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Approximate controllability and optimal controls of fractional evolution systems in abstract spaces
Advances in Difference Equations volume 2014, Article number: 322 (2014)
In this paper, under the assumption that the corresponding linear system is approximately controllable, we obtain the approximate controllability of semilinear fractional evolution systems in Hilbert spaces. The approximate controllability results are proved by means of the Hölder inequality, the Banach contraction mapping principle, and the Schauder fixed point theorem. We also discuss the existence of optimal controls for semilinear fractional controlled systems. Finally, an example is also given to illustrate the applications of the main results.
MSC:26A33, 49J15, 49K27, 93B05, 93C25.
During the past few decades, fractional differential equations have proved to be valuable tools in the modeling of many phenomena in viscoelasticity, electrochemistry, control, porous media, and electromagnetism, etc. Due to its tremendous scopes and applications, several monographs have been devoted to the study of fractional differential equations; see the monographs [1–5]. Controllability is a mathematical problem. Since approximately controllable systems are considered to be more prevalent and very often approximate controllability is completely adequate in applications, a considerable interest has been shown in approximate controllability of control systems consisting of a linear and a nonlinear part [6–10]. In addition, the problems associated with optimal controls for fractional systems in abstract spaces have been widely discussed [10–22]. Wang and Wei  obtained the existence and uniqueness of the PC-mild solution for one order nonlinear integro-differential impulsive differential equations with nonlocal conditions. Bragdi  established exact controllability results for a class of nonlocal quasilinear differential inclusions of fractional order in a Banach space. Machado et al.  considered the exact controllability for one order abstract impulsive mixed point-type functional integro-differential equations with finite delay in a Banach space. Approximate controllability for one order nonlinear evolution equations with monotone operators was attained in . By the well-known monotone iterative technique, Mu and Li  obtained existence and uniqueness results for fractional evolution equations without mixed type operators in nonlinearity.
Wang and Zhou  studied a class of fractional evolution equations of the following type:
where is the Caputo fractional derivative of order , −A is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators. A suitable α-mild solution of the semilinear fractional evolution equations is given, and the existence and uniqueness of α-mild solutions are also proved. Then by inducing a control term, the existence of an optimal pair of systems governed by a class of fractional evolution equations is also presented.
Mahmudov and Zorlu  considered the following semilinear fractional evolution system:
where is the Caputo fractional derivative of order , the state variable x takes values in a Hilbert space , A is the infinitesimal generator of a -semigroup of bounded operators on the Hilbert space , the control function u is given in , U is a Hilbert space, B is a bounded linear operator from U into . is a Volterra integral operator. They studied the approximate controllability of the above controlled system described by semilinear fractional integro-differential evolution equation by the Schauder fixed point theorem. Very recently, Wang et al.  researched nonlocal problems for fractional integro-differential equations via fractional operators and optimal controls, and they obtained the existence of mild solutions and the existence of optimal pairs of systems governed by fractional integro-differential equations with nonlocal conditions. Subsequently, Ganesh et al.  presented the approximate controllability results for fractional integro-differential equations studied in .
In this paper, we concern the following fractional semilinear integro-differential evolution equation with nonlocal initial conditions:
where denotes the Caputo derivative, , the state variable x takes values in a Hilbert space with the norm , is the infinitesimal generator of a -semigroup of uniformly bounded linear operators, that is, there exists such that for all , , . We denote by a Hilbert space of equipped with norm for all , which is equivalent to the graph norm of , . The control function u is given in , U is a Hilbert space, B is a bounded linear operator from U into . The Volterra integral operator H is defined by . The nonlinear term f and the nonlocal term g will be specified later.
Here, it should be emphasized that no one has investigated the approximate controllability and further the existence of optimal controls for the fractional evolution system (1.1) in a Hilbert space, and this is the main motivation of this paper. The main objective of this paper is to derive sufficient conditions for approximate controllability and existence of optimal controls for the abstract fractional equation (1.1). The considered system (1.1) is of a more general form, with a coefficient function in front of the nonlinear term. Finally, an example is also given to illustrate the applications of the theory. The previously reported results in [6, 15, 20] are only the special cases of our research.
The rest of this paper is organized as follows. In Section 2, we present some necessary preliminaries and lemmas. In Section 3, we prove the approximate controllability for the system (1.1). In Section 4, we study the existence of optimal controls for the Bolza problem. At last, an example is given to demonstrate the effectiveness of the main results in Section 5.
2 Preliminaries and lemmas
Unless otherwise specified, represents the norm, , is a Banach space equipped with supnorm given by for .
Let , here is the resolvent set of A. Define
It follows that each is an injective continuous endomorphism of . So we can define , which is a closed bijective linear operator in . It can be shown that has a dense domain and for . Moreover, , with , where , I is the identity in . We have for (with ), and the embedding is continuous. Moreover, has the following basic properties.
Lemma 2.1 (see )
and have the following properties:
, for each and .
