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On controllability of second order nonlinear impulsive differential systems
Nonlinear Analysis 71 (2009) 45–52
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On controllability of second order nonlinear impulsive differential systems R. Sakthivel a,∗ , N.I. Mahmudov b , J.H. Kim c a
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea
b
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Mersin 10, Turkey
c
Department of Mathematics, Yonsei University, Seoul 120-749, South Korea
article
info
Article history: Received 4 July 2007 Accepted 10 October 2008 Keywords: Controllability Fixed point theorem Nonlinear impulsive systems Nonlocal conditions
a b s t r a c t Many dynamical systems in physics, chemistry, biology and engineering sciences have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes. The mathematical models of such processes are called differential systems with impulse effect. This paper studies the exact controllability issue of certain types of second order nonlinear impulsive control differential systems. Sufficient conditions are formulated and proved for the exact controllability of such systems. Without imposing a compactness condition on the cosine family of operators, we establish controllability results by using a fixed point analysis approach. Finally, an example is presented to illustrate the utility of the proposed result. The results improve some recent results. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Many dynamical systems are characterized by the fact that at certain moments in their evolution they undergo rapid changes. In recent years the theory of impulsive differential systems has been emerging as an important area of investigation in applied sciences. The reason is that it is richer than the corresponding theory of classical differential equations and it is more adequate to represent some processes arising in various disciplines. The theory of impulsive systems provides a general framework for mathematical modeling of many real world phenomena. Moreover, these impulsive phenomena can also be found in fields such as information science, electronics, automatic control systems, robotics and telecommunications (see [19,24] and references therein). Control theory is an area of application-oriented mathematics which deals with basic principles underlying the analysis and design of control systems. It is well known that the study of controllability plays an important role in the control theory and engineering. In recent years, the study of impulsive control systems has received increasing interest, since dynamical systems with impulsive effects have great importance in applied sciences [see [27] and references therein]. This paper is concerned with the study of controllability of a second order nonlinear impulsive differential system of the form x00 (t ) = Ax(t ) + Bu(t ) + f (t , x(t ), x0 (t )), x(0) = x0 ,
t ∈ J = [0, b], t 6= tk ,
(1)
x0 (0) = y0
4x(tk ) = Ik1 (x(tk )),
4x0 (tk ) = Ik2 (x0 (tk+ )),
k = 1, 2 . . . , m
in a Banach space X . Here x : J → X is the state function, u(·) ∈ L2 (J , U ) is the control function, U is a Banach space,
∗
Corresponding author. Tel.: +82 31 2999 4527. E-mail address: [email protected] (R. Sakthivel).
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.10.029
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R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
A is the infinitesimal generator of a strongly continuous cosine family of linear operators (C (t ))t ∈R on X , B : U → X is a linear bounded operator, 0 = t0 < t1 < · · · < tm < tm+1 = b, 4x(tk ) = x(tk+ ) − x(tk− ), 4x0 (tk ) = x0 (tk+ ) − x0 (tk− ), and j
Ik : X → X , j = 1, 2, f : J × X × X → X are appropriate continuous functions to be specified later. In the last few years many papers have been published about the global and local controllability of nonlinear systems, in which the authors effectively used fixed point techniques. The problem of controllability of first order nonlinear systems has been studied by several researchers [2,5,9,12,20–23] and also first-order impulsive differential equations have received much attention [3,8,17,18]. Li et al., [18] established sufficient conditions for the controllability of impulsive functional differential systems with finite delay whereas in [2] Benchohra et al. studied controllability of impulsive differential inclusions in Banach spaces. Recently, Chang [8] studied the controllability of first order impulsive functional systems with infinite delay using Scahuder’s fixed point theorem. In many cases it is advantages to treat second order abstract differential equations directly rather than convert them to first order systems. The study of abstract deterministic secondorder evolution equations governed by the generator of a strongly continuous cosine family was initiated by Fattorini [10], and subsequently studied by Travis and Webb [25,26]. The controllability result for second order nonlinear systems with and without nonlocal conditions has been studied by many authors [4,7,13,14]. The results in those papers are only applicable to ordinary differential equations since the compactness assumption assumed on the cosine function is valid if and only if the underlying space is finite dimensional. However, to the authors’ knowledge the corresponding theory for controllability of second order impulsive differential equations in abstract spaces has not been explored. In this paper, we derive results on exact controllability of second order impulsive differential systems without imposing a compactness condition on the semigroup of cosine family. Further, the result is extended to study the controllability of the nonlinear second order impulsive system with nonlocal conditions described in the form x00 (t ) = Ax(t ) + Bu(t ) + f (t , x(t ), x(0) = x0 + p(x, x ),
x0 (t )), t ∈ J , t 6= tk ,
(2)
x (0) = y0 + q(x, x ) 4x(tk ) = Ik1 (x(tk )), 4x0 (tk ) = Ik2 (x0 (tk+ )), k = 1, 2 . . . , m, where p, q : C (J , X ) × C (J , X ) → X are appropriate functions to be specified later. 0
0
0
2. Preliminaries In this section, we review some basic concepts, notations and properties needed to establish our main results. Throughout this paper, A is the infinitesimal generator of a strongly continuous cosine family, (C (t ))t ∈R , of bounded linear operators defined on a Banach space X endowed with a norm k.k. We denote by (S (t ))t ∈R the sine function associated to (C (t ))t ∈R which is defined by t
Z
S (t )x :=
C (s)xds,
x ∈ X , t ∈ R.
0
˜ are positive constants such that kC (t )k ≤ M and kS (t )k ≤ M ˜ for every t ∈ J. Moreover, M and M [D(A)] is the space D(A) = {x ∈ X : C (t )x is twice continuously differentiable in t}, endowed with the norm kxkA = kxk + kAxk, x ∈ D(A). Define E = {x ∈ X : C (t )x is once continuously differentiable in t} endowed with the norm kxkE = kxk + sup kAS (t )xk,
x ∈ E,
0≤t ≤1
is a Banach space. The operator-valued function C (t ) AS (t )
h(t ) =
S (t ) C (t )
is a strongly continuous group of bounded linear operators on the space E × X generated by the operator A¯ =
0 A
I 0
defined on D(A) × E. From this, it follows that AS (t ) : E → X is a bounded linear operator and that AS (t )x → 0 as t → 0, for each x ∈ E. Furthermore, if x : [0, ∞) → X is locally integrable function, then y(t ) =
t
Z
S (t − s)x(s)ds 0
defines an E-valued continuous function which is a consequence of the fact that
Z t
Z
h( t − s ) 0
t
0 Z0 ds = t x(s)
S (t − s)x(s) ds C (t − s)x(s) ds
0
defines an (E × X )-valued continuous function.
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
47
The following properties are well known [25]: (i) (ii) (iii) (iv)
if x if x if x if x
∈ X then S (t )x ∈ E for every t ∈ R. ∈ E then S (t )x ∈ D(A), (d/dt )C (t )x = AS (t )x and (d2 /dt 2 )S (t )x = AS (t )x for every t ∈ R. ∈ D(A) then C (t )x ∈ D(A), and (d2 /dt 2 )C (t )x = AC (t )x = C (t )Ax for every t ∈ R. ∈ D(A) then S (t )x ∈ D(A), and (d2 /dt 2 )S (t )x = AS (t )x = S (t )Ax for every t ∈ R.
