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Approximate controllability of nonlinear impulsive differential systems
Vol. 60 (2007)
REPORTS ON MATHEMATICAL PHYSICS
No. 1
APPROXIMATE C O N T R O L L A B I L I T Y OF NONLINEAR I M P U L S I V E DIFFERENTIAL SYSTEMS R. SAKTHIVEL*, N. I. MAHMUDOV** and J. H. KIM* Department of Mathematics, Yonsei University, Seoul 120-749, South Korea Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Turkey (e-mails: [email protected], [email protected], [email protected]) (Received January 15, 2007)
Many practical systems in physical and biological sciences have impulsive dynamical behaviours during the evolution process which can be modeled by impulsive differential equations. This paper studies the approximate controllability issue for nonlinear impulsive differential and neutral functional differential equations in Hilbert spaces. Based on the semigroup theory and fixed point approach, sufficient conditions for approximate controllability of impulsive differential and neutral functional differential equations are established. Finally, two examples are presented to illustrate the utility of the proposed result. The results improve some recent results. Keywords: approximate controllability, impulsive differential equations, fixed point theorem.
1.
Introduction
Many systems in physics and biology exhibit impulsive dynamical behaviour due to sudden jumps at certain instants in the evolution process. Differential equations involving impulse effects occur in many applications: pharmacokinetics, the radiation of electromagnetic waves, population dynamics, biological systems, the abrupt increase of glycerol in fed-batch culture, bio-technology etc., [1, 5, 13, 21, 22]. For the basic theory of impulsive differential equations the reader can refer to Samoilenko and Perestyuk [21]. On the other hand, it is well known that control problems of deterministic equation are widely used in many branches of science and engineering. Moreover, the interest in impulsive control systems which is based on the theory of impulsive differential equations has grown in recent years due to its theoretical and practical significance. In [24] some recent developments on the impulsive control theory and its applications to nanoelectronics are discussed. In recent years, controllability problems for different kinds of dynamical systems have been considered in many publications [2-4, 6, 12]. The controllability of impulsive systems in Banach spaces is also very interesting and researchers are engaged in it. To our knowledge, there are few papers discussing controllability of impulsive differential equations on infinite dimensional Banach spaces. We refer to the paper of Benchohra et al. [2] who studied the controllability of impulsive differential inclusions by using a fixed point theorem for the condens[85]
86
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
ing maps due to Martelli. Li et al. [18] established sufficient conditions for the exact controllability of impulsive functional differential systems using Schaefer's fixed point theorem. Very recently, Chang [8] studied the exact controllability of impulsive functional systems with infinite delay. However, in order to establish these results, the invertibility of a controllability operator is imposed. It turns out that in practice it is rather difficult to verify this condition directly, see [7], and it fails in infinite-dimensional spaces when the semigroup generated by A is compact. Therefore, it is important, in fact, necessary to study the weaker concept of controllability, namely approximate controllability for impulsive differential systems. Klamka [16] investigated constrained approximate controllability problems for linear abstract dynamical systems with linear unbounded control operator and piecewise polynomial controls. Jeong and Roh [15] studied the approximate controllability for the semilinear retarded control system. The approximate controllability and approximate null controllability of control systems governed by a class of abstract semilinear integro-differential equations are discussed in [23]. Approximate controllability for semilinear control systems can be found in Mahmudov [19,20], Dauer and Mahmudov [10], and many other papers [11,23]. However, up to now approximate controllability problems for nonlinear dynamical systems with impulses have not been considered in the literature. In order to fill this gap, this paper studies the approximate controllability problem for nonlinear systems described by differential equation with impulses and control operator. In Section 3, we study the approximate controllability of the following impulsive control differential equation x'(t) = A x ( t ) + ( B u ) ( t ) + f ( t , x(t)),
t E J = [0, b] \ D,
(1)
x (0) = xo,
Ax(tk) = Ik(X(tk)),
k = 1 . . . . . m,
where A is the infinitesimal generator of a strongly continuous semigroup T ( t ) in a Hilbert space X, B is a linear bounded operator from a Hilbert space U into X, f : J × X ~ X is a nonlinear operator, the control u(.) 6 L2(J, U). Here D = {h,t2 . . . . . tm} C J, 0 = to < tl < "'" < t m < tm+l = b, Ik(k = 1,2 . . . . . m) is a nonlinear map and A x ( t k ) = x ( t ~ ) ) - x ( t ~ ) = x ( t +) - X ( t k ) . This represents the jump in the state x at time tk with Ik determining the size of the jump. It should be noted that if X is infinite dimensional, the semigroup is compact and B is bounded, then the infinite-dimensional linear control system x'(t) = A x ( t ) + ( B u ) ( t ) ,
t ~ [0, b],
(2)
x (0) = Xo,
is not exactly controllable [7]. In view of the above result the exact controllability results of impulsive neutral systems [17] hold only in finite-dimensional spaces. With the proceeding reason in this article, we study approximate controllability of neutral functional differential equations with impulses described in the form
APPROXIMATE CONTROLLABILITYOF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS 87 d ~-7[x(t) + g(t, xt))] = A[x(t) + g(t, xt)] + (Bu)(t) + F(t, xt),
x(t) =c/)(t),
t E J,
t 7~ tk
(3)
t ~ [ - h , 0],
x(t~) - x(t~) = [k(X(tk)),
k = 1 . . . . . m,
where A generates a strongly continuous semigroup T(t) on the Hilbert space X. The functions g, F, [k and ~b are given continuous functions to be specified later.
2.
Preliminaries
Let Xb(Xo; u) be the state value of (1) at terminal time b corresponding to the control u and the initial value Xo. Introduce the set
~t(b, xo) = {Xb(XO; U)(0) : u(-) ~ L2(J, U)}, which is called the reachable set of system (1) at terminal time b, its closure in X is denoted by ~ ( b , x0). DEFINITION 2.1. The system (1) is said to be approximately controllable on the interval J if 91(b, x 0 ) = X. It is convenient at this point to define operators
V~ =
T(b - s ) B B * T * ( b - s)ds, R(oe, Pob) = ( a l + P0b)-1.
($1) otR(c~, I'0b) ~ 0 as ~ --+ 0 + in the strong operator topology. The assumption ($1) holds if and only if the linear system (2) is approximately controllable on J, see [6]. Now for convenience, let us introduce the notation M r = max{liT(Oil • 0 < t < b}, k = max{l, M r M ~ , M r M s b } ,
ai -=
M s = IIBII,
3kMZrMs[[~.i[l,
bi =
I1~.i II = fo b IPvi(s)lds,
3Mrl[)~il[, ci m
d l = 3kMrMB(llxbl[ + Mrllxoll + M r ~ _ d k ) , k=l m
d2 = 3 M r Ilx0ll + M r ~
dk,
k=l
We introduce the following assumptions: (I) The semigroup T(t), t > 0, is compact.
d = max{dl, d2}.
