In this section, we impose the following assumptions:
(H1) is continuous and there exist such that
(3.1)
for all , , and .
(H2) , there exists a function and
such that
(3.2)
for each and , where .
(H3) is continuous and there exists a constant such that
(3.3)
for any .
(H4) The function defined by
(3.4)
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:
(3.5)
and define the operator by
(3.6)
Obviously is well defined on .
For , we have
(3.7)
By (H1)-(H3), Lemma 2.3 and the Hölder inequality, we have and
(3.8)
(3.9)
Then we can deduce that
(3.10)
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
(3.11)
Then
(3.12)
where
(3.13)
Therefore we have
(3.14)
Define
(3.15)
it follows that
(3.16)
By assumptions (H1)-(H3), it is easy to get as . Then
(3.17)
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:
-
(1)
For each , the function is measurable.
-
(2)
For each , the function is continuous.
-
(3)
For any , there exist functions such that
(3.18)
and there exists a constant such that
(3.19)
where has been specified in assumption (H2).
(H7) is completely continuous. For any , there exist constants such that
(3.20)
and there exists a constant such that
(3.21)
(H8) The following inequality holds:
(3.22)
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:
(3.23)
and define the operator by
(3.24)
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
(3.25)
where
(3.26)
(3.27)
Then we get
(3.28)
If operator is not bounded, for each , there would exist and such that
(3.29)
Dividing both sides by r and taking the lower as , we have
(3.30)
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
(3.31)
(3.32)
By the Lebesgue dominated convergence theorem, for each , we get
(3.33)
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 ,
(3.34)
Since is compact in and is bounded on , then the set is a relatively compact set in . On the other hand,
(3.35)
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 .
For ,
(3.36)
Form the Hölder inequality, Lemmas 2.1, 2.3, and assumption (H6), we obtain
(3.37)
From Lemma 2.4, we have
(3.38)
By (3.25), it is easy to see that
(3.39)
Similar to (3.39), we obtain
(3.40)
For , , it can easily be seen that . For , when is small enough, we have
(3.41)
and
(3.42)
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
(3.43)
Consequently, the sequence is bounded in , then there is a subsequence denoted by , which converges weakly to in .
It follows that
(3.44)
Now, by the compactness of the operator and (H7), it is easy to get as . Then
(3.45)
This proves the approximate controllability of (1.1). The proof is completed. □