Applied Mathematics and Computation 232 (2014) 1235–1241

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Controllability of a class of nonlinear neutral time-delaysystems

http://dx.doi.org/10.1016/j.amc.2014.01.0090096-3003/� 2014 Elsevier Inc. All rights reserved.

E-mail addresses: shaho@aut.ac.ir, s_sh_alaviani@yahoo.com

S.Sh. AlavianiDepartment of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran

a r t i c l e i n f o

Keywords:ControllabilityNonlinear systemsNeutral time-delay systemsFixed point theory

a b s t r a c t

In this paper, the problem of controllability for a class of time-varying nonlinear neutraltime-delay systems is considered. Sufficient conditions based on Krasnoselskii’s fixed pointtheorem are derived for the system to be globally and locally controllable. Eventually, someillustrative examples are given in order to present the results established.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Various practical systems such as car chasing, distributed networks, neural networks, oscillation in a vacuum tube [1],control of epidemics (like AIDS), the flip-flop circuit, control of economic growth [2], partial element equivalent circuit[3], and systems with lossless transmission lines [4, p. 289] can be modeled by nonlinear neutral time-delay systems. Theexistence of the time delay may cause instability or mal-performances in dynamical systems. Hence, the study of controlla-bility for such systems is important. Several authors have investigated controllability for nonlinear neutral time-delay sys-tems [2,5–11].

In [2, Ch. 12], sufficient conditions for controllability of nonlinear time-varying neutral delay systems whose differenceoperators are linear with respect to state have been given. In [5–11], several fixed point theorems have been used to derivesufficient conditions for controllability[5,6,8,9] and null-controllability[7] of nonlinear time-varying neutral time-delay sys-tems: the Banach fixed point theorem [5], the Fan fixed point theorem [6], the Schauder fixed point theorem [7], Sadovskii’sfixed point theorem [8], and Krasnoselskii’s fixed point theorem [9]. The difference operators are linear with respect to statesin [5–7] while in [8,9] they are nonlinear. The systems considered in [2,5–9] do not include control input in the differenceoperator; however, in [10,11], the considered nonlinear time-invariant system includes control input in the difference oper-ator which is linear with respect to both state and input, and sufficient conditions for null-controllability have been given.

In this paper, controllability problem for a class of time-varying nonlinear time-delay systems of neutral type is consid-ered. The system has nonlinear difference operator with respect to both state and input, so it is more general than thosedeemed in [2,5–11]. By using Krasnoselskii’s fixed point theorem which allows us to consider a more general system havingnonlinear difference operator with respect to control input, instead of a linear difference operator considered in [10,11], suf-ficient conditions for global and local controllability of the system are derived. The system has bounded functions for globalcontrollability conditions to be given. Although this paper cannot give any condition on global controllability of the systemwith unbounded functions, it may give some conditions on local controllability of the system. In some cases where sufficientconditions in [2,5–11] fail, sufficient conditions given here may be useful to give controllability conditions. In practice,electrical circuits with lossless transmission lines can be modeled by the system considered in this paper (see Example 3)

1236 S.Sh. Alaviani / Applied Mathematics and Computation 232 (2014) 1235–1241

and not by the systems in [2,5–11]. Therefore, the study of controllability for this kind of systems is very important anduseful.

This paper is organized as follows. In Section 2, the mathematical model of the system is presented. In Section 3, sufficientconditions for global and local controllability are given. Ultimately, some illustrative examples are provided to show the re-sults investigated.

Notation. N and Rn�m denote the set of all natural numbers and the n�m real matrix space, respectively. k � k represents thesupremum norm, that is, kxk ¼ supt2RjxðtÞj, where j � j denotes the infinity norm for x 2 Rn. Also, if AðtÞ is an n� n real matrixfunction, then the supremum norm of A(t) is defined by kxk ¼ supt2RjAðtÞj, where jAðtÞj ¼max16i6n

Pnj¼1jaijðtÞj. Cð½t1; t2� ! DÞ

represents the set of all continuous functions defined on ½t1; t2� into D. SnðaÞ ¼ fx2n : jxj 6 ag.

