The Hautus-Type Inequality for Abstract Fractional Cauchy Problems and Its Observability

Chen-Yu, Li

doi:https://doi.org/10.1155/2024/6179980

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Abstract Introduction Preliminaries Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 6179980 | https://doi.org/10.1155/2024/6179980

The Hautus-Type Inequality for Abstract Fractional Cauchy Problems and Its Observability

Li Chen-Yu¹

Academic Editor: Akbar Ali

Received13 Feb 2023

Revised07 Mar 2024

Accepted14 Mar 2024

Published23 Mar 2024

Abstract

In this paper, we investigate the observability of the fractional resolvent family, and we prove two main results: the first result shows a generalization of the Hautus-type test for observable exponentially stable semigroups to the fractional resolvent family and the second result shows the equivalence of the observability and the below boundedness of the linear operator on the wave packet when the generator conforms to a specific form.

1. Introduction

In this paper, we mainly investigate the observability of the following fractional differential equation in Hilbert space: is a closed densely defined linear operator on a Banach space X. Our main results can be listed as follows: First, we prove that the exact observability in finite time can deduce a Hautus type test with a constant depending on , and the assumptions are weaker than the classical one, for example [1], Theorem 6.5.3. Next, we show that if the resolvent is stable and observation operator is commuted with coefficient operator , then the Hautus type inequality is enough to prove the approximate observability. Second, we assume that , where is a negative self-adjoint operator and is a constant small enough. We prove that for such an operator , the exact observability of resolvent generated by can be deduced from the Hautus type test, and a perturbation result is given as a corollary. Finally, we show that if has a compact resolvent, then we can use the conditions satisfied by the spectrum of the operator to fully characterize its observability.

The properties of the fractional resolvent family have been studied very intensively over the past few years; for example, many stability, regularity, and continuity results concerning the fractional resolvent family can be found in [2] and references therein. Also, many pieces of literature on this topic provide applications for Caputo fractional calculus. In [3], authors investigate the stability of fractional evolution systems with memory. In [4–8], authors also did some quantitative analysis related to Caputo fractional calculus, and for many other results about fractional differential equations, we refer to [9–14].

The observability of semigroup is a classical topic, and there is a lot of literature on this subject; for example [1, 15], these two pieces of literature give extensive results on the observability of semigroups: the first one focuses on the elementary introduction and classical results and gives many applications and relations to wave equation or Schröndinger’s equation, see also [16], while the second one provides many ideals and open problems about this subject. Also, there are some works about observability not only on general Hilbert space but also on some measurable sets, for example [7, 17]. In addition to studying the subject itself, this concept is also a powerful tool for proving the stability of semigroups; for example, in the literature [18], the authors proved that the semigroup generated by is strongly polynomially stable by assuming that the operators satisfy a nonuniform Hatus-type test. In the literature [19], the observability and stabilization of magnetic Schrödinger equations have been investigated. If we concentrate on certain equations, observability can also be proved, for example [20].

Fractional order differential equations, and evolution systems containing fractional order differentials, are very common in nature, so the question of controllability and observability of this class of equations is also a matter of interest, and this is the motivation of this paper. Comparatively, the observability of fractional resolvent families has been much less studied, and many kinds of literature concentrate on fractional differential equations with boundary value conditions such as [14, 21]. The main difficulty in dealing with the fractional resolvent family is that the fractional derivative does not satisfy the chain rule and the fractional integral operator is a nonlocal operator; therefore, we use a different approach when dealing with fractional resolvent families, and the results are slightly different from the semigroup case.

This paper is organized as follows: in Section 2, some necessary definitions and results on observability, Mittag–Leffler functions, and fractional resolvent families are given. Section 3 is devoted to proving a Hautus-type necessary condition for exact observability. Also, Section 4 deals with the resolvent family generated by operator which conforms to a specific form and characterizes the observability by the spectrum of .

2. Preliminaries

In this paper, means the complex plane, the real line, and the set of all integers, respectively. For every complex number , being its real, imaginary part and being its angle. are two constants that may change from line to line, and the constants , vary with and . In all cases, we assume that and are Hilbert spaces and that is a densely defined, closed, linear operator on with , , and being its kernel, domain, and range, respectively. and signify the spectrum and resolvents set of , respectively, and is a linear bounded operator map to . Regularly, stands for the convolution on :and denote the Laplace transform of an exponentially bounded function , defined byif this integral is convergent. Also, denote the Fourier transform of a function , defined byif is defined on , then we extend it to by zero and still denote by if no confusion occurs.

