Research ArticleApproximate Controllability of Semilinear ImpulsiveEvolution Equations

Hugo Leiva

Departamento de Matematica Facultad de Ciencias Universidad de los Andes Merida 5101 Venezuela

Correspondence should be addressed to Hugo Leiva hleivaulave

Received 25 August 2014 Accepted 18 October 2014

Academic Editor Xiaofu Ji

Copyright copy 2015 Hugo Leiva This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We prove the approximate controllability of the following semilinear impulsive evolution equation 1199111015840 = 119860119911+119861119906(119905)+119865(119905 119911 119906) 119911 isin

119885 119905 isin (0 120591] 119911(0) = 1199110 119911(119905+

119896) = 119911(119905

minus

119896) + 119868119896(119905119896 119911(119905119896) 119906(119905119896)) 119896 = 1 2 3 119901 where 0 lt 119905

1lt 1199052lt 1199053lt sdot sdot sdot lt 119905

119901lt 120591 119885 and 119880 are

Hilbert spaces 119906 isin 1198712(0 120591 119880) 119861 119880 rarr 119885 is a bounded linear operator 119868

119896 119865 [0 120591] times 119885 times 119880 rarr 119885 are smooth functions and

119860 119863(119860) sub 119885 rarr 119885 is an unbounded linear operator in 119885 which generates a strongly continuous semigroup 119879(119905)119905ge0

sub 119885 Wesuppose that 119865 is bounded and the linear system is approximately controllable on [0 120575] for all 120575 isin (0 120591) Under these conditionswe prove the following statement the semilinear impulsive evolution equation is approximately controllable on [0 120591]

1 Introduction

There are many practical examples of impulsive control sys-tems a chemical reactor system with the quantities of differ-ent chemicals serving as the states variable a financial systemwith two state variables of the amount of money in a marketand the saving rates of a central bank and the growth ofa population diffusing throughout its habitat which is oftenmodeled by reaction-diffusion equation for which much hasbeen done under the assumption that the system parametersrelated to the population environment either are constantor change continuously However one may easily visualizesituations in nature where abrupt changes such as harvestingdisasters and instantaneous stoking may occur

This observation motivates us to study the approximatecontrollability of the following semilinear impulsive evolu-tion equation

1199111015840= 119860119911 + 119861119906 (119905) + 119865 (119905 119911 119906) 119911 isin 119885 119905 isin (0 120591]

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119868119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(1)

where 0 lt 1199051

lt 1199052

lt 1199053

lt sdot sdot sdot lt 119905119901

lt 120591 119885 and 119880 areHilbert spaces 119906 isin 119871

2(0 120591 119880) 119861 119880 rarr 119885 is a bounded

linear operator 119868119896 119865 [0 120591] times 119885 times 119880 rarr 119885 are smooth

functions and 119860 119863(119860) sub 119885 rarr 119885 is an unboundedlinear operator in 119885 which generates a strongly continuoussemigroup 119879(119905)

119905ge0sub 119885

Definition 1 (approximate controllability) System (1) is saidto be approximately controllable on [0 120591] if for every 119911

0 1199111isin

119885 120576 gt 0 there exists 119906 isin 1198712(0 120591 119880) such that the solution

119911(119905) of (1) corresponding to 119906 verifies (see Figures 1 and 2)

119911 (0) = 1199110

1003817100381710038171003817119911(120591) minus 119911

1

1003817100381710038171003817119885

lt 120576 (2)

We assume the following main hypotheses

(A) 119865 is a bounded function

(B) linear system without impulses (8) is approximatelycontrollable on [120591 minus 120575 120591] for all 0 lt 120575 lt 120591

That is the Gramian controllability operator

119876120575= 119866120575119866lowast

120575= int

120575

0

119879 (119905) 119861119861lowast119879lowast

(119905) 119889119905 (3)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 797439 7 pageshttpdxdoiorg1011552015797439

2 Abstract and Applied Analysis

z(0) = z0

z(120591) = z1

Figure 1

z(120591)

z(0) = z0

z1

120598

Figure 2

satisfies119876120575gt 0 for all 0 lt 120575 lt 120591 which is equivalent accord-

ing to (13) and Lemma 3(c) to the approximate controllabilityof linear system (8) on [120591 minus 120575 120591] for all 0 lt 120575 lt 120591

This paper has been motivated by the works done inBashirov and Ghahramanlou [1] Bashirov and Jneid [2] andBashirov et al [3] where a new technique to prove the con-trollability of evolution equations without impulses is usedavoiding fixed point theorems and the work done in [4]

The controllability of impulsive evolution equations hasbeen studied recently by several authors but most of themstudy the exact controllability only it is worth mentioningthat Chalishajar [5] studied the exact controllability of impul-sive partial neutral functional differential equations withinfinite delay Radhakrishnan and Balachandran [6] studiedthe exact controllability of semilinear impulsive integrodiffer-ential evolution systems with nonlocal conditions and Selviand Mallika Arjunan [7] studied the exact controllability forimpulsive differential systems with finite delay To our knowl-edge there are a few works on approximate controllability ofimpulsive semilinear evolution equations worth mentioningChen and Li [8] studied the approximate controllability ofimpulsive differential equations with nonlocal conditionsusing measure of noncompactness and Monch fixed pointtheorem and assuming that the nonlinear term 119891(119905 119911) doesnot depend on the control variable and Leiva andMerentes in[4] studied the approximate controllability of the semilinearimpulsive heat equation using the fact that the semigroupgenerated by Δ is compact

In this paper we are not assuming the compactness of thesemigroup 119879(119905)

119905ge0generated by the unbounded operator

119860 when this semigroup is compact we can consider weakercondition on the nonlinear perturbation 119865 and in the linear

part of the system without impulses Specifically we canassume the following hypotheses

(a)

119865 (119905 119911 119906)119885le 11988601199111205720

119885+ 11988701199061205730

119885+ 1198880

1003817100381710038171003817119868119896(119905 119911 119906)

1003817100381710038171003817119885

le 119886119896119911120572119896

119885+ 119887119896119906120573119896

119885+ 119888119896

119896 = 1 2 3 119901

(4)

with 12 le 120572119896lt 1 12 le 120573

119896lt 1 119896 = 0 1 2 3 119901

(b) the linear system is approximately controllable onlyon [0 120591]

This case is similar to the semilinear impulsive heatequations studied in [4] where the authors use conditions(a) and (b) the compactness of the semigroup generated bythe Laplacian operator Δ and Rothersquos fixed point theorem toprove the approximate controllability of the system on [0 120591]

When it comes to the wave equation the situation istotally different the semigroup generated by the linear part isnot compact it is in fact a group which can never be compactFurthermore if the control acts on a portion 120596 of the domainΩ for the spatial variable then the system is approximatelycontrollable only on [0 120591] for 120591 ge 2 which was proved in [9]where the following system governed by the wave equationswas studied

119910119905119905

= Δ119910 + 1120596119906 (119905 119909) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

(5)

whereΩ is a bounded domain inR119899 120596 is an open nonemptysubset of Ω 1

120596denotes the characteristic function of the set

120596 the distributed control 119906 isin 1198712([0 120591] 119871

2(Ω)) and 119910

0isin

1198672(Ω) cap 119867

1

0 1199101isin 1198712(Ω)

However if the control acts on the whole domain Ω itwas proved in [10] that the system is controllable [0 120591] forall 120591 gt 0 More specifically the authors studied the followingsystem

119910119905119905

= Δ119910 + 119906 (119905 119909) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

(6)

where Ω is a bounded domain in R119899 the distributed control119906 isin 119871

2([0 120591] 119871

2(Ω)) and 119910

0isin 1198672(Ω) cap 119867

1

0 1199101isin 1198712(Ω)

To justify the use of this new technique [1] in this paperwe consider as an application the semilinear impulsive wave

Abstract and Applied Analysis 3

equation with controls acting on the whole domainΩ so thathypotheses (A) and (B) hold

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909)

+ 119868119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(7)

where 0 lt 1199051

lt 1199052

lt 1199053

lt sdot sdot sdot lt 119905119901

lt 120591 Ω is a boundeddomain in R119899 the distributed control 119906 isin 119871

2([0 120591] 119871

2(Ω))

1199100isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891 are smooth functions

with 119891 being bounded

2 Controllability of the Linear Equationwithout Impulses

In this section we will present some characterization ofthe approximate controllability of the corresponding linearequations without impulses To this end we note that for all1199110isin 119885 and 119906 isin 119871

2(0 120591 119880) the initial value problem

1199111015840= 119860119911 + 119861119906 (119905) 119911 isin 119885

119911 (1199050) = 1199110

(8)

admits only one mild solution given by

119911 (119905) = 119911 (119905 1199050 1199110 119906)

= 119879 (119905) 1199110

+ int

119905

1199050

119879 (119905 minus 119904) 119861119906 (119904) 119889119904 119905 isin [1199050 120591] 0 le 119905

0le 120591

(9)

Definition 2 For system (8) one defines the following con-cept the controllability maps 119866

120591120575 1198712(120591 minus 120575 120591 119880) rarr 119885

119866120575 1198712(0 120575 119880) rarr 119885 defined by

119866120591120575119906 = int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906 (119904) 119889119904 119906 isin 1198712

(120591 minus 120575 120591 119880)

119866120575V = int

120575

0

119879 (119904) 119861V (119904) 119889119904 V isin 1198712

(0 120575 119880)

(10)

satisfy the following relation

119866120591120575119906 = int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906 (119904) 119889119904

= int

120575

0

119879 (119904) 119861119906 (120591 minus 119904) 119889119904

= 119866120575119906 (120591 minus sdot)

(11)

The adjoints of these operators 119866lowast

120591120575 119885 rarr 119871

2(120591 minus 120575 120591 119880)

119866lowast

120575 119885 rarr 119871

2(0 120575 119880) are given by

(119866lowast

120591120575119911) (119905) = 119861

lowast119879lowast

(120591 minus 119905) 119911 119905 isin [120591 minus 120575 120591]

(119866lowast

120575119909) (119905) = 119861

lowast119879lowast

(119905) 119911 119905 isin [0 120575]

(12)

The Gramian controllability operators are given by (3) and

119876120591120575

= 119866120591120575119866lowast

120591120575= int

120591

120591minus120575

119879 (120591 minus 119905) 119861119861lowast119879lowast

(120591 minus 119905) 119889119905 = 119876120575 (13)

The following lemma holds in general for a linearbounded operator 119866 119882 rarr 119885 between Hilbert spaces 119882

and 119885 (see [4 11 12])

Lemma 3 The following statements are equivalent to theapproximate controllability of the linear system (8) on [120591minus120575 120591]

(a) Range (119866120591120575) = 119885

(b) Ker(119866lowast120591120575) = 0

(c) ⟨119876120591120575119911 119911⟩ gt 0 119911 = 0 in 119885

(d) lim120572rarr0

+120572(120572119868 + 119876120591120575)minus1119911 = 0

(e) For all 119911 isin 119885 one has 119866120591120575119906120572

= 119911 minus 120572(120572119868 + 119876119879120575

)minus1119911

where

119906120572= 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911 120572 isin (0 1] (14)

So lim120572rarr0

119866120591120575119906120572

= 119911 and the error 119864120591120575119911 of this

approximation is given by the formula

119864120591120575119911 = 120572 (120572119868 + 119876

120591120575)minus1

119911 120572 isin (0 1] (15)

(f) Moreover if one considers for each V isin 1198712(120591 minus 120575 120591 119880)

the sequence of controls given by

119906120572= 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911

+ (V minus 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119866120591120575V) 120572 isin (0 1]

(16)

one gets that

119866120591120575119906120572= 119911 minus 120572 (120572119868 + 119876

119879120575)minus1

(119911 + 119866120591120575V)

lim120572rarr0

119866120591120575119906120572= 119911

(17)

with the error 119864120591120575119911 of this approximation given by the

formula

119864120591120575119911 = 120572 (120572119868 + 119876

120591120575)minus1

(119911 + 119866120591120575V) 120572 isin (0 1] (18)

Remark 4 The foregoing lemma implies that the family oflinear operators Γ

120572120591120575 119885 rarr 119882 defined for 0 lt 120572 le 1 by

Γ120572120591120575

119911 = 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911 (19)

is an approximate inverse for the right of the operator 119882 inthe sense that

lim120572rarr0

119866120591120575Γ120572120591120575

= 119868 (20)

in the strong topology

4 Abstract and Applied Analysis

Lemma5 119876120575gt 0 if and only if linear system (8) is controllable

on [120591 minus 120575 120591] Moreover given an initial state 1199100and a final

state 1199111 one can find a sequence of controls 119906120575

1205720lt120572le1

sub 1198712(120591 minus

120575 120591 119880)

119906120572= 119866lowast

120591120575(120572119868 + 119866

120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120591) 119910

0) 120572 isin (0 1]

(21)

such that the solutions 119910(119905) = 119910(119905 120591 minus 120575 1199100 119906120575

120572) of the initial

value problem

1199101015840= 119860119910 + 119861119906

120572(119905) 119910 isin 119885 119905 gt 0

119910 (120591 minus 120575) = 1199100

(22)

satisfy

lim120572rarr0

+

119910 (120591 120591 minus 120575 1199100 119906120572) = 1199111 (23)

that is

lim120572rarr0

+

119910 (120591)

= lim120572rarr0

+

119879 (120575) 1199100+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 = 119911

1

(24)

