Universidad Nacional de Chimborazo

NOVASINERGIA, 2020, Vol. 3, No. 1, diciembre-mayo, (37-44)

ISSN: 2631-2654

https://doi.org/10.37135/ns.01.05.04

Research Article

Controllability of semilinear evolution equations with impulses and

delays

Controlabilidad de ecuaciones de evolución semilineales con impulsos y retardos

Jesús Aponte

1

, Hugo Leiva

2

*

1

Departamento de Matemáticas, Escuela Superior Politécnica del Litoral. Guayaquil, Ecuador, 090604;

japonte@espol.edu.ec

2

Escuela de Ciencias Matemáticas y Computacionales, Universidad de Tecnología Experimental Yachay. San Miguel de

Urcuquí, Ecuador. 100115

* Correspondence: hleiva@yachaytech.edu.ec

Recibido 20 abril 2020; Aceptado 30 abril 2020; Publicado 01 junio 2020

Abstract:

For many control systems in real life, impulses and delays are intrinsic phenomena that

do not modify their controllability. So we conjecture that, under certain conditions,

perturbations of the system caused by abrupt changes and delays do not affect certain

properties such as controllability. In this regard, we show that under certain conditions,

the impulses and delays as perturbations do not destroy the controllability of systems

governed by evolution equations. As application, we consider a semi-linear wave

equation with impulses and delays.

Keywords:

Controllability, impulsive semilinear evolution equations, semilinear wave equation,

strongly continuous semigroup

Resumen:

Para muchos sistemas de control en la vida real, impulsos y retardos son fenómenos

intrínsecos que no modiﬁcan su controlabilidad. Así conjeturamos que, bajo ciertas

condiciones, perturbaciones del sistema causadas por cambios abruptos y retardos no

afectan ciertas propiedades como la controlabilidad. A este respecto, mostramos que

bajo ciertas condiciones, los impulsos y retardos como perturbaciones no destruyen la

controlabilidad de sistemas gobernados por ecuaciones de evolución. Como aplicación

consideramos una ecuación de ondas semilineal con impulsos y retardos.

Palabras clave:

Controlabilidad, ecuaciones de evolución semilineales, ecuación de onda semilineal,

semigrupos fuertemente continuos

1 Introduction

There are several practical examples of control

systems with impulses and delays: a chemical

reactor system with the quantities of different

chemicals serving as the state, a ﬁnancial system

with two state variables being the amount of money

in a market and the saving rates of a central bank,

and the growth of a population diffusing throughout

its habitat, often modeled by reaction-diffusion

equation. However, one may easily visualize

situations in nature where abrupt changes such as

http://novasinergia.unach.edu.ec

harvesting, disasters, or instantaneous stoking may

occur.

This paper has been motivated by the works done

by Hugo Leiva in Leiva (2015a,c,b) where the

approximate controllability of Semilinear Evolution

Equation with impulses was proved in the case of

non necessarily compact semigroup and bounded

non linear perturbation.

In this paper, we study a more general problem since

we consider the following semilinear evolution

equation with impulses and delays simultaneously

z

0

= Az + Bu(t) + F(t,z

t

,u), z ∈ Z, t ∈ (0,τ],

z(s) = φ(s), s ∈ [−r,0],

z(t

+

k

) = z(t

−

k

) + I

k

(t

k

,z(t

k

),u(t

k

)), k = 1, ..., p.

(1)

where 0 < t

1

< t

2

< t

3

< ·· · < t

p

< τ, Z and U

are Hilbert Spaces, u ∈ L

2

(0,τ;U), B : U −→ Z

is a bounded linear operator, standard notation z

t

deﬁnes a function from [−r,0] to Z by z

t

(s) = z(t +

s),−r ≤ s ≤ 0, I

k

: [0, τ] × Z ×U → Z, F : [0,τ] ×

C(−r,0;Z) ×U → Z are smooth functions, and A :

D(A) ⊂ Z → Z is an unbounded linear operator in

Z which generates a strongly continuous semigroup

{T (t)}

t≥0

⊂ Z non necessarily compact.

We assume the following hypotheses:

(H1) The linear system without impulses (6) is

approximately controllable on [τ− δ,τ] for all

0 < δ < τ.