For additional details concerning cosine function theory, we refer the reader to Fattorini [10] and Travis & Webb [25,26]. Additionally, for a bounded function ξ : J → X and t ∈ J, we employ the notation ξt for
ξt = sup{kξ (s)kX : s ∈ [0, t ]}. Let C ([0, b], X ) be the space of the bounded functions x : [0, b] → X such that x is continuous in t 6= tk , x is left continuous in each tk and x(tk+ ) exist. Also assume C ([0, b], X ) endowed with the uniform convergence norm. Finally C 1 ([0, b], X ) will be the space C 1 ([0, b], X ) = {(x, z ) : x, z ∈ C ([0, b], X ) and x0 (t ) = z (t ) for t 6= tk } provided with the norm k(x, z )k = kxkb + kz kb . Definition 2.1. Let xb (x0 , y0 ; u) be the state value of (1) at time b corresponding to the control u and the initial value x0 and y0 . Introduce the set R(b, x0 , y0 ) = {xb (x0 , y0 ; u)(0) : u(.) ∈ L2 (J , U )}. The system (1) is said to be exactly controllable on the interval J if R(b, x0 , y0 ) = X . Assume that the second order linear system x00 (t ) = Ax(t ) + Bu(t ), x(0) = x0 ,
t ∈J
(3)
x0 (0) = y0
is exactly controllable on J. It is convenient at this point to define the operator (see [20])
Γ0b
b
Z
S (b − s)BB∗ S ∗ (b − s)ds.
= 0
Note that the system (3) is exactly controllable iff there exists a γ > 0 such that
hΓ0b x, xi ≥ γ kxk2 ,
for all x ∈ X
Then
k(Γ0b )−1 k ≤
1
γ
.
Note 2.2. If X is infinite dimensional, the corresponding cosine family of semigroup C (t ) is compact and B is bounded, then the second order linear control system x00 (t ) = Ax(t ) + Bu(t ) is not exactly controllable. Definition 2.3. A function x(·) ∈ C 1 ([0, b], X ) is said to be a mild solution of the impulsive control problem (1) if the impulsive conditions in (1) are satisfied and x(t ) = C (t )x0 + S (t )y0 +
t
Z
S (t − s)[Bu(s) + f (s, x(s), x0 (s))]ds 0
+
X 0
C (t − tk )Ik1 (x(tk )) +
X
S (t − tk )Ik2 (x0 (tk+ )).
(4)
0
Now for convenience, let us introduce the notation K = kBk. In order to establish the controllability result, we introduce the following technical hypothesis: (I) f : J × X × X → X is a continuous function and there exist positive constants K1 and K2 such that
kf (t , x, y) − f (t , x0 , y0 )k ≤ K1 kx − x0 k + K2 ky − y0 k for every x, x0 , y and y0 ∈ X (II) The linear system (3) is exactly controllable.
48
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52 j
j
(III) The functions Ik : X × X are continuous and there exist positive constants L(Ik ) such that
kIkj (x) − Ikj (x0 )k ≤ L(Ikj )kx − x0 k for each x, x0 ∈ X . (IV) There exist functions G : [0, b] → $ (X ) (the Banach space of bounded linear operators from X into X ) Fk : X → X , k ∈ {1, 2, . . . , m} such that: (i) G(0) = 0, G(.) is strongly continuous and (d/dt )C (t )Ik1 (x) = G(t )Fk (x) for every x ∈ X and k ∈ {1, 2, . . . , m} (ii) for each k ∈ {1, 2, . . . , m} there exists a positive constant L(Fk ) such that kFk (x) − Fk (x0 )k ≤ L(Fk )kx − x0 k (V) Let
˜ + ηM ˜ + M )K1 b + φ1 = (β M
m X (ηML(Ik1 ) + kGkL(Fk ) + β ML(Ik1 )) k=1
and
˜ + ηM ˜ + M )K2 b + φ2 = (β M
m X ˜ + ηM ˜ + M )L(Ik2 ), (β M k=1
˜ 2 K 2 b. ˜ and β = 1 + M where η = γ MK M γ 1
1
2
3. Controllability result Theorem 3.1. Let x0 ∈ E , y0 ∈ X . If the conditions (I)–(V) and max{φ1 , φ2 } < 1 are satisfied, then the second order impulsive control system (1) is exactly controllable. Proof. For (x, z ) ∈ C 1 ([0, b] : X ), define the nonlinear operator
Φ (x, z ) = (Φ1 (x, z ), Φ2 (x, z )), where
Z t Φ1 (x, z )(t ) = C (t )x0 + S (t )y0 + S (t − s)[Bu(s) + f (s, x(s), x0 (s))]ds 0 X X + C (t − tk )Ik1 (x(tk )) + S (t − tk )Ik2 (x0 (tk+ )), 0
0 < tk < t
Z
t
Φ2 (x, z )(t ) = AS (t )x0 + C (t )y0 + C (t − s)[Bu(s) + f (s, x(s), x0 (s))]ds 0 X X + G(t − tk )Fk (x(tk )) + C (t − tk )Ik2 (x0 (tk+ )). 0
0
Define the feedback control function
" u(t ) = B S (b − t )( ∗ ∗
)
Γ0b −1
xb − C (b)x0 − S (b)y0 −
b
Z
S (b − s)f (s, x(s), x0 (s))ds
0
−
m X
C (b −
) (x(tk )) −
tk Ik1
k=1
m X
# S (b −
) (x (tk )) .
tk Ik2
0
+
(5)
k=1
Note that the control (5) transfers the system (4) from the initial state to the final state provided that the operator Φ has a fixed point. So if the operator Φ has a fixed point then the system (4) is exactly controllable. To prove the exact controllability, it is enough to show that the operator Φ has a fixed point in C 1 ([0, b], X ). The proof is based on the classical fixed point theorem for contractions. It follows from the assumptions that each Φi , i = 1, 2 is well defined and continuous. In order to prove that Φ is a contraction mapping on C 1 ([0, b] : X ), we take (x, z )(v, w) in C 1 ([0, b] : X ) From the conditions (I) and (III), we get 1 kΦ1 (x, z )(t ) − Φ1 (v, w)(t )k ≤ M˜ 2 K 2 b
"Z
γ
+
b
˜ [K1 kx − vk + K2 kx0 − v 0 k]ds M 0
m X k=1
ML( )kx(tk ) − v(tk )k + Ik1
m X k=1
# ˜ ( )kx (tk ) − v (tk )k ML Ik2
0
+
0
+
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52 t
Z
X
˜ [K1 kx − vk + K2 kx0 − v 0 k]ds + M
+
ML(Ik1 )kx(tk ) − v(tk )k
0
0
X
+
49
˜ (Ik2 )kx0 (tk+ ) − v 0 (tk+ )k. ML
0
Thus we have
kΦ1 (x, z )(t ) − Φ1 (v, w)(t )k ≤
"
1 1 + M˜ 2 K 2 b
˜ 1b + MK
γ
" +
m X
!# ML( )
kx − vk
Ik1
k=1
1 1 + M˜ 2 K 2 b
˜ 2b + M ˜ MK
γ
m X
!# L( )
kz − wk.