= max{ai, bi},
88
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
(II) The function f • J x X ~ X is continuous and there exist functions ) - i ( ' ) E L I ( J , R +) and h i ( ' ) E LI(X, R+), i = 1, 2 . . . . . q such that q
Ilf(t,x)ll
<
Z)~i(t)hi(x)
for all
( t , x ) ~ J x X.
i=1
(III) For each ot > 0,
(
~-~ --Ci sup{hi(x)
lira sup r r~oo-
" IIxll _< r
})
-- ~ .
i=1 O/
(IV) Ik ~ C ( X , X ) and there exists a constant dk such that IIIk(x)ll _< dk for each x E X ( k = 1 . . . . . . m). (V) The function f : J x X --~ X is continuous and uniformly bounded and there exists N > 0 such that Ilf(t,x)l[ < N for all ( t , x ) ~ J × X. In Section 3.1, it will be shown that the system (1) is approximately controllable if for all ot > 0 there exists a continuous function x(.) ~ C ( J , X ) such that u(t) = B * T * ( b - t)R(ot, Fg)p(x(-)),
X(t)
T(t)xo +
fo'
T(t - s)[Bu(s) + f(s, x(s)]ds +
(4) y~
O
T ( t - tk)Ik(x(tk)),
(5)
where p ( x ( . ) ) = xb - T ( b ) x o -
fo
m
T ( b - s ) f ( s , x ( s ) ) d s - ~_, T ( b - tk)Ik(x(tk)). k=l
3. Approximate controllability of impulsive systems 3.1. Impulsive differential systems THEOREM 3.1. Assume that conditions (I)-(IV) are satisfied, then for all 0 < ot < 1 the system (1) has a solution on J. Proof: The main aim in this section is to find conditions for solvability of system (4) and (5) for c~ > 0. In the Banach space C ( J , X ) consider a set Yr
=
{X(.) E C ( J , X ) [ x(O) = xo, [Ixll _< r},
where r is the positive constant. For a > 0, define the operator F~ on C ( J , X ) as F~(x) = z,
where
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS
v(t) = B*T*(b - t)R(c~, z(t) = T(t)xo +
ro~)p(x(.)),
fo T(t - s)[Bv(s)
p(x(.)) = Xb -- r(b)xo -
89
fo
+ f ( s , x(s))]ds + ~_. T(t - tk)Ik(x(tD), O
m T(b
T(b - s ) f ( s , x(s))ds - ~
- tk)Ik(X(tk)).
k=l
It will be shown that for all c~ > 0 the operator F~ from C(J, X) into itself has a fixed point. Step 1. For an arbitrary oe > 0 there is a positive constant r0 = r(o0 such that
Fa : rro----~ Yro. Let (i(r)= sup{hi(x):llxl[ < r,x • X}. By assumption (iii), there exists ro > 0 such that
d £ - + O/
ci --(i(ro) <_ ro.
i = 1 O/
If x(-) • Yro, then we obtain Hv(t)[[ _< 1MrMB(llXbl[ + MrllXoll + Mr k
fb Jo
q
m
i=1
k=l q
1 ( ~_~) < - M T M 8 Ilxbll + MTllXoll + Mr dk + 1 M r M g ~__, ll)~ill(i(ro) 13/
Ol
k=l
d
1 q~ ci
< - + - 3kol ~
i=l
ro
2.a --(i(ro) i=1 u
< -- 3k
and
d
Ilz(t)ll = g + MrMBbllvH + Mr
fot£ )~i(s)hi(x(s))ds i=1
<_ -~ + kllvl[ + -~ ~_(ci(i(ro) <_ '=
Finally we get
d+
cir.i(ro) i=1
+el[vii
< --
-3
II(F~x)(t)l[ = I[z(t)ll _< r0.
Thus F~ maps Yr0 into itself. Step 2. For each 0 < oe _< 1, the operator F~ maps Yro into a relatively compact subset of Yro" According to infinite-dimensional version of the Ascoli-Arzela theorem and Step 1 we need to prove that: (i) for an arbitrary t • J the set g(t) = {(F~x)(t) • x(.) • Yr0} is relatively compact,
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
90
(ii) for an arbitrary e > 0, there exist 6 > 0 such that I I ( F ~ x ) ( t l ) - ( F ~ x ) ( h ) l l < if Ilxll _< r, Itl - tzl < 3, tl, ta ~ J In the case t = 0 it is trivial, since V(0) = {x0}. So let t be a fixed real number, and let r be a given real number satisfying 0 < r < t. Define
( F d x ) ( t ) = T ( r ) z ( t - r). Since T ( t ) is compact and z ( t -
r) is bounded on Yro, the set
Vr(t) = { ( F d x ) ( t ) :x(-) C Yro} is a relatively compact set in X. That is, a finite set {Yi, 1 < i < n} in X exists .m. such that V~(t) C [,..J N ( y i , e/2),
i=1 where N(yi, ~/2) is an open ball in X with center at Yi and radius ~/2. On the other hand,
L
II(f~x)(t) - (F~x)(t)ll =
T(t - s)[Bv(s) + f(s, x(s))]ds
77
<_ - M r M B
O/
Xi(s)(i(ro)ds
\Ilxoll + Mrllx011
i=1
+ Zm T ( b - tk)Ik(x(tk)) ) r + M r ~ f t ~
k=l Consequently
? i(s)ds(i(ro) <_ 2"~5
i=1 n. V ( t ) C [ J N ( y i , e).
i=l Hence for each t 6 [0, b], V ( t ) is relatively compact in X. To prove (ii), we have to show that V = {(Fax)(.) Ix(.) ~ Yro} is equicontinuous on [0, b]. For 0 < h < t2 _< b, we have Ilz(q) - z(t2)ll _< liT(t0 - T(t2)llllxoll + M T M B
-4- M r +
fo
y~
ft,
IIv(s)llds
liT(t2 - s) - T ( h - s)ll IIv(s)llds
liT(t2 - tk) - T(tl - t~)llllIk(x(tk))ll
O
+
~
l i T ( t 2 - tDIIIIIk(x(tk))ll + Mr ftl t2 Zq) ~ i ( s ) h i ( x ( s ) ) d s
t 1
i= 1
+ MT fo tl liT(t2 - s) - T(tl - s)]] q X i ( s ) h i ( x ( s ) ) d s
i=1
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS 91
_< liT(t1) - T(t2)llllxoll + M T M B +
IIv(s)llds
~ liT(t2 - tk) - Z(tl - t~)lldk + y~, liT(t2 - tk)lldk O
fo"
[[r(tl - s) - T(tz - s)ll llw(s)llds
+ Mr
+ M r ~_, q ftl t2 ~,i(s)ds(i(ro) i=1
+ ~q f0 tl [[T(tl -- s) -- T(t2 - s)ll),i(s)ds(i(ro).