The sector is defined asfor and ]; if , then we write for convenience.

We now define the observability of the fractional resolvent family.

Definition 1. If generates a bounded fractional resolvent family , then the pair is exactly observable in time , and there is a constant such thatThe pair is approximately observable in time if

Definition 2. Let . The pair is exactly (or approximately) observable in infinite time if is bounded from below (or ).

Then, we recall the Mittag–Leffler function, which plays an important role in studying fractional differential equations. The properties and applications of this function can be found in the [13, 22, 23].

The Mittag–Leffler function is defined bywhere , is the Hankel contour which starts and ends at and encircles the disc counterclockwise. We write if there is no confusion. The Mittag–Leffler function satisfies the fractional differential equation.where is the Caputo derivative of -order (see [13, 24, 25]). The most useful properties of this function are the following integral:and their asymptotic expansion for and :whereas and be a integer bigger than 1. From the asymptotic expansion, one knows that as when .

Next, we define fractional resolvent families and list some properties here.

Definition 3 (see [24]) [Definition 3]. Let , a family is called an -times resolvent family generated by if the following conditions are satisfied:(a) is strongly continuous for and (b) for ; that is, and for (c)for , the resolvent equation holds for all , where .

It is well known that the -times resolvent family is uniquely determined by its generator . If generates an -times resolvent family , then the solution to the abstract fractional Cauchy problem (1) is given by .

Definition 4. An -times resolvent family is said to be exponentially bounded if there exists a constant and such that for every . is called bounded if can be taken as 0, i.e., for all .
Let ], an -times resolvent family is called the analytic of angle if admits an analytic extension to the sectorial sector . An analytic -times resolvent family is called bounded if is uniformly bounded for for any .

Lemma 5 (see [24]). Theorems 2.8 and 2.9. Let . Then, generates an -times resolvent family satisfying for every if and only if and

In this case, andfor every . In particular, if is bounded, then .

3. A Hautus-Type Necessary Condition for Exact Observability

The Hautus-type necessary condition for exponential stable semigroup can be found in many works of literature, such as [1], but the conditions we assume here are different.

Theorem 6. If generates a bounded fractional resolvent family ,

If pair is exactly observable in time , then for every and every ,where and .

Proof. Since pair is exactly observable in time , we only need to prove thatWe choose and , we denotethen we haveand then we conclude thatwhere .
Then, we haveBy using the property of convolutions, we haveSo if we denote and , then we have

Example 1. (1) Take , , . Then, it is well known that generates a stable -semigroup ,Thus, it is easy to see that generates a stable -times fractional resolvent family ,where is the Wright-type function, which can be found in many pieces of literature, such as [11, 22, 24].
Now, let , the andSo we haveThus, is exactly observable in time , . Then, by this theorem, we deduce that for every , the following equation is valid:

By using Theorem 6, we can prove the following result.

Proposition 7. If generates a stable fractional resolvent family with , there exists a constant such that for every ,

If commute with and equation (18) holds for some , then pair is approximately observable in infinite time.

Proof. Since and commute, we deduce thatThen, we denote , of course, is well-defined, and we know that is an invariant space of , and we need to prove that . Let be the restriction of on which also be a fractional resolvent family with generator , the restriction of on . Then, it is easy to prove thatNow, suppose we have equation (18) with constants and , then, for every and every ,or equivalently, for every ,It can be calculated directly that for every given , is uniformly bounded for . For , the uniform boundedness of can be deduced from the property of fractional resolvent family with estimate (31), and then, we obtain that is uniformly bounded for , then by [1], Lemma 6.5.5, .

Proposition 8. If pair is exactly observable in infinite time and is stable with rate and , then it is exactly observable in time for big enough.

Proof. For and , we haveSince there are constants , such thatThen, equation (36) readsIf big enough such that , for such , pair is exactly observable in time .