Ik2
(6)
k=1
Similarly, we have
" ˜ + M )K1 b + kΦ2 (x, z )(t ) − Φ2 (v, w)(t )k ≤ (ηM
m X
# (ηML( ) + kGkL(Fk )) kx − vk Ik1
k=1
" ˜ + M )K2 b + + (ηM
m X
# ˜ + M )L( ) kz − wk. (ηM Ik2
(7)
k=1
The above inequalities (6) and (7) and the assumption max{φ1 , φ2 } < 1 imply that Φ is a contraction mapping. Hence there exists a unique fixed point x(·) in C 1 ([0, b], X ) which is the solution of Eq. (4). Thus the system (1) is exactly controllable. 4. Impulsive nonlocal differential systems The nonlocal condition is a generalization of the classical initial condition. The first results concerning the existence and uniqueness of mild solutions to Cauchy problems with nonlocal conditions were studied by Byszewski [1]. Recently, the existence of mild and classical solutions for a class of second order nonlinear differential systems with nonlocal conditions has been studied in [15]. As indicated in [1,15] and the references therein, the nonlocal conditions can be applied in physics with better effect than the classical Cauchy problem, since more measurements are allowed. In this section, we consider the second order impulsive nonlocal system (2) governed by the evolution equation x(t ) = C (t )(x0 + p(x, x0 )) + S (t )(y0 + q(x, x0 )) +
t
Z
S (t − s)[Bua (s) + f (s, x(s), x0 (s))]ds 0
+
X 0
X
C (t − tk )Ik1 (x(tk )) +
S (t − tk )Ik2 (x0 (tk+ )).
(8)
0
The purpose of this section is to prove the exact controllability of the impulsive nonlocal system of the form (8) under the simple and fundamental assumption on the system operators, in particular that the corresponding linear system is appropriately controllable. Moreover, N and N˜ are positive constants such that kC (t )k ≤ N and kS (t )k ≤ N˜ for every t ∈ J. Concerning the operators f , p and q, assume the following hypothesis: (H1) The function f : J × X × X → X satisfies the following conditions; (i) The function f (t , ·) : X × X → X is continuous a.e.t ∈ J; (ii) The function f (·, x, y) : J → X is strongly measurable for each (x, y) ∈ X × X . (iii) There exists a constant k˜ f such that
kf (t , x1 , y1 ) − f (t , x2 , y2 )k ≤ k˜ f (kx1 − x2 k) + (ky1 − y2 k) for every x1 , x2 , y1 , y2 ∈ X . (H2) The functions p, q : C (I ; X ) × C (I ; X ) → X are continuous and there exist positive constants k˜ p , k˜ q such that kp(x1 , y1 ) − p(x2 , y2 )k ≤ k˜ p (kx1 − x2 k + ky1 − y2 k), kq(x1 , y1 ) − q(x2 , y2 )k ≤ k˜ q (kx1 − x2 k + ky1 − y2 k). for every x1 , x2 , y1 , y2 ∈ C (I ; X ). P Pm 1 ˜ k˜ f + m ˜ ˜ ˜ ˜ ˜˜ ˜ ˜ (H3) Let Q1 = τ (N k˜ p + N˜ k˜ q + Nb k=1 NL(Ik )) + (ρ kp + N kq + Nbkf + k=1 kGkL(Fk )) and Q2 = τ (N kp + N kq + Nbkf +
Pm
k=1
˜ (Ik2 )) + (ρ k˜ p + N k˜ q + Nbk˜ f + NL
Pm
k=1
˜ 2 and ρ = supt ∈J kAS (t )k. NL(Ik2 )), where τ = 1 + γ1 K 2 N˜2 b + γ1 N NK
Theorem 4.1. Assume that (II)–(IV) and (H1 )–(H3 ) hold. If max{Q1 , Q2 } < 1 are satisfied, then the second order impulsive nonlocal control system (8) is exactly controllable.
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R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
Proof. For (x, z ) ∈ C 1 ([0, b] : X ), define the nonlinear operator
˜ (x, z ) = (Φ3 (x, z ), Φ4 (x, z )), Φ where
Z t S (t − s)[Bua (s) + f (s, x(s), x0 (s))]ds Φ3 (x, z )(t ) = C (t )(x0 + p(x, x0 )) + S (t )(y0 + q(x, x0 )) + 0 X X + C (t − tk )Ik1 (x(tk )) + S (t − tk )Ik2 (x0 (tk+ )), 0
0 < tk < t
Z t C (t − s)[Bua (s) + f (s, x(s), x0 (s))]ds Φ4 (x, z )(t ) = AS (t )(x0 + p(x, x0 )) + C (t )(y0 + q(x, x0 )) + 0 X X + G(t − tk )Fk (x(tk )) + C (t − tk )Ik2 (x0 (tk+ )). 0
0
Define the feedback control function ua (t ) = B∗ S ∗ (b − t )(Γ0b )−1 [xb − C (b)(x0 + p(x, x0 )) − S (b)(y0 + q(x, x0 )) b
Z
S (b − s)f (s, x(s), x0 (s))ds −
− 0
m X
C (b − tk )Ik1 (x(tk )) −
k=1
m X
S (b − tk )Ik2 (x0 (tk+ ))].
k=1
˜ has a fixed point by employing the method used in the previous section. Then we One can easily prove that the operator Φ can show that the system (8) is exactly controllable by adopting the technique used in Theorem 3.1. Remark 4.2. Differential inclusions play an important role in characterizing many social, physical, biological and engineering problems. In particular, the problems in physics, especially in solid mechanics, where non-monotone and multivalued constitutive laws lead to differential inclusions. The above result can be extended to study the controllability of second order nonlinear impulsive differential inclusions by suitably introducing the multivalued map defined in [4,6,13]. 5. Application Example 5.1. Consider the following partial second-order impulsive nonlocal differential equation with control µ(t , ·) ∈ L2 [0, π]
∂ ∂t
∂ z (t , y) ∂t
∂ 2 z (t , y) ∂ z (t , y) = + µ(t , y) + g t , z (t , y), , ∂ y2 ∂t
y ∈ [0, π], t ∈ J ,
(9)
subject to the conditions: z (t , 0) = z (t , π ) = 0,
4z (tk )(y) =
tk
Z
t ∈J
ak (tk − s)z (s, y)ds
0
4z 0 (tk )(y) =
tk
Z
a˜ k (tk − s)z (s, y)ds
0
∂ z (s) (y)dµ( ¯ s) ∂t 0 n l X X ∂ z (0, y) = y1 (y) + αi z (ti , y) + βj z (sj , y), ∂t i =1 j =1
z (0, y) = y0 (y) +
Z
b
P
z (s),
where y0 , y1 ∈ X , µ(·) ¯ is a real function of bounded variation on J and ak , a˜ k ∈ C (R, R). Let X = L2 [0, π] and let A : X → X be an operator defined by Az = z 00 with domain D(A) = {z ∈ X : z and z 0 are absolutely continuous, z 00 ∈ X , z (0) = z (π ) = 0}. It is well known that A is the infinitesimal generator of a strongly continuous cosine family of operators on X . Here 0 < ti , sj < π , αi , βj ∈ R are prefixed numbers and g : J × X 2 → X , P : X × X → X satisfy the following conditions: (i) g (·) is continuous and there exists a constant Lg such that
|g (t , x1 , w1 ) − g (s, x2 , w2 )| ≤ Lg (|t − s| + |x1 − x2 | + |w1 − w2 |) , for every t , s ∈ I and every xi , wi ∈ X ;
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
51
(v) There exist lP such that
kP (x1 , w1 ) − P (x2 , w2 )kE ≤ lP (kx1 − x2 k + kw1 − w2 k),
xi , wi ∈ X .