3 Controllability of the SemilinearImpulsive System

In this section we will prove the main result of this paperthe approximate controllability of the semilinear impulsiveevolution equation given by (1) To this end for all 119911

0isin 119885

and 119906 isin 1198712(0 120591 119880) the initial value problem

1199111015840= 119860119911 + 119861119906 + 119865 (119905 119911 119906) 119905 isin (0 120591] 119905 = 119905

119896 119911 isin 119885

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119868119896(119905 119911 (119905

119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(25)

admits only one mild solution given by

119911119906(119905) = 119879 (119905) 119911

0+ int

119905

0

119879 (119905 minus 119904) 119861119906 (119904) 119889119904

+ int

119905

0

119879 (119905 minus 119904) 119865 (119904 119911119906(119904) 119906 (119904)) 119889119904

+ sum

0lt119905119896lt119905

119879 (119905 minus 119905119896) 119868119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119905 isin [0 120591]

(26)

Now we are ready to present and prove themain result of thispaper which is the approximate controllability of semilinearimpulsive equation (1)

Theorem 6 Under conditions (A) and (B) semilinear impul-sive system (1) is approximately controllable on [0 120591]

Proof Given an initial state 1199110 a final state 119911

1 and 120598 gt 0 we

want to find a control 119906120575120572

isin 1198712(0 120591 119880) steering the system

from 1199110to an 120598-neighborhood of 119911

1on time 120591 Specifically

lim120572rarr0

119879 (120591) 1199110+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119896(119911120572(119905119896) 119906120575

120572(119905119896)) = 119911

1

(27)

Consider any 119906 isin 1198712(0 120591 119880) and the corresponding solution

119911(119905) = 119911(119905 0 1199110 119906) of initial value problem (25) For 120572 isin

(0 1] we define the control 119906120575120572isin 1198712(0 120591 119880) as follows

119906120575

120572(119905) =

119906 (119905) if 0 le 119905 le 120591 minus 120575

119906120572(119905) if 120591 minus 120575 lt 119905 le 120591

(28)

where119906120572(119905) = 119861

lowast119879lowast

(120591 minus 119905) (120572119868 + 119866120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120575) 119911 (120591 minus 120575))

120591 minus 120575 lt 119905 le 120591

(29)Now assume that 0 lt 120575 lt 120591 minus 119905

119901 Then the corresponding

solution 119911120575

120572(119905) = 119911(119905 0 119911

0 119906120575

120572) of initial value problem (25) at

time 120591 can be written as follows

119911120575

120572(120591) = 119879 (120591) 119911

0+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

= 119879 (120575)

119879 (120591 minus 120575) 1199110+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591minus120575

119879 (120591 minus 120575 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

= 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(30)

Abstract and Applied Analysis 5

So

119911120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(31)

The corresponding solution 119910120575

120572(119905) = 119910(119905 120591 minus120575 119911(120591minus120575) 119906

120572) of

initial value problem (22) at time 120591 is given by

119910120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 (32)

Therefore

10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le int

120591

120591minus120575

119879 (120591 minus 119904)

10038171003817100381710038171003817119865 (119904 119911

120575

120572(119904) 119906120572(119904))

10038171003817100381710038171003817119889119904

(33)

Then putting 119872 = sup0le119905le120591

119879(119905) and 119870 = sup[0120591]times119885times119880

119865(119905

119911 119906) we obtain the following estimate10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le 119872119870120575 (34)

Hence10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 119872119870120575 +

10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817 (35)

From Lemma 5 there exists 120572 gt 0 such that10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le

120598

2

(36)

So taking 120575 lt min120591 minus 119905119901 1205982119872119870 we obtain for the corres-

ponding control 119906120575120572that

10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 120598 (37)

Geometrically the proof goes as shown in Figure 3This completes the proof of the theorem

4 Applications

As an application we will prove the approximate controllabil-ity of the following control system governed by the semilinearimpulsive wave equation

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

(38)

where Ω is a bounded domain in R119899 the distributed control119906 isin 119871

2([0 120591] 119871

2(Ω)) 119910

0isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891

are smooth functions with 119891 being bounded

z0

z(120591 minus 120575)

z(120591)

z120575120572(120591)

z1

y120575120572(120591)

120576

Figure 3

41 Abstract Formulation of the Problem In this part we willchoose the space where this problem will be set up as anabstract control system in a Hilbert space Let 119883 = 119871

2(Ω) =

1198712(ΩR) and consider the linear unbounded operator 119860

119863(119860) sub 119883 rarr 119883 defined by 119860120601 = minusΔ120601 where

119863(119860) = 1198672

(ΩR) cap 1198671

0(ΩR) (39)

Then the eigenvalues 120582119895of 119860 have finite multiplicity 120574

119895equal

to the dimension of the corresponding eigenspace and 0 lt

1205821lt 1205822lt sdot sdot sdot lt 120582

119899rarr infin Moreover consider the following

(a) There exists a complete orthonormal set 120601119895119896

ofeigenvectors of 119860(b) For all 119909 isin 119863(119860) we have

119860119909 =

infin

sum

119895=1

120582119895

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

=

infin

sum

119895=1

120582119895119864119895119909 (40)

where ⟨sdot sdot⟩ is the usual inner product in 1198712 and

119864119895119909 =

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

(41)

which means the set 119864119895infin

119895=1is a complete family of

orthogonal projections in119883 and119909 = suminfin

119895=1119864119895119909119909 isin 119883

(c) minus119860 generates an analytic semigroup 119890minus119860119905

givenby

119890minus119860119905

119909 =

infin

sum

119895=1

119890minus120582119895119905

119864119895119909 (42)

(d) The fractional powered spaces119883119903 are given by

119883119903= 119863 (119860

119903) = 119909 isin 119883

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

lt infin 119903 ge 0

(43)

6 Abstract and Applied Analysis

with the norm

119909119903=

10038171003817100381710038171198601199031199091003817100381710038171003817=

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

12

119909 isin 119883119903

119860119903119909 =

infin

sum

119899=1

120582119903

119899119864119899119909

(44)

Also for 119903 ge 0 we define 119885119903= 119883119903times 119883 which is a

Hilbert space endowed with the norm given by10038171003817100381710038171003817100381710038171003817

[119910

V]10038171003817100381710038171003817100381710038171003817119885119903

= radic10038171003817100381710038171199101003817100381710038171003817

2

119903+ V2 (45)

Then (1) can be written as abstract second order ordinarydifferential equations in 119885

12as follows

11991010158401015840= minus119860119910 + 119906 + 119891

119890(119905 119910 119910

1015840 119906) 119905 isin (0 120591] 119905 = 119905

119896

119910 (0) = 1199100 119910

1015840

(0) = 1199101

1199101015840(119905+

119896) = 1199101015840(119905minus

119896) + 119868119890

119896(119905119896 119910 (119905119896) 1199101015840(119905119896) 119906 (119905

119896))

119896 = 1 2 3 119901

(46)

where 119868119890

119896 119891119890 [0 120591] times 119885

12times 119880 rarr 119885

12are defined by

119868119890

119896(119905 119910 V 119906) (119909) = 119868

119896(119905 119910 (119909) V (119909) 119906 (119909))

119891119890

(119905 119911 119906) (119909) = 119891 (119905 119910 (119909) V (119909) 119906 (119909))

forall119909 isin Ω 119896 = 1 2 119901

(47)

With the change of variables 1199101015840

= V we can write secondorder equation (46) as a first order system of ordinarydifferential equations in the Hilbert space 119885

12= 11988312

times 119883

as follows1199111015840= A119911 + 119861119906 + 119865 (119905 119911 119906 (119905)) 119911 isin 119885

12 119905 isin (0 120591] 119905 = 119905

119896

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119869119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(48)

where 119906 isin 1198712([0 120591] 119880) 119880 = 119883 = 119871

2(Ω)

119911 = [119910

V] 119861 = [0

119868] A = [

0 119868119883

minus119860 0] (49)

is an unbounded linear operator with domain 119863(A) =

119863(119860) times 119863(11986012

) and 119869119896 119865 [0 120591] times 119885

12times 119880 rarr 119885

12are

defined by

119865 (119905 119911 119906) = [0

119891119890(119905 119910 V 119906)]

119869119896(119905 119911 119906) = [

0

119868119890

119896(119905 119908 V 119906)]

(50)

It is well known that the operatorA generates a stronglycontinuous group 119879(119905)

119905ge0in the space 119885 = 119885

12= 11988312

times119883

(see [13]) The following representation for this group can befound in [9] as Theorem 22

Proposition 7 The group 119879(119905)119905ge0

generated by the operatorA has the following representation

119879 (119905) 119911 =

infin

sum

119899119895=1

119890119860119895119905

119875119895119911 119911 isin 119885

12 119905 ge 0 (51)

where 119875119895119895ge0

is a complete family of orthogonal projections inthe Hilbert space 119885

12given by

119875119895= [

119864119895

0

0 119864119895

] 119895 ge 1

119860119895= 119877119895119875119895 119877119895= [

0 1

minus120582119895

0] 119895 ge 1

(52)

42 Approximate Controllability Now we are ready to for-mulate and prove the main result of this section which is theapproximate of the semilinear impulsive wave equation withbounded nonlinear perturbation

Theorem 8 Semilinear impulsive wave equation (38) isapproximately controllable on [0 120591]

Proof From [10] we know that the corresponding linearsystem without impulses

1199111015840= A119911 + 119861119906 119911 isin 119885

12 119905 isin (0 120591]

119911 (0) = 1199110

(53)

is exactly controllable on [0 120575] for all 0 lt 120575 lt 120591 On theother hand since 119891 is bounded there exists 119872 gt 0 such thatsup[0120591]timesRtimesRtimesR119891(119905 119910 V 119906) le 119872 Therefore we obtain

119865(119905 119911 119906)11988512

= int

Ω

1003816100381610038161003816119891 (119905 119910 (119909) V (119909) 119906 (119909))

1003816100381610038161003816

2

119889119909

12

le radic120583 (Ω)119872

(54)

for all (119905 119911 119906) isin [0 120591] times 11988512

times 119880 with 120583(Ω) the Lebesguemeasure of Ω Hence hypotheses (A) and (B) in Theorem 6are satisfied and we get the result

5 Final Remark

This technique can be applied to those systems where thelinear part does not generate a compact semigroup and iscontrollable on any [0 120575] for 120575 gt 0 and the nonlinearperturbation is bounded Example of such systems is thefollowing controlled thermoelastic plate equation whoselinear part was studied in [10]

119910119905119905+ Δ2119910 + 120572Δ120579 = 119906

1(119905 119909) + 119891

1(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

Abstract and Applied Analysis 7

120579119905minus 120573Δ120579 minus 120572Δ119910

119905= 1199062(119905 119909) + 119891

2(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

120579 = 119910 = Δ119910 = 0 on (0 120591) times 120597Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

1

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

120579 (119905+

119896 119909) = 120579 (119905

minus

119896 119909) + 119868

2

119896(119905119896 120579 (119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(55)

in the space 119885 = 1198831times 119883 times 119883 where Ω is a bounded domain

inR119899 the distributed controls 1199061 1199062isin 1198712([0 120591] 119871

2(Ω)) and

119868119894

119896 119891119894are smooth functions with 119891

119894 119894 = 1 2 being bounded

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV

References

[1] A E Bashirov and N Ghahramanlou ldquoOn partial approximatecontrollability of semilinear systemsrdquo Cogent Engineering vol1 no 1 2014

[2] A E Bashirov and M Jneid ldquoOn partial complete controlla-bility of semilinear systemsrdquo Abstract and Applied Analysis vol2013 Article ID 521052 8 pages 2013

[3] A E Bashirov N Mahmudov N Semı and H Etıkan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007

[4] H Leiva and N Merentes ldquoApproximate controllability of theimpul sive semilinear heat equationrdquo to appear in Journal ofMathematics and Applications

[5] D N Chalishajar ldquoControllability of impulsive partial neutralfunctional differential equation with infinite delayrdquo Interna-tional Journal of Mathematical Analysis vol 5 no 5ndash8 pp 369ndash380 2011

[6] B Radhakrishnan andK Balachandran ldquoControllability resultsfor semilinear impulsive integrodifferential evolution systemswith nonlocal conditionsrdquo Journal of Control Theory and Appli-cations vol 10 no 1 pp 28ndash34 2012

[7] S Selvi and M Mallika Arjunan ldquoControllability results forimpulsive differential systems with finite delayrdquo Journal ofNonlinear Science and Its Applications vol 5 no 3 pp 206ndash2192012

[8] L Chen and G Li ldquoApproximate controllability of impulsivedifferential equations with nonlocal conditionsrdquo InternationalJournal of Nonlinear Science vol 10 no 4 pp 438ndash446 2010

[9] H Leiva and N Merentes ldquoControllability of second-orderequations in 119871

2(Ω)rdquoMathematical Problems in Engineering vol

2010 Article ID 147195 11 pages 2010

[10] H Larez H Leiva and J Uzcategui ldquoControllability of blockdiagonal systems and applicationsrdquo International Journal of Sys-tems Control and Communications vol 3 no 1 2011

[11] R F Curtain andA J Pritchard InfiniteDimensional Linear Sys-tems vol 8 of Lecture Notes in Control and Information SciencesSpringer Berlin Germany 1978

[12] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory vol 21 of Text in Applied Mathe-matics Springer New York NY USA 1995

