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J. Nonlinear Sci. Appl. 5 (2012), 206–219
Research Article
Controllability results for impulsive differential
systems with finite delay
S. Selvia , M. Mallika Arjunanb,∗
a
b
Department of Mathematics, Muthayammal College of Arts & Science, Rasipuram- 637408, Tamil Nadu, India.
Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore- 641 114, Tamil Nadu, India
This paper is dedicated to Professor Ljubomir Ćirić
Communicated by Professor V. Berinde
Abstract
This paper establishes some sufficient conditions for controllability of impulsive functional differential equations with finite delay in a Banach space. The results are obtained by using the measures of noncompactness
and Monch fixed point theorem. Particularly, we do not assume the compactness of the evolution system.
c
Finally, an example is provided to illustrate the theory.2012
NGA. All rights reserved.
Keywords: Controllability, Impulsive differential equations, Measures of noncompactness, Semigroup
theory, Fixed point.
2010 MSC: Primary 93B05, 34A37, 34G20.
1. Introduction
Impulsive differential equations have become more important in recent years in some mathematical
models of real processes and phenomena studied in control, physics, chemistry, population dynamics, aeronautics and engineering. There has been a significant development in impulsive theory in recent years,
especially in the area of impulsive differential equations with fixed moments, see the monographs of Bainov
and Simeonov [3], Lakshmikantham et al. [14] and Samoilenko and Perestyuk [20] and the papers of
[1, 2, 5, 7, 8, 9, 10, 12, 23]. On the other hand, differential equations with delay was initiated about
existence and stability by Travis and Webb [21] and Webb [22]. Since such equations are often more realistic to describe natural phenomena than those without delay, they have been investigated in variant
∗
Corresponding author
Email addresses: sselvimaths@yahoo.com (S. Selvi), arjunphd07@yahoo.co.in (M. Mallika Arjunan)
Received 2011-10-12
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
207
aspects by many authors [2, 15]. The concept of controllability plays an important role in many areas of
applied mathematics. In recent years, significant progress has been made in the controllability of linear and
nonlinear deterministic systems [6, 11, 17, 19]. In [11], the author studied the controllability of impulsive
functional differential systems of the form
x0 (t) = A(t)x(t) + f (t, x(t)) + (Bu)(t), a.e. on [0, b],
∆x|t=ti = Ii (x(ti )), i = 1, 2, . . . , s,
x(0) + M (x) = x0 ,
where A(t) is a family of linear operators which generates an evolution operator U : ∆ = {(t, s) ∈ [0, b]×[0, b] :
0 ≤ s ≤ t ≤ b} → L(X), here X is a Banach space, L(X) is the space of all bounded linear operators in
X; f : [0, b] × X → X; 0 < t1 < · · · < ts < ts+1 = b; Ii : X → X, i = 1, 2, . . . , s, are impulsive functions;
M : P C([0, b], X) → X; B is a bounded linear operator from a Banach space V to X and the control
function u(·) is given in L2 ([0, b], V ). The results are obtained by using the measures of noncompactness
and Monch fixed point theorem.
Motivated by the above mentioned works [7, 11, 15, 23], the main purpose of this paper is to establish
the sufficient conditions for the controllability of impulsive differential system with finite delay of the form
x0 (t) = A(t)x(t) + f (t, xt ) + (Bu)(t),
(1.1)
t ∈ J = [0, b], t 6= ti , i = 1, 2, . . . , s,
∆x|t=ti = Ii (xti ), i = 1, 2, . . . , s,
x(t) = ϕ(t), t ∈ [−r, 0],
(1.2)
(1.3)
where A(t) is a family of linear operators which generates an evolution system {U (t, s) : 0 ≤ s ≤ t ≤ b}.
The state variable x(·) takes the values in the real Banach space X with norm k · k. The control function
u(·) is given in L2 (J, V ) a Banach space of admissible control functions with V as a Banach space. B is a
bounded linear operator from V into X. f : J × D → X is given function, where D = {ψ : [−r, 0] → X :
−
ψ(t) is continuous everywhere except for a finite number of points ti at which ψ(t+
i ) and ψ(ti ) exist and
−
ψ(ti ) = ψ(ti )}; Ii : D → X; i = 1, 2, . . . , s, are impulsive functions, 0 < t1 < t2 < · · · < ts < ts+1 = b,
−
∆ξ(ti ) is the jump of a function ξ at ti , which is defined by ∆ξ(ti ) = ξ(t+
i ) − ξ(ti ).
For any function x ∈ PC and any t ∈ J, xt denotes the function in D defined by
xt (θ) = x(t + θ),
θ ∈ [−r, 0].
where PC is defined in Preliminaries. Here xt (·) represents the history of the state from the time t − r
upto the present time t. Our approach here is based on semigroup theory, measures of noncompactness and
Monch fixed point theorem.
2. Preliminaries
In this section, we recall some basic definitions and lemmas which will be used to prove our main results
of this paper.
Let L1 ([0, b], X) the space of X-valued Bochner integrable functions on [0,b] with the norm kf kL1 =
Rb
0 kf (t)kdt. In ordernto define the solution of the problem (1.1)-(1.3), we consider the following space:
PC([−r, b], X) = x : [−r, b] → X such that x(·) is continuous except for a finite number of points ti at
o
−
−
which x(t+
)
and
x(t
)
exist
and
x(t
)
=
x(t
)
.
i
i
i
i
It is easy to verify that PC([−r, b], X) is a Banach space with the norm
kxkPC = sup{kx(t)k : t ∈ [−r, b]}.