By defining the operators f : J × X × X → X and p, q : C (I ; X ) × C (I ; X ) → X by f (t , w, v)(y) = g (t , w(y), v(y)) π
Z
p(z , v)(y) =
P (z (s), v(s))(y)dµ(s),
z , v ∈ C (I ; X ),
0 n X
q(z , v)(y) =
αi z (ti , y) +
i =1
k X
βj v(sj , y).
j =1
Also defining the maps Ik1 , Ik2 and B by
(Bu)(t )(y) = µ(t , y) Z π Ik1 (w)(y) = ak (s)w(s, y)ds 0 Z π Ik2 (w)(y) = a˜ k (s)w(s, y)ds, 0
the system (9) can be modelled as (2). It is known that the linearized system associated with (9) is exactly controllable (see [9]). Therefore system (9) can be written in the abstract form (2). Thus by Theorem 4.1, the second order system (9) is exactly controllable. Example 5.2. Now, from a practical viewpoint, we remark that the physical motivation for the study of Eqs. (1) and (2) is related to the partial differential equation governing the dynamical buckling of a hinged extensible beam, which is stretched or compressed by an axial force. Mathematical models of this phenomenon have been studied in the paper [11,16]. In this paper, we shall consider this type of equation with control variable. A mathematical model for this problem is an initial boundary value problem for the nonlinear hyperbolic control equation of the form
∂ ∂t
∂ z (t , y) ∂t
! Z L ∂ 2 z (t , y) ∂ 4 z (t , y) ∂ z (t , w) 2 ˆ ˆ = αˆ + µ( t , y ) − β + δ d w ∂ y4 ∂w ∂ y2 0 ∂ z (t , y) +g t , z (t , y), , y ∈ [0, L], t ∈ J ∂t
(10)
subject to the initial and boundary conditions defined as in Example 5.1. Here z (t , y) gives the deflection of the beam at ˆ δˆ > 0 are given parameters, µ and g are as in Example 5.1. point y at time t, L is the length of the beam, and α, ˆ β, Let X = L2 [0, L] and define the operator A : D(A) ⊂ X → X by Az (t , .) =
∂ 4 z (t , .) ∂ y4
with domain D(A) = {z ∈ X 4 [0, L] : z (0) = z (L) = z 00 (0) = z 00 (L) = 0}. It has been shown in [11] that A is a positive, self-adjoint operator on X . It is well known that A is the infinitesimal generator of a strongly continuous cosine family of operators on X . In order to write (10) in the abstract form (2), we need to express the various terms on the left side of (10) using certain fractional powers of A, as outlined in [11]; we recall the essential highlights of that discussion√ here for completeness. To begin, the eigenvalues of A are {λn = (nπ )4 |n ∈ N } with corresponding eigenvectors
{zn (s) =
2 L
Az =
sin(nπ s)|n ∈ N , s ∈ [0, L]}. Then A has the spectral representation
∞ X
λn hz , un iun .
n=1
Furthermore, since the fractional powers of A are positive and self-adjoint, the following are well-defined: 1
∞ X
1
∞ X
A2 z =
1
λn2 hz , un iun = −
n =1
A4 z =
n =1
1
λn4 hz , un iun =
∂ 2z . ∂ u2
∂z . ∂u
52
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
Also 1 4
1 4
1 4
kA z k = hA z , A z i = 2
∞ X n=1
2 Z L
∂z
λn hz , un i =
∂ u (t , w) dw. 1 2
2
0
Using these identifications together with the functions f , p, q, Ik1 , Ik2 , B defined as in Example 5.1, (10) can be written in the abstract form (2). In addition, one can use standard computations involving the inner product and properties of the fractional 1
1
powers of A to show that kA 4 z (t )k2 and kA 2 z (t )k2 are uniformly bounded on [0, b]. It is known that the linearized system associated with (10) is exactly controllable (see [9]). As such, (10) can be written in the abstract form (2) in X so that an application of Theorem 4.1 yields the exact controllability of (10) on [0, b]. Acknowledgements The authors would like to thank the referees for their many useful comments and constructive suggestions which substantially improved the manuscript. References [1] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Journal of Mathematical Analysis and Applications 162 (1991) 494–505. [2] M. Benchohra, A. Ouahab, Controllability results for functional semilinear differential inclusions in Frechet spaces, Nonlinear Analysis 61 (2005) 405–423. [3] M. Benchohra, L. Gorniewicz, S.K. Ntouyas, A. Ouahab, Controllability results for impulsive functional differential inclusions, Reports on Mathematical Physics 54 (2004) 211–228. [4] M. Benchohra, S.K. Ntouyas, Controllability of second order differential inclusion in Banach spaces with nonlocal conditions, Journal of Optimization Theory and Applications 107 (2000) 559–571. [5] A.E. Bashirov, N.I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM Journal on Control and Optimization 37 (1999) 1808–1821. [6] Y.K. Chang, W.T. Li, J.J. Nieto, Controllability of evolution differential inclusions in Banach spaces, Nonlinear Analysis 67 (2007) 623–632. [7] Y.K. Chang, W.T. Li, Controllability of second order differential and integrodifferential inclusions in Banach spaces, Journal of Optimization Theory and Applications 129 (2006) 77–87. [8] Y.K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons & Fractals 33 (2007) 1601–1609. [9] J.P. Dauer, N.I. Mahmudov, Controllability of some nonlinear systems in Hilbert spaces, Journal of Optimization Theory and Applications 123 (2004) 319–329. [10] H.O. Fattorini, Second order linear differential equations in Banach spaces, in: North Holland Mathematics Studies, vol. 108, Elsevier Science, North Holland, 1985. [11] W.E Fitzgibbon, Global existence and boundedness of solutions to the extensible beam equation, SIAM Journal of Mathematical Analysis 13 (1982) 739–745. [12] L. Gorniewicz, S.K. Ntouyas, D.O. Regan, Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Reports on Mathematical Physics 56 (2005) 437–470. [13] L. Gorniewicz, S.K. Ntouyas, D.O. Regan, Existence and controllability results for first- and second-order functional semilinear differential inclusions with nonlocal conditions, Numerical Functional Analysis and Optimization 28 (2007) 53–82. [14] R.K. George, D.N. Chalishajar, A.K. Nandakumaran, Controllability of second order semilinear neutral functional differential inclusions in Banach spaces, Mediterranean Journal of Mathematics 1 (2004) 463–477. [15] E.M. Hernandez, L.M. Pelicer, Existence results for a second-order abstract Cauchy problem with nonlocal conditions, Electronic Journal of Differential Equations 73 (2005) 1–17. [16] S. Kouemou-Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, Journal of Differential Equations 135 (1997) 299–314. [17] B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Analysis 60 (2005) 1533–1552. [18] M.L. Li, M.S. Wang, F.Q. Zhang, Controllability of impulsive functional differential systems in Banach spaces, Chaos, Solitons & Fractals 29 (2006) 175–181. [19] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [20] N.I. Mahmudov, M.A. Mckibben, Approximate controllability of second order neutral stochastic evolution equations, Dynamics of Continuous, Discrete and Impulsive Systems 13 (2006) 619–634. [21] N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization 42 (2003) 1604–1622. [22] R. Sakthivel, S.M. Anthoni, J.H. Kim, Existence and controllability result for semilinear evolution integrodifferential systems, Mathematical and Computer Modelling 41 (2005) 1005–1011. [23] R. Sakthivel, Q.H. Choi, S.M. Anthoni, Controllability result for nonlinear evolution integrodifferential systems, Applied Mathematics Letters 17 (2004) 1015–1023. [24] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [25] C.C. Travis, G.F. Webb, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston Journal of Mathematics 3 (1977) 555–567. [26] C.C. Travis, G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Mathematica Academia Scientia Hungaricae 32 (1978) 76–96. [27] T. Yang, Impulsive Systems and Control: Theory and Applications, Springer-Verlag, Berlin, Germany, 2001.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