[13] S P Chen and R Triggiani ldquoProof of extensions of two conjec-tures on structural damping for elastic systemsrdquo Pacific Journalof Mathematics vol 136 no 1 pp 15ndash55 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

z(0) = z0

z(120591) = z1

Figure 1

z(120591)

z(0) = z0

z1

120598

Figure 2

satisfies119876120575gt 0 for all 0 lt 120575 lt 120591 which is equivalent accord-

ing to (13) and Lemma 3(c) to the approximate controllabilityof linear system (8) on [120591 minus 120575 120591] for all 0 lt 120575 lt 120591

This paper has been motivated by the works done inBashirov and Ghahramanlou [1] Bashirov and Jneid [2] andBashirov et al [3] where a new technique to prove the con-trollability of evolution equations without impulses is usedavoiding fixed point theorems and the work done in [4]

The controllability of impulsive evolution equations hasbeen studied recently by several authors but most of themstudy the exact controllability only it is worth mentioningthat Chalishajar [5] studied the exact controllability of impul-sive partial neutral functional differential equations withinfinite delay Radhakrishnan and Balachandran [6] studiedthe exact controllability of semilinear impulsive integrodiffer-ential evolution systems with nonlocal conditions and Selviand Mallika Arjunan [7] studied the exact controllability forimpulsive differential systems with finite delay To our knowl-edge there are a few works on approximate controllability ofimpulsive semilinear evolution equations worth mentioningChen and Li [8] studied the approximate controllability ofimpulsive differential equations with nonlocal conditionsusing measure of noncompactness and Monch fixed pointtheorem and assuming that the nonlinear term 119891(119905 119911) doesnot depend on the control variable and Leiva andMerentes in[4] studied the approximate controllability of the semilinearimpulsive heat equation using the fact that the semigroupgenerated by Δ is compact

In this paper we are not assuming the compactness of thesemigroup 119879(119905)

119905ge0generated by the unbounded operator

119860 when this semigroup is compact we can consider weakercondition on the nonlinear perturbation 119865 and in the linear

part of the system without impulses Specifically we canassume the following hypotheses

(a)

119865 (119905 119911 119906)119885le 11988601199111205720

119885+ 11988701199061205730

119885+ 1198880

1003817100381710038171003817119868119896(119905 119911 119906)

1003817100381710038171003817119885

le 119886119896119911120572119896

119885+ 119887119896119906120573119896

119885+ 119888119896

119896 = 1 2 3 119901

(4)

with 12 le 120572119896lt 1 12 le 120573

119896lt 1 119896 = 0 1 2 3 119901

(b) the linear system is approximately controllable onlyon [0 120591]

This case is similar to the semilinear impulsive heatequations studied in [4] where the authors use conditions(a) and (b) the compactness of the semigroup generated bythe Laplacian operator Δ and Rothersquos fixed point theorem toprove the approximate controllability of the system on [0 120591]

When it comes to the wave equation the situation istotally different the semigroup generated by the linear part isnot compact it is in fact a group which can never be compactFurthermore if the control acts on a portion 120596 of the domainΩ for the spatial variable then the system is approximatelycontrollable only on [0 120591] for 120591 ge 2 which was proved in [9]where the following system governed by the wave equationswas studied

119910119905119905

= Δ119910 + 1120596119906 (119905 119909) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

(5)

whereΩ is a bounded domain inR119899 120596 is an open nonemptysubset of Ω 1

120596denotes the characteristic function of the set

120596 the distributed control 119906 isin 1198712([0 120591] 119871

2(Ω)) and 119910

0isin

1198672(Ω) cap 119867

1

0 1199101isin 1198712(Ω)

However if the control acts on the whole domain Ω itwas proved in [10] that the system is controllable [0 120591] forall 120591 gt 0 More specifically the authors studied the followingsystem

119910119905119905

= Δ119910 + 119906 (119905 119909) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

(6)

where Ω is a bounded domain in R119899 the distributed control119906 isin 119871

2([0 120591] 119871

2(Ω)) and 119910

0isin 1198672(Ω) cap 119867

1

0 1199101isin 1198712(Ω)

To justify the use of this new technique [1] in this paperwe consider as an application the semilinear impulsive wave

Abstract and Applied Analysis 3

equation with controls acting on the whole domainΩ so thathypotheses (A) and (B) hold

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909)

+ 119868119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(7)

where 0 lt 1199051

lt 1199052

lt 1199053

lt sdot sdot sdot lt 119905119901

lt 120591 Ω is a boundeddomain in R119899 the distributed control 119906 isin 119871

2([0 120591] 119871

2(Ω))

1199100isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891 are smooth functions

with 119891 being bounded

2 Controllability of the Linear Equationwithout Impulses

In this section we will present some characterization ofthe approximate controllability of the corresponding linearequations without impulses To this end we note that for all1199110isin 119885 and 119906 isin 119871

2(0 120591 119880) the initial value problem

1199111015840= 119860119911 + 119861119906 (119905) 119911 isin 119885

119911 (1199050) = 1199110

(8)

admits only one mild solution given by

119911 (119905) = 119911 (119905 1199050 1199110 119906)

= 119879 (119905) 1199110

+ int

119905

1199050

119879 (119905 minus 119904) 119861119906 (119904) 119889119904 119905 isin [1199050 120591] 0 le 119905

0le 120591

(9)

Definition 2 For system (8) one defines the following con-cept the controllability maps 119866

120591120575 1198712(120591 minus 120575 120591 119880) rarr 119885

119866120575 1198712(0 120575 119880) rarr 119885 defined by

119866120591120575119906 = int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906 (119904) 119889119904 119906 isin 1198712

(120591 minus 120575 120591 119880)

119866120575V = int

120575

0

119879 (119904) 119861V (119904) 119889119904 V isin 1198712

(0 120575 119880)

(10)

satisfy the following relation

119866120591120575119906 = int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906 (119904) 119889119904

= int

120575

0

119879 (119904) 119861119906 (120591 minus 119904) 119889119904

= 119866120575119906 (120591 minus sdot)

(11)

The adjoints of these operators 119866lowast

120591120575 119885 rarr 119871

2(120591 minus 120575 120591 119880)

119866lowast

120575 119885 rarr 119871

2(0 120575 119880) are given by

(119866lowast

120591120575119911) (119905) = 119861

lowast119879lowast

(120591 minus 119905) 119911 119905 isin [120591 minus 120575 120591]

(119866lowast

120575119909) (119905) = 119861

lowast119879lowast

(119905) 119911 119905 isin [0 120575]

(12)

The Gramian controllability operators are given by (3) and

119876120591120575

= 119866120591120575119866lowast

120591120575= int

120591

120591minus120575

119879 (120591 minus 119905) 119861119861lowast119879lowast

(120591 minus 119905) 119889119905 = 119876120575 (13)

The following lemma holds in general for a linearbounded operator 119866 119882 rarr 119885 between Hilbert spaces 119882

and 119885 (see [4 11 12])

Lemma 3 The following statements are equivalent to theapproximate controllability of the linear system (8) on [120591minus120575 120591]

(a) Range (119866120591120575) = 119885

(b) Ker(119866lowast120591120575) = 0

(c) ⟨119876120591120575119911 119911⟩ gt 0 119911 = 0 in 119885

(d) lim120572rarr0

+120572(120572119868 + 119876120591120575)minus1119911 = 0

(e) For all 119911 isin 119885 one has 119866120591120575119906120572

= 119911 minus 120572(120572119868 + 119876119879120575

)minus1119911

where

119906120572= 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911 120572 isin (0 1] (14)

So lim120572rarr0

119866120591120575119906120572

= 119911 and the error 119864120591120575119911 of this

approximation is given by the formula

119864120591120575119911 = 120572 (120572119868 + 119876

120591120575)minus1

119911 120572 isin (0 1] (15)

(f) Moreover if one considers for each V isin 1198712(120591 minus 120575 120591 119880)

the sequence of controls given by

119906120572= 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911

+ (V minus 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119866120591120575V) 120572 isin (0 1]

(16)

one gets that

119866120591120575119906120572= 119911 minus 120572 (120572119868 + 119876

119879120575)minus1

(119911 + 119866120591120575V)

lim120572rarr0

119866120591120575119906120572= 119911

(17)

with the error 119864120591120575119911 of this approximation given by the

formula

119864120591120575119911 = 120572 (120572119868 + 119876

120591120575)minus1

(119911 + 119866120591120575V) 120572 isin (0 1] (18)

Remark 4 The foregoing lemma implies that the family oflinear operators Γ

120572120591120575 119885 rarr 119882 defined for 0 lt 120572 le 1 by

Γ120572120591120575

119911 = 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911 (19)

is an approximate inverse for the right of the operator 119882 inthe sense that

lim120572rarr0

119866120591120575Γ120572120591120575

= 119868 (20)

in the strong topology

4 Abstract and Applied Analysis

Lemma5 119876120575gt 0 if and only if linear system (8) is controllable

on [120591 minus 120575 120591] Moreover given an initial state 1199100and a final

state 1199111 one can find a sequence of controls 119906120575

1205720lt120572le1

sub 1198712(120591 minus

120575 120591 119880)

119906120572= 119866lowast

120591120575(120572119868 + 119866

120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120591) 119910

0) 120572 isin (0 1]

(21)

such that the solutions 119910(119905) = 119910(119905 120591 minus 120575 1199100 119906120575

120572) of the initial

value problem

1199101015840= 119860119910 + 119861119906

120572(119905) 119910 isin 119885 119905 gt 0

119910 (120591 minus 120575) = 1199100

(22)

satisfy

lim120572rarr0

+

119910 (120591 120591 minus 120575 1199100 119906120572) = 1199111 (23)

that is

lim120572rarr0

+

119910 (120591)

= lim120572rarr0

+

119879 (120575) 1199100+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 = 119911

1

(24)

3 Controllability of the SemilinearImpulsive System

In this section we will prove the main result of this paperthe approximate controllability of the semilinear impulsiveevolution equation given by (1) To this end for all 119911

0isin 119885

and 119906 isin 1198712(0 120591 119880) the initial value problem

1199111015840= 119860119911 + 119861119906 + 119865 (119905 119911 119906) 119905 isin (0 120591] 119905 = 119905

119896 119911 isin 119885

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119868119896(119905 119911 (119905

119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(25)

admits only one mild solution given by

119911119906(119905) = 119879 (119905) 119911

0+ int

119905

0

119879 (119905 minus 119904) 119861119906 (119904) 119889119904

+ int

119905

0

119879 (119905 minus 119904) 119865 (119904 119911119906(119904) 119906 (119904)) 119889119904

+ sum

0lt119905119896lt119905

119879 (119905 minus 119905119896) 119868119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119905 isin [0 120591]

(26)

Now we are ready to present and prove themain result of thispaper which is the approximate controllability of semilinearimpulsive equation (1)

Theorem 6 Under conditions (A) and (B) semilinear impul-sive system (1) is approximately controllable on [0 120591]

Proof Given an initial state 1199110 a final state 119911

1 and 120598 gt 0 we

want to find a control 119906120575120572

isin 1198712(0 120591 119880) steering the system

from 1199110to an 120598-neighborhood of 119911

1on time 120591 Specifically

lim120572rarr0

119879 (120591) 1199110+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119896(119911120572(119905119896) 119906120575

120572(119905119896)) = 119911

1

(27)

Consider any 119906 isin 1198712(0 120591 119880) and the corresponding solution

119911(119905) = 119911(119905 0 1199110 119906) of initial value problem (25) For 120572 isin

(0 1] we define the control 119906120575120572isin 1198712(0 120591 119880) as follows

119906120575

120572(119905) =

119906 (119905) if 0 le 119905 le 120591 minus 120575

119906120572(119905) if 120591 minus 120575 lt 119905 le 120591

(28)

where119906120572(119905) = 119861

lowast119879lowast

(120591 minus 119905) (120572119868 + 119866120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120575) 119911 (120591 minus 120575))

120591 minus 120575 lt 119905 le 120591

(29)Now assume that 0 lt 120575 lt 120591 minus 119905

119901 Then the corresponding

solution 119911120575

120572(119905) = 119911(119905 0 119911

0 119906120575

120572) of initial value problem (25) at

time 120591 can be written as follows

119911120575

120572(120591) = 119879 (120591) 119911

0+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

= 119879 (120575)

119879 (120591 minus 120575) 1199110+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591minus120575

119879 (120591 minus 120575 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

= 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(30)

Abstract and Applied Analysis 5

So

119911120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(31)

The corresponding solution 119910120575

120572(119905) = 119910(119905 120591 minus120575 119911(120591minus120575) 119906

120572) of

initial value problem (22) at time 120591 is given by

119910120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 (32)

Therefore

10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le int

120591

120591minus120575

119879 (120591 minus 119904)

10038171003817100381710038171003817119865 (119904 119911

120575

120572(119904) 119906120572(119904))

10038171003817100381710038171003817119889119904

(33)

Then putting 119872 = sup0le119905le120591

119879(119905) and 119870 = sup[0120591]times119885times119880

119865(119905

119911 119906) we obtain the following estimate10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le 119872119870120575 (34)

Hence10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 119872119870120575 +

10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817 (35)

From Lemma 5 there exists 120572 gt 0 such that10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le

120598

2

(36)

So taking 120575 lt min120591 minus 119905119901 1205982119872119870 we obtain for the corres-

ponding control 119906120575120572that

10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 120598 (37)

Geometrically the proof goes as shown in Figure 3This completes the proof of the theorem

4 Applications

As an application we will prove the approximate controllabil-ity of the following control system governed by the semilinearimpulsive wave equation

