Article
Kyungpook Mathematical Journal 2021; 61(3): 559-581
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
Approximate Controllability for Semilinear Neutral Differential Systems in Hilbert Spaces
Jin-Mun Jeong*, Ah-Ran Park, Sang-Jin Son
Department of Applied Mathematics, Pukyong National University, Busan 48513, South Korea
e-mail : jmjeong@pknu.ac.kr and alanida@naver.com
Department of Applied Mathematics, Pukyong National University, Busan 48513, South Korea
e-mail : sangjinyeah@nate.com
Received: April 16, 2020; Revised: October 12, 2020; Accepted: November 16, 2020
Abstract
In this paper, we establish the existence of solutions and the approximate controllability for the semilinear neutral dierential control system under natural assumptions such as the local Lipschitz continuity of nonlinear term. First, we deal with the regularity of solutions of the neutral control system using fractional powers of operators. We also consider the approximate controllability for the semilinear neutral control equation, with a control part in place of a forcing term, using conditions for the range of the controller without the inequality condition as in previous results.
Keywords: approximate controllability, semilinear equation, neutral differential equation, local lipschitz continuity, controller operator, reachable set.
1. Introduction
In this paper, we are concerned with the global existence of solution and the approximate controllability for the semilinear neutral system in a Hilbert space
Here,
In the first part of this paper, we establish the well-posedness and regularity property for (1.1). The solvability for a class of semilinear functional differential equations has been studied by many authors as seen in Section 4.3.1 of Barbu [1] and [11, 13, 17]. Our approach is to obtain the
Next, based on the regularity for (1.1), we intend to establish the approximate controllability for the following semilinaer neutral control system with control part in place of a forcing term:
namely that the reachable set of trajectories of (1.2) is a dense subset of
As for the approximate controllability for semilinear control systems, we refer to [2, 3, 5, 7, 20, 23]. The controllability for neutral equations has been studied by many authors, for example, the controllability of neutral functional differential systems with unbounded delay in [5, 6, 15], neutral evolution integrodifferential systems with state dependent delay in [14, 18], impulsive neutral functional evolution integrodifferential systems with infinite delay in [19]. However, there are few literature works treating the systems with local Lipschitz continuity. As a sufficient condition for the approximate controllabilityl, Wang [24] assumed that the semigroup
In this paper, we no longer require the compact property of the semigroup and the uniform boundedness of the nonlinear term, but instead we need properties fractional power of operators and conditions for the range of the controller without the inequality condition as in previous results.
The paper is organized as follows. In Section 2, the results of general linear evolution equations besides notations and assumptions are stated. In Section 3, we will obtain that the regularity for parabolic linear equations can also be applicable to (1.1) with nonlinear terms satisfying local Lipschitz continuity. The approach used here is similar to that developed in [11, 12, 16] on the general semilinear evolution equations, which is an important role to extend the theory of practical nonlinear partial differential equations. Thereafter, we investigate the approximate controllability for the problem (1.2) in Section 4. In the proofs of the main theorems, we need conditions on the range of the controller without the inequality condition as in previous results(see [16, 25]) without conditions of the compact property of a semigroup and the uniform boundedness. Finally we give a simple example to which our main result can be applied.
2. Regularity for Linear Equations
If
For
Therefore, we assume that
Let
where
Then
is also denoted by
where
is the graph norm of
Thus we have the following sequence
where each space is dense in the next one which continuous injection.
Lemma 2.1. With the notations (2.3), (2.4), we have
where
It is also well known that
If
Here, we note that by using interpolation theory, we have
The semigroup generatedby
The following Lemma is from Lemma 3.6.2 of [21].
Lemma 2.2. There exists a constant
First of all, consider the following linear system
By virtue of Theorem 3.3 of [4](or Theorem 3.1 of [11], [21]), we have the following result on the corresponding linear equation of (2.6).
Proposition 2.3. Suppose that the assumptions for the principal operator
(1) For
where
(2) Let
where
Corollary 2.4. Suppose that
and
it follows that
From (2.3), (2.9), and (2.10) it holds that
So, if we take a constant
the proof is complete.
