1. Introduction
Control theory plays a vital role in mathematical exploration, serving as a foundation for system optimization and stability analysis [
1,
2,
3]. In the past few years, numerous academics have conducted research on the controllability of diverse dynamical systems utilizing a range of methodologies [
4,
5,
6,
7,
8]. Exact controllability means that the system can accurately reach the target state through deterministic control, while approximate controllability means that the system can approach the target state through appropriate random control. The control theory of stochastic differential equations plays an important role in risk management, stock trading, weather forecasting, disease control, etc., which can improve the quality and effectiveness of decision-making and reduce risks and costs [
9,
10,
11].
Compared with integer derivatives, fractional derivatives have wider applicability, more complete descriptive power, better disclosure of non-local properties, and more mathematical and physical applications. Therefore, the control theory of fractional stochastic differential equations has been garnering increasing attention from researchers. In [
12], Sakthivel et al. studied the approximate controllability of the Caputo FSEEs via the fixed point theorem. In [
13], Shu et al. studied the approximate controllability of the Riemann–Liouville FSEEs with order
by using the concepts related to sectorial operators and Mönch′s fixed point theorem. For further research on the approximate controllability of fractional differential equations, we recommend consulting [
14,
15,
16,
17,
18]. Ding and Li in [
19], studied the exact controllability of the Caputo FSEEs with order
by using measure of noncompactness and Mönch′s fixed-point theorem. For research achievements related to the exact controllability of fractional differential equations, we recommend readers refer to [
20,
21,
22].
The Hilfer fractional derivative can be regarded as a synthesis or extension of the Riemann–Liouville fractional derivative and the Caputo fractional derivative [
23]. When studying Hilfer fractional systems, we face a problem: their equivalent integral equations make sense only on open intervals. This limits our analysis, especially when trying to use the fixed-point theorem and Ascoli-Arzelà theorem to study the properties of systems. It is worth noting that, compared with reference [
24], the hypothesis conditions of this paper are weaker.
In order to avoid confusion, we will first introduce some basic notations and concepts. Let , , and be separable Hilbert spaces with norm . Moreover, is a complete probability space with normal filtration , where is a nonempty sample space, is a -algebra on , and is a probability measure defined on . The stochastic process is a -value Wiener process defined on . Moreover, this Wiener process has a nonnegative covariance operator with a finite trace, , where is orthogonal system satisfying .
We explore the HFSEEs:
In this equation,
represents the Hilfer fractional derivative with order
and type
. The Riemann–Liouville integral operator
with order
,
.
is the infinitesimal generator of a cosine family
consisting of strongly continuous and uniformly bounded linear operators. The stochastic process
is a
-value Wiener process defined on
. The control function
.
is a bounded linear operator and
.
and
are given.
.
To ensure a clear structure, the paper is divided into several parts.
Section 2 introduces fundamental information essential for our analysis. Following that,
Section 3 presents an approximate controllability result for problem (
1), while
Section 4 provides an exact controllability result for the same problem. In
Section 5, we validate the effectiveness of our findings with an example. Finally,
Section 6 summarizes the content discussed throughout the paper.
2. Preliminaries
represents the set of bounded linear operators mapping from
to
, where the norm is denoted as
. In particular, we use
to denote
.
represent a Banach space comprising square-integrable, strongly-measurable random variables. The norm
, where
.
denote the Banach space consisting of continuous mappings from
into
. Let
Lemma 1. (see [25]) If satisfies - (i)
For , is measurable ,
- (ii)
, then
Definition 1. (see [26]) The Riemann–Liouville fractional integral is defined as follows: Definition 2. (see [26]) The Riemann–Liouville fractional derivative is defined as follows: Definition 3. (see [26]) The Caputo fractional derivative is defined as follows:where the function is absolutely continuous and is continuous. Definition 4. (see [23]) The Hilfer fractional derivative is defined as follows:where , . Remark 1. (i) Especially, if , , then (ii) If , , then Let D be the bounded subset of Banach space X with the norm . The definition of the Kuratowski measure of noncompactness χ is as follows:where diam. Lemma 2. (see [27]) Let be Bochner integrable. If there exists such that for . Then Definition 5. (see [28]) The definition of Wright function is given by the following:which satisfies Definition 6. (see [29]) If a bounded linear operator maps , it is referred to as a strongly continuous cosine family if and only if - (i)
for all ,
- (ii)
,
- (iii)
is continuous for and .