On controllability of second order nonlinear impulsive differential systems R. Sakthivel a,∗ , N.I. Mahmudov b , J.H. Kim c a
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea
b
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Mersin 10, Turkey
c
Department of Mathematics, Yonsei University, Seoul 120-749, South Korea
article
info
Article history: Received 4 July 2007 Accepted 10 October 2008 Keywords: Controllability Fixed point theorem Nonlinear impulsive systems Nonlocal conditions
a b s t r a c t Many dynamical systems in physics, chemistry, biology and engineering sciences have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes. The mathematical models of such processes are called differential systems with impulse effect. This paper studies the exact controllability issue of certain types of second order nonlinear impulsive control differential systems. Sufficient conditions are formulated and proved for the exact controllability of such systems. Without imposing a compactness condition on the cosine family of operators, we establish controllability results by using a fixed point analysis approach. Finally, an example is presented to illustrate the utility of the proposed result. The results improve some recent results. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Many dynamical systems are characterized by the fact that at certain moments in their evolution they undergo rapid changes. In recent years the theory of impulsive differential systems has been emerging as an important area of investigation in applied sciences. The reason is that it is richer than the corresponding theory of classical differential equations and it is more adequate to represent some processes arising in various disciplines. The theory of impulsive systems provides a general framework for mathematical modeling of many real world phenomena. Moreover, these impulsive phenomena can also be found in fields such as information science, electronics, automatic control systems, robotics and telecommunications (see [19,24] and references therein). Control theory is an area of application-oriented mathematics which deals with basic principles underlying the analysis and design of control systems. It is well known that the study of controllability plays an important role in the control theory and engineering. In recent years, the study of impulsive control systems has received increasing interest, since dynamical systems with impulsive effects have great importance in applied sciences [see [27] and references therein]. This paper is concerned with the study of controllability of a second order nonlinear impulsive differential system of the form x00 (t ) = Ax(t ) + Bu(t ) + f (t , x(t ), x0 (t )), x(0) = x0 ,
t ∈ J = [0, b], t 6= tk ,
(1)
x0 (0) = y0
4x(tk ) = Ik1 (x(tk )),
4x0 (tk ) = Ik2 (x0 (tk+ )),
k = 1, 2 . . . , m
in a Banach space X . Here x : J → X is the state function, u(·) ∈ L2 (J , U ) is the control function, U is a Banach space,
∗
Corresponding author. Tel.: +82 31 2999 4527. E-mail address: [email protected] (R. Sakthivel).
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.10.029
46
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
A is the infinitesimal generator of a strongly continuous cosine family of linear operators (C (t ))t ∈R on X , B : U → X is a linear bounded operator, 0 = t0 < t1 < · · · < tm < tm+1 = b, 4x(tk ) = x(tk+ ) − x(tk− ), 4x0 (tk ) = x0 (tk+ ) − x0 (tk− ), and j
Ik : X → X , j = 1, 2, f : J × X × X → X are appropriate continuous functions to be specified later. In the last few years many papers have been published about the global and local controllability of nonlinear systems, in which the authors effectively used fixed point techniques. The problem of controllability of first order nonlinear systems has been studied by several researchers [2,5,9,12,20–23] and also first-order impulsive differential equations have received much attention [3,8,17,18]. Li et al., [18] established sufficient conditions for the controllability of impulsive functional differential systems with finite delay whereas in [2] Benchohra et al. studied controllability of impulsive differential inclusions in Banach spaces. Recently, Chang [8] studied the controllability of first order impulsive functional systems with infinite delay using Scahuder’s fixed point theorem. In many cases it is advantages to treat second order abstract differential equations directly rather than convert them to first order systems. The study of abstract deterministic secondorder evolution equations governed by the generator of a strongly continuous cosine family was initiated by Fattorini [10], and subsequently studied by Travis and Webb [25,26]. The controllability result for second order nonlinear systems with and without nonlocal conditions has been studied by many authors [4,7,13,14]. The results in those papers are only applicable to ordinary differential equations since the compactness assumption assumed on the cosine function is valid if and only if the underlying space is finite dimensional. However, to the authors’ knowledge the corresponding theory for controllability of second order impulsive differential equations in abstract spaces has not been explored. In this paper, we derive results on exact controllability of second order impulsive differential systems without imposing a compactness condition on the semigroup of cosine family. Further, the result is extended to study the controllability of the nonlinear second order impulsive system with nonlocal conditions described in the form x00 (t ) = Ax(t ) + Bu(t ) + f (t , x(t ), x(0) = x0 + p(x, x ),
x0 (t )), t ∈ J , t 6= tk ,
(2)
x (0) = y0 + q(x, x ) 4x(tk ) = Ik1 (x(tk )), 4x0 (tk ) = Ik2 (x0 (tk+ )), k = 1, 2 . . . , m, where p, q : C (J , X ) × C (J , X ) → X are appropriate functions to be specified later. 0
0
0
2. Preliminaries In this section, we review some basic concepts, notations and properties needed to establish our main results. Throughout this paper, A is the infinitesimal generator of a strongly continuous cosine family, (C (t ))t ∈R , of bounded linear operators defined on a Banach space X endowed with a norm k.k. We denote by (S (t ))t ∈R the sine function associated to (C (t ))t ∈R which is defined by t
Z
S (t )x :=
C (s)xds,
x ∈ X , t ∈ R.