(H2) The functions F, I

k

smooth enough and

kF(t,φ,u)k

Z

≤ akφ(−r)k + b,

u ∈ U, φ ∈ C(−r,0;Z). (2)

DEFINITION 1.1 (Approximate Controllability).

The system (1) is said to be approximately

controllable on [0,τ] if for every φ ∈ C(−r,0; Z),

z

1

∈ Z, ε > 0 there exists u ∈ L

2

(0,τ;U) such that

the solution z(t) of (1) corresponding to u satisﬁes:

kz(τ) − z

1

k

Z

< ε.

To address this problem we use a characterization

dense range linear operator from Leiva et al.

(2013), the approximate controllability of the

linear equation on [τ − δ,τ] for all τ > 0 and

the ideas presented in Bashirov et al. (2007),

Bashirov & Ghahramanlou (2013) and Bashirov

& Ghahramanlou (2014). The controllability of

impulsive evolution equations has been studied

recently by several authors, but most them study

the exact controllability only, e.g. in Chalishajar

(2011), studied the exact controllability of

impulsive partial neutral functional differential

equations with inﬁnite delay, Radhakrishnan &

Blachandran (2012) studied the exact controllability

of semilinear impulsive integro–differential

evolution systems with nonlocal conditions,

and Selvi & Arjunan (2012) studied the exact

controllability for impulsive differential systems

with ﬁnite delay. To the best of our knowledge,

there are a few works on approximate controllability

of impulsive semilinear evolution equations,

worth mentioning: Chen & Li (2010) studied

the approximate controllability of impulsive

differential equations with nonlocal conditions,

using measure of noncompactness and Monch’s

Fixed Point Theorem, and assuming that the

nonlinear term f (t,z) does not depend on the

control variable; Leiva & Merentes (2015) studied

the approximate controllability of the semilinear

impulsive heat equation using the fact that the

semigroup generated by ∆ is compact.

When it comes to the wave equation, the situation

is totally different: the semigroup generated by

the linear part is not compact; it is in fact a

group, which can never be compact. Furthermore,

if the control acts on a portion ω of the domain

Ω for the spatial variable, then the system is

approximately controllable only on [0, τ] for τ ≥ 2,

which was proved by Leiva & Merentes (2010).

More precisely, the following system governed by

the wave equations was studied.

y

tt

= ∆y + 1

ω

u(t,x), on (0,τ) × Ω;

y = 0, on (0,τ) × ∂Ω;

y(0,x) = y

0

(x), y

t

(0,x) = y

1

(x), in Ω.

(3)

where Ω is a bounded domain in R

n

, ω is an open

nonempty subset of Ω, 1

ω

denotes the characteristic

function of the set ω, the distributed control u ∈

L

2

([0,τ]; L

2

(Ω)) and y

0

∈ H

2

(Ω) ∩ H

1

0

,y

1

∈ L

2

(Ω).

However, if the control acts on the whole domain Ω,

it was proved in Larez et al. (2011) that the system

is controllable [0, τ], for all τ > 0. More speciﬁcally,

the authors studied the following system

y

tt

= ∆y + u(t,x), on (0,τ) × Ω;

y = 0, on (0,τ) × ∂Ω;

y(0,x) = y

0

(x), y

t

(0,x) = y

1

(x), in Ω,

(4)

where Ω is a bounded domain in R

n

, the distributed

control u ∈ L

2

([0,τ]; L

2

(Ω)) and y

0

∈ H

2

(Ω) ∩

H

1

0

,y

1

∈ L

2

(Ω).

To justify the use of this new technique (Bashirov

& Ghahramanlou, 2014), we consider as an

http://novasinergia.unach.edu.ec 38

application the following semilinear wave equation

with impulses, delays and controls acting on the

whole domain Ω, so that the hypotheses (H1) and

(H2) hold:

∂

2

y

∂

2

t

=

∆y + u(t,x)

+ f (t,y(t − r),

∂y

∂t

(t − r), u(t)),

on (0,τ) × Ω;

y = 0, on (0,τ) × ∂Ω;

y(s,x) = φ

0

(s,x),

∂y

∂t

(s,x) = φ

1

(s,x),

s ∈ [−r,0], x ∈ Ω;

y

t

(t

+

k

,x) =

y

t

(t

−

k

,x)

