We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.
In this paper, we study the fractional nonlocal integrodifferential system of the form
Basic researches in differential equations have showed that many phenomena in nature are modeled more accurately using fractional derivatives and integrals; for more detail, we can refer to [
Controllability is a fundamental concept in mathematical control theory and plays an important role in both finite and infinite dimensional spaces, that is, systems represented by ordinary differential equations and partial differential equations, respectively. So it is natural to extend this concept to dynamical systems represented by fractional differential equations. Several fractional partial differential equations and integrodifferential equations can be expressed abstractly in some Banach spaces, in many cases, the accurate analysis, design and assessment of systems subjected to realistic environments must take into account the potential of random loads and randomness in the system properties. Randomness is intrinsic to the mathematical formulation of many phenomena such as fluctuations in the stock market or noise in communication networks. Fu studied the controllability results of some kinds of neutral functional differential systems, see [
The existence results to evolution equations with nonlocal conditions in Banach space were studied first by Byszewski [
Deng [
In this paper, we introduce a new concept in the theory of Semigroup named “implicit evolution system” to show the reader “what is the main difference between the solutions of fractional (
Our paper is organized as follows. Section
The fractional integral of order
Riemann-Liouville derivative of order
(1) If
(2) The Caputo derivative of a constant is equal to zero.
(3) If
By a strong solution of the nonlocal Cauchy problem (
It is suitable to rewrite (
Let
Let
Let
The stability of
Let
Let
(H1) The family
(H2)
(H3) For
(H4) There is a constant
In the next section, we will introduce a new concept in the theory of semigroups.
Let
(1) If
(2) Since, in our case,
(3) For nonautonomous differential equations in a Banach space, the implicit evolution system is similar to our concept
(4) We can deduce that (
Further, we assume the following.
(H5) For every
(H6)
(H7) For every
(H8) The operator
(H9)
(H10)
(H11)
(H12) There exist positive constants
(H13) Further, there exists a constant
Let
Let
If
Assume the following. Conditions (H1) The functions There are numbers
Then, the problem (
Applying Theorem
According to (ii),
Set
Consider the following nonlocal Cauchy problem:
In next section, some results are obtained from Sakthivel et al. [
Consider fractional nonlocal evolution integrodifferential control system of the form
For all
We will say that system (
In order to prove the controllability result, in addition, we consider the following conditions.
(H14)
(H15) Let
(H16) The bounded linear operator
If hypotheses (H1)
Let
We define an operator
Clearly,
Now, we show that
We have
To illustrate the abstract results, we give the following examples.
Consider the nonlinear integropartial differential equation of fractional order
(i) The operator
(ii) All the coefficients
(iii) If the coefficients
There are numbers
The second example is concerned with the controllability result.
Consider the fractional nonlocal evolution integropartial differential control system of the form
Let us take
We define
Also, define
Assume that the linear operator
Let us assume that the nonlinear functions
All the conditions stated in Theorem 5.2 are satisfied. Hence, system (