4. Hautus-Type Tests for Exact Observability with a Special Generator

In this section, we focus on the fractional resolvent family generator of the following form:where is a negative self-adjoint operator and satisfies the following condition:where is the set consisting of all zeros of . Then, generates a bounded fractional resolvent family , and for every , we havewhere is the resolution of identity corresponding to .

This can be checked directly by the Laplace transform, dominant convergence theorem, and asymptotic behavior of the Mittag–Leffler function. More details about the resolution of identity can be found in [26], here we only list one of them.

Lemma 9 (see [26]). Theorem 13.24. Let be a resolution of identity on a set , then every measurable function corresponds a densely defined, closed operator with domain such that:

By using this lemma, estimate (11), and equation (40), we deduce that

Moreover, consider the operator for some , then generates a fractional resolvent family and

Theorem 10. If there exist constants and such that

If and , then pair is exactly observable in infinite time.

Proof. Let be the fractional resolvent family generated by , then, for every , we haveSince , thenthen by equation (46), we haveThen, Plancherel theorem implies thatthus,Combining equations (44) and (45) and , we havefor some constant . This completes the proof.

Remark 11. These results are also valid if is a normal operator with and for all and .

It should be noted that if a pair is exactly observable in time , then equation (18) is satisfied with constant and , and if and , then both and are bounded and then equation (46) is satisfied. So we conclude then the following corollary.

Corollary 12. Suppose small enough and . If pair is exactly observable in time , then pair is exactly observable in infinite time.

If operator has compact resolvent, then we can use spectral conditions to inscribe the observability of the fractional resolvent family.

Since is self-adjoint and has compact resolvent, then we denote by an orthonormal basis consisting of eigenvectors of and by , and the corresponding eigenvalues of and index set are a subset of .

For and , set

We call an element is a of of parameters and if can be represented as

We first give the following proposition.

Proposition 13. The following conditions are equivalent:(1) for all (2)The pair is approximately observable in infinite time

Proof. The proof of this proposition is the same as [1], Proposition 6.9.1, so we omit it here.
The following lemma is a direct extension of the [1], Lemma 6.9.4.

Lemma 14. For each and , we define the subspace ,

Let be the part of in . If is the nonincreasing functionthen

Now, we prove another main result of this section.

Theorem 15. If and small enough, , then the following statements are equivalent:(1)There exist such that for all and for every wave packet of of parameters and , denoted by , we have(2)Pair is exactly observable in time

Remark 16. Although Theorem 10 does not assume , we need this assumption to ensure the function is well-defined.

Proof. First, we show that (2) implies (1). Assume that pair is exactly observable in time , then by equation (18) we havethen choose such that and , we havethenThis proves the claim.
Next, we show that (1) implies (2). We first prove that (1) implies equation (46), then use Theorem 10 and Proposition 8 to get the desired result.
Take and represent it on the basis , denote by . Take , and such that (1) holds and decompose , whereThen, we haveBy using elementary inequality, we have, for every ,then we haveThen, by using Lemma 14 and (1), we haveSince is the restriction of in , thenCombining these equations, we havesince , choose and big enough, we haveThis completes the proof.

5. Discussion

This paper focuses on the observability of fractional differential equations, giving some sufficient and necessary conditions for determining that an equation is observable. In particular, we apply the Hautus-type inequality for the classical case to fractional differential equations, giving an analogous inequality. When the space is a Hilbert space and the operator is a self-adjoint operator, we give a condition that does not depend on the estimate of the resolvent of the operator, which is used to characterize the observability of the equations.

Since this paper focuses on abstract fractional Cauchy problems in general Banach spaces, the methods that are generally used to consider observability are not applicable, for example, considering the dual operator and using the controllability of dual problems to study observability [1]. Because abstract resolvent families have quite a few common features, we can use this approach in the future to study other problems, such as building similar Hautus-type inequalities for general k-regularized resolvent operator families, or even more general (b, l)-regularized resolvent operator families (see definitions of these operator families at [27] and reference therein). And because the study of the observability of equations now focuses on parabolic equations [7], using the resolvent family [28–30] to analyze the equations may provide a way to study nonparabolic equations.

Data Availability

No underlying data was collected or produced in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was supported by the NSF of China (no. 11971327).

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Copyright

Copyright © 2024 Li Chen-Yu. 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.

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