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

(38)

where Ω is a bounded domain in R119899 the distributed control119906 isin 119871

2([0 120591] 119871

2(Ω)) 119910

0isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891

are smooth functions with 119891 being bounded

z0

z(120591 minus 120575)

z(120591)

z120575120572(120591)

z1

y120575120572(120591)

120576

Figure 3

41 Abstract Formulation of the Problem In this part we willchoose the space where this problem will be set up as anabstract control system in a Hilbert space Let 119883 = 119871

2(Ω) =

1198712(ΩR) and consider the linear unbounded operator 119860

119863(119860) sub 119883 rarr 119883 defined by 119860120601 = minusΔ120601 where

119863(119860) = 1198672

(ΩR) cap 1198671

0(ΩR) (39)

Then the eigenvalues 120582119895of 119860 have finite multiplicity 120574

119895equal

to the dimension of the corresponding eigenspace and 0 lt

1205821lt 1205822lt sdot sdot sdot lt 120582

119899rarr infin Moreover consider the following

(a) There exists a complete orthonormal set 120601119895119896

ofeigenvectors of 119860(b) For all 119909 isin 119863(119860) we have

119860119909 =

infin

sum

119895=1

120582119895

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

=

infin

sum

119895=1

120582119895119864119895119909 (40)

where ⟨sdot sdot⟩ is the usual inner product in 1198712 and

119864119895119909 =

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

(41)

which means the set 119864119895infin

119895=1is a complete family of

orthogonal projections in119883 and119909 = suminfin

119895=1119864119895119909119909 isin 119883

(c) minus119860 generates an analytic semigroup 119890minus119860119905

givenby

119890minus119860119905

119909 =

infin

sum

119895=1

119890minus120582119895119905

119864119895119909 (42)

(d) The fractional powered spaces119883119903 are given by

119883119903= 119863 (119860

119903) = 119909 isin 119883

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

lt infin 119903 ge 0

(43)

6 Abstract and Applied Analysis

with the norm

119909119903=

10038171003817100381710038171198601199031199091003817100381710038171003817=

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

12

119909 isin 119883119903

119860119903119909 =

infin

sum

119899=1

120582119903

119899119864119899119909

(44)

Also for 119903 ge 0 we define 119885119903= 119883119903times 119883 which is a

Hilbert space endowed with the norm given by10038171003817100381710038171003817100381710038171003817

[119910

V]10038171003817100381710038171003817100381710038171003817119885119903

= radic10038171003817100381710038171199101003817100381710038171003817

2

119903+ V2 (45)

Then (1) can be written as abstract second order ordinarydifferential equations in 119885

12as follows

11991010158401015840= minus119860119910 + 119906 + 119891

119890(119905 119910 119910

1015840 119906) 119905 isin (0 120591] 119905 = 119905

119896

119910 (0) = 1199100 119910

1015840

(0) = 1199101

1199101015840(119905+

119896) = 1199101015840(119905minus

119896) + 119868119890

119896(119905119896 119910 (119905119896) 1199101015840(119905119896) 119906 (119905

119896))

119896 = 1 2 3 119901

(46)

where 119868119890

119896 119891119890 [0 120591] times 119885

12times 119880 rarr 119885

12are defined by

119868119890

119896(119905 119910 V 119906) (119909) = 119868

119896(119905 119910 (119909) V (119909) 119906 (119909))

119891119890

(119905 119911 119906) (119909) = 119891 (119905 119910 (119909) V (119909) 119906 (119909))

forall119909 isin Ω 119896 = 1 2 119901

(47)

With the change of variables 1199101015840

= V we can write secondorder equation (46) as a first order system of ordinarydifferential equations in the Hilbert space 119885

12= 11988312

times 119883

as follows1199111015840= A119911 + 119861119906 + 119865 (119905 119911 119906 (119905)) 119911 isin 119885

12 119905 isin (0 120591] 119905 = 119905

119896

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119869119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(48)

where 119906 isin 1198712([0 120591] 119880) 119880 = 119883 = 119871

2(Ω)

119911 = [119910

V] 119861 = [0

119868] A = [

0 119868119883

minus119860 0] (49)

is an unbounded linear operator with domain 119863(A) =

119863(119860) times 119863(11986012

) and 119869119896 119865 [0 120591] times 119885

12times 119880 rarr 119885

12are

defined by

119865 (119905 119911 119906) = [0

119891119890(119905 119910 V 119906)]

119869119896(119905 119911 119906) = [

0

119868119890

119896(119905 119908 V 119906)]

(50)

It is well known that the operatorA generates a stronglycontinuous group 119879(119905)

119905ge0in the space 119885 = 119885

12= 11988312

times119883

(see [13]) The following representation for this group can befound in [9] as Theorem 22

Proposition 7 The group 119879(119905)119905ge0

generated by the operatorA has the following representation

119879 (119905) 119911 =

infin

sum

119899119895=1

119890119860119895119905

119875119895119911 119911 isin 119885

12 119905 ge 0 (51)

where 119875119895119895ge0

is a complete family of orthogonal projections inthe Hilbert space 119885

12given by

119875119895= [

119864119895

0

0 119864119895

] 119895 ge 1

119860119895= 119877119895119875119895 119877119895= [

0 1

minus120582119895

0] 119895 ge 1

(52)

42 Approximate Controllability Now we are ready to for-mulate and prove the main result of this section which is theapproximate of the semilinear impulsive wave equation withbounded nonlinear perturbation

Theorem 8 Semilinear impulsive wave equation (38) isapproximately controllable on [0 120591]

Proof From [10] we know that the corresponding linearsystem without impulses

1199111015840= A119911 + 119861119906 119911 isin 119885

12 119905 isin (0 120591]

119911 (0) = 1199110

(53)

is exactly controllable on [0 120575] for all 0 lt 120575 lt 120591 On theother hand since 119891 is bounded there exists 119872 gt 0 such thatsup[0120591]timesRtimesRtimesR119891(119905 119910 V 119906) le 119872 Therefore we obtain

119865(119905 119911 119906)11988512

= int

Ω

1003816100381610038161003816119891 (119905 119910 (119909) V (119909) 119906 (119909))

1003816100381610038161003816

2

119889119909

12

le radic120583 (Ω)119872

(54)

for all (119905 119911 119906) isin [0 120591] times 11988512

times 119880 with 120583(Ω) the Lebesguemeasure of Ω Hence hypotheses (A) and (B) in Theorem 6are satisfied and we get the result

5 Final Remark

This technique can be applied to those systems where thelinear part does not generate a compact semigroup and iscontrollable on any [0 120575] for 120575 gt 0 and the nonlinearperturbation is bounded Example of such systems is thefollowing controlled thermoelastic plate equation whoselinear part was studied in [10]

119910119905119905+ Δ2119910 + 120572Δ120579 = 119906

1(119905 119909) + 119891

1(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

Abstract and Applied Analysis 7

120579119905minus 120573Δ120579 minus 120572Δ119910

119905= 1199062(119905 119909) + 119891

2(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

120579 = 119910 = Δ119910 = 0 on (0 120591) times 120597Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

1

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

120579 (119905+

119896 119909) = 120579 (119905

minus

119896 119909) + 119868

2

119896(119905119896 120579 (119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(55)

in the space 119885 = 1198831times 119883 times 119883 where Ω is a bounded domain

inR119899 the distributed controls 1199061 1199062isin 1198712([0 120591] 119871

2(Ω)) and

119868119894

119896 119891119894are smooth functions with 119891

119894 119894 = 1 2 being bounded

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV

References

[1] A E Bashirov and N Ghahramanlou ldquoOn partial approximatecontrollability of semilinear systemsrdquo Cogent Engineering vol1 no 1 2014

[2] A E Bashirov and M Jneid ldquoOn partial complete controlla-bility of semilinear systemsrdquo Abstract and Applied Analysis vol2013 Article ID 521052 8 pages 2013

[3] A E Bashirov N Mahmudov N Semı and H Etıkan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007

[4] H Leiva and N Merentes ldquoApproximate controllability of theimpul sive semilinear heat equationrdquo to appear in Journal ofMathematics and Applications

[5] D N Chalishajar ldquoControllability of impulsive partial neutralfunctional differential equation with infinite delayrdquo Interna-tional Journal of Mathematical Analysis vol 5 no 5ndash8 pp 369ndash380 2011

[6] B Radhakrishnan andK Balachandran ldquoControllability resultsfor semilinear impulsive integrodifferential evolution systemswith nonlocal conditionsrdquo Journal of Control Theory and Appli-cations vol 10 no 1 pp 28ndash34 2012

[7] S Selvi and M Mallika Arjunan ldquoControllability results forimpulsive differential systems with finite delayrdquo Journal ofNonlinear Science and Its Applications vol 5 no 3 pp 206ndash2192012

[8] L Chen and G Li ldquoApproximate controllability of impulsivedifferential equations with nonlocal conditionsrdquo InternationalJournal of Nonlinear Science vol 10 no 4 pp 438ndash446 2010

[9] H Leiva and N Merentes ldquoControllability of second-orderequations in 119871

2(Ω)rdquoMathematical Problems in Engineering vol

2010 Article ID 147195 11 pages 2010

[10] H Larez H Leiva and J Uzcategui ldquoControllability of blockdiagonal systems and applicationsrdquo International Journal of Sys-tems Control and Communications vol 3 no 1 2011

[11] R F Curtain andA J Pritchard InfiniteDimensional Linear Sys-tems vol 8 of Lecture Notes in Control and Information SciencesSpringer Berlin Germany 1978

[12] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory vol 21 of Text in Applied Mathe-matics Springer New York NY USA 1995

[13] S P Chen and R Triggiani ldquoProof of extensions of two conjec-tures on structural damping for elastic systemsrdquo Pacific Journalof Mathematics vol 136 no 1 pp 15ndash55 1989

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

equation with controls acting on the whole domainΩ so thathypotheses (A) and (B) hold

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909)

+ 119868119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(7)

where 0 lt 1199051

lt 1199052

lt 1199053

lt sdot sdot sdot lt 119905119901

lt 120591 Ω is a boundeddomain in R119899 the distributed control 119906 isin 119871

2([0 120591] 119871

2(Ω))

1199100isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891 are smooth functions

with 119891 being bounded

2 Controllability of the Linear Equationwithout Impulses

In this section we will present some characterization ofthe approximate controllability of the corresponding linearequations without impulses To this end we note that for all1199110isin 119885 and 119906 isin 119871

2(0 120591 119880) the initial value problem

1199111015840= 119860119911 + 119861119906 (119905) 119911 isin 119885

119911 (1199050) = 1199110

(8)

admits only one mild solution given by

119911 (119905) = 119911 (119905 1199050 1199110 119906)

= 119879 (119905) 1199110

+ int

119905

1199050

119879 (119905 minus 119904) 119861119906 (119904) 119889119904 119905 isin [1199050 120591] 0 le 119905

0le 120591

(9)

Definition 2 For system (8) one defines the following con-cept the controllability maps 119866

120591120575 1198712(120591 minus 120575 120591 119880) rarr 119885

119866120575 1198712(0 120575 119880) rarr 119885 defined by

119866120591120575119906 = int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906 (119904) 119889119904 119906 isin 1198712

(120591 minus 120575 120591 119880)

119866120575V = int

120575

0

119879 (119904) 119861V (119904) 119889119904 V isin 1198712

(0 120575 119880)

(10)

satisfy the following relation

119866120591120575119906 = int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906 (119904) 119889119904

= int

120575

0

119879 (119904) 119861119906 (120591 minus 119904) 119889119904

= 119866120575119906 (120591 minus sdot)

(11)

The adjoints of these operators 119866lowast

120591120575 119885 rarr 119871

2(120591 minus 120575 120591 119880)

119866lowast

120575 119885 rarr 119871

2(0 120575 119880) are given by

(119866lowast

120591120575119911) (119905) = 119861

lowast119879lowast

(120591 minus 119905) 119911 119905 isin [120591 minus 120575 120591]

(119866lowast

120575119909) (119905) = 119861

lowast119879lowast

(119905) 119911 119905 isin [0 120575]

(12)

The Gramian controllability operators are given by (3) and

119876120591120575

= 119866120591120575119866lowast

120591120575= int

120591

120591minus120575

119879 (120591 minus 119905) 119861119861lowast119879lowast

(120591 minus 119905) 119889119905 = 119876120575 (13)

The following lemma holds in general for a linearbounded operator 119866 119882 rarr 119885 between Hilbert spaces 119882

and 119885 (see [4 11 12])

Lemma 3 The following statements are equivalent to theapproximate controllability of the linear system (8) on [120591minus120575 120591]

(a) Range (119866120591120575) = 119885

(b) Ker(119866lowast120591120575) = 0

(c) ⟨119876120591120575119911 119911⟩ gt 0 119911 = 0 in 119885

(d) lim120572rarr0

+120572(120572119868 + 119876120591120575)minus1119911 = 0

(e) For all 119911 isin 119885 one has 119866120591120575119906120572

= 119911 minus 120572(120572119868 + 119876119879120575

)minus1119911

where

119906120572= 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911 120572 isin (0 1] (14)

So lim120572rarr0

119866120591120575119906120572

= 119911 and the error 119864120591120575119911 of this

approximation is given by the formula

119864120591120575119911 = 120572 (120572119868 + 119876

120591120575)minus1

119911 120572 isin (0 1] (15)

(f) Moreover if one considers for each V isin 1198712(120591 minus 120575 120591 119880)

the sequence of controls given by

119906120572= 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911

+ (V minus 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119866120591120575V) 120572 isin (0 1]