The family of operators is defined as follows: The operator is defined as the generator of a cosine family , which is strongly continuous. It satisfies the following equation:where . This paper discusses a strongly continuous cosine family in which consists of uniformly bounded linear operators. Consequently, there exists a constant satisfying for .
Definition 7. (see [30]) is an adapted stochastic process, , the mild solution of problem (
1)
is defined as follows:where Lemma 3. (see [30]) The following inequality holds for any and . Lemma 4. (see [30]) The following formula is true for and any .Moreover, Lemma 5. (Schauder′s fixed point theorem, see [31]) Let V be a closed, convex, and nonempty subset of a Banach space X. Let be a continuous mapping such that is a relatively compact subset of X. Then, Φ has at least one fixed point in V. Lemma 6. (Mönch′s fixed point theorem, see [32]) Let V be a closed convex subset of a Banach space X and . Assume that is a continuous map that satisfies Mönch′s condition, i.e., for is countable and is compact. Then, Φ has at least one fixed point in V. Let
be the state value of system (
1) at time
h with control
u and reachable set
.
Definition 8. (see [33]) The fractional stochastic control system (1) is said to be - (i)
approximate controllability on the interval if ;
- (ii)
exact controllability on the interval if .
Lemma 7. (see [33]) For any , there exists an −adapted stochastic process such that and . In order to present the main result of this paper, the following assumption is required:
: satisfies the Caristi condition: for , is Lebesgue measurable and for each , is continuous.
: satisfies the Caristi condition: for , is -measurable and , for each , is continuous.
: For and each , there exists that satisfies
where ∨ means the maximum of the two.
Define mapping
:
where
If
has a fixed-point
, then
is a mild solution for problem (
1).
As the Ascoli–Arzelà theorem is applicable only to finite closed intervals. Hence, it is necessary to transform Equation (
4).
Let , we define for . It can be easily seen that .
Introduce the operator
as follows:
where
So,
has a fixed point that is equivalent to
’s fixed point.
3. Approximate Controllability
We introduce a controllability matrix:
and
denote the adjoint of
B and
, respectively. According to Lemma 3, it becomes apparent that
is linear and bounded.
Let
. We define the control function
as follows:
where
By Definition 8, we can establish that the system (
1) is approximate controllability on the interval
if and only if there exists
, where
represents the mild solution to system (
1) corresponding to
. To prove this, our initial step is to demonstrate the existence of a mild solution for system (
1) under the condition
.
Because
, then, operator
in (
5) becomes
To demonstrate the approximate controllability outcome, the subsequent assumption is necessary:
: is a compact semigroup and for any .
: There exists a constant
, such that
: as in the strong operator topology.
By the fact form , we have
where
is defined in Lemma 8 of this article and
is a constant.
Let
Obviously,
and
are convex, nonempty and closed.
Next, we will establish several lemmas that are pertinent to main result.
Lemma 8. Suppose that and are satisfied for . Then Proof. By Lemma 3, (
2), Hölder′s inequality and assumption
,
, we have
□
Theorem 1. If and hold. Then, there is at least one mild solution to problem (
1)
in . Proof. Now, we divide this part of the proof into the following steps:
Step 1: is equicontinuous for .
From [
30], we obtain that
is equicontinuous. Next, we prove that
is equicontinuous.
When
, by Lemma 3, Lemma 8, (
2),
and Hölder′s inequality, we can obtain
When
, by
inequality, we can obtain
Now, we prove
. By
inequality, we have
where
By Hölder′s inequality and
, we have
Because
so, by Lemma 4 and Hölder′s inequality, we have
It is obvious that
. Hence
.
Next, we prove the
. By
inequality, we have
where
By Lemma 8 and Hölder′s inequality, we have
Because
so, by Hölder′s inequality and Lemma 4, we have
It’s obvious that
. Hence
as
.
From [
30], we can obtain
as
.
Through the above analysis, , for . To sum up, is equicontinuous for .
Step 2: is continuous.