+ I

k

(t,y(t

k

,x),y

t

(t

k

,x),u(t

k

,x)),

x ∈ Ω,

where 0 < t

1

< t

2

< t

3

< ·· · < t

p

< τ, Ω is a

bounded domain in R

n

, the distributed control u ∈

L

2

([0,τ]; L

2

(Ω)), φ

0

∈ C(−r,0;H

2

(Ω) ∩ H

1

0

),φ

1

∈

C(−r,0;L

2

(Ω)) and the nonlinear functions f ,I

k

:

[0,τ] × R × R × R → R are smooth enough and

| f (t, y,v,u)| ≤ a

0

p

y

2

+ v

2

+ b

0

, y, v, u ∈ R. (5)

2 Controllability of the Linear

Equation

In this section we present some characterization of

the approximate controllability of the corresponding

linear equations without impulses and delays. To

this end, we note that for all z

0

∈ Z and u ∈

L

2

(0,τ;U) the initial value problem

(

z

0

= Az + Bu(t), z ∈ Z;

z(t

0

) = z

0

,

(6)

admits only one mild solution given by

z(t) = z(t,t

0

,z

0

,u) =

T (t)z

0

+

Z

t

t

0

T (t −s)Bu(s)ds, t ∈ [t

0

,τ], 0 ≤ t

0

≤ τ.

(7)

(See for example (Curtain & Zwart, 1995; Leiva,

2003)).

DEFINITION 2.1. For system (6) we deﬁne the

following concept: The controllability map G

τδ

:

L

2

(τ − δ,τ;U)) → Z deﬁned by

G

τδ

u =

Z

τ

τ−δ

T (τ − s)Bu(s)ds, u ∈ L

2

(τ − δ,τ;U),

(8)

The adjoint of this operator G

∗

τδ

: Z → L

2

(τ−δ,τ;U)

is given by

(G

∗

τδ

z)(t) = B

∗

T

∗

(τ −t)z, t ∈ [τ − δ,τ].

The Gramian controllability operators are given by:

Q

τδ

= G

τδ

G

∗

τδ

=

Z

τ

τ−δ

T (τ−t)BB

∗

T

∗

(τ −t)dt. (9)

The following lemma holds in general for a linear

bounded operator G : W → Z between Hilbert

spaces W and Z (Bashirov et al., 2007; Leiva et al.,

2013; Curtain & Pritchard, 2010; Curtain & Zwart,

1995).

LEMMA 2.1. The following statements are

equivalent to the approximate controllability of the

linear system (6) on [τ − δ,τ].

a) Range(G

τδ

) = Z.

b) Ker(G

∗

τδ

) = {0}.

c) hQ

τδ

z,zi > 0, z 6= 0 in Z.

d) lim

α→0

+

α(αI + Q

τδ

)

−1

z = 0.

e) For all z ∈ Z, we have G

τδ

u

α

= z − α(αI +

Q

T δ

)

−1

z, where

u

α

= G

∗

τδ

(αI + Q

τδ

)

−1

z, α ∈ (0,1].

So, lim

α→0

G

τδ

u

α

= z and the error E

τδ

z of this

approximation is given by the formula

E

τδ

z = α(αI + Q

τδ

)

−1

z, α ∈ (0, 1].

f) Moreover, if we consider for each v ∈ L

2

(τ −

δ,τ;U)) the sequence of controls given by

u

α

= G

∗

τδ

(αI + Q

τδ

)

−1

z+

(v − G

∗

τδ

(αI + Q

τδ

)

−1

G

τδ

v), α ∈ (0,1],

we get that:

G

τδ

u

α

= z − α(αI + Q

T δ

)

−1

(z + G

τδ

v)

and

lim

α→0

G

τδ

u

α

= z,

with the error E

τδ

z of this approximation given

by the formula

E

τδ

z = α(αI + Q

τδ

)

−1

(z + G

τδ

v), α ∈ (0, 1].

REMARK 2.1. The foregoing lemma implies that the

family of linear operators

Γ

ατδ

: Z → W , deﬁned for 0 < α ≤ 1 by

Γ

ατδ

z = G

∗

τδ

(αI + Q

τδ

)

−1

z, (10)

is an approximate right inverse of the operator W ,

in the sense that

lim

α→0

G

τδ

Γ

ατδ

= I. (11)

in the strong topology.

http://novasinergia.unach.edu.ec 39

LEMMA 2.2. Leiva et al. (2013) Q

τδ

> 0 if and only

if the linear system (6) is controllable on [τ − δ,τ].