(6)
i=1
Thus the right-hand side of (6) does not depend on particular choices of x(-) and tends to zero as t t - t 2 ~ 0, since the compactness of T ( t ) for t > 0 implies the continuity in the uniform operator topology. So we obtain the equicontinuity of V. We considered here only the case 0 < tl < t2, since other cases h < t2 < 0 or tl < 0 < t2 are very simple. Thus F~[Yro] is equicontinuous and also bounded. By the Ascoli-Arzela theorem F~[Yro] is relatively compact in C ( J , X). It is easy to show that for all o~ > 0, F,~ is continuous on C ( J , X). Hence from Schauder's fixed point theorem F~ has a fixed point. Thus the problem (1) has a solution on J. [] THEOREM 3.2. A s s u m e that linear system (2) is approximately controllable on J. If the conditions (I), (III)-(V) are satisfied then the system (1) is approximately controllable. Proof: Let ~ ( . ) be a fixed point of F~ in Yro" Any fixed point of F~ is a mild solution of (1) under the control fi~(t) = B * T * ( b - t)R(ot,
and satisfies the inequality 2c~(b) = Xb + otR(u,
By the condition (V), fo b I I f 2~(s))ll2ds ( s , _
N2b
and consequently the sequence { f ( s , Yc~(s))} is-bounded in L2(J, X). Then there is a subsequence denoted by {f(s,;c~(s))}, that weakly converges to, say, f ( s ) in L2(J, X). Now, thanks to the compactness of an operator l(.) --+ fo T(. - s ) l ( s ) d s : L2(J, X ) --+ C ( J , X ) we obtain that IIP(:~) - wll =
fo b T ( b - s ) [ l l f ( s , ;c=(s)) - f (s)]ds
< sup II 0
f0'
T ( b - s ) [ f ( s , Yc~(s)) - f ( s ) l d s l l ~
0
92
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
as ot ~ 0 +, where
w = T(b)xo + fo b T(b - s ) f ( s ) d s - Xo. Then from
II~(b)
<_ II~R(~, rb)(w)ll + II~R(~, rb)ll IIP(2~) -- wll
-xbll
_< II~R(~, r~)(w)ll + I I p ( ~ ) - wll -~ o as c~ ~ 0 +. This proves the approximate controllability of (1).
[]
3.2. Neutral impulsive systems: A natural generalization of impulsive ordinary differential equations are the impulsive neutral functional differential equations. Impulsive neutral functional differential equations describe models of real processes and phenomena where both dependence on the past and momentary disturbances are observed. In recent years, the interest in impulsive neutral systems has been growing rapidly due to their successful applications in practical fields such as circuit theory, bioengineering, chemical technology and so on. In this section, it will be proved that the system (3) is approximately controllable if for all ot > 0 there exists a continuous function x(.) ~ C([-r, b], X) such that
Ua(t)
B*T*(b -
x(t) =
jo
t ) R (ot, r0b)[Xb -- T(b)[q~(0) + g(0, ~b)] + g(b, Xb)
n
r(b
s)f(s, xs)ds -
r(b
-
tk)ik(x(t[))j,]
k=l
r(t)[q~(o)
+ Z
-
+ g ( O , 4,)1
- g(t, xt) +
T(t - s)[Bu(s) + F(s, x~)]ds
T(t-tk)[~(x(t~)),
O
where [k : X ~ X, 0 = to < tl < . . . < tm < tm+l = b, x(t~) and x(t~) represent the right and left limits of x(t) at t = tk, respectively. Here C = C ( [ - h , 0], X) is the Banach space of all continuous functions cp : [ - h , 0] ~ X endowed with the norm II~b[I= sup{kb(0)l : - h < 0 < 0}. Also for any continuous function x defined on [ - h , b ] - {tl . . . . . tm} and t E J, we have xt E C for t ~ [0, b],xt(0) = x(t + 0 ) for 0 6 [ - h , 0]. For each k = 0, 1. . . . . m let Jk = [tk, tk+l]. Here C(J~, X) is the Banach space of all continuous functions from J~ into X with the norm I[xllj~ = sup{lx(t)l : t ~ Jk}. We consider the space C ( [ - h , b], X) = {x : [ - h , b ] -+ X :xk c C(Jk, X), k = 0 . . . . . m and there exist x(t +) and x(t~) with x(tk) = x(&), k = 1 . . . . . m} which
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS
93
is the Banach space with the norm
Ilxllc(t-h,b],S) = max{llx~llJ~, k = 0 . . . . . m}, where x~ is the restriction of x to Jk, k = 0 . . . . m. Concerning the operators F and g, assume the following hypothesis: (i) There exist functions ~-i(') 6 LI(J, R +) and h i ( ' ) C L I ( C , R + ) , . . . . q, such that
i = 1, 2,
q
IlF(t,~P)ll
_<
Z)~i(t)]~i(~9)
for all
(t,q~) 6 J x C.
i=1
(ii) For each ot > 0 lim sup
r
--
r-+~\
-- sup{/~i(cp) i=1 19l
"
114~11_< r}
)
= c~.
(iii) The function g : J × C ~ X is completely continuous and uniformly bounded, there exists a constant M3 > 0 such that IIg(t, 4~)11 _< M3, (iv)
t c J,
~b E C.
[k ~ C(X, X) and there exists a constant dh such that Illk(x)ll _< dh for each x E X ( k = l . . . . . m).
(v) The function F : J × C ~ X is continuous and uniformly bounded and there exists N1 > 0 such that [IF(t, q~)ll -< N1 for all (t, ~p) E J x C. THEOREM 3.3. Assume that linear system (2) is approximately controllable on [0, b]. If the semigroup T(t) is compact and conditions (i)-(v) are satisfied, then system (3) is approximately controllable.
Proof: For ot > 0, define the nonlinear operator F~ on C([-h, b], X) as F~(x) : z, where Ua(t) = B*T*(b z(t)
- t)R(ot, rb)p(x(.)),
g(t, xt) + f t T(t Jo T(t--tk)[k(x(t~)),
-----T(t)[4~(O) + g(O, ~b)] -
+ Z
- s ) [ B u a ( S ) q-
F(s, x,)]ds
O
zo(O)=dp(O),
0 c [ - h , 0],
p(x(.)) = Xb -- T(b)[q~(0) + g(0, ~b)] + g(b, Xb) -- fo b T(b - s)F(s, xs)ds -
-
~ k=l
T(b - tk)[k(X(t~)).
94
R. S A K T H I V E L , N. I. M A H M U D O V and J. H. K I M
One can easily prove that if for all ot > 0, the operator F~ has a fixed point by employing the method used in the previous section, then we can show that the system (3) is approximately controllable by adopting the technique used in Theorem 3.2. [] REMARK 3.1. The problems in physics, especially in solid mechanics, where one has nonmonotone and multi-valued constitutive laws lead to differential inclusions [9]. The above result can be extended to study the controllability of nonlinear neutral impulsive differential inclusions by suitably introducing the multivalued map defined in [2, 12]. .