2. System description

Consider the following nonlinear neutral time-delay system:

ddtðxðtÞ � hðt; xðt � s0ðtÞÞ; . . . ; xðt � spðtÞÞ; uðtÞÞÞ ¼ gðt; xðtÞ; xðt � s0ðtÞÞ; . . . ; xðt � spðtÞÞ;uðtÞÞ þ BðtÞuðtÞ; t P t0xðtÞ

¼ uðtÞ; �s 6 t 6 t0; ð1Þ

where xðtÞ 2 Rn is the state of the system, uðtÞ 2 Rm is the control input, t represents the time parameter, and t0 is the initialtime. The delays siðtÞ; i ¼ 0;1;2; . . . ; p, are continuous functions and �s ¼ inf t P t0

i ¼ 0;1;2; . . . ; p

ðt � siðtÞÞ. uðtÞ is a continuous

initial function. It is assumed that h and g are continuous functions with appropriate dimensions and satisfy a Lipschitz con-dition in their arguments except and that BðtÞ 2 Rn�m is a continuous matrix function.

The purpose of this paper is to study controllability problem for system (1) and to derive sufficient conditions for thesystem to be controllable. In next section, some criteria for controllability of the system are provided. Let usdefine Cð½�s; t0� ! RnÞ � X1;Cð½�s; t1� ! RnÞ � X2;Cð½t0; t1� ! RmÞ � U;Cð½�s; t0� ! SnðvÞÞ � X3, for some v P 0, and Cð½t0; t1� !SmðwÞÞ � U1, for some w P 0.

3. Criteria for controllability

If the functions h and g are continuous and satisfy a Lipschitz condition, and if one specifies an initial function u 2 X1,then a unique solution for system (1) exists for t > t0. This solution will be denoted by xðt; t0;u;uÞ. Now, the concept of con-trollability for system (1) is defined.

Definition 1. System (1) is said to be globally Euclidean controllable on ½t0; t1� if for each u 2 X1 and xf 2 Rn there exist afinite time t1 > t0 and a control function u 2 U such that the solution xðt; t0;u;uÞ of (1) exists and satisfies xðt1; t0;u;uÞ ¼ xf .The system is globally Euclidean null-controllable on ½t0; t1� if in this definition xf ¼ 0.

Next, a theorem for global Euclidean controllability of system (1) is given.

Theorem 1. Consider system (1). If the following conditions hold

(1) The functions h and g are continuous and globally Lipschitz with respect to their arguments except t.(2) BðtÞ has differentiable elements on the interval ½t0; t1�.(3) Hðt0; t1Þ ¼

R t1t0

BðsÞBT ðsÞds is nonsingular for t1 > t0.(4) The functions h and g are bounded by some constants N > 0 and L > 0, respectively, for all x 2 X2 and u 2 U.(5) kh ð:;u1Þ � h ð:;u2Þk 6 bk u1 � u2k, where rb < 0:5;r ¼ kBT H�1ðt0; t1Þk; 8u1;u2 2 U.

Then system (1) is globally Euclidean controllable.

Proof. The proof of Theorem 1 is based on Krasnoselskii’s fixed point theorem introduced in the following lemma. h

Lemma 1 [12]. Let M be a closed convex nonempty subset of a Banach space (S, ||�||). Suppose that D and E map M into S such that

(i) Dyþ Ez 2 M ðy; z 2 MÞ.(ii) D is continuous and DM is contained in a compact set.

(iii) E is a contraction with constant a < 1.

Then there is a z 2 M such that Dzþ Ez ¼ z.

S.Sh. Alaviani / Applied Mathematics and Computation 232 (2014) 1235–1241 1237

Now, let u 2 X1 be an arbitrary initial function, then (1) can be written in the following integral form:

xðtÞ ¼

uðtÞ; �s 6 t 6 t0

uðt0Þ�hðt0;uðt0 � s0ðt0ÞÞ; . . . ;uðt0 � spðt0ÞÞ;uðt0ÞÞþhðt; xðt � s0ðtÞÞ; . . . ; xðt � spðtÞÞ; uðtÞÞþR t

t0gðs; xðsÞ; xðs� s0ðsÞÞ; . . . ; xðs� spðsÞÞ; uðsÞÞds

þR t

t0BðsÞuðsÞds; t P t0:

8>>>>>>>>><>>>>>>>>>:

ð2Þ

Let S ¼ fu : u 2 U anduisboundedg, then ðS; k:kÞ is a Banach space. Define M ¼ fu : u 2 S; kuk 6 Kg for some K > 0, then M is abounded closed convex subset of the Banach space S. Now, define operator W for each u 2 M by