Moreover, given an initial state y

0

and a ﬁnal state

z

1

we can ﬁnd a sequence of controls {u

δ

α

}

0<α≤1

⊂

L

2

(τ − δ,τ;U)

u

α

= G

∗

τδ

(αI +G

τδ

G

∗

τδ

)

−1

(z

1

−T (τ)y

0

), α ∈ (0,1],

such that the solutions y(t) = y(t, τ−δ,y

0

,u

δ

α

) of the

initial value problem

(

y

0

= Ay + Bu

α

(t), y ∈ Z, t > 0;

y(τ − δ) = y

0

,

(12)

satisfy

lim

α→0

+

y(τ,τ − δ,y

0

,u

α

) = z

1

,

i.e.,

lim

α→0

+

y(τ) =

lim

α→0

+

T (δ)y

0

+

Z

τ

τ−δ

T (τ − s)Bu

α

(s)ds

= z

1

.

3 Controllability of the Semilinear

Equation

In this section we prove the main result of this

paper, that is, the approximate controllability of the

semilinear impulsive evolution equation given by

(1). To this end, for all φ ∈ C and u ∈ C(0,τ;U)

the initial value problem

z

0

= Az + Bu + F(t,z

t

,u(t)), z ∈ Z, t ≥ 0;

z(s) = φ(s), s ∈ [−r,0];

z(t

+

k

) = z(t

−

k

) + I

k

(t

k

,z(t

k

),u(t

k

)), k = 1, ..., p,

(13)

admits only one mild solution z ∈ PC(−r,τ; Z) given

by

z(t) =

T (t)φ(0)+

Z

t

0

T (t − s)Bu(s) ds

+

Z

t

0

T (t − s)F(s,z

s

,u(s)) ds

+

∑

0<t

k

<t

T (t − t

k

)I

k

(t

k

,z(t

k

),u(t

k

))

t ∈ [0, τ];

φ(t), t ∈ [−r,0].

(14)

Now, we are ready to present and prove the

main result of this paper, which is the interior

approximate controllability of heat equation with

delays (1).

THEOREM 3.1. Under conditions (H1) and (H2),

the semilinear system (1) with impulses and delays

is approximately controllable on [0, τ].

Proof. Given an initial state φ, a ﬁnal state z

1

and

ε > 0, we want to ﬁnd a control u

δ

α

∈ L

2

(0,τ;U)

steering the system from φ(0) to an ε-neighborhood

of z

1

at time τ. In other word, there exists

control u

δ

α

∈ L

2

(0,τ; U) such that corresponding of

solutions z

δ,α

of (1) satisﬁes:

kz

δ,α

(τ) − z

1

k ≤ ε.

In fact, consider any u ∈ L

2

(0,τ;U) and the

corresponding solution z(t) = z(t,0,z

0

,u) of the

initial value problem (13). For α ∈ (0,1] we deﬁne

the control u

δ

α

∈ L

2

(0,τ;U) as

u

δ

α

(t) =

(

u(t), if 0 ≤ t ≤ τ − δ;

u

α

(t), if τ − δ < t ≤ τ,

where

u

α

(t) =

B

∗

T

∗

(τ −t)(αI + G

τδ

G

∗

τδ

)

−1

(z

1

− T (δ)z(τ − δ)),

τ − δ < t ≤ τ.

Now, assume that 0 < δ < τ − t

p

. Then the

corresponding solution z

δ

α

(t) = z(t,0, z

0

,u

δ

α

) of the

initial value problem (13) at time τ can be written as

follows:

z

δ,α

(τ) =

T (τ)φ(0) +

Z

τ

0

T (τ − s)Bu

δ

α

(s)ds

+

Z

τ

0

T (τ − s)F(s,z

δ,α

s

,u

δ

α

(s))ds

+

∑

0<t

k

<τ

T (τ −t

k

)I

k

(z

δ,α

(t

k

),u

δ

α

(t

k

))

= T (δ)

T (τ − δ)φ(0) +

Z

τ−δ

0

T (τ − δ − s)Bu

δ

α

(s)ds

+

Z

τ−δ

0

T (τ − δ − s)F(s,z

δ,α

s

,u

δ

α

(s))ds

+

∑

0<t

k

<τ−δ

T (τ − δ −t

k

)I

k

(z

δ,α

(t

k

),u

δ

α

(t

k

))