Let
be a sequence, which is convergent to
z in
, then
Because
,
, by
and
, we have
Using
, we can obtain
As
is integrable for
, we can use the Lebesgue dominated convergence theorem to derive
By use (
6), (
8) and (
9), we can obtain
Because, from Lemma 8, we have
By using the Lebesgue dominated convergence theorem, we can obtain
So, by using (
8)–(
10), for each
, we obtain
Therefore,
is continuous.
Step 3: .
For
, by
, Lemma 4 and (
7), we have
For
, since
, we have
Therefore, we have .
Step 4: is completely continuous.
It is evident that problem (
1) has a mild solution
if and only if
has a fixed-point
. Based on Step 2 and Step 3, it can be concluded that the operator
is continuous. It is clear that
is completely continuous if
is relatively compact in
. From Step 1,
is equicontinuous. According to the Ascoli-Azelà theorem, to prove that
is completely continuous, we need to show that
is relatively compact in
for
. However, it is clear that
is relatively compact. Now, we will demonstrate that
is relatively compact in
for
.
When
and
, we have the following definition for
on
:
As
is compact, then
is also compact. Therefore, for any
and any
, it follows that
is relatively compact in
for
. Additionally, for any
, we can conclude that:
In order to prove that
, we first need to establish that
,
Because
for any
and (
7), we have
By utilizing Definition 5, we obtain
Considering
, by utilizing
and
for any
, we can deduce that
so, we can get
By the Lebegue dominated convergence theorem, we derive that
as
or
. Thus,
. Similar, we can get
and
. So,
is relatively compact in
. Thus,
is completely continuous.
By using Schauder′s fixed-point theorem, It can be inferred that
possesses at least one fixed point
. Let
for
, thus
□
The following theorem justifies the approximate controllability results of system (
1).
Theorem 2. Assume that and are fulfilled. Then, the system (
1)
is approximately controllable on . Proof. For
, according to Theorem 1, it follows that
has a fixed point in
when the control function
. Let
be the fixed-point of
. Then
where
Taking into consideration
and Lemma 7, simple calculation yields
From
, it follows that there exist two subsequences, which we will still denote by
and
, and these subsequences weakly converge to
and
, respectively. Therefore
By using
and the Lebesque dominated convergence theorem, it follows that
This proves that the system (
1) is approximately controllable on the interval
. □
4. Exact Controllability
To establish the exact controllability of the system (
1), the following hypotheses are necessary:
H1. The linear operatorit is bounded and invertible,.
H2. For any bounded set , there exists contant , such that H3. Assume that the following inequality holds,where For any
, we define the control function
as follows:
By Definition 8, we can conclude that the system (
1) is exactly controllable on
if and only if there exists
, where
is the mild solution to system (
1) corresponding to
. To prove this, we only need to prove that system (
1) has a mild solution when
. Next, let us make some preparations for applying the Mönch′s fixed point theorem.
As
, the operator
in (
5) becomes
From form
, we have
where
is defined in Lemma 9 of this article and
is a constants.
Let
Obviously,
and
are convex, nonempty and closed.
Lemma 9. Suppose that – are satisfied. Then Proof. By Lemma 3, Hölder′s inequality and assumption
,
, we have
□
Theorem 3. Suppose that and are satisfied, and is noncompact. Then, the system (
1)
is exact controllability on . Proof. Similar to Step 1, 2, and 3 in Theorem 1, by applying Lemma 9, we can verify that is equicontinuous, continuous, and that . Next, we will prove that satisfies Mönch′s condition. Suppose that and . Suppose that and . Then, we have for .
By Lemmas 2 and 3 and
, we have
For any
, by using Lemma 3 and (
2), we can derive that
Thus, by (
13), (
) and referring to [
13], we have
Combined with the above calculations, we have
By
, we can conclude that
and that
is relatively compact. According to the Mönch′s fixed point theorem,
has at least one fixed point
. This fixed point
is a mild solution of the fractional stochastic control system (
1) when the control function is taken as
. Furthermore, it satisfies
for any
. Therefore, we can conclude that system (1) has exact controllability on
. □
Theorem 4. Suppose that and are satisfied, and is compact. Then, the system (
1)
is exact controllability on . Proof. The proof follows a similar approach to that of Theorem 1. □