0
˜ are positive constants such that kC (t )k ≤ M and kS (t )k ≤ M ˜ for every t ∈ J. Moreover, M and M [D(A)] is the space D(A) = {x ∈ X : C (t )x is twice continuously differentiable in t}, endowed with the norm kxkA = kxk + kAxk, x ∈ D(A). Define E = {x ∈ X : C (t )x is once continuously differentiable in t} endowed with the norm kxkE = kxk + sup kAS (t )xk,
x ∈ E,
0≤t ≤1
is a Banach space. The operator-valued function C (t ) AS (t )
h(t ) =
S (t ) C (t )
is a strongly continuous group of bounded linear operators on the space E × X generated by the operator A¯ =
0 A
I 0
defined on D(A) × E. From this, it follows that AS (t ) : E → X is a bounded linear operator and that AS (t )x → 0 as t → 0, for each x ∈ E. Furthermore, if x : [0, ∞) → X is locally integrable function, then y(t ) =
t
Z
S (t − s)x(s)ds 0
defines an E-valued continuous function which is a consequence of the fact that
Z t
Z
h( t − s ) 0
t
0 Z0 ds = t x(s)
S (t − s)x(s) ds C (t − s)x(s) ds
0
defines an (E × X )-valued continuous function.
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
47
The following properties are well known [25]: (i) (ii) (iii) (iv)
if x if x if x if x
∈ X then S (t )x ∈ E for every t ∈ R. ∈ E then S (t )x ∈ D(A), (d/dt )C (t )x = AS (t )x and (d2 /dt 2 )S (t )x = AS (t )x for every t ∈ R. ∈ D(A) then C (t )x ∈ D(A), and (d2 /dt 2 )C (t )x = AC (t )x = C (t )Ax for every t ∈ R. ∈ D(A) then S (t )x ∈ D(A), and (d2 /dt 2 )S (t )x = AS (t )x = S (t )Ax for every t ∈ R.
For additional details concerning cosine function theory, we refer the reader to Fattorini [10] and Travis & Webb [25,26]. Additionally, for a bounded function ξ : J → X and t ∈ J, we employ the notation ξt for
ξt = sup{kξ (s)kX : s ∈ [0, t ]}. Let C ([0, b], X ) be the space of the bounded functions x : [0, b] → X such that x is continuous in t 6= tk , x is left continuous in each tk and x(tk+ ) exist. Also assume C ([0, b], X ) endowed with the uniform convergence norm. Finally C 1 ([0, b], X ) will be the space C 1 ([0, b], X ) = {(x, z ) : x, z ∈ C ([0, b], X ) and x0 (t ) = z (t ) for t 6= tk } provided with the norm k(x, z )k = kxkb + kz kb . Definition 2.1. Let xb (x0 , y0 ; u) be the state value of (1) at time b corresponding to the control u and the initial value x0 and y0 . Introduce the set R(b, x0 , y0 ) = {xb (x0 , y0 ; u)(0) : u(.) ∈ L2 (J , U )}. The system (1) is said to be exactly controllable on the interval J if R(b, x0 , y0 ) = X . Assume that the second order linear system x00 (t ) = Ax(t ) + Bu(t ), x(0) = x0 ,
t ∈J
(3)
x0 (0) = y0
is exactly controllable on J. It is convenient at this point to define the operator (see [20])
Γ0b
b
Z
S (b − s)BB∗ S ∗ (b − s)ds.
= 0
Note that the system (3) is exactly controllable iff there exists a γ > 0 such that
hΓ0b x, xi ≥ γ kxk2 ,
for all x ∈ X
Then
k(Γ0b )−1 k ≤
1
γ
.
Note 2.2. If X is infinite dimensional, the corresponding cosine family of semigroup C (t ) is compact and B is bounded, then the second order linear control system x00 (t ) = Ax(t ) + Bu(t ) is not exactly controllable. Definition 2.3. A function x(·) ∈ C 1 ([0, b], X ) is said to be a mild solution of the impulsive control problem (1) if the impulsive conditions in (1) are satisfied and x(t ) = C (t )x0 + S (t )y0 +
t
Z
S (t − s)[Bu(s) + f (s, x(s), x0 (s))]ds 0
+
X 0
C (t − tk )Ik1 (x(tk )) +
X
S (t − tk )Ik2 (x0 (tk+ )).
(4)
0
Now for convenience, let us introduce the notation K = kBk. In order to establish the controllability result, we introduce the following technical hypothesis: (I) f : J × X × X → X is a continuous function and there exist positive constants K1 and K2 such that
kf (t , x, y) − f (t , x0 , y0 )k ≤ K1 kx − x0 k + K2 ky − y0 k for every x, x0 , y and y0 ∈ X (II) The linear system (3) is exactly controllable.
48
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52 j
j
(III) The functions Ik : X × X are continuous and there exist positive constants L(Ik ) such that
kIkj (x) − Ikj (x0 )k ≤ L(Ikj )kx − x0 k for each x, x0 ∈ X . (IV) There exist functions G : [0, b] → $ (X ) (the Banach space of bounded linear operators from X into X ) Fk : X → X , k ∈ {1, 2, . . . , m} such that: (i) G(0) = 0, G(.) is strongly continuous and (d/dt )C (t )Ik1 (x) = G(t )Fk (x) for every x ∈ X and k ∈ {1, 2, . . . , m} (ii) for each k ∈ {1, 2, . . . , m} there exists a positive constant L(Fk ) such that kFk (x) − Fk (x0 )k ≤ L(Fk )kx − x0 k (V) Let
˜ + ηM ˜ + M )K1 b + φ1 = (β M
m X (ηML(Ik1 ) + kGkL(Fk ) + β ML(Ik1 )) k=1
and
˜ + ηM ˜ + M )K2 b + φ2 = (β M
m X ˜ + ηM ˜ + M )L(Ik2 ), (β M k=1
˜ 2 K 2 b. ˜ and β = 1 + M where η = γ MK M γ 1
1
2
3. Controllability result Theorem 3.1. Let x0 ∈ E , y0 ∈ X . If the conditions (I)–(V) and max{φ1 , φ2 } < 1 are satisfied, then the second order impulsive control system (1) is exactly controllable. Proof. For (x, z ) ∈ C 1 ([0, b] : X ), define the nonlinear operator
Φ (x, z ) = (Φ1 (x, z ), Φ2 (x, z )), where
Z t Φ1 (x, z )(t ) = C (t )x0 + S (t )y0 + S (t − s)[Bu(s) + f (s, x(s), x0 (s))]ds 0 X X + C (t − tk )Ik1 (x(tk )) + S (t − tk )Ik2 (x0 (tk+ )), 0
0 < tk < t
Z
t
Φ2 (x, z )(t ) = AS (t )x0 + C (t )y0 + C (t − s)[Bu(s) + f (s, x(s), x0 (s))]ds 0 X X + G(t − tk )Fk (x(tk )) + C (t − tk )Ik2 (x0 (tk+ )). 0
0
Define the feedback control function
" u(t ) = B S (b − t )( ∗ ∗
)
Γ0b −1
xb − C (b)x0 − S (b)y0 −
b
Z
S (b − s)f (s, x(s), x0 (s))ds
0
−
m X
C (b −
) (x(tk )) −
tk Ik1
k=1
m X
# S (b −
) (x (tk )) .