(16)

one gets that

119866120591120575119906120572= 119911 minus 120572 (120572119868 + 119876

119879120575)minus1

(119911 + 119866120591120575V)

lim120572rarr0

119866120591120575119906120572= 119911

(17)

with the error 119864120591120575119911 of this approximation given by the

formula

119864120591120575119911 = 120572 (120572119868 + 119876

120591120575)minus1

(119911 + 119866120591120575V) 120572 isin (0 1] (18)

Remark 4 The foregoing lemma implies that the family oflinear operators Γ

120572120591120575 119885 rarr 119882 defined for 0 lt 120572 le 1 by

Γ120572120591120575

119911 = 119866lowast

120591120575(120572119868 + 119876

120591120575)minus1

119911 (19)

is an approximate inverse for the right of the operator 119882 inthe sense that

lim120572rarr0

119866120591120575Γ120572120591120575

= 119868 (20)

in the strong topology

4 Abstract and Applied Analysis

Lemma5 119876120575gt 0 if and only if linear system (8) is controllable

on [120591 minus 120575 120591] Moreover given an initial state 1199100and a final

state 1199111 one can find a sequence of controls 119906120575

1205720lt120572le1

sub 1198712(120591 minus

120575 120591 119880)

119906120572= 119866lowast

120591120575(120572119868 + 119866

120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120591) 119910

0) 120572 isin (0 1]

(21)

such that the solutions 119910(119905) = 119910(119905 120591 minus 120575 1199100 119906120575

120572) of the initial

value problem

1199101015840= 119860119910 + 119861119906

120572(119905) 119910 isin 119885 119905 gt 0

119910 (120591 minus 120575) = 1199100

(22)

satisfy

lim120572rarr0

+

119910 (120591 120591 minus 120575 1199100 119906120572) = 1199111 (23)

that is

lim120572rarr0

+

119910 (120591)

= lim120572rarr0

+

119879 (120575) 1199100+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 = 119911

1

(24)

3 Controllability of the SemilinearImpulsive System

In this section we will prove the main result of this paperthe approximate controllability of the semilinear impulsiveevolution equation given by (1) To this end for all 119911

0isin 119885

and 119906 isin 1198712(0 120591 119880) the initial value problem

1199111015840= 119860119911 + 119861119906 + 119865 (119905 119911 119906) 119905 isin (0 120591] 119905 = 119905

119896 119911 isin 119885

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119868119896(119905 119911 (119905

119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(25)

admits only one mild solution given by

119911119906(119905) = 119879 (119905) 119911

0+ int

119905

0

119879 (119905 minus 119904) 119861119906 (119904) 119889119904

+ int

119905

0

119879 (119905 minus 119904) 119865 (119904 119911119906(119904) 119906 (119904)) 119889119904

+ sum

0lt119905119896lt119905

119879 (119905 minus 119905119896) 119868119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119905 isin [0 120591]

(26)

Now we are ready to present and prove themain result of thispaper which is the approximate controllability of semilinearimpulsive equation (1)

Theorem 6 Under conditions (A) and (B) semilinear impul-sive system (1) is approximately controllable on [0 120591]

Proof Given an initial state 1199110 a final state 119911

1 and 120598 gt 0 we

want to find a control 119906120575120572

isin 1198712(0 120591 119880) steering the system

from 1199110to an 120598-neighborhood of 119911

1on time 120591 Specifically

lim120572rarr0

119879 (120591) 1199110+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119896(119911120572(119905119896) 119906120575

120572(119905119896)) = 119911

1

(27)

Consider any 119906 isin 1198712(0 120591 119880) and the corresponding solution

119911(119905) = 119911(119905 0 1199110 119906) of initial value problem (25) For 120572 isin

(0 1] we define the control 119906120575120572isin 1198712(0 120591 119880) as follows

119906120575

120572(119905) =

119906 (119905) if 0 le 119905 le 120591 minus 120575

119906120572(119905) if 120591 minus 120575 lt 119905 le 120591

(28)

where119906120572(119905) = 119861

lowast119879lowast

(120591 minus 119905) (120572119868 + 119866120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120575) 119911 (120591 minus 120575))

120591 minus 120575 lt 119905 le 120591

(29)Now assume that 0 lt 120575 lt 120591 minus 119905

119901 Then the corresponding

solution 119911120575

120572(119905) = 119911(119905 0 119911

0 119906120575

120572) of initial value problem (25) at

time 120591 can be written as follows

119911120575

120572(120591) = 119879 (120591) 119911

0+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

= 119879 (120575)

119879 (120591 minus 120575) 1199110+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591minus120575

119879 (120591 minus 120575 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

= 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(30)

Abstract and Applied Analysis 5

So

119911120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(31)

The corresponding solution 119910120575

120572(119905) = 119910(119905 120591 minus120575 119911(120591minus120575) 119906

120572) of

initial value problem (22) at time 120591 is given by

119910120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 (32)

Therefore

10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le int

120591

120591minus120575

119879 (120591 minus 119904)

10038171003817100381710038171003817119865 (119904 119911

120575

120572(119904) 119906120572(119904))

10038171003817100381710038171003817119889119904

(33)

Then putting 119872 = sup0le119905le120591

119879(119905) and 119870 = sup[0120591]times119885times119880

119865(119905

119911 119906) we obtain the following estimate10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le 119872119870120575 (34)

Hence10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 119872119870120575 +

10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817 (35)

From Lemma 5 there exists 120572 gt 0 such that10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le

120598

2

(36)

So taking 120575 lt min120591 minus 119905119901 1205982119872119870 we obtain for the corres-

ponding control 119906120575120572that

10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 120598 (37)

Geometrically the proof goes as shown in Figure 3This completes the proof of the theorem

4 Applications

As an application we will prove the approximate controllabil-ity of the following control system governed by the semilinearimpulsive wave equation

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

(38)

where Ω is a bounded domain in R119899 the distributed control119906 isin 119871

2([0 120591] 119871

2(Ω)) 119910

0isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891

are smooth functions with 119891 being bounded

z0

z(120591 minus 120575)

z(120591)

z120575120572(120591)

z1

y120575120572(120591)

120576

Figure 3

41 Abstract Formulation of the Problem In this part we willchoose the space where this problem will be set up as anabstract control system in a Hilbert space Let 119883 = 119871

2(Ω) =

1198712(ΩR) and consider the linear unbounded operator 119860

119863(119860) sub 119883 rarr 119883 defined by 119860120601 = minusΔ120601 where

119863(119860) = 1198672

(ΩR) cap 1198671

0(ΩR) (39)

Then the eigenvalues 120582119895of 119860 have finite multiplicity 120574

119895equal

to the dimension of the corresponding eigenspace and 0 lt

1205821lt 1205822lt sdot sdot sdot lt 120582

119899rarr infin Moreover consider the following

(a) There exists a complete orthonormal set 120601119895119896

ofeigenvectors of 119860(b) For all 119909 isin 119863(119860) we have

119860119909 =

infin

sum

119895=1

120582119895

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

=

infin

sum

119895=1

120582119895119864119895119909 (40)

where ⟨sdot sdot⟩ is the usual inner product in 1198712 and

119864119895119909 =

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

(41)

which means the set 119864119895infin

119895=1is a complete family of

orthogonal projections in119883 and119909 = suminfin

119895=1119864119895119909119909 isin 119883

(c) minus119860 generates an analytic semigroup 119890minus119860119905

givenby

119890minus119860119905

119909 =

infin

sum

119895=1

119890minus120582119895119905

119864119895119909 (42)

(d) The fractional powered spaces119883119903 are given by

119883119903= 119863 (119860

119903) = 119909 isin 119883

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

lt infin 119903 ge 0

(43)

6 Abstract and Applied Analysis

with the norm

119909119903=

10038171003817100381710038171198601199031199091003817100381710038171003817=

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

12

119909 isin 119883119903

119860119903119909 =

infin

sum

119899=1

120582119903

119899119864119899119909

(44)

Also for 119903 ge 0 we define 119885119903= 119883119903times 119883 which is a

Hilbert space endowed with the norm given by10038171003817100381710038171003817100381710038171003817

[119910

V]10038171003817100381710038171003817100381710038171003817119885119903

= radic10038171003817100381710038171199101003817100381710038171003817

2

119903+ V2 (45)

Then (1) can be written as abstract second order ordinarydifferential equations in 119885

12as follows

11991010158401015840= minus119860119910 + 119906 + 119891

119890(119905 119910 119910

1015840 119906) 119905 isin (0 120591] 119905 = 119905

119896

119910 (0) = 1199100 119910

1015840

(0) = 1199101

1199101015840(119905+

119896) = 1199101015840(119905minus

119896) + 119868119890

119896(119905119896 119910 (119905119896) 1199101015840(119905119896) 119906 (119905

119896))

119896 = 1 2 3 119901

(46)

where 119868119890

119896 119891119890 [0 120591] times 119885

12times 119880 rarr 119885

12are defined by

119868119890

119896(119905 119910 V 119906) (119909) = 119868

119896(119905 119910 (119909) V (119909) 119906 (119909))

119891119890

(119905 119911 119906) (119909) = 119891 (119905 119910 (119909) V (119909) 119906 (119909))

forall119909 isin Ω 119896 = 1 2 119901

(47)

With the change of variables 1199101015840

= V we can write secondorder equation (46) as a first order system of ordinarydifferential equations in the Hilbert space 119885

12= 11988312

times 119883

as follows1199111015840= A119911 + 119861119906 + 119865 (119905 119911 119906 (119905)) 119911 isin 119885

12 119905 isin (0 120591] 119905 = 119905

119896

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119869119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(48)

where 119906 isin 1198712([0 120591] 119880) 119880 = 119883 = 119871

2(Ω)

119911 = [119910

V] 119861 = [0

119868] A = [

0 119868119883

minus119860 0] (49)

is an unbounded linear operator with domain 119863(A) =

119863(119860) times 119863(11986012

) and 119869119896 119865 [0 120591] times 119885

12times 119880 rarr 119885

12are

defined by

119865 (119905 119911 119906) = [0

119891119890(119905 119910 V 119906)]

119869119896(119905 119911 119906) = [

0

119868119890

119896(119905 119908 V 119906)]

(50)

It is well known that the operatorA generates a stronglycontinuous group 119879(119905)

119905ge0in the space 119885 = 119885

12= 11988312

times119883

(see [13]) The following representation for this group can befound in [9] as Theorem 22

Proposition 7 The group 119879(119905)119905ge0

generated by the operatorA has the following representation

119879 (119905) 119911 =

infin

sum

119899119895=1

119890119860119895119905

119875119895119911 119911 isin 119885

12 119905 ge 0 (51)

where 119875119895119895ge0

is a complete family of orthogonal projections inthe Hilbert space 119885

12given by

119875119895= [

119864119895

0

0 119864119895

] 119895 ge 1

119860119895= 119877119895119875119895 119877119895= [

0 1

minus120582119895

0] 119895 ge 1

(52)

42 Approximate Controllability Now we are ready to for-mulate and prove the main result of this section which is theapproximate of the semilinear impulsive wave equation withbounded nonlinear perturbation

Theorem 8 Semilinear impulsive wave equation (38) isapproximately controllable on [0 120591]

Proof From [10] we know that the corresponding linearsystem without impulses

1199111015840= A119911 + 119861119906 119911 isin 119885

12 119905 isin (0 120591]

119911 (0) = 1199110

(53)

is exactly controllable on [0 120575] for all 0 lt 120575 lt 120591 On theother hand since 119891 is bounded there exists 119872 gt 0 such thatsup[0120591]timesRtimesRtimesR119891(119905 119910 V 119906) le 119872 Therefore we obtain

119865(119905 119911 119906)11988512

= int

Ω

1003816100381610038161003816119891 (119905 119910 (119909) V (119909) 119906 (119909))

1003816100381610038161003816

2

119889119909

12

le radic120583 (Ω)119872

(54)

for all (119905 119911 119906) isin [0 120591] times 11988512

times 119880 with 120583(Ω) the Lebesguemeasure of Ω Hence hypotheses (A) and (B) in Theorem 6are satisfied and we get the result

5 Final Remark

This technique can be applied to those systems where thelinear part does not generate a compact semigroup and iscontrollable on any [0 120575] for 120575 gt 0 and the nonlinearperturbation is bounded Example of such systems is thefollowing controlled thermoelastic plate equation whoselinear part was studied in [10]

119910119905119905+ Δ2119910 + 120572Δ120579 = 119906

1(119905 119909) + 119891

1(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

Abstract and Applied Analysis 7

120579119905minus 120573Δ120579 minus 120572Δ119910

119905= 1199062(119905 119909) + 119891

2(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

120579 = 119910 = Δ119910 = 0 on (0 120591) times 120597Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

1

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

120579 (119905+

119896 119909) = 120579 (119905

minus

119896 119909) + 119868

2

119896(119905119896 120579 (119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(55)

in the space 119885 = 1198831times 119883 times 119883 where Ω is a bounded domain

inR119899 the distributed controls 1199061 1199062isin 1198712([0 120591] 119871

2(Ω)) and

119868119894

119896 119891119894are smooth functions with 119891

119894 119894 = 1 2 being bounded

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV

References

[1] A E Bashirov and N Ghahramanlou ldquoOn partial approximatecontrollability of semilinear systemsrdquo Cogent Engineering vol1 no 1 2014

[2] A E Bashirov and M Jneid ldquoOn partial complete controlla-bility of semilinear systemsrdquo Abstract and Applied Analysis vol2013 Article ID 521052 8 pages 2013

