator given by where denotes the generator of the semigroup with domain , the vector-valued Sobolev space, and the Dirac distribution, that is, for each . Finally, define as follows: and denote In [15, VI.7], it is shown that generates a -semigroup on . In the sequel, we denote We now rewrite the Volterra system (2) into an equivalent Cauchy system as follows: Note that it was early observed in  (see also ) that the solutions of (21) are given by , where is the projection mapping to (see [15, VI.7]) for the case and  for , the space of bounded and uniformly continuous functions. That is, for all .
Observe that the resolvent of (21) is exponentially bounded; this extends the result in [35, when ] which has been obtained by a direct method. Indeed, the semigroup is always exponentially bounded (see, e.g., [15, Proposition, 5.5 p.39]); thus there exist constants and such that . Making use of the fact that , we deduce that is also exponentially bounded and that . Thus it is permissible to consider its Laplace transform.
By virtue of Proposition 5, the Laplace transform of is well defined and it is given by for all and .
The following lemma is quite useful.
Lemma 7. (i) For and , we have
(ii) For , we have where is the resolvent of .
Proof. (i) Invoking (28) (see [6, page 339] for more information on ) we have
Thus, using (28) for all , we have
which implies (31).
(ii) For each with , from [15, Lemmas VI.7.23-24], we know that Using we get, since , that is invertible in and by, for example, [31, Lemma A.4.2], we have Finally, a direct computation yields
Here is finally the result about strong convergence of the complex inversion formula.
Proposition 8. Let be the exponentially bounded resolvent family for (2) on . Suppose that is a UMD space. Then there exists , such that for all and one has in , uniformly in from compact subsets of .
Proof. Since is a , the product space is a (see [45, 46]). Applying [19, Theorem 1] to the semigroup , we obtain for all and , where the limit exists uniformly with respect to in . Taking in (40) and by virtue of Lemma 7(i), a straightforward computation shows that for all we have for all , uniformly for .
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 . As for the semigroups’ case, it would be of interest to see whether, for scalar Volterra integral systems, this convergence holds true for (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].
The next result is a version of [38, Proposition 2.3] for resolvent family.
Proposition 9. Let be exponentially bounded and let be the exponentially bounded resolvent family for (3). If generates an analytic -semigroup in , then for all , and the integral converges in as , hence, in for all .
Proof. The proof is more or less the same for the semigroups’ case. Assume first that . Then for all and using the resolvent identity we obtain Let be the boundary of the half-disc defined by . Then Cauchy’s theorem yields Since generates an analytic semigroup on , the set is uniformly bounded in (see ) and, hence, is uniformly bounded in . It follows that is uniformly bounded in . 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 ]), the integral on the right-hand side of the above equality converges to zero as . Using the last equality, the density of in , and the fact that the integral is uniformly bounded in , an elementary equicontinuity argument guarantees the convergence in for every thanks to Proposition 6. Now, let and for some . Then we get Since , we conclude that converges in .
Corollary 10. Let be the generator of an analytic -semigroup in and let 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 .
5. 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 where is the extrapolation space with respect to (see, e.g., ) and is another Banach space. It is further assumed that is exponentially bounded and that the uncontrolled system (i.e., (21)) admits an exponentially bounded resolvent family .
Since the resolvent of (21) commutes with the operator , then it can be easily seen that the restriction to of , the solution of (21), is strongly continuous. Moreover, since (see Proposition 5) solves for each and replacing . Likewise, has a unique bounded extension to for each and is also strongly continuous, and it solves (21) in with replacing .
The mild solution of (47) is formally given by the variation of the constant formula which is actually the classical solution if , , 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 and every or .
In the same spirit of the semigroups’ case, the following are the most natural definitions of the -admissibility for resolvent families.
Definition 11. Let and .(i) is called (infinite-time) an -admissible operator for , if there exists a constant , such that (ii) is called a finite-time -admissible operator for if there exist and a constant , such that
Note that the definition of (infinite-time) -admissible control operator for was introduced in  when and implies the finite-time -admissibility condition considered in . Our definitions of finite- and infinite-time -admissible control operator for correspond to that of the semigroups and also imply that of  when . It is well known that : finite-time -admissibility and the uniform finite-time -admissibility, which means that for all there exists a constant , such that for all , are equivalent for semigroups and : finite-time -admissibility and the infinite-time -admissibility are equivalent for exponentially stable semigroups. We emphasize that, in , the authors have found an example (i.e., (2) with generator of exponentially stable semigroup) for which finite-time -admissibility and infinite-time -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., - are still true for resolvent families. In [37, Corollary 5.4 and Proposition 5.6], partial answers were given to these problems when .
It has been observed in [36, for ] (resp., [37, for ]) that (resp., ) admissibility of is equivalent to the fact that there exists a constant , such that for all (the space of functions in with compact support).
Of course, -admissibility of guarantees that the operator , given by possesses an extension to a linear bounded operator from to . We denote this extension again by .
Thanks to Proposition 5, the Laplace-transform of is well defined and it is given by In the sequel we use the following notations.
For , , , and we set 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 UMD-property. The result encompasses Hilbert control spaces, but the proposition below yields the criterion’s necessity.
Theorem 12. Let . Then the following assertions hold. (i)If is uniformly bounded on for some then is finite-time -admissible for .(ii)Assume that is a UMD space, , and . Then is -admissible for if and only if is uniformly bounded on .
Proof. Part (i). The proof of this is similar to that of [38, Proposition 3.2]. Notice that, for any and constant input , we have . By Proposition 6, we have
Thanks to [48, Lemma 1] the integral 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 for , it suffices to prove that for any step function with compact support that does not contain zero, the following uniform estimate
holds for some .
So, consider a step function with compact support that does not contain zero. Then there exists such that Since is uniformly bounded with for some and for all , then is uniformly bounded as (see [48, Corollary 3]) and we have for some and for all Invoking (56), an elementary equicontinuity argument, and vector-valued dominated convergence theorem in , respectively, we obtain which implies via (59) that for all In other words, we have due to the fact that commute with (see [48, Theorem 7]).
Passing to the limit (i.e., ) in the above inequality, we deduce The same argument also shows that which implies that the sequence is a Cauchy sequence in . Moreover, By virtue of , and the fact that satisfies (63), we find which completes the proof of (i).
Part (ii): Assume that is -admissible for . Let , with . By Fubini’s theorem we have The use of Fubini theorem in this chain of equalities is justified by the fact that the maps and belong to , and to , respectively.
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 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 , we obtain for , which implies that is a Cauchy sequence in . Thus converges in as , for all and that is uniformly bounded in according to Banach-Steinhaus’s theorem. Furthermore, using once again in , we obtain converges to in as . By virtue of with continuous injection and the fact that converges in as , we deduce that converges to in as , for all .
Assume that is uniformly bounded on . For and we have and for some .
Thanks to (i) we deduce that is finite-time -admissible (for time ) for . By examining the proof of (i) we deduce that is -admissible for . This ends the proof.
Remark 13. (1) The result of Theorem 12 may extend to systems of Volterra integral equations (3) provided that takes value in for all . One may of course ask whether it is a severe restriction to consider only . This will be a subject of a forthcoming work.
(2) We can retrieve the result stated in Proposition 8 by using [38, Remark 3.5 (ii)]. Indeed, let , then is infinite-time -admissible for with . In fact, for and , consider the input , we set: with Thanks to [38, Remark 3.5] we have with and as for all .
Passing to the limit in (76) as and invoking [38, Corollary 4.2] we obtain with In other words, for we have Using Lemma 7(i) we obtain