Application
EXAMPLE 4.1. Consider a control system governed by the heat equation with impulses: O
82
~-~Z(t, y) = -~v2z(t, y) + Bu(t, y) + f ( t , z(t, y)), Az(tk, y) = --z(tk, y), z(t, O) = z(t, Jr) = O, z(O, y) : zo(y),
y E (0, 1),
0 < y < rr,
(7)
k = 1. . . . . m,
t E J,
y E [0, Jr].
Let X = L2[0, Jr] and let A : X --+ X be an operator defined by Az = z" with domain D(A) = {w ~ X : w and w f are absolutely continuous, w" 6 X, w(0) = w(rr) = 0}. Then oo I
Aw = - - ~ n 2 ( W , en)en,
W ~ D(A),
n=l
where e.(y) = (2/Tr)l/Zsiny, 0 _< y < Jr, n = 1,2 . . . . . . Let Z0 E L2[0, rr], Ik(x(tk)) =--z(tk, y), and f be Lipschitz continuous and satisfies linear growth conditions. It is well known that A generates compact semigroup T(t) in X and is given by oo
T(t)w = Z
e-
n2
(W, en)en,
WE X.
n=l
Define an infinite-dimensional space fx)
(3o
U={u'u=Zunenl~U2
n=2
with the norm defined by Ilullu = (Y~,nC~=2Un )2 1/2. Define a mapping B from U to X as follows: oo
Bu = 2u2el -q- ~ n=2
Unen.
}
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS 95
Because off compactness of the semigroup T(t) generated by A, the associated linear system is not exactly controllable but it is approximately controllable [7]. Therefore, system (7) can be written in the abstract form (1). By Theorem 3.2, system (7) is approximately controllable on [0, b]. EXAMPLE 4.2. We consider the following controlled neutral differential equation with impulses: 0 02 -~[z(t, y) - I~l(t, z(t - h, y))] = -~v2[z(t, y) - #l(t, z(t - h, y))] +/z(t,y)+/z2(t,z(t-h,y)),
z(t +, y) - z(t~-, y) = [k(Z(tk-, y)), z(t, O) = z(t, 1) = 0, z(t, y) =d~(t, y),
0
< 1, (8)
k = 1 . . . . . m, t > 0, - h < t < O,
where 4> is continuous and [k ~ C(R, R). Let g(t, wt)(y) =/Zl(t, w ( t - y)), F(t, wt)(y) = lz2(t, w ( t - y)) and (Bu)(t)(y) = /z(t, y), y E (0, 1). Take X = L2[0, 1] and define A : X ~ X by
Aw--
d2w dy 2
with domain
D(A) =
dw w c X " w and d~- are absolutely continuous,
dZw
dw dw I --dy2 c X, -~--y(0) = -d-~--y(1) = 0 ]. The operator A has the eigenvalues ~., = - n 2 n -2, n >_ 0, and the corresponding eigenvectors en(y) = 2 (1/2) cos(nzry) for n >_ 1, eo = 1, from an orthonormal basis for L2(0, 1). It is well known that A generates a compact semigroup T(t), t > 0, in X and is given by
T ( t ) w = fo 1 w ( v ) d v + End=12e-n2jr2t cos(nTrY) fo' cos(nTry)w(v)dv,
w c X.
Further, the functions #1,/~2 " [0, 1] >( [0, 1] ~ [0, 1] are continuous and there exist constants kl,k2 such that I[#a(t, w(t --Y))[I -< f:l and [[/Zz(t, w(t - y))[[
96
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
Acknowledgements The work of RS and JHK was supported by Brain Korea 21 project at Yousei University, 2006. REFERENCES [I] D.D. Bainov, S. I. Kostadinov and A. D. Myshkis: Asymptotic equivalence of abstract impulsive differential equations, Int. J. Theoret. Phys. 35 (1996), 383-393. [2] M. Benchohra, L. G6rniewicz, S. K. Ntouyas and A. Ouahab: Controllability results for impulsive functional differential inclusions, Rep. Math. Phys. 54 (2004), 211-228. [3] M. Benchohra and A. Ouahab: Controllability results for functional semilinear differential inclusions in Frechet spaces, Nonlinear Analysis 61 (2005), 405-423. [4] M. Benchohra, L. G6rniewicz, S. K. Ntouyas and A. Ouahab: Controllability results for nondensely defined semilinear functional differential equations, Zeitschrift fiir Analysis und ihre Anwendungen 25 (2006), 311-325. [5] G. Ballinger and X. Liu: Boundness for impulsive delay differential equations and applications to population growth models, Nonlinear Analysis 53 (2003) 1041-1062. [6] A. E. Bashirov and N. I. Mahmudov: On concepts of controllability for deterministic and stochastic systems, SlAM Journal on Control and Optimization 37 (1999), 1808-1821. [7] R. Curtain and H. J. Zwart: An Introduction to Infinite Dimensional Linear Systems Theory, Springer, New York 1995. [8] Y. K. Chang: Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons & Fractals 33 (2007), 1601-1609. [9] G. Duvaut and J. L. Lions: Inequalities in Mechanics and Physics, Springer, Berlin 1976. [10] J. P. Dauer and N. 1. Mahmudov: Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal Applications 273 (2002), 310-327. [11] V. N. Do: A note on approximate controllability of semilinear systems, Systems and Control Letters 12 (1989), 365-371. [12] L. G6rniewicz, S. K. Ntouyas and D. O'Regan: Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Rep. Math. Phys. 56 (2005), 437-470. [13] C. Gao, K. Li, E. Feng and Z. Xiu: Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos, Solitons & Fractals 28 (2006), 271-277. [14] Y. Hino, S. Murakami and T. Naito: Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Springer, Berlin, Vol. 1473 (1991). [15] J. M. Jeong and H, H. Roh: Approximate controllability for semilinear retarded systems, J. Math. Anal. Applications 321 (2006), 961-975. [16] J. Klamka: Constrained approximate controllability, IEEE Transactions on Automatic Control 45 (2000), 1745-1749. [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 and E Q. Zhang: Controllability of impulsive functional differential systems in Banach spaces, Chaos, Solitons & Fractals 29 (2006), 175-181. [19] 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. [20] N. I. Mahmudov: Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Analysis, to appear. [21] A. M. Samoilenko and N. A. Perestyuk: Impulsive Differential Equations, World Scientific, Singapore 1995. [22] S. Tang and L. Chen: Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biology 44 (2002), 185-199. [23] L. Wang: Approximate controllability and approximate null controllability of semilinear systems, Commun. Pure and Applied Analysis 5 (2006), 953-962. [24] T. Yang: Impulsive Systems and Control: Theory and Applications, Springer, Berlin 2001.