, for each and .
For every , is bounded in , and there exists such that(2.2)
is a bounded linear operator for , and there exists such that .
Definition 2.1 The fractional integral of order q with the lower limit zero for a function f is defined as
provided that the right side is point-wise defined on , where is the gamma function.
Definition 2.2 The Riemann-Liouville derivative of the order q with the lower limit zero for a function can be written as
Definition 2.3 The Caputo derivative of the order q for a function can be written as
If , then(2.6)
The Caputo derivative of a constant equals zero.
If f is an abstract function with values in , then the integrals which appear in Definitions 2.1, 2.2, and 2.3 are taken in Bochner’s sense.
Definition 2.4 A solution is said to be a mild solution of the system (1.1), we mean that for any , the following integral equation holds:
is a probability density function defined on , that is,
Definition 2.5 The system (1.1) is said to be approximately controllable on if , that is, given an arbitrary , it is possible to steer from the point to within a distance for all points in the state space at time b. Here , is called the reachable set of the system (1.1) at terminal time b, is the state value at terminal time b corresponding to the control u and the initial value , represents its closure in .
where denotes the adjoint of B and is the adjoint of . Obviously, is a linear bounded operator. We define the following linear fractional control system:
Lemma 2.2 (see )
The linear fractional control system (2.14) is approximately controllable on if and only if as in the strong operator topology.
Lemma 2.3 (see )
The operators and have the following properties:
For fixed , and are linear and bounded operators, that is, for any ,(2.15)
and are strongly continuous.
For every , and are also compact if is compact.
For any , and , we have(2.16)
For fixed and any , we have(2.17)
and are uniformly continuous, that is, for each fixed and , there exists such that(2.18)
Lemma 2.4 (see )
For and we have .
Lemma 2.5 (Schauder’s fixed point theorem)
If B is a closed bounded and convex subset of a Banach space and is completely continuous, then Q has a fixed point in B.
3 Approximate controllability
In this section, we impose the following assumptions:
(H1) is continuous and there exist such that
for all , , and .
(H2) , there exists a function and
for each and , where .
(H3) is continuous and there exists a constant such that
for any .
(H4) The function defined by
satisfies for all , where .
Theorem 3.1 Assume that conditions (H1)-(H4) are satisfied. In addition, the functions f and g are bounded and the linear system (2.14) is approximately controllable on . Then the fractional system (1.1) is approximately controllable on .
Proof For arbitrary , define a control function as follows:
and define the operator by
Obviously is well defined on .
For , we have
By (H1)-(H3), Lemma 2.3 and the Hölder inequality, we have and
Then we can deduce that
From (H4) and the contraction mapping principle, we conclude that the operator has a fixed point in . Since f and g are bounded, for definiteness and without loss of generality, let be a fixed point of in , where . From the boundedness of , there is a subsequence denoted by which converges weakly to x as , and as . Then . Any fixed point is a mild solution of (1.1) under the control
Therefore we have
it follows that
By assumptions (H1)-(H3), it is easy to get as . Then
This proves the approximate controllability of (1.1). □
In order to obtain approximate controllability results by the Schauder fixed point theorem, we pose the following conditions:
(H5) is a compact analytic semigroup in .
(H6) There exist constants such that and f satisfies:
For each , the function is measurable.
For each , the function is continuous.
For any , there exist functions such that(3.18)
and there exists a constant such that
where has been specified in assumption (H2).
(H7) is completely continuous. For any , there exist constants such that
and there exists a constant such that
(H8) The following inequality holds:
where will be specified in the following theorem.
Theorem 3.2 Assume that conditions (H3), (H5)-(H8) are satisfied. In addition, the linear system (2.14) is approximately controllable on . Then the fractional system (1.1) is approximately controllable on .
Proof For , we set . For arbitrary , define the control function as follows:
and define the operator by
We divide the proof into five steps.
Step 1: maps bounded sets into bounded sets, that is, for arbitrary , there is a positive constant such that .
Let , from (2.12), (2.13), and (3.23), we have
Then we get
If operator is not bounded, for each , there would exist and such that
Dividing both sides by r and taking the lower as , we have
which is a contradiction to (H8). Then maps bounded sets into bounded sets.
Step 2. is continuous.
Let and as . From assumptions (H6)-(H7), for each , we have
By the Lebesgue dominated convergence theorem, for each , we get
which implies that is continuous.
Step 3. For each , the set is relatively compact in .
The case is trivial, is compact in (see (H7)). So let be a fixed real number, and let h be given a real number satisfied . For any , define ,
Since is compact in and is bounded on , then the set is a relatively compact set in . On the other hand,
This implies that there are relatively compact sets arbitrarily close to the set for each . Then , is relatively compact in . Since it is compact at , we have the relatively compactness of in for all .
Step 4. is an equicontinuous family of functions on .