For our convenience let PC = PC([−r, b], X) and J0 = [0, t1 ]; Ji = (ti , ti+1 ], i = 1, 2, . . . , s.
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
208
Definition 2.1. Let E + be the positive cone of an order Banach space (E, ≤). A function Φ defined on the set
of all bounded subsets of the Banach space X with values in E + is called a measure of noncompactness(MNC)
on X if Φ(coΩ) = Φ(Ω) for all bounded subsets Ω ⊆ X, where coΩ stands for the closed convex hull of Ω.
The MNC Φ is said:
(1) Monotone if for all bounded subsets Ω1 , Ω2 of X we have: (Ω1 ⊆ Ω2 ) ⇒ (Φ(Ω1 ) ≤ Φ(Ω2 ));
(2) Nonsingular if Φ({a} ∪ Ω) = Φ(Ω) for every a ∈ X, Ω ⊂ X;
(3) Regular if Φ(Ω) = 0 if and only if Ω is relatively compact in X.
One of the most examples of MNC is the noncompactness measure of Hausdorff β defined on each bounded
subset Ω of X by
β(Ω) = inf{ > 0; Ω can be covered by a finite number of balls of radii smaller than }
It is well known that MNC β enjoys the above properties and other properties see [4, 13]: For all bounded
subsets Ω, Ω1 , Ω2 of X,
(4) β(Ω1 + Ω2 ) ≤ β(Ω1 ) + β(Ω2 ), where Ω1 + Ω2 = {x + y : x ∈ Ω1 , y ∈ Ω2 };
(5) β(Ω1 ∪ Ω2 ) ≤ max{β(Ω1 ), β(Ω2 )};
(6) β(λΩ) ≤ |λ|β(Ω) for any λ ∈ R;
(7) If the map Q : D(Q) ⊆ X → Z is Lipschitz continuous with constant k, then βZ (QΩ) ≤ kβ(Ω) for
any bounded subset Ω ⊆ D(Q), where Z is a Banach space.
Definition 2.2. A two parameter family of bounded linear operators U (t, s), 0 ≤ s ≤ t ≤ b on X is called
an evolution system if the following two conditions are satisfied:
(i) U (s, s) = I, U (t, r)U (r, s) = U (t, s) for 0 ≤ s ≤ r ≤ t ≤ b;
(ii) (t, s) → U (t, s) is strongly continuous for 0 ≤ s ≤ t ≤ b.
Since the evolution system U (t, s) is strongly continuous on the compact operator set J × J, then there
exists M1 > 0 such that kU (t, s)k ≤ M1 for any (t, s) ∈ J × J. More details about evolution system can be
found in Pazy [18].
Definition 2.3. A function x(·) ∈ PC is said to be a mild solution of the system (1.1) − (1.3) if, x(t) = ϕ(t)
on [−r, 0]; ∆x|t=ti = Ii (xti ), i = 1, 2 . . . , s; the restriction of x(·) to the interval Ji (i = 1, 2, . . . , s) is
continuous and the following integral equation is satisfied.
Z t
h
i
X
x(t) = U (t, 0)ϕ(0) +
U (t, s) Bu(s) + f (s, xs ) ds +
U (t, ti )Ii (xti ), t ∈ J.
0
0<ti <t
Definition 2.4. The system (1.1) − (1.3) is said to be controllable on the interval J if, for every initial
function ϕ ∈ D and x1 ∈ X, there exists a control u ∈ L2 (J, V ) such that the mild solution x(·) of (1.1)−(1.3)
satisfies x(b) = x1 .
1
∞
Definition 2.5. A countable set {fn }∞
n=1 ⊂ L ([0, b], X) is said to be semicompact if the sequence {fn }n=1
1
+
is relatively compact in X for almost all t ∈ [0, b] and if there is a function µ ∈ L ([0, b], R ) satisfying
supkfn (t)k ≤ µ(t) for a.e. t ∈ [0, b].
n≥1
Lemma 2.1. ([4]) If W ⊂ C([a, b], X) is bounded and equicontinuous, then β(W (t)) is continuous for
t ∈ [a, b] and
β(W ) = sup{β(W (t)), t ∈ [a, b]},
where
W (t) = {x(t) : x ∈ W } ⊆ X.
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
209
Lemma 2.2. ([23]) If W ⊂ PC([a, b], X) is bounded and piecewise equicontinuous on [a, b] then β(W (t)) is
piecewise continuous for t ∈ [a, b] and
β(W ) = sup{β(W (t)), t ∈ [a, b]}.
1
+
Lemma 2.3. ([17]) Let {fn }∞
n=1 be a sequence of functions in L ([0, b], R ). Assume that there exist
µ, η ∈ L1 ([0, b], R+ ) satisfying supkfn (t)k ≤ µ(t) and β({fn (t)}∞
n=1 ) ≤ η(t) a.e. t ∈ [0, b],
n≥1
then for all t ∈ [0, b], we have
β
n Z
t
Z t
o
η(s)ds.
U (t, s)fn (s)ds : n ≥ 1
≤ 2M1
0
0
Rt
1
Lemma 2.4. ([17]) Let (Gf )(t) = 0 U (t, s)f (s)ds, If {fn }∞
n=1 ⊂ L ([0, b], X) is semicompact, then the set
∞
{Gfn }n=1 is relatively compact in C([0, b], X) and moreover if fn * f0 , then for all t ∈ [0, b],
(Gfn )(t) → (Gf0 )(t), as n → ∞.