REPORTS ON MATHEMATICAL PHYSICS
No. 1
APPROXIMATE C O N T R O L L A B I L I T Y OF NONLINEAR I M P U L S I V E DIFFERENTIAL SYSTEMS R. SAKTHIVEL*, N. I. MAHMUDOV** and J. H. KIM* Department of Mathematics, Yonsei University, Seoul 120-749, South Korea Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Turkey (e-mails: [email protected], [email protected], [email protected]) (Received January 15, 2007)
Many practical systems in physical and biological sciences have impulsive dynamical behaviours during the evolution process which can be modeled by impulsive differential equations. This paper studies the approximate controllability issue for nonlinear impulsive differential and neutral functional differential equations in Hilbert spaces. Based on the semigroup theory and fixed point approach, sufficient conditions for approximate controllability of impulsive differential and neutral functional differential equations are established. Finally, two examples are presented to illustrate the utility of the proposed result. The results improve some recent results. Keywords: approximate controllability, impulsive differential equations, fixed point theorem.
1.
Introduction
Many systems in physics and biology exhibit impulsive dynamical behaviour due to sudden jumps at certain instants in the evolution process. Differential equations involving impulse effects occur in many applications: pharmacokinetics, the radiation of electromagnetic waves, population dynamics, biological systems, the abrupt increase of glycerol in fed-batch culture, bio-technology etc., [1, 5, 13, 21, 22]. For the basic theory of impulsive differential equations the reader can refer to Samoilenko and Perestyuk [21]. On the other hand, it is well known that control problems of deterministic equation are widely used in many branches of science and engineering. Moreover, the interest in impulsive control systems which is based on the theory of impulsive differential equations has grown in recent years due to its theoretical and practical significance. In [24] some recent developments on the impulsive control theory and its applications to nanoelectronics are discussed. In recent years, controllability problems for different kinds of dynamical systems have been considered in many publications [2-4, 6, 12]. The controllability of impulsive systems in Banach spaces is also very interesting and researchers are engaged in it. To our knowledge, there are few papers discussing controllability of impulsive differential equations on infinite dimensional Banach spaces. We refer to the paper of Benchohra et al. [2] who studied the controllability of impulsive differential inclusions by using a fixed point theorem for the condens[85]
86
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
ing maps due to Martelli. Li et al. [18] established sufficient conditions for the exact controllability of impulsive functional differential systems using Schaefer's fixed point theorem. Very recently, Chang [8] studied the exact controllability of impulsive functional systems with infinite delay. However, in order to establish these results, the invertibility of a controllability operator is imposed. It turns out that in practice it is rather difficult to verify this condition directly, see [7], and it fails in infinite-dimensional spaces when the semigroup generated by A is compact. Therefore, it is important, in fact, necessary to study the weaker concept of controllability, namely approximate controllability for impulsive differential systems. Klamka [16] investigated constrained approximate controllability problems for linear abstract dynamical systems with linear unbounded control operator and piecewise polynomial controls. Jeong and Roh [15] studied the approximate controllability for the semilinear retarded control system. The approximate controllability and approximate null controllability of control systems governed by a class of abstract semilinear integro-differential equations are discussed in [23]. Approximate controllability for semilinear control systems can be found in Mahmudov [19,20], Dauer and Mahmudov [10], and many other papers [11,23]. However, up to now approximate controllability problems for nonlinear dynamical systems with impulses have not been considered in the literature. In order to fill this gap, this paper studies the approximate controllability problem for nonlinear systems described by differential equation with impulses and control operator. In Section 3, we study the approximate controllability of the following impulsive control differential equation x'(t) = A x ( t ) + ( B u ) ( t ) + f ( t , x(t)),
t E J = [0, b] \ D,
(1)
x (0) = xo,
Ax(tk) = Ik(X(tk)),
k = 1 . . . . . m,
where A is the infinitesimal generator of a strongly continuous semigroup T ( t ) in a Hilbert space X, B is a linear bounded operator from a Hilbert space U into X, f : J × X ~ X is a nonlinear operator, the control u(.) 6 L2(J, U). Here D = {h,t2 . . . . . tm} C J, 0 = to < tl < "'" < t m < tm+l = b, Ik(k = 1,2 . . . . . m) is a nonlinear map and A x ( t k ) = x ( t ~ ) ) - x ( t ~ ) = x ( t +) - X ( t k ) . This represents the jump in the state x at time tk with Ik determining the size of the jump. It should be noted that if X is infinite dimensional, the semigroup is compact and B is bounded, then the infinite-dimensional linear control system x'(t) = A x ( t ) + ( B u ) ( t ) ,
t ~ [0, b],
(2)
x (0) = Xo,
is not exactly controllable [7]. In view of the above result the exact controllability results of impulsive neutral systems [17] hold only in finite-dimensional spaces. With the proceeding reason in this article, we study approximate controllability of neutral functional differential equations with impulses described in the form
APPROXIMATE CONTROLLABILITYOF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS 87 d ~-7[x(t) + g(t, xt))] = A[x(t) + g(t, xt)] + (Bu)(t) + F(t, xt),
x(t) =c/)(t),
t E J,
t 7~ tk
(3)
t ~ [ - h , 0],
x(t~) - x(t~) = [k(X(tk)),
k = 1 . . . . . m,
where A generates a strongly continuous semigroup T(t) on the Hilbert space X. The functions g, F, [k and ~b are given continuous functions to be specified later.
2.
Preliminaries
Let Xb(Xo; u) be the state value of (1) at terminal time b corresponding to the control u and the initial value Xo. Introduce the set
~t(b, xo) = {Xb(XO; U)(0) : u(-) ~ L2(J, U)}, which is called the reachable set of system (1) at terminal time b, its closure in X is denoted by ~ ( b , x0). DEFINITION 2.1. The system (1) is said to be approximately controllable on the interval J if 91(b, x 0 ) = X. It is convenient at this point to define operators
V~ =
T(b - s ) B B * T * ( b - s)ds, R(oe, Pob) = ( a l + P0b)-1.
($1) otR(c~, I'0b) ~ 0 as ~ --+ 0 + in the strong operator topology. The assumption ($1) holds if and only if the linear system (2) is approximately controllable on J, see [6]. Now for convenience, let us introduce the notation M r = max{liT(Oil • 0 < t < b}, k = max{l, M r M ~ , M r M s b } ,
ai -=
M s = IIBII,
3kMZrMs[[~.i[l,
bi =
I1~.i II = fo b IPvi(s)lds,
3Mrl[)~il[, ci m
d l = 3kMrMB(llxbl[ + Mrllxoll + M r ~ _ d k ) , k=l m
d2 = 3 M r Ilx0ll + M r ~
dk,
k=l
We introduce the following assumptions: (I) The semigroup T(t), t > 0, is compact.
d = max{dl, d2}.
