1. Introduction
Recently, fractional differential equations have been considered greatly by research community in various aspects due to its salient features for real world problems (see [
1,
2,
3,
4,
5,
6,
7]). Controllability problems for different kinds of dynamical systems have been studied by several authors (see [
8,
9,
10,
11,
12,
13,
14,
15]) and references therein. Thus, the dynamical systems must be treated by the weaker concept of controllability, namely approximate controllability (see [
16,
17,
18,
19,
20,
21]). However, up to now, no work has been reported yet regarding the null controllability of Sobolev type nonlinear fractional delay integro-differential system with the impulsive condition of order
. Motivated by these facts, we study the existence of the mild solution for Sobolev-type impulsive fractional differential equations and we also discuss the sufficient conditions for approximate controllability and null controllability of the same problem.
2. Preliminaries
Let be a Banach space of all bounded linear operators from X into X equipped with the norm and let be the Banach space of all continuous functions from J to X equipped with the norm Moreover, is the closed ball in X with center at x and radius
Definition 1. ([22]). The linear space of all functions from into the Banach space X with a semi-norm is known as phase space The fundamental axioms on are the following:
- (A)
If is a continuous function on J and then for every the following conditions hold:
- (i)
- (ii)
,
- (iii)
where is a constant, is continuous, is locally bounded, and , are independent of .
- (B)
For the function in (A), is continuous function for
- (C)
The space is complete.
To obtain our results, we assume that the abstract fractional integro-differential problem
has an associated
q-resolvent operator of bounded linear operator
on
The operators and satisfy the following hypotheses:
- (I)
A and are closed linear operators,
- (II)
and is bijective,
- (III)
is continuous.
Here, (I) and (II) together with the closed graph theorem imply the boundedness of the linear operator Furthermore, generates a strongly continuous semigroup of bonded linear operators in X.
Definition 2. ([23]). A family of bounded linear operators on X is said to be resolvent operator for Equation (1) if the following conditions are verified: - (a)
The function is strongly continuous and for all and
- (b)
For ,
In this paper, we have considered the following conditions.
(P1) The operator is a closed linear operator with dense in
Let For some for each there is a positive constant such that for each where and for all
(P2) For all is a closed linear operator, and is strongly measurable on for each There exists such that exists for and
for all and Moreover, the operator
has an analytical extension to such that
for all and as
(P3) There exists a subspace dense in and a positive constant such that and
For and then
For
are the paths
and
oriented counterclockwise. In addition,
We now define the operator family by
Definition 3. ([23]). Let we define the family by Lemma 1. ([23]). Assume that conditions – are fulfilled. Then, there exists a unique q-resolvent operator for the problem (2). Lemma 2. ([23]). The function is strongly continuous. Lemma 3. ([23]). If is compact for some then and are compact for all We denote by the fractional power of operator .
Lemma 4. ([23]). Suppose that the conditions (P1)–(P3) are satisfied. Let and such that there exists positive number such that:for If then 3. Existence Solution
In this section, we investigate the existence of mild solution of Sobolev type of fractional integro- differential equation with finite delay and impulsive conditions in the following form:
where the state
x takes values in a Banach space
and
are closed linear operators on
represent the Caputo derivative of order
. The history
given by
belongs to some abstract phase space
, defined later,
and
are appropriate functions,
are prefixed points and
is the jump of the solutions at impulse points
which is defined by
.
To establish the result, we need the following hypotheses:
Hypothesis 1 (H1). is the infinitesimal generator of a resolvent operator in X and there exists constant such that Hypothesis 2 (H2). The functions and are continuous and there exist positive constants and such that Hypothesis 3 (H3). for every and there exist a function such that: In addition, there exists a constant such that
Hypothesis 4 (H4). There exists a constant such that Definition 4. ([23]). We say that is a mild solution of the system on if it satisfies Theorem 1. If Hypotheses (H1)–(H4) are satisfied, then the system has a unique mild solution on J provided that Proof. Consider the operator
on
defined as follows:
□
We want to prove that the operator has a fixed point.
First, we show that
maps
into itself. For
,
Thus, maps into itself.
Next, we prove that
is contraction on
For,
, we obtain
Then, is a contraction mapping on . Next, we will prove that is completely continuous
Let,
be a sequence in
and
such that
we want to prove that
as
Since the functions
and
G are continuous, i.e., then, as
, the following are satisfied:
Therefore, is continuous.
Next, we show that
is equicontinuous on
J for any
. Let
and
be sufficiently small; then,
It is known that the right-hand side of (5) tends to zero as . Hence, is completely continuous on J. By using a fixed point theorem, has a unique fixed point on J. Therefore, system (3) has a unique mild solution on J.