)

+

Z

τ

τ−δ

T (τ − s)Bu

δ

α

(s)ds

+

Z

τ

τ−δ

T (τ − s)F(s,z

δ,α

s

),u

δ

α

(s))ds

= T (δ)z(τ − δ) +

Z

τ

τ−δ

T (τ − s)Bu

α

(s)ds

+

Z

τ

τ−δ

T (τ − s)F(s,z

δ,α

s

,u

α

(s))ds.

http://novasinergia.unach.edu.ec 40

Thus,

z

δ,α

(τ) = T (δ)z(τ − δ) +

Z

τ

τ−δ

T (τ − s)Bu

α

(s)ds

+

Z

τ

τ−δ

T (τ − s)F(s,z

δ,α

s

,u

α

(s))ds.

The corresponding solution y

δ

α

(t) = y(t,τ − δ,z(τ −

δ),u

α

) of the initial value problem (12) at time τ is

given by:

y

δ

α

(τ) = T (δ)z(τ − δ) +

Z

τ

τ−δ

T (τ − s)Bu

α

(s)ds.

Therefore,

kz

δ,α

(τ) − y

δ

α

(τ)k

≤

Z

τ

τ−δ

kT (τ − s)k{akz

δ,α

(s − r)k+b}ds.

If we take 0 < δ < r and τ − δ ≤ s ≤ τ, then s − r ≤

τ − r < τ − δ and

z

δ,α

(s − r) = z(s − r).

Thus, there exists δ small enough such that 0 < δ <

min{r,τ −t

p

} and

kz

δ,α

(τ) − y

δ,α

(τ)k

≤

Z

τ

τ−δ

kT (τ − s)k{akz(s − r)k + b} ds <

ε

2

.

Hence,

kz

δ,α

(τ) − z

1

k

≤

Z

τ

τ−δ

kT (τ − s)k{akz

δ,α

(s − r)k+b}ds

+ ky

δ,α

(τ) − z

1

k

=

Z

τ

τ−δ

kT (τ − s)k{akz(s − r)k + b} ds

+ ky

δ,α

(τ) − z

1

k

<

ε

2

+

ε

2

< ε.

Geometrically, the proof goes as follows:

CONTROLLABILITY OF EVOLUTION EQUATIONS WITH IMPULSES AND DELAY. 7

if we take 0 < δ < r and τ − δ ≤ s ≤ τ, then s − r ≤ τ − r < τ − δ and

z

δ,α

(s − r) = z(s − r).

Therefore, there exists δ small enough such that 0 < δ < min{r, τ − t

p

} and

kz

δ,α

(τ) − y

δ,α

(τ)k ≤

Z

τ

τ −δ

kT (τ − s)k{akz(s − r)k + b}ds <

ǫ

2

.

Hence,

kz

δ,α

(τ) − z

1

k ≤

Z

τ

τ −δ

kT (τ − s)k{akz

δ,α

(s − r)k + b}ds + ky

δ,α

(τ) − z

1

k

=

Z

τ

τ −δ

kT (τ − s)k{akz(s − r)k + b}ds + ky

δ,α

(τ) − z

1

k

<

ǫ

2

+

ǫ

2

< ǫ.

Geometrically, the proof goes as follows:

z

0

z(τ − δ)

z

δ,α

(

s −

r) = z

(s −

r)

z(τ )

z

δ

α

(τ )

y

δ

α

(τ )

z

1

ε

This completes the proof of the theor em.

This completes the proof of the theorem.

4 Applications

As an application, we prove the approximate

controllability of the following control system

governed by the semilinear wave equation with

impulses and delays

∂

2

y

∂

2

t

=

∆y + u(t,x)

+ f (t,y(t − r),

∂y

∂t

(t − r), u(t)),

on (0,τ) × Ω;

y = 0, on (0,τ) × ∂Ω;

y(s,x) = φ

0

(s,x),

∂y

∂t

(s,x) = φ

1

(s,x),

s ∈ [−r,0], x ∈ Ω;

y

t

(t

+

k

,x) =

y

t

(t

−

k

,x)

+ I

k

(t,y(t

k

,x),y

t

(t

k

,x),u(t

k

,x)),

x ∈ Ω,

(15)

where 0 < t

1

< t

2

< t

3

< ·· · < t

p

< τ, Ω is a

bounded domain in R

n

, the distributed control u ∈

L

2

([0,τ]; L

2

(Ω)), φ

0

∈ C(−r,0;H

2

(Ω) ∩ H

1

0

),φ

1

∈

C(−r,0;L

2

(Ω)) and the nonlinear functions f ,I

k

:

[0,τ] × R × R × R → R are smooth enough and f

satisﬁes (5).