tk Ik2
0
+
(5)
k=1
Note that the control (5) transfers the system (4) from the initial state to the final state provided that the operator Φ has a fixed point. So if the operator Φ has a fixed point then the system (4) is exactly controllable. To prove the exact controllability, it is enough to show that the operator Φ has a fixed point in C 1 ([0, b], X ). The proof is based on the classical fixed point theorem for contractions. It follows from the assumptions that each Φi , i = 1, 2 is well defined and continuous. In order to prove that Φ is a contraction mapping on C 1 ([0, b] : X ), we take (x, z )(v, w) in C 1 ([0, b] : X ) From the conditions (I) and (III), we get 1 kΦ1 (x, z )(t ) − Φ1 (v, w)(t )k ≤ M˜ 2 K 2 b
"Z
γ
+
b
˜ [K1 kx − vk + K2 kx0 − v 0 k]ds M 0
m X k=1
ML( )kx(tk ) − v(tk )k + Ik1
m X k=1
# ˜ ( )kx (tk ) − v (tk )k ML Ik2
0
+
0
+
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52 t
Z
X
˜ [K1 kx − vk + K2 kx0 − v 0 k]ds + M
+
ML(Ik1 )kx(tk ) − v(tk )k
0
0
X
+
49
˜ (Ik2 )kx0 (tk+ ) − v 0 (tk+ )k. ML
0
Thus we have
kΦ1 (x, z )(t ) − Φ1 (v, w)(t )k ≤
"
1 1 + M˜ 2 K 2 b
˜ 1b + MK
γ
" +
m X
!# ML( )
kx − vk
Ik1
k=1
1 1 + M˜ 2 K 2 b
˜ 2b + M ˜ MK
γ
m X
!# L( )
kz − wk.
Ik2
(6)
k=1
Similarly, we have
" ˜ + M )K1 b + kΦ2 (x, z )(t ) − Φ2 (v, w)(t )k ≤ (ηM
m X
# (ηML( ) + kGkL(Fk )) kx − vk Ik1
k=1
" ˜ + M )K2 b + + (ηM
m X
# ˜ + M )L( ) kz − wk. (ηM Ik2
(7)
k=1
The above inequalities (6) and (7) and the assumption max{φ1 , φ2 } < 1 imply that Φ is a contraction mapping. Hence there exists a unique fixed point x(·) in C 1 ([0, b], X ) which is the solution of Eq. (4). Thus the system (1) is exactly controllable. 4. Impulsive nonlocal differential systems The nonlocal condition is a generalization of the classical initial condition. The first results concerning the existence and uniqueness of mild solutions to Cauchy problems with nonlocal conditions were studied by Byszewski [1]. Recently, the existence of mild and classical solutions for a class of second order nonlinear differential systems with nonlocal conditions has been studied in [15]. As indicated in [1,15] and the references therein, the nonlocal conditions can be applied in physics with better effect than the classical Cauchy problem, since more measurements are allowed. In this section, we consider the second order impulsive nonlocal system (2) governed by the evolution equation x(t ) = C (t )(x0 + p(x, x0 )) + S (t )(y0 + q(x, x0 )) +
t
Z
S (t − s)[Bua (s) + f (s, x(s), x0 (s))]ds 0
+
X 0
X
C (t − tk )Ik1 (x(tk )) +
S (t − tk )Ik2 (x0 (tk+ )).
(8)
0
The purpose of this section is to prove the exact controllability of the impulsive nonlocal system of the form (8) under the simple and fundamental assumption on the system operators, in particular that the corresponding linear system is appropriately controllable. Moreover, N and N˜ are positive constants such that kC (t )k ≤ N and kS (t )k ≤ N˜ for every t ∈ J. Concerning the operators f , p and q, assume the following hypothesis: (H1) The function f : J × X × X → X satisfies the following conditions; (i) The function f (t , ·) : X × X → X is continuous a.e.t ∈ J; (ii) The function f (·, x, y) : J → X is strongly measurable for each (x, y) ∈ X × X . (iii) There exists a constant k˜ f such that
kf (t , x1 , y1 ) − f (t , x2 , y2 )k ≤ k˜ f (kx1 − x2 k) + (ky1 − y2 k) for every x1 , x2 , y1 , y2 ∈ X . (H2) The functions p, q : C (I ; X ) × C (I ; X ) → X are continuous and there exist positive constants k˜ p , k˜ q such that kp(x1 , y1 ) − p(x2 , y2 )k ≤ k˜ p (kx1 − x2 k + ky1 − y2 k), kq(x1 , y1 ) − q(x2 , y2 )k ≤ k˜ q (kx1 − x2 k + ky1 − y2 k). for every x1 , x2 , y1 , y2 ∈ C (I ; X ). P Pm 1 ˜ k˜ f + m ˜ ˜ ˜ ˜ ˜˜ ˜ ˜ (H3) Let Q1 = τ (N k˜ p + N˜ k˜ q + Nb k=1 NL(Ik )) + (ρ kp + N kq + Nbkf + k=1 kGkL(Fk )) and Q2 = τ (N kp + N kq + Nbkf +
Pm
k=1
˜ (Ik2 )) + (ρ k˜ p + N k˜ q + Nbk˜ f + NL
Pm
k=1
˜ 2 and ρ = supt ∈J kAS (t )k. NL(Ik2 )), where τ = 1 + γ1 K 2 N˜2 b + γ1 N NK
Theorem 4.1. Assume that (II)–(IV) and (H1 )–(H3 ) hold. If max{Q1 , Q2 } < 1 are satisfied, then the second order impulsive nonlocal control system (8) is exactly controllable.
50
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
Proof. For (x, z ) ∈ C 1 ([0, b] : X ), define the nonlinear operator
˜ (x, z ) = (Φ3 (x, z ), Φ4 (x, z )), Φ where
Z t S (t − s)[Bua (s) + f (s, x(s), x0 (s))]ds Φ3 (x, z )(t ) = C (t )(x0 + p(x, x0 )) + S (t )(y0 + q(x, x0 )) + 0 X X + C (t − tk )Ik1 (x(tk )) + S (t − tk )Ik2 (x0 (tk+ )), 0
0 < tk < t
Z t C (t − s)[Bua (s) + f (s, x(s), x0 (s))]ds Φ4 (x, z )(t ) = AS (t )(x0 + p(x, x0 )) + C (t )(y0 + q(x, x0 )) + 0 X X + G(t − tk )Fk (x(tk )) + C (t − tk )Ik2 (x0 (tk+ )). 0
0
Define the feedback control function ua (t ) = B∗ S ∗ (b − t )(Γ0b )−1 [xb − C (b)(x0 + p(x, x0 )) − S (b)(y0 + q(x, x0 )) b
Z
S (b − s)f (s, x(s), x0 (s))ds −
− 0
m X
C (b − tk )Ik1 (x(tk )) −
k=1
m X
S (b − tk )Ik2 (x0 (tk+ ))].