4. Approximate Controllability
We will establish a set of sufficient conditions for approximate controllability of impulsive delay fractional differential equation in the following form:
where the control function
the Banach space of admissible control functions with
U a Banach space and
B is a bounded linear operator from
U into
X.
Definition 5. We say that is a mild solution of system (6) if it satisfies In order to study the approximate controllability for system (6), we introduce the following linear fractional differential system:
We define the operators associated with (6) as follows:
Let be the state value of (6) at terminal state T, corresponding to the control u and the initial value . Denote by the reachable set of system (6) at terminal time T, its closure in X is denoted by
Definition 6. System is said to be approximately controllable on the interval if
Lemma 5. ([24]). The linear system (7) is approximate controllable on if and only if the operator as in the strong operator topology. To establish the result, we need the following addition.
Hypothesis 5 (H5). There exist a constant such that
where
Theorem 2. If hypotheses (H1)–(H3) and (H5) are satisfied, then system (6) has a mild solution on J provided that Proof. Consider the operator
on
as follows:
where
□
We want to prove that the operator
has a fixed point. This fixed point is then a mild solution of system (6). We show that
maps
into itself,
,
Thus, maps into itself.
Next, for
, we obtain
Then, is a contraction mapping and hence there exist a unique fixed point such that . Hence, any fixed point of is a mild solution of (6) on J.
Theorem 3. Assume that hypotheses (H1)–(H5) hold. Further, if the functions
are uniformly bounded and the resolvent operator and compact operators, then system (6) is approximately controllable on J.
Proof. Let
be a fixed point of
in
. Any fixed point of
is a mild solution of (6) on
J under the control
where
and satisfies
Thus, the Lebesgue dominated convergence theorem and the compactness of
yield
Then, system (6) is approximate controllable. □
5. Exact Null Controllability
In this section, we investigate the exact null controllability of fractional nonlinear differential equation of the system (6).
To study the exact null controllability of (6), we consider the fractional linear system
associated with system (6).
Define the operator where has a bounded inverse operator with values in and,
Definition 7. ([25].) The linear system (14) is said to be exactly null controllable on J if Definition 8. ([26].) System (6) is said to be exactly null controllable if there is a such that the solution x of the system (6) satisfies Definition 9. ([26].) Suppose that the linear system is exactly null controllable on . Then, the linear operator is bounded. In this section, we need the following assumption.
Hypothesis 6 (H6). The linear system (14) is exactly null controllable on . Through this section, set
Now, we are able to state and prove our main results.
Theorem 4. Assume assumptions (H1)–(H3) and (H6) are satisfied. Then, the system (6) is exactly null controllable provided that: Proof. For an arbitrary
x, define the operator
φ on
as follows:
where
It will be shown that the operator φ from into itself has a fixed point. □
Step 1. The control is bounded on .
Step 2. There exist such that φ sends into itself.
If
we have
Hence, φ maps into itself.
Step 3. We prove is continuous on J for any .
Let
and
be sufficiently small, then,
Clearly, from the continuity of the operators and , the right-hand side of (15) tends to zero as .
Hence, is continuous on .
Step 4. We prove that is a contraction on X.
Let
for
be fixed; then,
Hence, φ is a contraction on . From the Banach fixed point theorem, φ has a unique fixed point. Therefore, system (6) is exact null controllable on J.
6. Application
Consider the following fractional delay partial differential equation of fractional order:
where
is a Caputo fractional derivative of order of order
. The operators
and
introduced as follows:
where each domain
and
is given by
are absolutely continuous, }.
Then, A and μ can be written respectively as:
The operator is self-adjoint and has the eigenvalues , with the corresponding normalized eigenvectors
We define the bounded operator by .
In addition, we define the following functions:
Now, the conditions (H1)–(H3) and (H5) are satisfied.
Hence, all the hypotheses of Theorems 2 and 3 are satisfied and
Thus, the fractional delay partial differential equation with impulsive condition (16) is approximately controllable on J.
7. Conclusions
This paper dealt with a Sobolev type nonlinear fractional delay integro-differential system with an impulsive condition. By using fractional calculus and fixed point theorems with the resolvent operator, we proved the existence of a mild solution for a Sobolev type nonlinear fractional delay integro-differential system with an impulsive condition. In addition, we established the necessary conditions for approximate controllability and null controllability of a Sobolev type nonlinear fractional delay integro-differential system with an impulsive condition. In the end, we provided an example in a fractional integro-partial differential equation to illustrate our results.