= max{ai, bi},
88
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
(II) The function f • J x X ~ X is continuous and there exist functions ) - i ( ' ) E L I ( J , R +) and h i ( ' ) E LI(X, R+), i = 1, 2 . . . . . q such that q
Ilf(t,x)ll
<
Z)~i(t)hi(x)
for all
( t , x ) ~ J x X.
i=1
(III) For each ot > 0,
(
~-~ --Ci sup{hi(x)
lira sup r r~oo-
" IIxll _< r
})
-- ~ .
i=1 O/
(IV) Ik ~ C ( X , X ) and there exists a constant dk such that IIIk(x)ll _< dk for each x E X ( k = 1 . . . . . . m). (V) The function f : J x X --~ X is continuous and uniformly bounded and there exists N > 0 such that Ilf(t,x)l[ < N for all ( t , x ) ~ J × X. In Section 3.1, it will be shown that the system (1) is approximately controllable if for all ot > 0 there exists a continuous function x(.) ~ C ( J , X ) such that u(t) = B * T * ( b - t)R(ot, Fg)p(x(-)),
X(t)
T(t)xo +
fo'
T(t - s)[Bu(s) + f(s, x(s)]ds +
(4) y~
O
T ( t - tk)Ik(x(tk)),
(5)
where p ( x ( . ) ) = xb - T ( b ) x o -
fo
m
T ( b - s ) f ( s , x ( s ) ) d s - ~_, T ( b - tk)Ik(x(tk)). k=l
3. Approximate controllability of impulsive systems 3.1. Impulsive differential systems THEOREM 3.1. Assume that conditions (I)-(IV) are satisfied, then for all 0 < ot < 1 the system (1) has a solution on J. Proof: The main aim in this section is to find conditions for solvability of system (4) and (5) for c~ > 0. In the Banach space C ( J , X ) consider a set Yr
=
{X(.) E C ( J , X ) [ x(O) = xo, [Ixll _< r},
where r is the positive constant. For a > 0, define the operator F~ on C ( J , X ) as F~(x) = z,
where
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS
v(t) = B*T*(b - t)R(c~, z(t) = T(t)xo +
ro~)p(x(.)),
fo T(t - s)[Bv(s)
p(x(.)) = Xb -- r(b)xo -
89
fo
+ f ( s , x(s))]ds + ~_. T(t - tk)Ik(x(tD), O
m T(b
T(b - s ) f ( s , x(s))ds - ~
- tk)Ik(X(tk)).
k=l
It will be shown that for all c~ > 0 the operator F~ from C(J, X) into itself has a fixed point. Step 1. For an arbitrary oe > 0 there is a positive constant r0 = r(o0 such that
Fa : rro----~ Yro. Let (i(r)= sup{hi(x):llxl[ < r,x • X}. By assumption (iii), there exists ro > 0 such that
d £ - + O/
ci --(i(ro) <_ ro.
i = 1 O/
If x(-) • Yro, then we obtain Hv(t)[[ _< 1MrMB(llXbl[ + MrllXoll + Mr k
fb Jo
q
m
i=1
k=l q
1 ( ~_~) < - M T M 8 Ilxbll + MTllXoll + Mr dk + 1 M r M g ~__, ll)~ill(i(ro) 13/
Ol
k=l
d
1 q~ ci
< - + - 3kol ~
i=l
ro
2.a --(i(ro) i=1 u
< -- 3k
and
d
Ilz(t)ll = g + MrMBbllvH + Mr
fot£ )~i(s)hi(x(s))ds i=1
<_ -~ + kllvl[ + -~ ~_(ci(i(ro) <_ '=
Finally we get
d+
cir.i(ro) i=1
+el[vii
< --
-3
II(F~x)(t)l[ = I[z(t)ll _< r0.
Thus F~ maps Yr0 into itself. Step 2. For each 0 < oe _< 1, the operator F~ maps Yro into a relatively compact subset of Yro" According to infinite-dimensional version of the Ascoli-Arzela theorem and Step 1 we need to prove that: (i) for an arbitrary t • J the set g(t) = {(F~x)(t) • x(.) • Yr0} is relatively compact,
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
90
(ii) for an arbitrary e > 0, there exist 6 > 0 such that I I ( F ~ x ) ( t l ) - ( F ~ x ) ( h ) l l < if Ilxll _< r, Itl - tzl < 3, tl, ta ~ J In the case t = 0 it is trivial, since V(0) = {x0}. So let t be a fixed real number, and let r be a given real number satisfying 0 < r < t. Define
( F d x ) ( t ) = T ( r ) z ( t - r). Since T ( t ) is compact and z ( t -
r) is bounded on Yro, the set
Vr(t) = { ( F d x ) ( t ) :x(-) C Yro} is a relatively compact set in X. That is, a finite set {Yi, 1 < i < n} in X exists .m. such that V~(t) C [,..J N ( y i , e/2),
i=1 where N(yi, ~/2) is an open ball in X with center at Yi and radius ~/2. On the other hand,
L
II(f~x)(t) - (F~x)(t)ll =
T(t - s)[Bv(s) + f(s, x(s))]ds
77
<_ - M r M B
O/
Xi(s)(i(ro)ds
\Ilxoll + Mrllx011
i=1
+ Zm T ( b - tk)Ik(x(tk)) ) r + M r ~ f t ~
k=l Consequently
? i(s)ds(i(ro) <_ 2"~5
i=1 n. V ( t ) C [ J N ( y i , e).
i=l Hence for each t 6 [0, b], V ( t ) is relatively compact in X. To prove (ii), we have to show that V = {(Fax)(.) Ix(.) ~ Yro} is equicontinuous on [0, b]. For 0 < h < t2 _< b, we have Ilz(q) - z(t2)ll _< liT(t0 - T(t2)llllxoll + M T M B
-4- M r +
fo
y~
ft,
IIv(s)llds
liT(t2 - s) - T ( h - s)ll IIv(s)llds
liT(t2 - tk) - T(tl - t~)llllIk(x(tk))ll
O
+
~
l i T ( t 2 - tDIIIIIk(x(tk))ll + Mr ftl t2 Zq) ~ i ( s ) h i ( x ( s ) ) d s
t 1
i= 1
+ MT fo tl liT(t2 - s) - T(tl - s)]] q X i ( s ) h i ( x ( s ) ) d s
i=1
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS 91
_< liT(t1) - T(t2)llllxoll + M T M B +
IIv(s)llds
~ liT(t2 - tk) - Z(tl - t~)lldk + y~, liT(t2 - tk)lldk O
fo"
[[r(tl - s) - T(tz - s)ll llw(s)llds
+ Mr
+ M r ~_, q ftl t2 ~,i(s)ds(i(ro) i=1
+ ~q f0 tl [[T(tl -- s) -- T(t2 - s)ll),i(s)ds(i(ro).