k=1
˜ has a fixed point by employing the method used in the previous section. Then we One can easily prove that the operator Φ can show that the system (8) is exactly controllable by adopting the technique used in Theorem 3.1. Remark 4.2. Differential inclusions play an important role in characterizing many social, physical, biological and engineering problems. In particular, the problems in physics, especially in solid mechanics, where non-monotone and multivalued constitutive laws lead to differential inclusions. The above result can be extended to study the controllability of second order nonlinear impulsive differential inclusions by suitably introducing the multivalued map defined in [4,6,13]. 5. Application Example 5.1. Consider the following partial second-order impulsive nonlocal differential equation with control µ(t , ·) ∈ L2 [0, π]
∂ ∂t
∂ z (t , y) ∂t
∂ 2 z (t , y) ∂ z (t , y) = + µ(t , y) + g t , z (t , y), , ∂ y2 ∂t
y ∈ [0, π], t ∈ J ,
(9)
subject to the conditions: z (t , 0) = z (t , π ) = 0,
4z (tk )(y) =
tk
Z
t ∈J
ak (tk − s)z (s, y)ds
0
4z 0 (tk )(y) =
tk
Z
a˜ k (tk − s)z (s, y)ds
0
∂ z (s) (y)dµ( ¯ s) ∂t 0 n l X X ∂ z (0, y) = y1 (y) + αi z (ti , y) + βj z (sj , y), ∂t i =1 j =1
z (0, y) = y0 (y) +
Z
b
P
z (s),
where y0 , y1 ∈ X , µ(·) ¯ is a real function of bounded variation on J and ak , a˜ k ∈ C (R, R). Let X = L2 [0, π] and let A : X → X be an operator defined by Az = z 00 with domain D(A) = {z ∈ X : z and z 0 are absolutely continuous, z 00 ∈ X , z (0) = z (π ) = 0}. It is well known that A is the infinitesimal generator of a strongly continuous cosine family of operators on X . Here 0 < ti , sj < π , αi , βj ∈ R are prefixed numbers and g : J × X 2 → X , P : X × X → X satisfy the following conditions: (i) g (·) is continuous and there exists a constant Lg such that
|g (t , x1 , w1 ) − g (s, x2 , w2 )| ≤ Lg (|t − s| + |x1 − x2 | + |w1 − w2 |) , for every t , s ∈ I and every xi , wi ∈ X ;
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
51
(v) There exist lP such that
kP (x1 , w1 ) − P (x2 , w2 )kE ≤ lP (kx1 − x2 k + kw1 − w2 k),
xi , wi ∈ X .
By defining the operators f : J × X × X → X and p, q : C (I ; X ) × C (I ; X ) → X by f (t , w, v)(y) = g (t , w(y), v(y)) π
Z
p(z , v)(y) =
P (z (s), v(s))(y)dµ(s),
z , v ∈ C (I ; X ),
0 n X
q(z , v)(y) =
αi z (ti , y) +
i =1
k X
βj v(sj , y).
j =1
Also defining the maps Ik1 , Ik2 and B by
(Bu)(t )(y) = µ(t , y) Z π Ik1 (w)(y) = ak (s)w(s, y)ds 0 Z π Ik2 (w)(y) = a˜ k (s)w(s, y)ds, 0
the system (9) can be modelled as (2). It is known that the linearized system associated with (9) is exactly controllable (see [9]). Therefore system (9) can be written in the abstract form (2). Thus by Theorem 4.1, the second order system (9) is exactly controllable. Example 5.2. Now, from a practical viewpoint, we remark that the physical motivation for the study of Eqs. (1) and (2) is related to the partial differential equation governing the dynamical buckling of a hinged extensible beam, which is stretched or compressed by an axial force. Mathematical models of this phenomenon have been studied in the paper [11,16]. In this paper, we shall consider this type of equation with control variable. A mathematical model for this problem is an initial boundary value problem for the nonlinear hyperbolic control equation of the form
∂ ∂t
∂ z (t , y) ∂t
! Z L ∂ 2 z (t , y) ∂ 4 z (t , y) ∂ z (t , w) 2 ˆ ˆ = αˆ + µ( t , y ) − β + δ d w ∂ y4 ∂w ∂ y2 0 ∂ z (t , y) +g t , z (t , y), , y ∈ [0, L], t ∈ J ∂t
(10)
subject to the initial and boundary conditions defined as in Example 5.1. Here z (t , y) gives the deflection of the beam at ˆ δˆ > 0 are given parameters, µ and g are as in Example 5.1. point y at time t, L is the length of the beam, and α, ˆ β, Let X = L2 [0, L] and define the operator A : D(A) ⊂ X → X by Az (t , .) =
∂ 4 z (t , .) ∂ y4
with domain D(A) = {z ∈ X 4 [0, L] : z (0) = z (L) = z 00 (0) = z 00 (L) = 0}. It has been shown in [11] that A is a positive, self-adjoint operator on X . It is well known that A is the infinitesimal generator of a strongly continuous cosine family of operators on X . In order to write (10) in the abstract form (2), we need to express the various terms on the left side of (10) using certain fractional powers of A, as outlined in [11]; we recall the essential highlights of that discussion√ here for completeness. To begin, the eigenvalues of A are {λn = (nπ )4 |n ∈ N } with corresponding eigenvectors
{zn (s) =
2 L
Az =
sin(nπ s)|n ∈ N , s ∈ [0, L]}. Then A has the spectral representation
∞ X
λn hz , un iun .
n=1
Furthermore, since the fractional powers of A are positive and self-adjoint, the following are well-defined: 1
∞ X
1
∞ X
A2 z =
1
λn2 hz , un iun = −
n =1
A4 z =
n =1
1
λn4 hz , un iun =
∂ 2z . ∂ u2
∂z . ∂u
52
R. Sakthivel et al. / Nonlinear Analysis 71 (2009) 45–52
Also 1 4
1 4
1 4
kA z k = hA z , A z i = 2
∞ X n=1
2 Z L
∂z
λn hz , un i =
∂ u (t , w) dw. 1 2
2
0
Using these identifications together with the functions f , p, q, Ik1 , Ik2 , B defined as in Example 5.1, (10) can be written in the abstract form (2). In addition, one can use standard computations involving the inner product and properties of the fractional 1
1
powers of A to show that kA 4 z (t )k2 and kA 2 z (t )k2 are uniformly bounded on [0, b]. It is known that the linearized system associated with (10) is exactly controllable (see [9]). As such, (10) can be written in the abstract form (2) in X so that an application of Theorem 4.1 yields the exact controllability of (10) on [0, b]. Acknowledgements The authors would like to thank the referees for their many useful comments and constructive suggestions which substantially improved the manuscript. References [1] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Journal of Mathematical Analysis and Applications 162 (1991) 494–505. [2] M. Benchohra, A. Ouahab, Controllability results for functional semilinear differential inclusions in Frechet spaces, Nonlinear Analysis 61 (2005) 405–423. [3] M. Benchohra, L. Gorniewicz, S.K. Ntouyas, A. 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