Form the Hölder inequality, Lemmas 2.1, 2.3, and assumption (H6), we obtain
From Lemma 2.4, we have
By (3.25), it is easy to see that
Similar to (3.39), we obtain
For , , it can easily be seen that . For , when is small enough, we have
Since we have assumption (H5), , in t is continuous in the uniformly operator topology, it can easily be seen that and tend to zero independently of as , . It is clear that , , as . Then is equicontinuous and bounded. By the Ascoli-Arzela theorem, is relatively compact in . Hence is a completely continuous operator. From the Schauder fixed point theorem, has a fixed point, that is, the fractional control system (1.1) has a mild solution on .
Step 5. Similar to the proof in Theorem 3.1, it is easy to show that the semilinear fractional system (1.1) is approximately controllable on .
Since the nonlinear term f is bounded, for any , there exists a constant such that
Consequently, the sequence is bounded in , then there is a subsequence denoted by , which converges weakly to in .
It follows that
Now, by the compactness of the operator and (H7), it is easy to get as . Then
This proves the approximate controllability of (1.1). The proof is completed. □
4 Optimal controls
In this section, we assume that is another separable reflexive Banach space from which the controls u take the values. We define the admissible control set , , where the multifunction is measurable, represents a class of nonempty closed and convex subsets of , and , E is a bounded set of .
We consider the following controlled system:
where , , it is easy to see that for all .
Let denote a mild solution of the system (4.1) corresponding to . Here we consider the Bolza problem (P), which means that we shall find an optimal pair such that
We list here some suitable hypotheses on the operator C, ϕ, and l as follows:
The functional is Borel measurable.
is sequentially lower semicontinuous on for almost all .
is convex on for each and almost all .
There exist , , and such that .
The functional is continuous and nonnegative.
The following inequality holds:(4.4)
Theorem 4.1 Assume that assumptions (H3), (H5)-(H7), and (HL) are satisfied. Then the Bolza problem (P) admits at least one optimal pair on provided that
Proof Firstly, we show that the system (4.1) has a mild solution corresponding to u given by the following integral equation:
From Lemmas 2.1, 2.3, and (3.11), we have
where is the norm of Banach space . Meanwhile, assumptions (H5)-(H7) and (HL) are satisfied. Similar to the proof of Theorem 3.2, we can verify that the system (4.1) has a mild solution corresponding to u easily.
Secondly, we discuss the existence of optimal controls. If , there is nothing to prove. We assume that . Using condition (HL), we know
here , is a constant.
By the definition of an infimum, there exists a minimizing sequence of the feasible pair , where , such that as . Since , is bounded in , there exists a subsequence, relabeled as , and satisfies
in . Since is closed and convex, by the Marzur lemma, we have .
Let be a mild solution of the system (4.1) corresponding to (), then satisfies the following integral equation:
Noting that is a bounded continuous operator from I into , we have
Furthermore, is bounded in , so there exists a subsequence, relabeled such that
We denote the operators and by
Since is bounded, similar to the proof of Theorem 3.2, it is easy to see that is bounded. It is easy to verify that is compact and equicontinuous in . Due to the Ascoli-Arzela theorem, is relatively compact in . Obviously, is linear and continuous, then is a strongly continuous operator, and we obtain
Similarly, we can verify is a strongly continuous operator and
Now, we turn to considering the following controlled system:
By Theorem 3.2, the above system has a mild solution corresponding to , and
From Lemma 2.3 and (H3), we obtain
Similarly, we have
By (4.15), (4.16), and the Lebesgue dominated convergence theorem, we can deduce that as .
For each , , we have
Noting (4.5), we get
Furthermore, we can infer that
Using the uniqueness of the limit, we have
which is just a mild solution of the system (4.1) corresponding to .
Since , using assumption (HL) and the Balder theorem, we have
This implies that J attains its minimum at . □
Example 5.1 Consider optimal controls for the following fractional controlled system:
with the cost function
where , , , and .
Let . The operator is defined by with , then A generates a compact, analytic semigroup of uniformly bounded linear operator. Clearly, assumption (H5) is satisfied. Moreover, the eigenvalues of A are and the corresponding normalized eigenvectors are , .
Define the control function such that . It means that going from into is measurable. Set where . We restrict the admissible controls to be all the such that .
Let , where and the operator is given by
for each and .
Suppose that is a Banach space equipped with the supnorm , , . Define by
where is defined by
Obviously, we have
The system (5.1) can be transformed into
with the cost function
and we can verify (HL)(1)-(5) are satisfied. It is also not difficult to see that
then there exists and such that (3.19) holds, and conditions (H6) is satisfied. Meanwhile, it comes from the example in  that g is a completely continuous operator from and there exist constants and such that
Let , , it is easy to verify that (H3) and (H7) hold. Since , condition (H8) and condition (HL)(6) are satisfied automatically. By Theorem 4.1, we can conclude that the system (5.1) has at least one optimal pair while the condition holds.
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