The following fixed-point theorem, a nonlinear alternative of Monch type, plays a key role in our proof
of controllability of the system (1.1) − (1.3).
Lemma 2.5. ([16, Theorem 2.2]) Let D be a closed convex subset of a Banach space X and 0 ∈ D. Assume
that F : D → X is a continuous map which satisfies Monch’s condition, that is (M ⊆ D is countable,
M ⊆ co({0} ∪ F (M )) ⇒ M is compact ). Then F has a fixed point in D.
3. Controllability Results
In this section, we present and prove the controllability results for the problem (1.1) − (1.3). In order to
prove the main theorem of this section, we list the following hypotheses:
(H1) A(t) is a family of linear operators, A(t) : D(A) → X, D(A) not depending on t and dense subset of X,
generating an equicontinuous evolution system {U (t, s) : 0 ≤ s ≤ t ≤ b}, i.e., (t, s) → {U (t, s)x : x ∈
B} is equicontinuous for t > 0 and for all bounded subsets B and M1 = sup{kU (t, s)k : (t, s) ∈ J × J}.
(H2) The function f : J × D → X satisfies:
(i) For a.e. t ∈ J, the function f (t, ·) : D → X is continuous and for all ϕ ∈ D, the function
f (·, ϕ) : J → X is strongly measurable.
(ii) For every positive integer r, there exists αr ∈ L1 ([0, b]; R+ ) such that
sup kf (t, ϕ)k ≤ αr (t) for a.e. t ∈ J,
kϕkD ≤r
and
Z
lim inf
r→∞
0
b
αr (t)
dt = σ < ∞.
r
(iii) There exists integrable function η : [0, b] → [0, ∞) such that
β(f (t, D)) ≤ η(t) sup β(D(θ))
for a.e. t ∈ J and D ⊂ D,
−r≤θ≤0
where D(θ) = {v(θ) : v ∈ D}.
(H3) The linear operator W : L2 (J, V ) → X is defined by
Z b
Wu =
U (t, s)Bu(s)ds such that
0
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
210
(i) W has an invertible operator W −1 which take values in
L2 (J, V )\kerW , and there exist positive constants M2 and M3 such that
kBk ≤ M2 , kW −1 k ≤ M3 .
(ii) There is KW ∈ L1 (J, R+ ) such that, for every bounded set Q ⊂ X,
β(W −1 Q)(t) ≤ KW (t)β(Q).
(H4) Ii : D → X, i = 1, 2 . . . , s, be a continuous operator such that:
(i) There are nondecreasing functions Li : R+ → R+ such that
kIi (x)k ≤ Li (kxkD ) i = 1, 2 . . . , s,
and
lim inf
ρ→∞
Li (ρ)
= λi < ∞,
ρ
x ∈ D,
i = 1, 2 . . . , s.
(ii) There exist constants Ki ≥ 0 such that,
β(Ii (S)) ≤ Ki sup β(S(θ)),
i = 1, 2 . . . , s,
−r≤θ≤0
for every bounded subset S of D.
(H5) The following estimation holds true:
h
N = (M1 +
2M12 M2 kKW kL1 )
s
X
i
Ki + (2M1 + 4M12 M2 kKW kL1 )kηkL1 < 1.
i=1
Theorem 3.1. Assume that the hypotheses (H1)−(H5) are satisfied. Then the impulsive differential system
(1.1) − (1.3) is controllable on J provided that,
1
M1 (1 + M1 M2 M3 b 2 )(σ +
s
X
λi ) < 1.
(3.1)
i=1
Proof. Using the hypothesis (H3)(i), for every x ∈ PC([−r, b], X), define the control
Z b
h
i
X
−1
x1 − U (b, 0)ϕ(0) −
U (b, s)f (s, xs )ds −
U (b, ti )Ii (xti ) (t).
ux (t) = W
0
0<ti <b
We shall now show that when using this control the operator defined by

ϕ(t), t ∈ [−r, 0],


Z t



(F x)(t) = U (t, 0)ϕ(0) + 0 U (t, s)[f (s, xs ) + (Bux )(s)]ds
X



U (t, ti )Ii (xti ), t ∈ J,

 +
0<ti <t
has a fixed point. This fixed point is then a solution of (1.1) − (1.3). Clearly x(b) = (F x)(b) = x1 , which
implies the system (1.1) − (1.3) is controllable. We rewrite the problem (1.1) − (1.3) as follows:
For ϕ ∈ D, we define ϕ̂ ∈ PC by
(
U (t, 0)ϕ(0), t ∈ J,
ϕ̂(t) =
ϕ(t), t ∈ [−r, 0].