ðWuÞðtÞ ¼ �BTðtÞH�1ðt0; t1Þðuðt0Þ � xf � hðt0;uðt0 � s0ðt0ÞÞ; . . . ;uðt0 � spðt0ÞÞ;uðt0ÞÞ þ hðt1; xuðt1

� s0ðt1ÞÞ; . . . ; xuðt1 � spðt1ÞÞ; uðt1ÞÞ þZ t1

t0

gðs; xuðsÞ; xuðs� s0ðsÞÞ; . . . ; xuðs� spðsÞÞ;uðsÞÞdsÞ: t0

6 t 6 t1; ð3Þ

where xf 2 Rn is arbitrary. Applying Lemma 1 needs to construct two mappings D and E such that

ðWuÞðtÞ ¼ ðDuÞðtÞ þ ðEuÞðtÞ;

where

ðDuÞðtÞ ¼ �BTðtÞH�1ðt0; t1ÞZ t1

t0

gðs; xuðsÞ; xuðs� s0ðsÞÞ; . . . ; xuðs� spðsÞÞ;uðsÞÞds; t0 6 t 6 t1; ð4Þ

ðEuÞðtÞ ¼ �BTðtÞH�1ðt0; t1Þðuðt0Þ � xf � hðt0;uðt0 � s0ðt0ÞÞ; . . . ;uðt0 � spðt0ÞÞ;uðt0ÞÞ þ hðt1; xuðt1

� s0ðt1ÞÞ; . . . ; xuðt1 � spðt1ÞÞ;uðt1ÞÞÞ; t0 6 t 6 t1; ð5Þ

in which Hðt0; t1Þ ¼R t1

t0BðsÞBTðsÞds. The nonsingularity of matrix H is one of the conditions which will be imposed on system

(1) to obtain a controllability condition. xuðtÞ is the solution of (1) corresponding to the control input uðtÞ and the initial func-tion uðtÞ; �s 6 t 6 t0. Since h and g satisfy a Lipschitz condition, such a solution can be shown to exist. Furthermore, this solu-tion is unique and is defined for all t 2 ½�s; t1�. Assume that the functions h and g are bounded by some constants N > 0 andL > 0, respectively, for all x 2 X2 and u 2 U. Thus we have that

kDuk ¼ supt06t6t1jðDuÞðtÞj 6 rLðt1 � t0Þ; ð6Þ

kEuk ¼ supt06t6t1jðEuÞðtÞj 6 rðkuk þ jxf j þ 2NÞ; ð7Þ

where kr ¼ BT H�1ðt0; t1Þk.If we take

K ¼ rðkuk þ jxf j þ 2N þ Lðt1 � t0ÞÞ; ð8Þ

then operator Duþ Eu maps the set M into itself. Therefore, condition (i) of Lemma 1 is satisfied.Due to continuity of function g, continuity of the solution xuðtÞ with respect to the parameter u, and continuity of BðtÞ,

operator D is continuous. Let uk 2 S be a sequence with kukk 6 J where k 2 N and J > 0, then we obtain kDukk 6 l for somepositive constants l. It can be checked that

jðDukÞðtÞ � ðDukÞð~tÞj 6 jBTðtÞ � BTð~tÞjjH�1ðt0; t1ÞjLðt1 � t0Þ 6Xm

j¼1

Xn

i¼1

jbTijðtÞ � bT

ijð~tÞj !

jH�1ðt0; t1ÞjLðt1 � t0Þ

6 mn�bjt � ~tjjH�1ðt0; t1ÞjLðt1 � t0Þ ¼ Q jt � ~tj; ð9Þ

where �b ¼max 1 6 i 6 n1 6 j 6 m

supt06t6t1j dbT

ijðtÞdt j. So the sequence Duk is uniformly bounded and equi-continuous. As a matter of

fact, the Arzela–Ascoli theorem [13, p. 109] implies that Duk uniformly converges to a continuous function. Therefore, theoperator D is compact, and condition (ii) of Lemma 1 is satisfied.

Since h is Lipschitz with respect to u, we have that

khð:;u1Þ � hð:;u2Þk 6 bku1 � u2k; b P 0; 8u1; u2 2 M:

If rb < 0:5, then operator E is a contraction. So condition (iii) of Lemma 1 is satisfied, and according to the Lemma,Duþ Eu ¼ u has a solution u�, i.e.