4.1 Abstract Formulation of the

Problem

First we choose the space where this problem will

be set up as an abstract control system in a Hilbert

space. Let X = L

2

(Ω) = L

2

(Ω,R) and consider

the linear unbounded operator A : D(A) ⊂ X → X

deﬁned by Aφ = −∆φ, where

D(A) = H

2

(Ω,R) ∩ H

1

0

(Ω,R). (16)

Then the eigenvalues λ

j

of A have ﬁnite multiplicity

γ

j

equal to the dimension of the corresponding

eigenspace and 0 < λ

1

< λ

2

< ·· · < λ

n

→ ∞.

Moreover,

a) there exists a complete orthonormal set {φ

j,k

} of

eigenvectors of A;

b) for all x ∈ D(A) we have

Ax =

∞

∑

j=1

λ

j

γ

j

∑

k=1

< x,φ

j,k

> φ

j,k

=

∞

∑

j=1

λ

j

E

j

x,

(17)

where < ·,· > is the usual inner product in L

2

and

E

j

x =

γ

j

∑

k=1

< x,φ

j,k

> φ

j,k

, (18)

http://novasinergia.unach.edu.ec 41

which means the set {E

j

}

∞

j=1

is a complete

family of orthogonal projections in X and x =

∞

∑

j=1

E

j

x, x ∈ X;

c) −A generates an analytic semigroup {e

−At

}

given by

e

−At

x =

∞

∑

j=1

e

−λ

j

t

E

j

x; (19)

d) the fractional powered spaces X

r

are given by:

X

r

= D(A

r

) = {x ∈ X :

∞

∑

n=1

λ

2r

n

kE

n

xk

2

< ∞}, r ≥ 0,

with the norm

kxk

r

= kA

r

xk =

(

∞

∑

n=1

λ

2r

n

kE

n

xk

2

)

1/2

,x ∈ X

r

, and

A

r

x =

∞

∑

n=1

λ

r

n

E

n

x. (20)

Also, for r ≥ 0 we deﬁne Z

r

= X

r

× X, which is

a Hilbert space endowed with the norm:

y

v

Z

r

=

q

kyk

2

r

+ kvk

2

.

Then, the equations (1) can be written as an abstract

second order ordinary differential equations in Z

1/2

as follows

y

00

=

− Ay + u

+ f

e

(t,y(t − r),y

0

(t − r), u),

t ∈ (0, τ], t 6= t

k

,

y(s) = φ

0

(s), y

0

(s) = φ

1

(s), s ∈ [−r,0],

y

0

(t

+

k

) =

y

0

(t

−

k

) + I

e

k

(t

k

,y(t

k

),y

0

(t

k

),u(t

k

)),

k = 1, ..., p,

(21)

where

I

e

k

: [0,τ] × Z

1/2

×U → Z

1/2

and

f

e

: [0,τ] ×C

0

×C

1

×U → Z

1/2

with C

0

= C(−r,0;Z

1/2

) and C

1

= C(−r,0;Z) are

deﬁned by

I

e

k

(t,y,v,u)(x) =

I

k

(t,y(x), v(x),u(x)), ∀x ∈ Ω, k = 1,2,..., p,

f

e

(t,φ

0

,φ

1

,u)(x) =

f (t,φ

0

(−r,x),φ

1

(−r,x),u(x)),

∀x ∈ Ω,

φ

0

φ

1

∈ C

0

×C

1

.

