Abstract
In this paper, abstract results concerning the approximate controllability of semilinear evolution systems in a separable reflexive Banach space are obtained. An approximate controllability result for semilinear systems is obtained by means of Schauder’s fixedpoint theorem under the compactness assumption of the linear operator involved. It is also proven that the controllability of the linear system implies the controllability of the associated semilinear system. Then the obtained results are applied to derive sufficient conditions for the approximate controllability of the semilinear fractional integrodifferential equations in Banach spaces and heat equations.
1 Introduction
The problems of controllability of infinite dimensional nonlinear (fractional) systems
were studied widely by many authors; see [16] and the references therein. The approximate controllability of nonlinear systems
when the semigroup
In recent years, controllability problems for various types of nonlinear fractional dynamical systems in infinite dimensional spaces have been considered in many publications. An extensive list of these publications focused on the complete and approximate controllability of the fractional dynamical systems can be found (see [15,7,947]). A pioneering work has been reported by Bashirov and Mahmudov [17], Dauer and Mahmudov [28] and Mahmudov [31]. Sakthivel et al.[40] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. Klamka [2326] derived a set of sufficient conditions for constrained local controllability near the origin for semilinear dynamical control systems. Wang and Zhou [3] investigated the complete controllability of fractional evolution systems without assuming the compactness of characteristic solution operators. Sukavanam and Kumar [47] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and Schauder’s fixedpoint theorem.
Consider an abstract semilinear equation
and define the following sets:
Here Y, X are separable reflexive Banach spaces and V is a Hilbert space,
reachable from
transfers equation (2) from
where
in the strong operator topology as
The same idea is now used to investigate the controllability of semilinear system
(1). To do so, for each
where
One can see that if the operator
if
It is clear that the fixed points of the nonlinear operator
To the best of our knowledge, the approximate controllability problem for semilinear
abstract systems in Banach spaces has not been investigated yet. Motivated by this
consideration, in this paper we study the approximate controllability of semilinear
abstract systems in Banach spaces. The approximate controllability of (1) is derived
under the compactness assumption of the linear operator involved. We prove that the
approximate controllability of linear system (2) implies the approximate controllability
of semilinear system (1) under some assumptions. On the other hand, it is known that
if the operator L is compact, then
In Section 2 an abstract result concerning the approximate controllability of semilinear system (1) is obtained. It is proven that the controllability of (2) implies the controllability of (1). Finally, these abstract results are applied to the approximate controllability of semilinear fractional integrodifferential equations. These equations serve as an abstract formulation of a fractional partial integrodifferential equation arising in various applications such as viscoelasticity, heat equations and many other physical phenomena.
2 Approximate controllability of semilinear systems
Let X be a separable reflexive Banach space and let
is bijective, demicontinuous, i.e., continuous from X with a strong topology into
An operator
for all
Lemma 1[31]
For every
has a unique solution
Theorem 2[31]
Let Γ be a symmetric operator. Then the following three conditions are equivalent:
(i) Γ is positive, that is,
(ii) For all
(iii) For all
We impose the following assumptions:
(A1)
(A2)
(A3) For all
Note that the condition (A3) holds if and only if
Definition 3 System (1) is approximately controllable if
Theorem 4Assume (A1)(A3) hold. Then semilinear system (1) is approximate controllability.
Proof Step 1. Show that the operator
Assume that
and
Thus we proved that
Step 2. Assume
So,
By the assumptions (A1) and (A2), the operator F is continuous bounded and L is compact. So, there exists a subsequence, still denoted by
Then we can extract a subsequence, still denoted by
for some
since Γ is positive. So,
Thus
3 Fractional integrodifferential equations
The purpose of this section is to establish sufficient conditions for the approximate controllability of certain classes of abstract fractional integrodifferential equations of the form
where the state variable x takes values in a separable reflexive Banach space X;
Definition 5 The fractional integral of order α with the lower limit 0 for a function f is defined as
provided the righthand side is pointwise defined on
Definition 6 RiemannLiouville derivative of order α with the lower limit 0 for a function
Definition 7 The Caputo derivative of order α for a function
Remark 8
(1) If
(2) The Caputo derivative of a constant is equal to zero.
(3) If f is an abstract function with values in X, then the integrals which appear in the above definitions are taken in Bochner’s sense.
For basic facts about fractional integrals and fractional derivatives, one can refer to [49].
In order to define the concept of a mild solution for problem (10), we associate problem (10) to the integral equation
where
and
Lemma 9[34]
For any fixed
Definition 10 A solution
Let
Definition 11 System (10) is said to be approximately controllable on J if
Consider the following linear fractional differential system:
The approximate controllability for linear fractional system (12) is a natural generalization of the approximate controllability of a linear firstorder control system. It is convenient at this point to introduce the controllability operator associated with (12) as
where
Proposition 12If
is compact from
Proof According to the infinite dimensional version of the AscoliArzela theorem, we need to show that
(i) for arbitrary
(ii) for arbitrary
To prove (i), fix
Since
One can estimate
and
where we have used the equality
Consequently,
To prove (ii), note that, for
Applying the Hölder inequality, we obtain
It is clear that
Theorem 13Suppose
Proof Let
for
4 Application
Consider the partial differential system of the form
where
where
Then
where
Then
Now if
5 Conclusion
In this paper, abstract results concerning the approximate controllability of semilinear evolution systems in a separable reflexive Banach space are obtained. An approximate controllability result for semilinear systems is obtained by means of Schauder’s fixedpoint theorem under the compactness assumption. It is also proven that the controllability of the linear system implies the controllability of the associated semilinear system. Then the obtained results are applied to derive sufficient conditions for the approximate controllability of the semilinear fractional integrodifferential equations in Banach spaces. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a wide class of fractional deterministic and stochastic differential equations.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The author would like to thank the reviewers for their valuable comments and helpful suggestions that improved the note’s quality.
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