[3] A E Bashirov N Mahmudov N Semı and H Etıkan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007

[4] H Leiva and N Merentes ldquoApproximate controllability of theimpul sive semilinear heat equationrdquo to appear in Journal ofMathematics and Applications

[5] D N Chalishajar ldquoControllability of impulsive partial neutralfunctional differential equation with infinite delayrdquo Interna-tional Journal of Mathematical Analysis vol 5 no 5ndash8 pp 369ndash380 2011

[6] B Radhakrishnan andK Balachandran ldquoControllability resultsfor semilinear impulsive integrodifferential evolution systemswith nonlocal conditionsrdquo Journal of Control Theory and Appli-cations vol 10 no 1 pp 28ndash34 2012

[7] S Selvi and M Mallika Arjunan ldquoControllability results forimpulsive differential systems with finite delayrdquo Journal ofNonlinear Science and Its Applications vol 5 no 3 pp 206ndash2192012

[8] L Chen and G Li ldquoApproximate controllability of impulsivedifferential equations with nonlocal conditionsrdquo InternationalJournal of Nonlinear Science vol 10 no 4 pp 438ndash446 2010

[9] H Leiva and N Merentes ldquoControllability of second-orderequations in 119871

2(Ω)rdquoMathematical Problems in Engineering vol

2010 Article ID 147195 11 pages 2010

[10] H Larez H Leiva and J Uzcategui ldquoControllability of blockdiagonal systems and applicationsrdquo International Journal of Sys-tems Control and Communications vol 3 no 1 2011

[11] R F Curtain andA J Pritchard InfiniteDimensional Linear Sys-tems vol 8 of Lecture Notes in Control and Information SciencesSpringer Berlin Germany 1978

[12] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory vol 21 of Text in Applied Mathe-matics Springer New York NY USA 1995

[13] S P Chen and R Triggiani ldquoProof of extensions of two conjec-tures on structural damping for elastic systemsrdquo Pacific Journalof Mathematics vol 136 no 1 pp 15ndash55 1989

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Lemma5 119876120575gt 0 if and only if linear system (8) is controllable

on [120591 minus 120575 120591] Moreover given an initial state 1199100and a final

state 1199111 one can find a sequence of controls 119906120575

1205720lt120572le1

sub 1198712(120591 minus

120575 120591 119880)

119906120572= 119866lowast

120591120575(120572119868 + 119866

120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120591) 119910

0) 120572 isin (0 1]

(21)

such that the solutions 119910(119905) = 119910(119905 120591 minus 120575 1199100 119906120575

120572) of the initial

value problem

1199101015840= 119860119910 + 119861119906

120572(119905) 119910 isin 119885 119905 gt 0

119910 (120591 minus 120575) = 1199100

(22)

satisfy

lim120572rarr0

+

119910 (120591 120591 minus 120575 1199100 119906120572) = 1199111 (23)

that is

lim120572rarr0

+

119910 (120591)

= lim120572rarr0

+

119879 (120575) 1199100+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 = 119911

1

(24)

3 Controllability of the SemilinearImpulsive System

In this section we will prove the main result of this paperthe approximate controllability of the semilinear impulsiveevolution equation given by (1) To this end for all 119911

0isin 119885

and 119906 isin 1198712(0 120591 119880) the initial value problem

1199111015840= 119860119911 + 119861119906 + 119865 (119905 119911 119906) 119905 isin (0 120591] 119905 = 119905

119896 119911 isin 119885

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119868119896(119905 119911 (119905

119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(25)

admits only one mild solution given by

119911119906(119905) = 119879 (119905) 119911

0+ int

119905

0

119879 (119905 minus 119904) 119861119906 (119904) 119889119904

+ int

119905

0

119879 (119905 minus 119904) 119865 (119904 119911119906(119904) 119906 (119904)) 119889119904

+ sum

0lt119905119896lt119905

119879 (119905 minus 119905119896) 119868119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119905 isin [0 120591]

(26)

Now we are ready to present and prove themain result of thispaper which is the approximate controllability of semilinearimpulsive equation (1)

Theorem 6 Under conditions (A) and (B) semilinear impul-sive system (1) is approximately controllable on [0 120591]

Proof Given an initial state 1199110 a final state 119911

1 and 120598 gt 0 we

want to find a control 119906120575120572

isin 1198712(0 120591 119880) steering the system

from 1199110to an 120598-neighborhood of 119911

1on time 120591 Specifically

lim120572rarr0

119879 (120591) 1199110+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119896(119911120572(119905119896) 119906120575

120572(119905119896)) = 119911

1

(27)

Consider any 119906 isin 1198712(0 120591 119880) and the corresponding solution

119911(119905) = 119911(119905 0 1199110 119906) of initial value problem (25) For 120572 isin

(0 1] we define the control 119906120575120572isin 1198712(0 120591 119880) as follows

119906120575

120572(119905) =

119906 (119905) if 0 le 119905 le 120591 minus 120575

119906120572(119905) if 120591 minus 120575 lt 119905 le 120591

(28)

where119906120572(119905) = 119861

lowast119879lowast

(120591 minus 119905) (120572119868 + 119866120591120575119866lowast

120591120575)minus1

(1199111minus 119879 (120575) 119911 (120591 minus 120575))

120591 minus 120575 lt 119905 le 120591

(29)Now assume that 0 lt 120575 lt 120591 minus 119905

119901 Then the corresponding

solution 119911120575

120572(119905) = 119911(119905 0 119911

0 119906120575

120572) of initial value problem (25) at

time 120591 can be written as follows

119911120575

120572(120591) = 119879 (120591) 119911

0+ int

120591

0

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

0

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591

119879 (120591 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

= 119879 (120575)

119879 (120591 minus 120575) 1199110+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591minus120575

0

119879 (120591 minus 120575 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

+ sum

0lt119905119896lt120591minus120575

119879 (120591 minus 120575 minus 119905119896) 119868119890

119896(119911120575

120572(119905119896) 119906120575

120572(119905119896))

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120575

120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120575

120572(119904)) 119889119904

= 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(30)

Abstract and Applied Analysis 5

So

119911120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(31)

The corresponding solution 119910120575

120572(119905) = 119910(119905 120591 minus120575 119911(120591minus120575) 119906

120572) of

initial value problem (22) at time 120591 is given by

119910120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 (32)

Therefore

10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le int

120591

120591minus120575

119879 (120591 minus 119904)

10038171003817100381710038171003817119865 (119904 119911

120575

120572(119904) 119906120572(119904))

10038171003817100381710038171003817119889119904

(33)

Then putting 119872 = sup0le119905le120591

119879(119905) and 119870 = sup[0120591]times119885times119880

119865(119905

119911 119906) we obtain the following estimate10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le 119872119870120575 (34)

Hence10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 119872119870120575 +

10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817 (35)

From Lemma 5 there exists 120572 gt 0 such that10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le

120598

2

(36)

So taking 120575 lt min120591 minus 119905119901 1205982119872119870 we obtain for the corres-

ponding control 119906120575120572that

10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 120598 (37)

Geometrically the proof goes as shown in Figure 3This completes the proof of the theorem

4 Applications

As an application we will prove the approximate controllabil-ity of the following control system governed by the semilinearimpulsive wave equation

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

(38)

where Ω is a bounded domain in R119899 the distributed control119906 isin 119871

2([0 120591] 119871

2(Ω)) 119910

0isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891

are smooth functions with 119891 being bounded

z0

z(120591 minus 120575)

z(120591)

z120575120572(120591)

z1

y120575120572(120591)

120576

Figure 3

41 Abstract Formulation of the Problem In this part we willchoose the space where this problem will be set up as anabstract control system in a Hilbert space Let 119883 = 119871

2(Ω) =

1198712(ΩR) and consider the linear unbounded operator 119860

119863(119860) sub 119883 rarr 119883 defined by 119860120601 = minusΔ120601 where

119863(119860) = 1198672

(ΩR) cap 1198671

0(ΩR) (39)

Then the eigenvalues 120582119895of 119860 have finite multiplicity 120574

119895equal

to the dimension of the corresponding eigenspace and 0 lt

1205821lt 1205822lt sdot sdot sdot lt 120582

119899rarr infin Moreover consider the following

(a) There exists a complete orthonormal set 120601119895119896

ofeigenvectors of 119860(b) For all 119909 isin 119863(119860) we have

119860119909 =

infin

sum

119895=1

120582119895

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

=

infin

sum

119895=1

120582119895119864119895119909 (40)

where ⟨sdot sdot⟩ is the usual inner product in 1198712 and

119864119895119909 =

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

(41)

which means the set 119864119895infin

119895=1is a complete family of

orthogonal projections in119883 and119909 = suminfin

119895=1119864119895119909119909 isin 119883

(c) minus119860 generates an analytic semigroup 119890minus119860119905

givenby

119890minus119860119905

119909 =

infin

sum

119895=1

119890minus120582119895119905

119864119895119909 (42)

(d) The fractional powered spaces119883119903 are given by

119883119903= 119863 (119860

119903) = 119909 isin 119883

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

lt infin 119903 ge 0

(43)

6 Abstract and Applied Analysis

with the norm

119909119903=

10038171003817100381710038171198601199031199091003817100381710038171003817=

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

12

119909 isin 119883119903

119860119903119909 =

infin

sum

119899=1

120582119903

119899119864119899119909

(44)

Also for 119903 ge 0 we define 119885119903= 119883119903times 119883 which is a

Hilbert space endowed with the norm given by10038171003817100381710038171003817100381710038171003817

[119910

V]10038171003817100381710038171003817100381710038171003817119885119903

= radic10038171003817100381710038171199101003817100381710038171003817

2

119903+ V2 (45)

Then (1) can be written as abstract second order ordinarydifferential equations in 119885

12as follows

11991010158401015840= minus119860119910 + 119906 + 119891

119890(119905 119910 119910

1015840 119906) 119905 isin (0 120591] 119905 = 119905

119896

119910 (0) = 1199100 119910

1015840

(0) = 1199101

1199101015840(119905+

119896) = 1199101015840(119905minus

119896) + 119868119890

119896(119905119896 119910 (119905119896) 1199101015840(119905119896) 119906 (119905

119896))

119896 = 1 2 3 119901

(46)

where 119868119890

119896 119891119890 [0 120591] times 119885

12times 119880 rarr 119885

12are defined by

119868119890

119896(119905 119910 V 119906) (119909) = 119868

119896(119905 119910 (119909) V (119909) 119906 (119909))

119891119890

(119905 119911 119906) (119909) = 119891 (119905 119910 (119909) V (119909) 119906 (119909))

forall119909 isin Ω 119896 = 1 2 119901

(47)

With the change of variables 1199101015840

= V we can write secondorder equation (46) as a first order system of ordinarydifferential equations in the Hilbert space 119885

12= 11988312

times 119883

as follows1199111015840= A119911 + 119861119906 + 119865 (119905 119911 119906 (119905)) 119911 isin 119885

12 119905 isin (0 120591] 119905 = 119905

119896

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119869119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(48)

where 119906 isin 1198712([0 120591] 119880) 119880 = 119883 = 119871

2(Ω)

119911 = [119910

V] 119861 = [0

119868] A = [

0 119868119883

minus119860 0] (49)

is an unbounded linear operator with domain 119863(A) =

119863(119860) times 119863(11986012

) and 119869119896 119865 [0 120591] times 119885

12times 119880 rarr 119885

12are

defined by

119865 (119905 119911 119906) = [0

119891119890(119905 119910 V 119906)]

119869119896(119905 119911 119906) = [

0

119868119890

119896(119905 119908 V 119906)]

(50)

It is well known that the operatorA generates a stronglycontinuous group 119879(119905)

119905ge0in the space 119885 = 119885

12= 11988312

times119883

(see [13]) The following representation for this group can befound in [9] as Theorem 22

Proposition 7 The group 119879(119905)119905ge0

generated by the operatorA has the following representation

119879 (119905) 119911 =

infin

sum

119899119895=1

119890119860119895119905

119875119895119911 119911 isin 119885

12 119905 ge 0 (51)

where 119875119895119895ge0

is a complete family of orthogonal projections inthe Hilbert space 119885

12given by

119875119895= [

119864119895

0

0 119864119895

] 119895 ge 1

119860119895= 119877119895119875119895 119877119895= [

0 1

minus120582119895

0] 119895 ge 1

(52)

42 Approximate Controllability Now we are ready to for-mulate and prove the main result of this section which is theapproximate of the semilinear impulsive wave equation withbounded nonlinear perturbation

Theorem 8 Semilinear impulsive wave equation (38) isapproximately controllable on [0 120591]

Proof From [10] we know that the corresponding linearsystem without impulses

1199111015840= A119911 + 119861119906 119911 isin 119885

12 119905 isin (0 120591]

119911 (0) = 1199110

(53)

is exactly controllable on [0 120575] for all 0 lt 120575 lt 120591 On theother hand since 119891 is bounded there exists 119872 gt 0 such thatsup[0120591]timesRtimesRtimesR119891(119905 119910 V 119906) le 119872 Therefore we obtain

119865(119905 119911 119906)11988512

= int

Ω

1003816100381610038161003816119891 (119905 119910 (119909) V (119909) 119906 (119909))

1003816100381610038161003816

2

119889119909

12

le radic120583 (Ω)119872

(54)

for all (119905 119911 119906) isin [0 120591] times 11988512

times 119880 with 120583(Ω) the Lebesguemeasure of Ω Hence hypotheses (A) and (B) in Theorem 6are satisfied and we get the result