With the change of variables y

0

= v, we can write the

second order equation (21) as a ﬁrst order system of

ordinary differential equations in the Hilbert space

Z

1/2

= X

1/2

× X as follows:

z

0

=

Az + Bu

+ F(t,z(t − r),u(t)),

z ∈ Z

1/2

, t ∈ (0,τ], t 6= t

k

;

z(s) = φ(s), s ∈ [−r,0];

z(t

+

k

) =

z(t

−

k

) + J

k

(t

k

,z(t

k

),u(t

k

)),

k = 1, ..., p,

(22)

where u ∈ L

2

([0,τ];U) and C = C

0

× C

1

=

C(−r,0;Z

1/2

),

φ =

φ

0

φ

1

∈C, z =

y

v

, B =

0

I

z and A =

0 I

X

−A 0

(23)

is an unbounded linear operator with domain

D(A) = D(A) × D(A

1/2

) and

J

k

: [0,τ] × Z

1/2

× U → Z

1/2

, F : [0,τ] × C ×U →

Z

1/2

are deﬁned by:

F(t,φ,u) =

0

f

e

(t,φ

0

,φ

1

,u)

and J

k

(t,z, u) =

0

I

e

k

(t,y,v,u)

. (24)

The following result follows from condition (5)

PROPOSITION 4.1. Under the conditions (5) the

functions F satisfy:

kF(t,φ,u)k

Z

1/2

≤ ˜a

0

kφ(−r)k

Z

1/2

+ b

0

. (25)

It is well known that the operator A generates a

strongly continuous group

{

T (t)

}

t≥0

in the space

Z = Z

1/2

= X

1/2

× X (Chen & Triggiani, 1989).

Now, using Lemma 2.1 from Leiva (2003) or

Lemma 3.1 from Carrasco & Leiva (2007), one can

get the following representation for this group.

PROPOSITION 4.2. The group

{

T (t)

}

t≥0

generated

by the operator A has the following representation

T (t)z =

∞

∑

n j=1

e

A

j

t

P

j

z, z ∈ Z

1/2

, t ≥ 0, (26)

where

P

j

j≥0

is a complete family of orthogonal

projections in the Hilbert space Z

1/2

given by

P

j

=

E

j

0

0 E

j

, j ≥ 1, (27)

and

A

j

= R

j

P

j

, R

j

=

0 1

−λ

j

0

, j ≥ 1. (28)

http://novasinergia.unach.edu.ec 42

4.2 Approximate Controllability

Now, we are ready to formulate and prove the main

result of the this section, which is the approximate

of the semilinear impulsive wave equation with

bounded nonlinear perturbation.

THEOREM 4.1. The semilinear wave equation

(15) with impulses and delays is approximately

controllable on [0,τ].

Proof. From Larez et al. (2011), we know that the

corresponding linear system without impulses

(

z

0

= Az + Bu, z ∈ Z

1/2

, t ∈ (0,τ];

z(0) = z

0

,

(29)

is controllable on [τ − δ,τ] for all 0 < δ < τ. On

the other hand, the hypothesis (H1) and (H2) in

Theorem 3.1 are satisﬁed, and we get the result.

5 Final Remark

This technique can be applied to those systems

where the linear part does not generate a compact

semigroup, are controllable on any [0,δ] for δ >

0, and the nonlinear perturbation is bounded. An

example of such systems is the following controlled

thermoelastic plate equation whose linear part was

studied in Larez et al. (2011).

y

tt

+ ∆

2

y + α∆θ

=u

1

(t,x) + f

1

(t,y,y

t

,θ, u(t)),

on (0,τ) × Ω,

θ

t

− β∆θ − α∆y

t

= u

2

(t,x) + f

2

(t,y,y

t

,θ, u(t)),

on (0,τ) × Ω,

θ = y = ∆y = 0, on (0,τ) × ∂Ω,

y

t

(t

+

k

,x) =

y

t

(t

−

k

,x)

+ I

1

k

(t,y(t

k

,x),y

t

(t

k

,x),u(t

k

,x)),

x ∈ Ω,

θ(t

+

k

,x) =

θ(t

−

k

,x)+I

2

k

(t

k

,θ(t

k

,x),u(t

k

,x)),

x ∈ Ω,

(30)

in the space Z = X

1

× X × X, where Ω is a

bounded domain in R

n

, the distributed controls

u

1

,u

2

∈ L

2

([0,τ]; L

2

(Ω)) and I

i

k

, f

i

are smooth

functions with f

i

,i = 1, 2 bounded. Of course,

for ﬁnite-dimensional control systems, all these

results are valid for exact controllability; so from

the point of view of applications, we can study

real life control systems governed by ordinary

differential equations in ﬁnite-dimensional spaces,

with impulses and delays.

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