3. Semilinear Differential Equations
From now on, we establish the following results on the local solvability of the following equation;
where
Lemma 3.1. For any
We give the following assumptions.
Assumption(A). Let
be a continuous function. Then there exists a constant
Assumption(F). Let
hold for any
Assumption(G). Let
(i) For any
(ii) There exist positive constants
for all
(iii)
for
Let us rewrite
hold for
Lemma 3.2. Let us assume Assumptions (F),(G) and (A) for
and
Moreover, we have
Theorem 3.3. Let Assumptions (F), (G) and (A) be satisfied. Assume that
To provea local solution, we will use the successive iteration method. First, put
and define
By virtue of Proposition 2.3, we have
that
where
Putting
by (2.11) of Corollary 2.4, we have
Let
Then From Assumption (G), (A), (3.3) and (3.4), we have
Set
Then by Assumption (G), (3.2) and (3.4),
Put
Then for any
By induction, it can be shown that for all
Hence, from the equation
Put
In a similar way to (3.11) and (3.12) and Assumption (F), we can observe that the inequality
Choose
From now on, we give a norm estimation of the solution of (3.1) and establish the global existence of solutions with the aid of norm estimations by similar argument using (3.1) and (iii) of Assumption (G).
Theorem 3.4. Under the Assumptions (A), (F) and (G), there exists a unique solution
for any
where
where
By (3.9), we have
that
where
Thus, Moreover, there exists a constant
Now from
and
it follows that
Hence, we cansolve the equation in
From the following result, we obtain that the solution mapping is continuous, which is useful for physical applications of the given equation.
Corollary 3.5. Let the Assumptions (A), (F) and (G) be satisfied and
Then the solution
is continuous.
Let
Then, by virtue of 2) of Proposition 2.1, we get
Set
Then, we have
Hence, by (3.17), (3.18) and (iii) of Assumption (G), we see that
This implies that
Repeating this processwe conclude that
4. Controllability
Let
Let
Definition 4.1 The system (4.1) is said to be approximately controllable in the time interval
In order to obtain results of controllability, we need the stronger hypotheses than those of Section 3:
Assumption (A1). Let
be a continuous function. Then there exists a constant
Assumption (F1). Let
(i)
(ii)
hold for any
Assumption (G1). Let
(i) (i) and (iii) of Assumption (G) in Section 3 are satisfied.
(ii) There exists positive constants
for all
We define the linear operator
Assumption (S). For any
where
Here, we remark that Assumptions (A1), (F1) and (G1) are actually sufficient conditions for Assumptions (A), (F) and (G), respectively. So, if
is continuous.
Lemma 4.2. Let
for
Then, we see
Here, by Assumptions (F1) and(G1), he following inequalities hold:
and
Thus from (4.4)-(4.6) and using Gronwall's inequality, it follows that
Therefore, (4.2) holds.
Theorem 4.3. Under the Assumptions (A1), (F1), (G1) and (S), the system (4.1) is approximately controllable on
Then we will show that
Noting that
the solution of (4.1) is represented as
where
As
for instance, take
Let
it follows
We can also choose
and by Assumption (S)
for
Choose a constant
According to a simple calculation of (4.7), from Lemma 4.1 we have
where
Thus, in view of (4.9) and Assumption (S), we see
Put
for
Then, we have
By proceeding this process, the following holds
it follows that
Therefore, by virtue of Assumption (F1), there exists
From (4.8), (4.9) it follows that
By choosing choose
putting
Therefore, for
and
Thus the system (4.1) is approximately controllable on
5. Example
Let
Consider the following semilinear neutral differential control system in Hilbert space
where
Let
Then
The eigenvalue and the eigenfunction of
(a1)
Moreover, there exists a constant
(a2) Let
In particular,
The nonlinear mapping that appears on the control system for a diffusion and reaction process in an enzyme membrane is defined as
Then since
we can see that
Define
Then it can be checked that Assumption (G) is satisfied. Indeed, for
where
such that
for a constant
Hence we have
It is immediately seen that Assumption (G1) has been satisfied. A simple example of the controller operator
Then as seen in [16], for a given
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