5 Final Remark

This technique can be applied to those systems where thelinear part does not generate a compact semigroup and iscontrollable on any [0 120575] for 120575 gt 0 and the nonlinearperturbation is bounded Example of such systems is thefollowing controlled thermoelastic plate equation whoselinear part was studied in [10]

119910119905119905+ Δ2119910 + 120572Δ120579 = 119906

1(119905 119909) + 119891

1(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

Abstract and Applied Analysis 7

120579119905minus 120573Δ120579 minus 120572Δ119910

119905= 1199062(119905 119909) + 119891

2(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

120579 = 119910 = Δ119910 = 0 on (0 120591) times 120597Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

1

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

120579 (119905+

119896 119909) = 120579 (119905

minus

119896 119909) + 119868

2

119896(119905119896 120579 (119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(55)

in the space 119885 = 1198831times 119883 times 119883 where Ω is a bounded domain

inR119899 the distributed controls 1199061 1199062isin 1198712([0 120591] 119871

2(Ω)) and

119868119894

119896 119891119894are smooth functions with 119891

119894 119894 = 1 2 being bounded

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV

References

[1] A E Bashirov and N Ghahramanlou ldquoOn partial approximatecontrollability of semilinear systemsrdquo Cogent Engineering vol1 no 1 2014

[2] A E Bashirov and M Jneid ldquoOn partial complete controlla-bility of semilinear systemsrdquo Abstract and Applied Analysis vol2013 Article ID 521052 8 pages 2013

[3] A E Bashirov N Mahmudov N Semı and H Etıkan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007

[4] H Leiva and N Merentes ldquoApproximate controllability of theimpul sive semilinear heat equationrdquo to appear in Journal ofMathematics and Applications

[5] D N Chalishajar ldquoControllability of impulsive partial neutralfunctional differential equation with infinite delayrdquo Interna-tional Journal of Mathematical Analysis vol 5 no 5ndash8 pp 369ndash380 2011

[6] B Radhakrishnan andK Balachandran ldquoControllability resultsfor semilinear impulsive integrodifferential evolution systemswith nonlocal conditionsrdquo Journal of Control Theory and Appli-cations vol 10 no 1 pp 28ndash34 2012

[7] S Selvi and M Mallika Arjunan ldquoControllability results forimpulsive differential systems with finite delayrdquo Journal ofNonlinear Science and Its Applications vol 5 no 3 pp 206ndash2192012

[8] L Chen and G Li ldquoApproximate controllability of impulsivedifferential equations with nonlocal conditionsrdquo InternationalJournal of Nonlinear Science vol 10 no 4 pp 438ndash446 2010

[9] H Leiva and N Merentes ldquoControllability of second-orderequations in 119871

2(Ω)rdquoMathematical Problems in Engineering vol

2010 Article ID 147195 11 pages 2010

[10] H Larez H Leiva and J Uzcategui ldquoControllability of blockdiagonal systems and applicationsrdquo International Journal of Sys-tems Control and Communications vol 3 no 1 2011

[11] R F Curtain andA J Pritchard InfiniteDimensional Linear Sys-tems vol 8 of Lecture Notes in Control and Information SciencesSpringer Berlin Germany 1978

[12] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory vol 21 of Text in Applied Mathe-matics Springer New York NY USA 1995

[13] S P Chen and R Triggiani ldquoProof of extensions of two conjec-tures on structural damping for elastic systemsrdquo Pacific Journalof Mathematics vol 136 no 1 pp 15ndash55 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

So

119911120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904

+ int

120591

120591minus120575

119879 (120591 minus 119904) 119865 (119904 119911120575

120572(119904) 119906120572(119904)) 119889119904

(31)

The corresponding solution 119910120575

120572(119905) = 119910(119905 120591 minus120575 119911(120591minus120575) 119906

120572) of

initial value problem (22) at time 120591 is given by

119910120575

120572(120591) = 119879 (120575) 119911 (120591 minus 120575) + int

120591

120591minus120575

119879 (120591 minus 119904) 119861119906120572(119904) 119889119904 (32)

Therefore

10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le int

120591

120591minus120575

119879 (120591 minus 119904)

10038171003817100381710038171003817119865 (119904 119911

120575

120572(119904) 119906120572(119904))

10038171003817100381710038171003817119889119904

(33)

Then putting 119872 = sup0le119905le120591

119879(119905) and 119870 = sup[0120591]times119885times119880

119865(119905

119911 119906) we obtain the following estimate10038171003817100381710038171003817119911120575

120572(120591) minus 119910

120575

120572(120591)

10038171003817100381710038171003817le 119872119870120575 (34)

Hence10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 119872119870120575 +

10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817 (35)

From Lemma 5 there exists 120572 gt 0 such that10038171003817100381710038171003817119910120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le

120598

2

(36)

So taking 120575 lt min120591 minus 119905119901 1205982119872119870 we obtain for the corres-

ponding control 119906120575120572that

10038171003817100381710038171003817119911120575

120572(120591) minus 119911

1

10038171003817100381710038171003817le 120598 (37)

Geometrically the proof goes as shown in Figure 3This completes the proof of the theorem

4 Applications

As an application we will prove the approximate controllabil-ity of the following control system governed by the semilinearimpulsive wave equation

119910119905119905

= Δ119910 + 119906 (119905 119909) + 119891 (119905 119910 119910119905 119906 (119905)) on (0 120591) times Ω

119910 = 0 on (0 120591) times 120597Ω

119910 (0 119909) = 1199100(119909) 119910

119905(0 119909) = 119910

1(119909) in Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

(38)

where Ω is a bounded domain in R119899 the distributed control119906 isin 119871

2([0 120591] 119871

2(Ω)) 119910

0isin 1198672(Ω)cap119867

1

0 1199101isin 1198712(Ω) and 119868

119896 119891

are smooth functions with 119891 being bounded

z0

z(120591 minus 120575)

z(120591)

z120575120572(120591)

z1

y120575120572(120591)

120576

Figure 3

41 Abstract Formulation of the Problem In this part we willchoose the space where this problem will be set up as anabstract control system in a Hilbert space Let 119883 = 119871

2(Ω) =

1198712(ΩR) and consider the linear unbounded operator 119860

119863(119860) sub 119883 rarr 119883 defined by 119860120601 = minusΔ120601 where

119863(119860) = 1198672

(ΩR) cap 1198671

0(ΩR) (39)

Then the eigenvalues 120582119895of 119860 have finite multiplicity 120574

119895equal

to the dimension of the corresponding eigenspace and 0 lt

1205821lt 1205822lt sdot sdot sdot lt 120582

119899rarr infin Moreover consider the following

(a) There exists a complete orthonormal set 120601119895119896

ofeigenvectors of 119860(b) For all 119909 isin 119863(119860) we have

119860119909 =

infin

sum

119895=1

120582119895

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

=

infin

sum

119895=1

120582119895119864119895119909 (40)

where ⟨sdot sdot⟩ is the usual inner product in 1198712 and

119864119895119909 =

120574119895

sum

119896=1

⟨119909 120601119895119896

⟩120601119895119896

(41)

which means the set 119864119895infin

119895=1is a complete family of

orthogonal projections in119883 and119909 = suminfin

119895=1119864119895119909119909 isin 119883

(c) minus119860 generates an analytic semigroup 119890minus119860119905

givenby

119890minus119860119905

119909 =

infin

sum

119895=1

119890minus120582119895119905

119864119895119909 (42)

(d) The fractional powered spaces119883119903 are given by

119883119903= 119863 (119860

119903) = 119909 isin 119883

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

lt infin 119903 ge 0

(43)

6 Abstract and Applied Analysis

with the norm

119909119903=

10038171003817100381710038171198601199031199091003817100381710038171003817=

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

12

119909 isin 119883119903

119860119903119909 =

infin

sum

119899=1

120582119903

119899119864119899119909

(44)

Also for 119903 ge 0 we define 119885119903= 119883119903times 119883 which is a

Hilbert space endowed with the norm given by10038171003817100381710038171003817100381710038171003817

[119910

V]10038171003817100381710038171003817100381710038171003817119885119903

= radic10038171003817100381710038171199101003817100381710038171003817

2

119903+ V2 (45)

Then (1) can be written as abstract second order ordinarydifferential equations in 119885

12as follows

11991010158401015840= minus119860119910 + 119906 + 119891

119890(119905 119910 119910

1015840 119906) 119905 isin (0 120591] 119905 = 119905

119896

119910 (0) = 1199100 119910

1015840

(0) = 1199101

1199101015840(119905+

119896) = 1199101015840(119905minus

119896) + 119868119890

119896(119905119896 119910 (119905119896) 1199101015840(119905119896) 119906 (119905

119896))

119896 = 1 2 3 119901

(46)

where 119868119890

119896 119891119890 [0 120591] times 119885

12times 119880 rarr 119885

12are defined by

119868119890

119896(119905 119910 V 119906) (119909) = 119868

119896(119905 119910 (119909) V (119909) 119906 (119909))

119891119890

(119905 119911 119906) (119909) = 119891 (119905 119910 (119909) V (119909) 119906 (119909))

forall119909 isin Ω 119896 = 1 2 119901

(47)

With the change of variables 1199101015840

= V we can write secondorder equation (46) as a first order system of ordinarydifferential equations in the Hilbert space 119885

12= 11988312

times 119883

as follows1199111015840= A119911 + 119861119906 + 119865 (119905 119911 119906 (119905)) 119911 isin 119885

12 119905 isin (0 120591] 119905 = 119905

119896

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119869119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(48)

where 119906 isin 1198712([0 120591] 119880) 119880 = 119883 = 119871

2(Ω)

119911 = [119910

V] 119861 = [0

119868] A = [

0 119868119883

minus119860 0] (49)

is an unbounded linear operator with domain 119863(A) =

119863(119860) times 119863(11986012

) and 119869119896 119865 [0 120591] times 119885

12times 119880 rarr 119885

12are

defined by

119865 (119905 119911 119906) = [0

119891119890(119905 119910 V 119906)]

119869119896(119905 119911 119906) = [

0

119868119890

119896(119905 119908 V 119906)]

(50)

It is well known that the operatorA generates a stronglycontinuous group 119879(119905)

119905ge0in the space 119885 = 119885

12= 11988312

times119883

(see [13]) The following representation for this group can befound in [9] as Theorem 22

Proposition 7 The group 119879(119905)119905ge0

generated by the operatorA has the following representation

119879 (119905) 119911 =

infin

sum

119899119895=1

119890119860119895119905

119875119895119911 119911 isin 119885

12 119905 ge 0 (51)

where 119875119895119895ge0

is a complete family of orthogonal projections inthe Hilbert space 119885

12given by

119875119895= [

119864119895

0

0 119864119895

] 119895 ge 1

119860119895= 119877119895119875119895 119877119895= [

0 1

minus120582119895

0] 119895 ge 1

(52)

42 Approximate Controllability Now we are ready to for-mulate and prove the main result of this section which is theapproximate of the semilinear impulsive wave equation withbounded nonlinear perturbation

Theorem 8 Semilinear impulsive wave equation (38) isapproximately controllable on [0 120591]

Proof From [10] we know that the corresponding linearsystem without impulses

1199111015840= A119911 + 119861119906 119911 isin 119885

12 119905 isin (0 120591]

119911 (0) = 1199110

(53)

is exactly controllable on [0 120575] for all 0 lt 120575 lt 120591 On theother hand since 119891 is bounded there exists 119872 gt 0 such thatsup[0120591]timesRtimesRtimesR119891(119905 119910 V 119906) le 119872 Therefore we obtain

119865(119905 119911 119906)11988512

= int

Ω

1003816100381610038161003816119891 (119905 119910 (119909) V (119909) 119906 (119909))

1003816100381610038161003816

2

119889119909

12

le radic120583 (Ω)119872

(54)

for all (119905 119911 119906) isin [0 120591] times 11988512

times 119880 with 120583(Ω) the Lebesguemeasure of Ω Hence hypotheses (A) and (B) in Theorem 6are satisfied and we get the result

5 Final Remark

This technique can be applied to those systems where thelinear part does not generate a compact semigroup and iscontrollable on any [0 120575] for 120575 gt 0 and the nonlinearperturbation is bounded Example of such systems is thefollowing controlled thermoelastic plate equation whoselinear part was studied in [10]

119910119905119905+ Δ2119910 + 120572Δ120579 = 119906

1(119905 119909) + 119891

1(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

Abstract and Applied Analysis 7

120579119905minus 120573Δ120579 minus 120572Δ119910

119905= 1199062(119905 119909) + 119891

2(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

120579 = 119910 = Δ119910 = 0 on (0 120591) times 120597Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

1

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

120579 (119905+

119896 119909) = 120579 (119905

minus

119896 119909) + 119868

2

119896(119905119896 120579 (119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(55)

in the space 119885 = 1198831times 119883 times 119883 where Ω is a bounded domain

inR119899 the distributed controls 1199061 1199062isin 1198712([0 120591] 119871

2(Ω)) and

119868119894

119896 119891119894are smooth functions with 119891

119894 119894 = 1 2 being bounded

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV

References

[1] A E Bashirov and N Ghahramanlou ldquoOn partial approximatecontrollability of semilinear systemsrdquo Cogent Engineering vol1 no 1 2014

[2] A E Bashirov and M Jneid ldquoOn partial complete controlla-bility of semilinear systemsrdquo Abstract and Applied Analysis vol2013 Article ID 521052 8 pages 2013