1238 S.Sh. Alaviani / Applied Mathematics and Computation 232 (2014) 1235–1241

u�ðtÞ ¼ �BTðtÞH�1ðt0; t1Þðuðt0Þ � xf � hðt0;uðt0 � s0ðt0ÞÞ; . . . ;uðt0 � spðt0ÞÞ;u�ðt0ÞÞ þ hðt1; xu� ðt1

� s0ðt1ÞÞ; . . . ; xu� ðt1 � spðt1ÞÞ;u�ðt1ÞÞ þZ t1

t0

gðs; xu� ðsÞ; xu� ðs� s0ðsÞÞ; . . . ; xu� ðs� spðsÞÞ;u�ðsÞÞdsÞ; t0 6 t 6 t1;

ð10Þ

where xu� ðtÞ is given by the following:

xu� ðtÞ ¼

uðtÞ; �s 6 t 6 t0

uðt0Þ�hðt0;uðt0 � s0ðt0ÞÞ; . . . ;uðt0 � spðt0ÞÞ;u�ðt0ÞÞþhðt; xu� ðt � s0ðtÞÞ; . . . ; xu� ðt � spðtÞÞ;u�ðtÞÞþR t

t0gðs; xðsÞ; xu� ðs� s0ðsÞÞ; . . . ; xu� ðs� spðsÞÞ;u�ðsÞÞds

þR t

t0BðsÞu�ðsÞds: t P t0:

8>>>>>>>>><>>>>>>>>>:

ð11Þ

Substituting (10) for (11) yields xu� ðt1Þ ¼ xf . Since u and xf are arbitrary, system (1) is globally Euclidean controllable. Thusthe proof is complete.

In general, functions h and g are not defined for all x 2 X2 and u 2 U but rather for some smaller subsets. For such cases,some conditions for local Euclidean controllability are very useful. Now we have the following definitions.

Definition 2. System (1) is said to be locally Euclidean controllable with constraint on ½t0; t1� if for each u 2 X3 andxf 2 Snðz1Þ, for some finite z1 P 0, there exist a finite time t1 > t0 and a control function u 2 U1 such that the solutionxðt; t0;u;uÞ of (1) exists and satisfies xðt1; t0;u; uÞ ¼ xf . The system is locally Euclidean null-controllable with constraint on½t0; t1� if in this definition xf ¼ 0.

Definition 3. The domain of null-controllability of system (1) is the set of all initial functions u such that there exist a finitetime t1 > t0 and a control input u such that xðt1; t0;u;uÞ ¼ 0.

Definition 4. The Euclidean attainable set of system (1) at time t is the set }ðt; t0;uÞ ¼ fxðt; t0;u;uÞ : x is a solution of (1)with some admissible controls}.

Definition 5. The Euclidean reachable set of system (1) is the set of all points in Rn that can be attained in the same finitetime t1 from initial function u ¼ 0 by some admissible controls.

Definition 6. System (1) is said to be approximately Euclidean controllable if the closure of the reachable set is dense in Rn.Next, a theorem for local Euclidean controllability of system (1) is given.

Theorem 2. Consider system (1). If the conditions 2 and 3 of Theorem 1 are satisfied and additionally we have that

(I) The functions h and g are continuous and locally Lipschitz with respect to their arguments except t.(II) h and g are defined and bounded by some constants N > 0 and L > 0, respectively, for all x 2 Cð½�s; t1� ! SnðvÞÞ and

u 2 Cð½t0; t1� ! SmðwÞÞ where v and w are certain fixed constants.(III) For some 0 6 r; z 6 v ,

rðr þ zþ 2N þ Lðt1 � t0ÞÞ 6 w; ð12Þ

r þ 2N þ Lðt1 � t0Þ þ awðt1 � t0Þ 6 v; ð13Þ

where a ¼ supt06t6t1jBðtÞj.

(IV) khð:;u1Þ � hð:;u2Þk 6 bku1 � u2k where rb < 0:5; r ¼ kBT H�1ðt0; t1Þk;8u1;u2 2 Cð½t0; t1� ! SmðwÞÞ.

Then system (1) is locally Euclidean controllable with constraint.The proof is similar to that of Theorem 1. An allowable control is any continuous function u : ½t0; t1� ! SmðwÞ. Condition

(12) is to guarantee the existence of a solution of system (1) for allowable controls, and condition (13) is to guarantee theexistence of an allowable control such that the solution of (1) implies xðt1; t0;u;uÞ ¼ xf .