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
211
Then ϕ̂ ∈ PC. Let x(t) = y(t) + ϕ̂(t), t ∈ [−r, b]. It is easy to see that y satisfies y0 = 0 and
Z
t
U (t, s)[f (s, ys + ϕ̂s ) + Buy (s)]ds +
y(t) =
0
X
U (t, ti )Ii (yti + ϕ̂ti ),
0<ti <t
where
Z b
h
uy (s) = W −1 x1 − U (b, 0)ϕ(0) −
U (b, s)f (s, ys + ϕ̂s )ds
0
−
s
X
i
U (b, ti )Ii (yti + ϕ̂ti ) (s),
i=1
if and only if x satisfies
Z
t
U (t, s)[f (s, xs ) + Bux (s)]ds +
x(t) = U (t, 0)ϕ(0) +
0
X
U (t, ti )Ii (xti ),
0<ti <t
and x(t) = ϕ(t), t ∈ [−r, 0]. Define PC 0 = {y ∈ PC : y0 = 0}. Let G : PC 0 → PC 0 be an operator defined
by

0, t ∈ [−r, 0],


Z


 t
U (t, s)[f (s, ys + ϕ̂s ) + Buy (s)]ds
(Gy)(t) =
(3.2)
0
X



U (t, ti )Ii (yti + ϕ̂ti ), t ∈ J.

 +
0<ti <t
Obviously the operator F has a fixed point is equivalent to G has one. So it turns out to prove G has a
fixed point.
Let G = G1 + G2 ,
where
X
(G1 y)(t) =
U (t, ti )Ii (yti + ϕ̂ti ),
(3.3)
0<ti <t
t
Z
(G2 y)(t) =
U (t, s)[f (s, ys + ϕ̂s ) + Buy (s)]ds.
(3.4)
0
Step 1: There exists a positive number q ≥ 1 such that G(Bq ) ⊆ Bq , where Bq = {y ∈ PC 0 : kykPC ≤ q}.
Suppose the contrary. Then for each positive integer q, there exists a function y q (·) ∈ Bq but G(y q ) ∈
/ Bq .
i.e., kG(y q )(t)k > q for some t ∈ J.
We have from (H1) − (H4),
q < k(Gy q )(t)k
Z b
s
X
≤ M1
kf (s, ysq + ϕ̂s ) + Buyq (s)kds + M1
Li (kytqi + ϕ̂ti kD )
0
Z
i=1
b
≤ M1
Z
kBuyq (s)kds + M1
αq0 (s)ds + M1
0
Z
≤ M1
b
0
b
Li (q 0 )
i=1
1
αq0 (s)ds + M1 M2 b 2 kuyq kL2 + M1
0
s
X
s
X
i=1
Li (q 0 ),
(3.5)
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
212
where
Z b
s
i
h
X
αq0 (s)ds + M1
Li (q 0 ) .
≤ M3 kx1 k + M1 kϕkD + M1
kuyq kL2
0
(3.6)
i=1
Hence by (3.5),
b
Z
Z
h
1
2
αq0 (s)ds + M1 M2 b M3 kx1 k + M1 kϕkD + M1
q < M1
0
+ M1
b
αq0 (s)ds
0
s
X
s
i
X
Li (q 0 ) + M1
Li (q 0 )
i=1
i=1
b
1
≤ (1 + M1 M2 M3 b 2 )M1
hZ
αq0 (s)ds +
0
s
X
i
Li (q 0 ) + M,
i=1
1
where M = M1 M2 M3 b 2 (kx1 k + M1 kϕkD ) is independent of q and q 0 = q + kϕ̂kPC .
Dividing both sides by q and noting that q 0 = q + kϕ̂kPC → ∞ as q → ∞. We obtain
R b α 0 (s)ds R b α 0 (s)ds q 0 q
q
lim inf 0
= lim inf 0
.
= σ,
q→+∞
q→+∞
q
q0
q
s
Ps L (q 0 ) Ps L (q 0 ) q 0 X
i=1 i
i=1 i
lim inf
= lim inf
.
=
λi .
q→+∞
q→+∞
q
q0
q
i=1
Thus we have
1
2
1 ≤ M1 (1 + M1 M2 M3 b )(σ +
s
X
λi ).
i=1
This contradicts (3.1). Hence for some positive number q, G(Bq ) ⊆ Bq .
Step 2: G : PC 0 → PC 0 is continuous.
(n) → y in PC . Then there is a number q > 0 such that ky (n) (t)k ≤ q for all
Let {y (n) (t)}∞
0
n=1 ⊆ PC 0 with y
(n)
n and a.e. t ∈ J, so y (n) ∈ Bq and y ∈ Bq . By (H2 )(i), f (t, yt + ϕ̂t ) → f (t, yt + ϕ̂t ) for each t ∈ J. By
(n)
(n)
(H2 )(ii), kf (t, yt + ϕ̂t ) − f (t, yt + ϕ̂t )k < 2αq0 (t) and by (H4 ), Ii (yti + ϕ̂ti ) → Ii (yti + ϕ̂ti ), i = 1, 2, . . . , s.
Then we have
s
X
(n)
kG1 y (n) − G1 ykPC ≤ M1
(3.7)
kIi (yti + ϕ̂ti ) − Ii (yti + ϕ̂ti )k.
i=1
and
kG2 y (n) − G2 ykPC
Z b
Z b
≤ M1
kf (s, ys(n) + ϕ̂s ) − f (s, ys + ϕ̂s )kds + M1 M2
kuy(n) (s) − uy (s)kds
0
0
Z b
1
≤ M1
kf (s, ys(n) + ϕ̂s ) − f (s, ys + ϕ̂s )kds + M1 M2 b 2 ku(n)
y − uy kL2 ,
(3.8)
0
where
Z b
h
ku(n)
−
u
k
≤
M
M
kf (s, ys(n) + ϕ̂s ) − f (s, ys + ϕ̂s )kds
2
y L
3
1
y
0
+ M1
s
X
i=1
i
(n)
kIi (yti + ϕ̂ti ) − Ii (yti + ϕ̂ti )k .