(6)
i=1
Thus the right-hand side of (6) does not depend on particular choices of x(-) and tends to zero as t t - t 2 ~ 0, since the compactness of T ( t ) for t > 0 implies the continuity in the uniform operator topology. So we obtain the equicontinuity of V. We considered here only the case 0 < tl < t2, since other cases h < t2 < 0 or tl < 0 < t2 are very simple. Thus F~[Yro] is equicontinuous and also bounded. By the Ascoli-Arzela theorem F~[Yro] is relatively compact in C ( J , X). It is easy to show that for all o~ > 0, F,~ is continuous on C ( J , X). Hence from Schauder's fixed point theorem F~ has a fixed point. Thus the problem (1) has a solution on J. [] THEOREM 3.2. A s s u m e that linear system (2) is approximately controllable on J. If the conditions (I), (III)-(V) are satisfied then the system (1) is approximately controllable. Proof: Let ~ ( . ) be a fixed point of F~ in Yro" Any fixed point of F~ is a mild solution of (1) under the control fi~(t) = B * T * ( b - t)R(ot,
and satisfies the inequality 2c~(b) = Xb + otR(u,
By the condition (V), fo b I I f 2~(s))ll2ds ( s , _
N2b
and consequently the sequence { f ( s , Yc~(s))} is-bounded in L2(J, X). Then there is a subsequence denoted by {f(s,;c~(s))}, that weakly converges to, say, f ( s ) in L2(J, X). Now, thanks to the compactness of an operator l(.) --+ fo T(. - s ) l ( s ) d s : L2(J, X ) --+ C ( J , X ) we obtain that IIP(:~) - wll =
fo b T ( b - s ) [ l l f ( s , ;c=(s)) - f (s)]ds
< sup II 0
f0'
T ( b - s ) [ f ( s , Yc~(s)) - f ( s ) l d s l l ~
0
92
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
as ot ~ 0 +, where
w = T(b)xo + fo b T(b - s ) f ( s ) d s - Xo. Then from
II~(b)
<_ II~R(~, rb)(w)ll + II~R(~, rb)ll IIP(2~) -- wll
-xbll
_< II~R(~, r~)(w)ll + I I p ( ~ ) - wll -~ o as c~ ~ 0 +. This proves the approximate controllability of (1).
[]
3.2. Neutral impulsive systems: A natural generalization of impulsive ordinary differential equations are the impulsive neutral functional differential equations. Impulsive neutral functional differential equations describe models of real processes and phenomena where both dependence on the past and momentary disturbances are observed. In recent years, the interest in impulsive neutral systems has been growing rapidly due to their successful applications in practical fields such as circuit theory, bioengineering, chemical technology and so on. In this section, it will be proved that the system (3) is approximately controllable if for all ot > 0 there exists a continuous function x(.) ~ C([-r, b], X) such that
Ua(t)
B*T*(b -
x(t) =
jo
t ) R (ot, r0b)[Xb -- T(b)[q~(0) + g(0, ~b)] + g(b, Xb)
n
r(b
s)f(s, xs)ds -
r(b
-
tk)ik(x(t[))j,]
k=l
r(t)[q~(o)
+ Z
-
+ g ( O , 4,)1
- g(t, xt) +
T(t - s)[Bu(s) + F(s, x~)]ds
T(t-tk)[~(x(t~)),
O
where [k : X ~ X, 0 = to < tl < . . . < tm < tm+l = b, x(t~) and x(t~) represent the right and left limits of x(t) at t = tk, respectively. Here C = C ( [ - h , 0], X) is the Banach space of all continuous functions cp : [ - h , 0] ~ X endowed with the norm II~b[I= sup{kb(0)l : - h < 0 < 0}. Also for any continuous function x defined on [ - h , b ] - {tl . . . . . tm} and t E J, we have xt E C for t ~ [0, b],xt(0) = x(t + 0 ) for 0 6 [ - h , 0]. For each k = 0, 1. . . . . m let Jk = [tk, tk+l]. Here C(J~, X) is the Banach space of all continuous functions from J~ into X with the norm I[xllj~ = sup{lx(t)l : t ~ Jk}. We consider the space C ( [ - h , b], X) = {x : [ - h , b ] -+ X :xk c C(Jk, X), k = 0 . . . . . m and there exist x(t +) and x(t~) with x(tk) = x(&), k = 1 . . . . . m} which
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS
93
is the Banach space with the norm
Ilxllc(t-h,b],S) = max{llx~llJ~, k = 0 . . . . . m}, where x~ is the restriction of x to Jk, k = 0 . . . . m. Concerning the operators F and g, assume the following hypothesis: (i) There exist functions ~-i(') 6 LI(J, R +) and h i ( ' ) C L I ( C , R + ) , . . . . q, such that
i = 1, 2,
q
IlF(t,~P)ll
_<
Z)~i(t)]~i(~9)
for all
(t,q~) 6 J x C.
i=1
(ii) For each ot > 0 lim sup
r
--
r-+~\
-- sup{/~i(cp) i=1 19l
"
114~11_< r}
)
= c~.
(iii) The function g : J × C ~ X is completely continuous and uniformly bounded, there exists a constant M3 > 0 such that IIg(t, 4~)11 _< M3, (iv)
t c J,
~b E C.
[k ~ C(X, X) and there exists a constant dh such that Illk(x)ll _< dh for each x E X ( k = l . . . . . m).
(v) The function F : J × C ~ X is continuous and uniformly bounded and there exists N1 > 0 such that [IF(t, q~)ll -< N1 for all (t, ~p) E J x C. THEOREM 3.3. Assume that linear system (2) is approximately controllable on [0, b]. If the semigroup T(t) is compact and conditions (i)-(v) are satisfied, then system (3) is approximately controllable.
Proof: For ot > 0, define the nonlinear operator F~ on C([-h, b], X) as F~(x) : z, where Ua(t) = B*T*(b z(t)
- t)R(ot, rb)p(x(.)),
g(t, xt) + f t T(t Jo T(t--tk)[k(x(t~)),
-----T(t)[4~(O) + g(O, ~b)] -
+ Z
- s ) [ B u a ( S ) q-
F(s, x,)]ds
O
zo(O)=dp(O),
0 c [ - h , 0],
p(x(.)) = Xb -- T(b)[q~(0) + g(0, ~b)] + g(b, Xb) -- fo b T(b - s)F(s, xs)ds -
-
~ k=l
T(b - tk)[k(X(t~)).
94
R. S A K T H I V E L , N. I. M A H M U D O V and J. H. K I M
One can easily prove that if for all ot > 0, the operator F~ has a fixed point by employing the method used in the previous section, then we can show that the system (3) is approximately controllable by adopting the technique used in Theorem 3.2. [] REMARK 3.1. The problems in physics, especially in solid mechanics, where one has nonmonotone and multi-valued constitutive laws lead to differential inclusions [9]. The above result can be extended to study the controllability of nonlinear neutral impulsive differential inclusions by suitably introducing the multivalued map defined in [2, 12]. .