S.Sh. Alaviani / Applied Mathematics and Computation 232 (2014) 1235–1241 1239

Remark 1. Substituting z ¼ 0 for (12), we obtain the domain of null-controllability Cð½�s; t0� ! SnðrÞÞ; similarly for r ¼ 0, oneobtains the Euclidean reachable set SnðzÞ.

Remark 2. Substituting r ¼ kuk for (12) and (13), we obtain the Euclidean attainable set }ðt1; t0Þ ¼ SnðzÞ.

Remark 3. For controllability of the LTI neutral delay system ddt ðxðtÞ þ A0xðt � hÞÞ ¼ AxðtÞ þ A1xðt � hÞ þ BuðtÞ, reader is referred

to [2,14,15] where necessary and sufficient conditions for controllability of this system are provided.

4. Examples

Example 1. Consider the system

ddtðx1ðtÞ � 2 tanhðx1ðt � 1Þ þ quðtÞÞÞ ¼ Arctanðtx1ðt � 1Þx2ðt � 2Þx4ðt � 2ÞuðtÞÞ þ uðtÞ;

ddt

x2ðtÞ �x2ðt � 1Þ

1þ x22ðt � 1Þ

� �¼ cosðtx1ðt � 1Þx3ðt � 2ÞÞ þ tuðtÞ;

ddtðx3ðtÞ þ cos x1ðt � 1Þx3ðt � 2Þð ÞÞ ¼ sinðx1ðt � 1Þx4ðt � 2ÞuðtÞÞ þ t2uðtÞ;

ddt

x4ðtÞ þx4ðt � 2Þ

1þ x24ðt � 2Þ

� �¼ tanhðx1ðt � 1Þx2ðt � 2Þx3ðt � 1Þx4ðt � 2ÞÞ þ t3uðtÞ;0 6 t 6 1;

xðtÞ ¼

0:010:010:010:01

0BBB@

1CCCA:� 2 6 t 6 0:

Hence BðtÞ ¼

1tt2

t3

0BB@

1CCA and Hð0;1Þ ¼

1 12

13

14

12

13

14

15

13

14

15

16

14

15

16

17

0BB@

1CCA. Taking q ¼ 0:0005sinðtÞ, we have that r ¼ 384; b ¼ 0:001;N ¼ 2, and L ¼ p

2.

So the conditions of Theorem 1 are satisfied, and the system is globally Euclidean controllable. If q ¼ 0, then the conditions of[9] fail because j2ðtanhðxÞ � tanhðyÞÞj jx� yj8x; y 2 R, and [2,5–8,10,11] cannot give any condition.

Example 2. Consider the system

ddtðx1ðtÞ � 0:01x2

2ðt � 1Þu1ðtÞu2ðtÞÞ ¼1

40cosðtÞx2

2ðt � 2Þu21ðtÞu2ðtÞ þ u1ðtÞ;

ddtðx2ðtÞ � 0:01 sinðtÞx3

1ðt � 2Þu22ðtÞÞ ¼

140

x1ðt � 2Þx2ðt � 1Þu1ðtÞ þ tu2ðtÞ;0 6 t 6 1;

xðtÞ ¼0:010:01

� �;�2 6 t 6 0;

where x 2 Cð½�2;1� ! S2ð2ÞÞ, and u 2 Cð½0;1� ! S2ð1ÞÞ. It is clear that [2,5–11] cannot give any condition. Here, we have

BðtÞ ¼ 1 00 t

� �and Hð0;1Þ ¼ 1 0

0 13

� �. Hence a ¼ 1;r ¼ 3; L ¼ 0:1;N ¼ 0:08, and b ¼ 0:08. So the conditions of Theorem 2

except III are satisfied. It remains to check the following inequalities:

rðr þ zþ 2N þ Lðt1 � t0ÞÞ ¼ 3ðr þ zþ 0:16þ 0:1Þ 6 1) r þ z 611

150;

r þ 2N þ Lðt1 � t0Þ þ awðt1 � t0Þ ¼ r þ 0:16þ 0:1þ 1 6 2) r 6 0:74:

R

E(t)

C1i=g(v)i

lossless transmission line

x=0 x=l

+

-

Fig. 1. A circuit with lossless transmission line.

1240 S.Sh. Alaviani / Applied Mathematics and Computation 232 (2014) 1235–1241

It is seen that for r þ z 6 11150 the above inequalities are satisfied. Therefore, the system is locally Euclidean controllable with

constraint. In fact, the domain of null-controllability is Cð½�2;0� ! S2ð 11150ÞÞ, the Euclidean reachable set is S2ð 11

150Þ, and theEuclidean attainable set at time t1 ¼ 1 is S2ð 19

300Þ.