(3.9)
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
213
Observing (3.7) − (3.9) and by dominated convergence theorem we have that,
kGy (n) − GykPC ≤ kG1 y (n) − GykPC + kG2 y (n) − G2 ykPC → 0,
as n → +∞.
That is G is continuous.
Step 3: G(Bq ) is equicontinuous on every Ji , i = 1, 2, . . . , s. That is G(Bq ) is piecewise equicontinuous on
J.
Indeed for t1 , t2 ∈ Ji , t1 < t2 and y ∈ Bq , we deduce that
k(Gy)(t2 ) − (Gy)(t1 )k
Z t1
≤
kU (t2 , s) − U (t1 , s)kkf (s, ys + ϕ̂s ) + Buy (s)kds
0
Z t2
kU (t2 , s)kf (s, ys + ϕ̂s ) + Buy (s)kds
+
t1
t1
Z
t1
Z
kU (t2 , s) − U (t1 , s)kαq0 (s)ds +
≤
h
kU (t2 , s) − U (t1 , s)kM2 M3 kx1 k
0
0
Z
+ M1 kϕ(0)k + M1
b
αq0 ds + M1
0
Z
t2
s
X
Z
i
Li (q ) ds +
0
Z
kU (t2 , s)kM2 M3 kx1 k + M1 kϕ(0)k + M1
+
kU (t2 , s)kαq0 (s)ds
t1
i=1
h
t2
t1
b
αq0 ds + M1
0
s
X
i
Li (q 0 ) ds.
(3.10)
i=1
By the equicontinuity of U (·, s) and the absolute continuity of the Lebesgue integral, we can see that the
right hand side of (3.10) tends to zero and independent of y as t2 → t1 . Hence G(Bq ) is equicontinuous on
Ji (i = 1, 2, . . . , s).
Step 4: The Monch’s condition holds.
Suppose W ⊆ Bq is countable and W ⊆ co({0} ∪ G(W )). We shall show that β(W ) = 0, where β is the
Hausdorff MNC.
Without loss of generality, we may assume that W = {y (n) }∞
n=1 . Since G maps Bq into an equicontinuous
family, G(W ) is equicontinuous on Ji . Hence W ⊆ co({0} ∪ G(W )) is also equicontinuous on every Ji .
By (H4 )(ii), we have
β({G1 y (n) (t)}∞
n=1 )
o∞ n X
(n)
=β
U (t, ti )Ii (yti + ϕ̂ti )
0<ti <t
≤ M1
s
X
n=1
(n)
β({Ii (yti + ϕ̂ti )}∞
n=1 )
i=1
≤ M1
≤ M1
s
X
i=1
s
X
i=1
Ki sup β({y (n) (ti + θ) + ϕ̂(ti + θ)}∞
n=1 )
−r≤θ≤0
Ki sup β({y (n) (τi )}∞
n=1 ).
0≤τi ≤ti
(3.11)
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
214
By Lemma 2.3 and from (H2)(iii), (H3)(ii) and (H4)(ii), we have that
o∞ h n Z b
(n)
U
(b,
s)f
(s,
y
+
ϕ̂
)ds
βV ({uy(n) (s)}∞
)
≤
K
(s)
β
s
W
s
n=1
n=1
0
+β
s
n X
o∞ i
(n)
U (b, ti )Ii (yti + ϕ̂ti ))
n=1
i=1
Z b
h
η(s) sup β({y (n) (s + θ) + ϕ̂(s + θ)}∞
≤ KW (s) 2M1
n=1 )ds
−r≤θ≤0
0
+ M1
s
X
Ki sup β({y (n) (ti + θ) + ϕ̂(ti + θ)}∞
n=1 )
−r≤θ≤0
i=1
h
≤ KW (s) 2M1
b
Z
η(s) sup β({y (n) (τ )}∞
n=1 )ds
0≤τ ≤s
0
+ M1
s
X
i
i
Ki sup β({y (n) (τi )}∞
)
n=1 .
i=1
(3.12)
0≤τi ≤ti
This implies that
β({G2 y (n) (t)}∞
n=1 )
n Z t
o∞ n Z t
o∞ (n)
≤β
U (t, s)f (s, ys + ϕ̂s )ds
+β
U (t, s)Buy(n) (s)ds
n=1
n=1
0
0
Z b
≤ 2M1
η(s) sup β({y (n) (s + θ) + ϕ̂(s + θ)}∞
n=1 )ds
−r≤θ≤0
b
0
Z
βV ({uy(n) (s)}∞
+ 2M1 M2
n=1 )ds
0
Z b
Z b
(n)
∞
2
≤ 2M1
η(s) sup β({y (τ )}n=1 )ds + 4M1 M2
KW (s)ds
×
Z
0
b
0≤τ ≤s
0
η(s) sup β({y (n) (τ )}∞
n=1 )ds
0≤τ ≤s
0
+ 2M12 M2
Z
b
s
X
KW (s)ds
Ki sup β({y (n) (τi )}∞
n=1 ,
0
i=1
(3.13)
0≤τi ≤ti
for each t ∈ J. From (3.11) and (3.13) we obtain that
β({Gy (n) (t)}∞
n=1 )
(n)
≤ β({G1 y (n) (t)}∞
(t)}∞
n=1 ) + β({G2 y
n=1 )
Z b
s
X
2
≤ M1
Ki sup β({y (n) (τi )}∞
)
+
2M
+
4M
M
K
(s)ds
1
2
W
n=1
1
i=1
Z b
×
0≤τi ≤ti
η(s) sup β({y (n) (τ )}∞
n=1 )ds
0
+
2M12 M2
0≤τ ≤s
Z
b
KW (s)ds
0
for each t ∈ J.