Application
EXAMPLE 4.1. Consider a control system governed by the heat equation with impulses: O
82
~-~Z(t, y) = -~v2z(t, y) + Bu(t, y) + f ( t , z(t, y)), Az(tk, y) = --z(tk, y), z(t, O) = z(t, Jr) = O, z(O, y) : zo(y),
y E (0, 1),
0 < y < rr,
(7)
k = 1. . . . . m,
t E J,
y E [0, Jr].
Let X = L2[0, Jr] and let A : X --+ X be an operator defined by Az = z" with domain D(A) = {w ~ X : w and w f are absolutely continuous, w" 6 X, w(0) = w(rr) = 0}. Then oo I
Aw = - - ~ n 2 ( W , en)en,
W ~ D(A),
n=l
where e.(y) = (2/Tr)l/Zsiny, 0 _< y < Jr, n = 1,2 . . . . . . Let Z0 E L2[0, rr], Ik(x(tk)) =--z(tk, y), and f be Lipschitz continuous and satisfies linear growth conditions. It is well known that A generates compact semigroup T(t) in X and is given by oo
T(t)w = Z
e-
n2
(W, en)en,
WE X.
n=l
Define an infinite-dimensional space fx)
(3o
U={u'u=Zunenl~U2
n=2
with the norm defined by Ilullu = (Y~,nC~=2Un )2 1/2. Define a mapping B from U to X as follows: oo
Bu = 2u2el -q- ~ n=2
Unen.
}
APPROXIMATE CONTROLLABILITY OF NONLINEAR IMPULSIVE DIFFERENTIAL SYSTEMS 95
Because off compactness of the semigroup T(t) generated by A, the associated linear system is not exactly controllable but it is approximately controllable [7]. Therefore, system (7) can be written in the abstract form (1). By Theorem 3.2, system (7) is approximately controllable on [0, b]. EXAMPLE 4.2. We consider the following controlled neutral differential equation with impulses: 0 02 -~[z(t, y) - I~l(t, z(t - h, y))] = -~v2[z(t, y) - #l(t, z(t - h, y))] +/z(t,y)+/z2(t,z(t-h,y)),
z(t +, y) - z(t~-, y) = [k(Z(tk-, y)), z(t, O) = z(t, 1) = 0, z(t, y) =d~(t, y),
0
< 1, (8)
k = 1 . . . . . m, t > 0, - h < t < O,
where 4> is continuous and [k ~ C(R, R). Let g(t, wt)(y) =/Zl(t, w ( t - y)), F(t, wt)(y) = lz2(t, w ( t - y)) and (Bu)(t)(y) = /z(t, y), y E (0, 1). Take X = L2[0, 1] and define A : X ~ X by
Aw--
d2w dy 2
with domain
D(A) =
dw w c X " w and d~- are absolutely continuous,
dZw
dw dw I --dy2 c X, -~--y(0) = -d-~--y(1) = 0 ]. The operator A has the eigenvalues ~., = - n 2 n -2, n >_ 0, and the corresponding eigenvectors en(y) = 2 (1/2) cos(nzry) for n >_ 1, eo = 1, from an orthonormal basis for L2(0, 1). It is well known that A generates a compact semigroup T(t), t > 0, in X and is given by
T ( t ) w = fo 1 w ( v ) d v + End=12e-n2jr2t cos(nTrY) fo' cos(nTry)w(v)dv,
w c X.
Further, the functions #1,/~2 " [0, 1] >( [0, 1] ~ [0, 1] are continuous and there exist constants kl,k2 such that I[#a(t, w(t --Y))[I -< f:l and [[/Zz(t, w(t - y))[[
96
R. SAKTHIVEL, N. I. MAHMUDOV and J. H. KIM
Acknowledgements The work of RS and JHK was supported by Brain Korea 21 project at Yousei University, 2006. REFERENCES [I] D.D. Bainov, S. I. Kostadinov and A. D. Myshkis: Asymptotic equivalence of abstract impulsive differential equations, Int. J. Theoret. Phys. 35 (1996), 383-393. [2] M. Benchohra, L. G6rniewicz, S. K. Ntouyas and A. Ouahab: Controllability results for impulsive functional differential inclusions, Rep. Math. Phys. 54 (2004), 211-228. [3] M. Benchohra and A. Ouahab: Controllability results for functional semilinear differential inclusions in Frechet spaces, Nonlinear Analysis 61 (2005), 405-423. [4] M. Benchohra, L. G6rniewicz, S. K. Ntouyas and A. Ouahab: Controllability results for nondensely defined semilinear functional differential equations, Zeitschrift fiir Analysis und ihre Anwendungen 25 (2006), 311-325. [5] G. Ballinger and X. Liu: Boundness for impulsive delay differential equations and applications to population growth models, Nonlinear Analysis 53 (2003) 1041-1062. [6] A. E. Bashirov and N. I. Mahmudov: On concepts of controllability for deterministic and stochastic systems, SlAM Journal on Control and Optimization 37 (1999), 1808-1821. [7] R. Curtain and H. J. Zwart: An Introduction to Infinite Dimensional Linear Systems Theory, Springer, New York 1995. [8] Y. K. Chang: Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons & Fractals 33 (2007), 1601-1609. [9] G. Duvaut and J. L. Lions: Inequalities in Mechanics and Physics, Springer, Berlin 1976. [10] J. P. Dauer and N. 1. Mahmudov: Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal Applications 273 (2002), 310-327. [11] V. N. Do: A note on approximate controllability of semilinear systems, Systems and Control Letters 12 (1989), 365-371. [12] L. G6rniewicz, S. K. Ntouyas and D. O'Regan: Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Rep. Math. Phys. 56 (2005), 437-470. [13] C. Gao, K. Li, E. Feng and Z. Xiu: Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos, Solitons & Fractals 28 (2006), 271-277. [14] Y. Hino, S. Murakami and T. Naito: Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Springer, Berlin, Vol. 1473 (1991). [15] J. M. Jeong and H, H. Roh: Approximate controllability for semilinear retarded systems, J. Math. Anal. Applications 321 (2006), 961-975. [16] J. Klamka: Constrained approximate controllability, IEEE Transactions on Automatic Control 45 (2000), 1745-1749. [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 and E Q. Zhang: Controllability of impulsive functional differential systems in Banach spaces, Chaos, Solitons & Fractals 29 (2006), 175-181. [19] 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. [20] N. I. Mahmudov: Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Analysis, to appear. [21] A. M. Samoilenko and N. A. Perestyuk: Impulsive Differential Equations, World Scientific, Singapore 1995. [22] S. Tang and L. Chen: Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biology 44 (2002), 185-199. [23] L. Wang: Approximate controllability and approximate null controllability of semilinear systems, Commun. Pure and Applied Analysis 5 (2006), 953-962. [24] T. Yang: Impulsive Systems and Control: Theory and Applications, Springer, Berlin 2001.