### 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 ^{2}

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 ^{*}_{1}^{*}_{2}

For ^{*}

Therefore, we assume that

Let

where

Then ^{*}

is also denoted by

where

is the graph norm of _{0}>0

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 ^{*}_{2}=0

If ^{2}(0,T;X)

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 _{1}

(2) Let

where _{1}

**Corollary 2.4.** Suppose that _{2}

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 _{α}

We give the following assumptions.

**Assumption(A)**. Let

be a continuous function. Then there exists a constant _{a}

**Assumption(F)**. Let

hold for any

**Assumption(G)**. Let _{g}

(i) For any

(ii) There exist positive constants _{g}

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 _{1}

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

_{3}

where _{3}

_{0}]_{0}>0_{1}_{2}

where

By (3.9), we have

that

where _{1}

Thus, Moreover, there exists a constant _{3}

Now from

and

it follows that

Hence, we cansolve the equation in _{0},2T_{0}]_{0})_{0}]_{1})

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.

_{i}

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 _{a}

**Assumption (F1)**. Let

(i)

(ii)

hold for any

**Assumption (G1)**. Let _{g}

(i) (i) and (iii) of Assumption (G) in Section 3 are satisfied.

(ii) There exists positive constants _{g}

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 _{1}_{2}

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 _{1}

According to a simple calculation of (4.7), from Lemma 4.1 we have

where _{3}

Thus, in view of (4.9) and Assumption (S), we see

Put _{3}

for _{2}

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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