0
s
X
i=1
Ki sup β({y (n) (τi )}∞
n=1 ) ,
0≤τi ≤ti
(3.14)
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
215
Since W and G(W ) are equicontinuous on every Ji , according to Lemma 2.2, the inequality (3.14) implies
that,
β({Gy (n) }∞
n=1 )
s
h
i
X
2
≤ M1
Ki + (2M1 + 4M1 M2 kKW kL1 )kηkL1 β({y (n) }∞
n=1 )
i=1
+
[2M12 M2 kKW kL1
s
X
Ki ]β({y (n) }∞
n=1 )
i=1
s
h
i
X
2
= (M1 + 2M1 M2 kKW kL1 )
Ki + (2M1 + 4M12 M2 kKW kL1 )kηkL1 β({y (n) }∞
n=1 )
i=1
=
N β({y (n) }∞
n=1 ).
That is β(GW ) ≤ N β(W ), where N is defined in (H5). Thus from the Monch’s condition, we get that
β(W ) ≤ β(co({0} ∪ G(W )) = β(G(W )) ≤ N β(W ),
since N < 1, which implies that β(W ) = 0. So we have that W is relatively compact in PC 0 . In the view of
Lemma 2.5, i.e., Monch’s fixed point theorem, we conclude that G has a fixed point y in W . Then x = y + ϕ̂
is a fixed point of F in PC and thus the system (1.1) − (1.3) is controllable on [0, b].
Remark 3.1. Note that if f is compact or Lipschitz continuous, then (H2)(iii) is automatically satisfied.
In the following, by using another MNC, we will prove the result of the Theorem 3.1 in the case there is
no equicontinuity of the evolution system U (t, s) and hypothesis (H5). Here we assume that the impulsive
operators Ii are compact. So, instead of (H4), we give the hypothesis (H4)0 :
(H40 ) Ii : D → X, i = 1, 2 . . . , s, be a continuous compact operator such that, there are nondecreasing
functions Li : R+ → R+ satisfying
kIi (x)k ≤ Li (kxkD )
and
lim inf
ρ→∞
i = 1, 2 . . . , s,
Li (ρ)
= λi < ∞,
ρ
x ∈ D,
i = 1, 2 . . . , s.
Theorem 3.2. Let {A(t)}t∈[0,b] be a family of linear operators that generates a strongly continuous evolution
system {U (t, s) : (t, s) ∈ J × J}. Assume that the hypothesis (H2), (H3) and (H40 ) are satisfied. Then the
impulsive differential system (1.1) − (1.3) is controllable on J.
Proof. In the view of Theorem 3.1, we should only prove that the function G : PC 0 → PC 0 given by the
formula (3.2) satisfies the Monch’s condition.
For this purpose, let W ⊆ Bq be countable and W ⊆ co({0} ∪ G(W )). We shall prove that W is relatively
compact.
We will denote by Φ the following MNC in PC 0 defined by (see[13]),
Φ(Ω) = max (α(E), modc (E)).
(3.15)
E∈∆(Ω)
for all bounded subsets of Ω of PC 0 , where ∆(Ω) is the set of countable subsets of Ω, α is the real MNC
defined by,
α(E) = sup e−Lt β(E(t)),
t∈[0,b]
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
216
with E(t) = {x(t) : x ∈ E}, L is a constant that we shall choose appropriately.
mod c (E) is the modulus of equicontinuity of the function set E given by the formula
mod c (E) = lim sup max
max
δ→0 x∈E 0≤i≤s t1 ,t2 ∈Ji ,kt1 −t2 k<δ
kx(t1 ) − x(t2 )k.
It was proved in [13] that Φ is well defined. (i.e., there is E0 ∈ ∆(Ω) which achieves the maximum in (3.15))
and is a monotone, nonsingular and regular MNC.
Let us choose a constant L > 0, such that
Z t
2
p = (2M1 + 4M1 M2 kKW kL1 ) sup
η(s)e−L(t−s) ds < 1,
(3.16)
t∈[0,b] 0
where M1 = sup{kU (t, s)k : (t, s) ∈ J × J} and η is the integrable function in the hypothesis (H2).
Let Gy = G1 y + G2 y as defined in theorem (3.1). From the regularity of Φ, it is enough to prove that
Φ(W ) = (0, 0). Since Φ(G(W )) is a maximum, let {z (n) }∞
n=1 ⊆ G(W ) be the denumerable set which
(n)
∞
achieves its maximum. Then there exists a set {y }n=1 ⊆ W such that
z (n) (t) = (Gy (n) )(t) = (G1 y (n) )(t) + (G2 y (n) )(t), for all n ≥ 1, t ∈ [0, b].
(3.17)
Now we give an estimation for α({z (n) }∞
n=1 ). Since Ii (·) is compact, we get
β({(G1 y (n) )(t)}∞
n=1 ) = 0, for t ∈ [0, b].