[3] A E Bashirov N Mahmudov N Semı and H Etıkan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007

[4] H Leiva and N Merentes ldquoApproximate controllability of theimpul sive semilinear heat equationrdquo to appear in Journal ofMathematics and Applications

[5] D N Chalishajar ldquoControllability of impulsive partial neutralfunctional differential equation with infinite delayrdquo Interna-tional Journal of Mathematical Analysis vol 5 no 5ndash8 pp 369ndash380 2011

[6] B Radhakrishnan andK Balachandran ldquoControllability resultsfor semilinear impulsive integrodifferential evolution systemswith nonlocal conditionsrdquo Journal of Control Theory and Appli-cations vol 10 no 1 pp 28ndash34 2012

[7] S Selvi and M Mallika Arjunan ldquoControllability results forimpulsive differential systems with finite delayrdquo Journal ofNonlinear Science and Its Applications vol 5 no 3 pp 206ndash2192012

[8] L Chen and G Li ldquoApproximate controllability of impulsivedifferential equations with nonlocal conditionsrdquo InternationalJournal of Nonlinear Science vol 10 no 4 pp 438ndash446 2010

[9] H Leiva and N Merentes ldquoControllability of second-orderequations in 119871

2(Ω)rdquoMathematical Problems in Engineering vol

2010 Article ID 147195 11 pages 2010

[10] H Larez H Leiva and J Uzcategui ldquoControllability of blockdiagonal systems and applicationsrdquo International Journal of Sys-tems Control and Communications vol 3 no 1 2011

[11] R F Curtain andA J Pritchard InfiniteDimensional Linear Sys-tems vol 8 of Lecture Notes in Control and Information SciencesSpringer Berlin Germany 1978

[12] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory vol 21 of Text in Applied Mathe-matics Springer New York NY USA 1995

[13] S P Chen and R Triggiani ldquoProof of extensions of two conjec-tures on structural damping for elastic systemsrdquo Pacific Journalof Mathematics vol 136 no 1 pp 15ndash55 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

with the norm

119909119903=

10038171003817100381710038171198601199031199091003817100381710038171003817=

infin

sum

119899=1

1205822119903

119899

10038171003817100381710038171198641198991199091003817100381710038171003817

2

12

119909 isin 119883119903

119860119903119909 =

infin

sum

119899=1

120582119903

119899119864119899119909

(44)

Also for 119903 ge 0 we define 119885119903= 119883119903times 119883 which is a

Hilbert space endowed with the norm given by10038171003817100381710038171003817100381710038171003817

[119910

V]10038171003817100381710038171003817100381710038171003817119885119903

= radic10038171003817100381710038171199101003817100381710038171003817

2

119903+ V2 (45)

Then (1) can be written as abstract second order ordinarydifferential equations in 119885

12as follows

11991010158401015840= minus119860119910 + 119906 + 119891

119890(119905 119910 119910

1015840 119906) 119905 isin (0 120591] 119905 = 119905

119896

119910 (0) = 1199100 119910

1015840

(0) = 1199101

1199101015840(119905+

119896) = 1199101015840(119905minus

119896) + 119868119890

119896(119905119896 119910 (119905119896) 1199101015840(119905119896) 119906 (119905

119896))

119896 = 1 2 3 119901

(46)

where 119868119890

119896 119891119890 [0 120591] times 119885

12times 119880 rarr 119885

12are defined by

119868119890

119896(119905 119910 V 119906) (119909) = 119868

119896(119905 119910 (119909) V (119909) 119906 (119909))

119891119890

(119905 119911 119906) (119909) = 119891 (119905 119910 (119909) V (119909) 119906 (119909))

forall119909 isin Ω 119896 = 1 2 119901

(47)

With the change of variables 1199101015840

= V we can write secondorder equation (46) as a first order system of ordinarydifferential equations in the Hilbert space 119885

12= 11988312

times 119883

as follows1199111015840= A119911 + 119861119906 + 119865 (119905 119911 119906 (119905)) 119911 isin 119885

12 119905 isin (0 120591] 119905 = 119905

119896

119911 (0) = 1199110

119911 (119905+

119896) = 119911 (119905

minus

119896) + 119869119896(119905119896 119911 (119905119896) 119906 (119905

119896)) 119896 = 1 2 3 119901

(48)

where 119906 isin 1198712([0 120591] 119880) 119880 = 119883 = 119871

2(Ω)

119911 = [119910

V] 119861 = [0

119868] A = [

0 119868119883

minus119860 0] (49)

is an unbounded linear operator with domain 119863(A) =

119863(119860) times 119863(11986012

) and 119869119896 119865 [0 120591] times 119885

12times 119880 rarr 119885

12are

defined by

119865 (119905 119911 119906) = [0

119891119890(119905 119910 V 119906)]

119869119896(119905 119911 119906) = [

0

119868119890

119896(119905 119908 V 119906)]

(50)

It is well known that the operatorA generates a stronglycontinuous group 119879(119905)

119905ge0in the space 119885 = 119885

12= 11988312

times119883

(see [13]) The following representation for this group can befound in [9] as Theorem 22

Proposition 7 The group 119879(119905)119905ge0

generated by the operatorA has the following representation

119879 (119905) 119911 =

infin

sum

119899119895=1

119890119860119895119905

119875119895119911 119911 isin 119885

12 119905 ge 0 (51)

where 119875119895119895ge0

is a complete family of orthogonal projections inthe Hilbert space 119885

12given by

119875119895= [

119864119895

0

0 119864119895

] 119895 ge 1

119860119895= 119877119895119875119895 119877119895= [

0 1

minus120582119895

0] 119895 ge 1

(52)

42 Approximate Controllability Now we are ready to for-mulate and prove the main result of this section which is theapproximate of the semilinear impulsive wave equation withbounded nonlinear perturbation

Theorem 8 Semilinear impulsive wave equation (38) isapproximately controllable on [0 120591]

Proof From [10] we know that the corresponding linearsystem without impulses

1199111015840= A119911 + 119861119906 119911 isin 119885

12 119905 isin (0 120591]

119911 (0) = 1199110

(53)

is exactly controllable on [0 120575] for all 0 lt 120575 lt 120591 On theother hand since 119891 is bounded there exists 119872 gt 0 such thatsup[0120591]timesRtimesRtimesR119891(119905 119910 V 119906) le 119872 Therefore we obtain

119865(119905 119911 119906)11988512

= int

Ω

1003816100381610038161003816119891 (119905 119910 (119909) V (119909) 119906 (119909))

1003816100381610038161003816

2

119889119909

12

le radic120583 (Ω)119872

(54)

for all (119905 119911 119906) isin [0 120591] times 11988512

times 119880 with 120583(Ω) the Lebesguemeasure of Ω Hence hypotheses (A) and (B) in Theorem 6are satisfied and we get the result

5 Final Remark

This technique can be applied to those systems where thelinear part does not generate a compact semigroup and iscontrollable on any [0 120575] for 120575 gt 0 and the nonlinearperturbation is bounded Example of such systems is thefollowing controlled thermoelastic plate equation whoselinear part was studied in [10]

119910119905119905+ Δ2119910 + 120572Δ120579 = 119906

1(119905 119909) + 119891

1(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

Abstract and Applied Analysis 7

120579119905minus 120573Δ120579 minus 120572Δ119910

119905= 1199062(119905 119909) + 119891

2(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

120579 = 119910 = Δ119910 = 0 on (0 120591) times 120597Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

1

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

120579 (119905+

119896 119909) = 120579 (119905

minus

119896 119909) + 119868

2

119896(119905119896 120579 (119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(55)

in the space 119885 = 1198831times 119883 times 119883 where Ω is a bounded domain

inR119899 the distributed controls 1199061 1199062isin 1198712([0 120591] 119871

2(Ω)) and

119868119894

119896 119891119894are smooth functions with 119891

119894 119894 = 1 2 being bounded

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV

References

[1] A E Bashirov and N Ghahramanlou ldquoOn partial approximatecontrollability of semilinear systemsrdquo Cogent Engineering vol1 no 1 2014

[2] A E Bashirov and M Jneid ldquoOn partial complete controlla-bility of semilinear systemsrdquo Abstract and Applied Analysis vol2013 Article ID 521052 8 pages 2013

[3] A E Bashirov N Mahmudov N Semı and H Etıkan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007

[4] H Leiva and N Merentes ldquoApproximate controllability of theimpul sive semilinear heat equationrdquo to appear in Journal ofMathematics and Applications

[5] D N Chalishajar ldquoControllability of impulsive partial neutralfunctional differential equation with infinite delayrdquo Interna-tional Journal of Mathematical Analysis vol 5 no 5ndash8 pp 369ndash380 2011

[6] B Radhakrishnan andK Balachandran ldquoControllability resultsfor semilinear impulsive integrodifferential evolution systemswith nonlocal conditionsrdquo Journal of Control Theory and Appli-cations vol 10 no 1 pp 28ndash34 2012

[7] S Selvi and M Mallika Arjunan ldquoControllability results forimpulsive differential systems with finite delayrdquo Journal ofNonlinear Science and Its Applications vol 5 no 3 pp 206ndash2192012

[8] L Chen and G Li ldquoApproximate controllability of impulsivedifferential equations with nonlocal conditionsrdquo InternationalJournal of Nonlinear Science vol 10 no 4 pp 438ndash446 2010

[9] H Leiva and N Merentes ldquoControllability of second-orderequations in 119871

2(Ω)rdquoMathematical Problems in Engineering vol

2010 Article ID 147195 11 pages 2010

[10] H Larez H Leiva and J Uzcategui ldquoControllability of blockdiagonal systems and applicationsrdquo International Journal of Sys-tems Control and Communications vol 3 no 1 2011

[11] R F Curtain andA J Pritchard InfiniteDimensional Linear Sys-tems vol 8 of Lecture Notes in Control and Information SciencesSpringer Berlin Germany 1978

[12] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory vol 21 of Text in Applied Mathe-matics Springer New York NY USA 1995

[13] S P Chen and R Triggiani ldquoProof of extensions of two conjec-tures on structural damping for elastic systemsrdquo Pacific Journalof Mathematics vol 136 no 1 pp 15ndash55 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

120579119905minus 120573Δ120579 minus 120572Δ119910

119905= 1199062(119905 119909) + 119891

2(119905 119910 119910

119905 120579 119906 (119905))

on (0 120591) times Ω

120579 = 119910 = Δ119910 = 0 on (0 120591) times 120597Ω

119910119905(119905+

119896 119909) = 119910

119905(119905minus

119896 119909) + 119868

1

119896(119905 119910 (119905

119896 119909) 119910

119905(119905119896 119909) 119906 (119905

119896 119909))

119909 isin Ω

120579 (119905+

119896 119909) = 120579 (119905

minus

119896 119909) + 119868

2

119896(119905119896 120579 (119905119896 119909) 119906 (119905

119896 119909)) 119909 isin Ω

(55)

in the space 119885 = 1198831times 119883 times 119883 where Ω is a bounded domain

inR119899 the distributed controls 1199061 1199062isin 1198712([0 120591] 119871

2(Ω)) and

119868119894

119896 119891119894are smooth functions with 119891

119894 119894 = 1 2 being bounded

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV

References

[1] A E Bashirov and N Ghahramanlou ldquoOn partial approximatecontrollability of semilinear systemsrdquo Cogent Engineering vol1 no 1 2014

[2] A E Bashirov and M Jneid ldquoOn partial complete controlla-bility of semilinear systemsrdquo Abstract and Applied Analysis vol2013 Article ID 521052 8 pages 2013

[3] A E Bashirov N Mahmudov N Semı and H Etıkan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007

[4] H Leiva and N Merentes ldquoApproximate controllability of theimpul sive semilinear heat equationrdquo to appear in Journal ofMathematics and Applications

[5] D N Chalishajar ldquoControllability of impulsive partial neutralfunctional differential equation with infinite delayrdquo Interna-tional Journal of Mathematical Analysis vol 5 no 5ndash8 pp 369ndash380 2011

[6] B Radhakrishnan andK Balachandran ldquoControllability resultsfor semilinear impulsive integrodifferential evolution systemswith nonlocal conditionsrdquo Journal of Control Theory and Appli-cations vol 10 no 1 pp 28ndash34 2012

[7] S Selvi and M Mallika Arjunan ldquoControllability results forimpulsive differential systems with finite delayrdquo Journal ofNonlinear Science and Its Applications vol 5 no 3 pp 206ndash2192012

[8] L Chen and G Li ldquoApproximate controllability of impulsivedifferential equations with nonlocal conditionsrdquo InternationalJournal of Nonlinear Science vol 10 no 4 pp 438ndash446 2010

[9] H Leiva and N Merentes ldquoControllability of second-orderequations in 119871

2(Ω)rdquoMathematical Problems in Engineering vol

2010 Article ID 147195 11 pages 2010

[10] H Larez H Leiva and J Uzcategui ldquoControllability of blockdiagonal systems and applicationsrdquo International Journal of Sys-tems Control and Communications vol 3 no 1 2011

[11] R F Curtain andA J Pritchard InfiniteDimensional Linear Sys-tems vol 8 of Lecture Notes in Control and Information SciencesSpringer Berlin Germany 1978

[12] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory vol 21 of Text in Applied Mathe-matics Springer New York NY USA 1995

[13] S P Chen and R Triggiani ldquoProof of extensions of two conjec-tures on structural damping for elastic systemsrdquo Pacific Journalof Mathematics vol 136 no 1 pp 15ndash55 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of