Example 3. Consider the electrical circuit in Fig. 1. The differential equations for the current iðx; tÞ and the voltage vðx; tÞ are

L@i@t¼ � @v

@x;

C@v@t¼ � @i

@x;

where L > 0 and C > 0 are the parameters of the line, and the boundary conditions are

EðtÞ ¼ vð0; tÞ þ Rið0; tÞ at x ¼ 0; ð14Þ

iðl; tÞ ¼ C1ddt

vðl; tÞ þ gðvðl; tÞÞ at x ¼ l; ð15Þ

where R P 0 and C1 > 0, and gð:Þ is a continuous function. The general solutions of the partial differential equations are

vðx; tÞ ¼ kðx� ltÞ þ wðxþ ltÞ; ð16Þ

iðx; tÞ ¼ 1d

kðx� ltÞ � 1d

wðxþ ltÞ; ð17Þ

where l ¼ ðLCÞ�12 and d ¼ ðLCÞ

12. If we set x ¼ 0 in (16) and (17), and substitute ið0; tÞ and vð0; tÞ for the boundary condition

(14), we obtain

EðtÞ ¼ kð�ltÞ þ wðltÞ þ Rd

kð�ltÞ � Rd

wðltÞ: ð18Þ

Replacing t by t � ll in (18) yields

kðl� ltÞ ¼ ddþ R

� �E t � l

l

� �þ R� d

Rþ d

� �wðlt � lÞ: ð19Þ

Let us define a new variable yðtÞ ¼ wðlþ ltÞ, then

y t � 2ll

� �¼ wðlt � lÞ: ð20Þ

If we set x ¼ l in (16) and (17), and use (19) and (20), we obtain

vðl; tÞ ¼ kðl� ltÞ þ wðlþ ltÞ ¼ ddþ R

� �E t � l

l

� �þ R� d

Rþ d

� �y t � 2l

l

� �þ yðtÞ; ð21Þ

iðl; tÞ ¼ 1d

kðl� ltÞ � 1d

wðlþ ltÞ ¼ 1dþ R

� �E t � l

l

� �þ R� d

dðRþ dÞ

� �y t � 2l

l

� �� 1

dyðtÞ: ð22Þ

Substituting (21) and (22) for (15), and if we set uðtÞ ¼ Eðt � llÞ as the control input, we finally obtain

S.Sh. Alaviani / Applied Mathematics and Computation 232 (2014) 1235–1241 1241

ddt

yðtÞ þ R� dRþ d

� �y t � 2l

l

� �þ d

dþ R

� �uðtÞ

� �¼ � 1

C1d

� �yðtÞ þ R� d

C1dðRþ dÞ

� �y t � 2l

l

� �

� 1C1

g yðtÞ þ R� dRþ d

� �y t � 2l

l

� �þ d

dþ R

� �uðtÞ

� �

þ 1C1ðdþ RÞ

� �uðtÞ: ð23Þ

Obviously 1C1ðdþRÞ–0, so the results of [10,11] fail here; however, Theorem 2 can give some conditions for local Euclidean con-

trollability with constraint of system (23). It is stressed that if R ¼ RðtÞ is time-varying, that makes system (23) be time-vary-ing, the conditions of this paper are still applicable.

5. Conclusions

In this paper, the problem of Euclidean controllability for a class of time-varying nonlinear neutral time-delay systemswhich is more general than those considered in the literature is considered. Sufficient conditions based on Krasnoselskii’sfixed point theorem for global and local Euclidean controllability of the system are derived. For the system with unboundedfunctions, although the conditions given in this paper cannot be applied for global Euclidean controllability, they may beapplied for local Euclidean controllability with constraint. In some cases where sufficient conditions given in the literaturefail to give results for controllability, sufficient conditions here may be useful to give controllability conditions. Finally, someillustrative examples are given to show the results of this study. In the last example, it is shown that an electrical circuit withlossless transmission line can be modeled by the system considered in this paper; therefore, the motivation of this work canbe clearly presented in that example. Future works may be needed to relax or omit some of the conditions given, e.g., thecondition that the matrix B(t) be continuous instead of continuously differentiable.

Acknowledgments

The author should thank the reviewers for their useful comments and his best friend Amin Ramezanifar for giving somereferences for completing the paper.

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