(3.18)
From (3.12), (3.13), noticing that Ki = 0, as Ii is compact, we have that
β({(G2 y (n) )(t)}∞
n=1 )
Z t
≤ 2M1
η(s) sup β({y (n) (τ )}∞
n=1 )ds
0≤τ ≤s
0
+
4M12 M2 kKW kL1
Z
t
η(s) sup β({y (n) (τ )}∞
n=1 )ds
0≤τ ≤s
0
≤ (2M1 + 4M12 M2 kKW kL1 )
Z
t
≤ (2M1 + 4M12 M2 kKW kL1 )
Z
0
η(s) sup β({y (n) (τ )}∞
n=1 )ds
0≤τ ≤s
0
t
η(s)eLs sup (e−Lt β({y (n) (t)}∞
n=1 ))ds
t∈[0,b]
= (2M1 + 4M12 M2 kKW kL1 )α({y (n) }∞
n=1 )
t
Z
η(s)eLs ds, for t ∈ [0, b].
0
From (3.18) and (3.19), it follows that
−Lt
α({z (n) }∞
β {(G1 y (n) )(t) + (G2 y (n) )(t)}∞
n=1 ) = sup e
n=1
t∈[0,b]
−Lt
≤ sup e
(2M1 +
4M12 M2 kKW kL1 )α({y (n) }∞
n=1 )
2
= α({y (n) }∞
n=1 )(2M1 + 4M1 M2 kKW kL1 ) sup
Z
t∈[0,b] 0
=
t
η(s)eLs ds
0
t∈[0,b]
α({y (n) }∞
n=1 )p.
Z
t
η(s)e−L(t−s) ds
(3.19)
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
217
Therefore, we have that
(n) ∞
α({y (n) }∞
}n=1 ) ≤ α({y (n) }∞
n=1 ) ≤ α(W ) ≤ α(co({0} ∪ G(W ))) = α({z
n=1 )p.
From (3.16), we obtain that
(n) ∞
α({y (n) }∞
}n=1 ) = 0.
n=1 ) = α(W ) = α({z
From the definition of α, we have
(n)
β({y (n) (t)}∞
(t)}∞
n=1 ) = β({z
n=1 ) = 0, for every t ∈ [0, b].
(3.20)
From (3.12) and (3.20), noticing that Ki = 0 in (3.12), we get that
(n)
β({f (t, yt
+ ϕ̂t ) + (Buy(n) )(t)}∞
n=1 )
≤ η(t) sup β({y (n) (t + θ) + ϕ̂(t + θ)}∞
n=1 )
−r≤θ≤0
Z
b
+ 2M1 M2 KW (s)
η(s) sup β({y (n) (τ )}∞
n=1 )ds
0≤τ ≤s
0
≤ η(t) sup β({y
(n)
0≤τ ≤t
(τ )}∞
n=1 )
Z
+ 2M1 M2 KW (s)
0
(n)
That is, {f (t, yt
b
η(s) sup β({yn (τ )}∞
n=1 )ds = 0,
0≤τ ≤s
+ ϕ̂t ) + Buy(n) (t)}∞
n=1 is relatively compact for almost all t ∈ [0, b] in X. Moreover, from
(n)
the fact that {y (n) }∞
+ ϕ̂t ) + Buy(n) (t)}∞
n=1
n=1 ⊆ Bq , by (H2)(ii) and (3.6), it is easy to see that {f (t, yt
is uniformly integrable for a.e. t ∈ [0, b]. So {f (·, y (n) + ϕ̂) + Buy(n) }∞
is
semicompact
according
to
the
n=1
(n)
∞
Definition 2.5. By applying Lemma 2.4, we have that G2 ({y }n=1 ) is relatively compact in PC 0 .
On the other hand, by the strong continuity of U (t, s) and the compactness of Ii , we can easily verify
(n) }∞ is also relatively compact in PC . Since
that G1 ({y (n) }∞
0
n=1 ) is relatively compact. Then by (3.17), {z
n=1
Φ is a monotone, nonsingular, regular MNC, from Monch’s condition, we have that
Φ(W ) ≤ Φ(co({0} ∪ G(W ))) = Φ({zn }∞
n=1 ) = (0, 0).
Therefore, W is relatively compact in PC 0 . This completes the the proof.
4. Example
In this section, we give an example to illustrate our results above.
Example 4.1. Consider the impulsive partial system of the form
∂
∂
z(t, ξ) =
z(t, ξ) + m(ξ)u(t, ξ) + F (t, z(t − r, ξ)),
∂t
∂ξ
for ξ ∈ [0, π], t ∈ [0, b], t 6= ti , i = 1, 2, . . . , s,
z(t+
i , ξ)
−
z(t−
i , ξ)
=
Ii (z(t−
i , ξ)),
ξ ∈ (0, π], i = 1, 2, . . . , s,
(4.1)
(4.2)
z(t, 0) = z(t, π) = 0, t ∈ [0, b],
(4.3)
z(t, ξ) = ϕ(t, ξ), t ∈ [−r, 0], ξ ∈ [0, π],
(4.4)
where r > 0, Ii > 0, i = 1, 2, . . . , s, ϕ ∈ D = {ψ : [−r, b] × [0, π] → R; ψ is continuous everywhere except for
a countable number of points at which ψ(s− ), ψ(s+ ) exists with ψ(s− ) = ψ(s)}, 0 = t0 < t1 < t2 < · · · <
−
ts+1 = b, z(t+
i ) = lim(h,ξ)→(0+ ,ξ) z(ti +h, ξ), z(ti ) = lim(h,ξ)→(0− ,ξ) z(ti +h, ξ), F : [0, b]×R → R, B : X → X.
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
218
Let
x(t)(ξ) = z(t, ξ), t ∈ [0, b], ξ ∈ [0, π],
−
Ii (x(t−
i ))(ξ) = Ii z(ti , ξ), ξ ∈ [0, π], i = i, 2, . . . , s,
F (t, ϕ)(s) = F (t, ϕ(θ, ξ)), θ ∈ [−r, 0], ξ ∈ [0, π],
ϕ(θ)(ξ) = ϕ(θ, ξ), θ ∈ [−r, 0], ξ ∈ [0, π],
(Bu)(ξ) = m(ξ)u(ξ), ξ ∈ [0, π].
Take X = L2 [0, π] and define A(t) ≡ A : D(A) ⊂ X → X by Aw = w0 with domain D(A) = {w ∈ X : w0 ∈
X, w(ξ) = w(0) = 0}. It is well known that A is an infinitesimal generator of a semigroup T (t) defined by
T (t)w(s) = w(t + s) for each w ∈ X. T (t) is not a compact semigroup on X and β(T (t)D) ≤ β(D), where
β is the Hausdorff MNC.
Then, the system (4.1) − (4.4) is the abstract formulation of the system (1.1) − (1.3). We can conclude
that the system (4.1) − (4.4) is controllable on [0,b].
Acknowledgements
The first author thanks to Mr. Muthuvel Ramaswamy, Secretary, Muthayammal College of Arts and
Science, Rasipuram- 637 408, Tamil Nadu, India, for his constant encouragements and support for this
research work.
The second author dedicates this paper to Silver Jubilee Year Celebrations of Karunya University,
Coimbatore-641 114, Tamil Nadu, India. And also the authors wish to thank Dr. Paul Dhinakaran, Chancellor, Dr. Paul P. Appasamy, ViceChancellor, and Dr(Mrs). Anne Mary Fernandez, Registrar, of Karunya
University, Coimbatore, for their constant encouragements and support for this research work.
References
[1] A. Anguraj and M. Mallika Arjunan, Existence and uniqueness of mild and classical solutions of impulsive
evolution equations, Electron J. Differential Equations, 111 (2005), 1-8. 1
[2] A. Anguraj and M. Mallika Arjunan, Existence results for an impulsive neutral integro-differential equations in
Banach spaces, Nonlinear Stud., 16(1)(2009), 33-48. 1
[3] D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman
Scientific and Technical Group, England, 1993. 1
[4] J. Banas and K. Goebel, Measure of Noncompactness in Banach Spaces, in: Lecture Notes in Pure and Applied
Matyenath, Marcel Dekker, New York, 1980. 2.1, 2.1
[5] M. Benchohra, J. Henderson and S.K. Ntouyas, Existence results for impulsive multivalued semilinear neutral
functional inclusions in Banach spaces, J. Math. Anal. Appl., 263(2001), 763-780. 1
[6] L. Chen and G. Li, Approximate controllability of impulsive differential equations with nonlocal conditions, Int.
J. Nonlinear Sci., 10(2010), 438-446. 1
[7] Z. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal.,
72(2010), 1104-1109. 1
[8] Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions,
J. Funct. Anal., (258)(2010), 1709-1727. 1
[9] E. Hernandez, M. Pierri and G. Goncalves, Existence results for an impulsive abstract partial differential equation
with state-dependent delay, Comput. Math. Appl., 52(2006), 411-420. 1
[10] E. Hernandez M., M. Rabello and H. Henriaquez, Existence of solutions for impulsive partial neutral functional
differential equations, J. Math. Anal. Appl., 331(2007), 1135-1158. 1
[11] S. Ji, G. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl.Math.
Comput., 217(2011), 6981-6989. 1
[12] S. Ji and S. Wen, Nonlocal cauchy problem for Impulsive differential equations in Banach spaces, Int. J. Nonlinear
Sci., 10(2010), 88-95. 1
[13] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions
in Banach Spaces, De Gruyter, 2001. 2.1, 3, 3
[14] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. 1
S. Selvi, M. Mallika Arjunan, J. Nonlinear Sci. Appl. 5 (2012), 206–219
219
[15] M. Li, M. Wang and F. Zhang, Controllability of impulsive functional differential systems in Banach spaces,
Chaos, Solitons and Fractals, 29(2006), 175-181. 1
[16] H. Monch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach
spaces, Nonlinear Anal., 4(1980), 985-999. 2.5
[17] V. Obukhovski and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach
space with a noncompact semigroup, Nonlinear Anal., 70(2009), 3424-3436. 1, 2.3, 2.4
[18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag,
Newyork, 1983. 2
[19] B. Radhakrishnan and K. Balachandran, Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay, Nonlinear Anal., 5(2011), 655-670. 1
[20] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. 1
[21] C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math.
Soc., 200(1974), 395-418. 1
[22] G.Webb, An abstract semilinear Volterra integrodifferential equations, Proc. Amer. Math. Soc., 69(1978), 255260. 1
[23] R. Ye, Existence of solutions for impulsive partial neutral functional differential equation with infinite delay,
Nonlinear Anal., 73(2010), 155-162. 1, 2.2