Journal
of Applied Mathematics
and Stochastic Analysis, 12:3
(1999), 265-277.
CONTROLLABILITY OF SECOND ORDER
SEMILINEAR ORDINARY DIFFERENTIAL
SYSTEMS IN BANACH SPACES
K. BALACHANDRAN and S. MARSHAL ANTHONI
Bharathiar University
Department of Mathematics
Coimbatore 61 05, India
(Received January, 1998; Revised September, 1998)
Sufficient conditions for controllability of semilinear second order ordinary
differential systems in Banach spaces are established using the theory of
strongly continuous cosine families. The results obtained are based on the
Schaefer theorem.
Key words: Controllability, Semilinear Ordinary Differential System,
The Schaefer Theorem.
AMS subject classifications: 93B05.
1. Introduction
Controllability of linear and nonlinear systems represented by ordinary differential
equations in finite dimensional spaces has been extensively studied. Several authors
extended the concept to infinite dimensional systems in Banach spaces with bounded
operators. Chukwu and Lenhart [3] studied controllability of nonlinear systems in abstract spaces. Naito [7, 8] studied controllability of semilinear systems and nonlinear
Volterra integrodifferential systems. Quinn and Carmichael [11] showed that a controllability problem in Banach spaces can be converted into a fixed-point problem for
a single-valued mapping. Balachandran et al. [1] established sufficient conditions for
controllability of nonlinear integrodifferential systems in Banach spaces.
In many cases it is advantageous to treat second order abstract differential equaFor example,
tions directly rather than convert them to first order systems.
for
establishing
abstract
differential
used
second
order
the
equations
Fitzgibbon [4]
of an
motion
the
transverse
of
of
solutions
an
the boundedness
equation governing
is
second
order
abstract
of
equations the
extensible beam. A useful tool for the study
of the basic
of
some
make
use
theory of strongly continuous cosine families. We will
order
second
systems can
ideas from cosine family theory [14, 15]. A motivation for
order
second
of
be found in [5, 6]. Recently, Park et al. [10] discussed controllability
fixed
the
point theoSchauder
nonlinear systems in Banach spaces with the help of
order
second
semilinear
of
rem. The purpose of this paper is to study controllability
theorem.
differential systems in Banach spaces by using the Schaefer
Printed in the U.S.A.
(C)1999 by North
Atlantic Science Publishing Company
265
K. BALACHANDRAN and S. MARSHAL ANTHONI
266
2. Prehminaries
We consider the semilinear second order control system
Ax(t) + f(t, x(t), x’(t)) + Bu(t),
x"(t)
t
J
[0, T],
(1)
where the state x(. takes values in a Banach space X, x0, Y0 E X, A is the infinitesimal generator of the strongly continuous cosine family C(t)(t E ), of bounded
linear operators in X, f is a nonlinear mapping from J x X x X to X, B is a bounded
linear operator from V to X and the control function u(.) is given in L2(J,U), a
Banach space of admissible control functions, with U also being a Banach space.
Definition 1: [14] A one parameter family C(t), tEN, of bounded linear
operators in the Banach space X is called a strongly continuous cosine family iff
(i)
C(s + t) + C(s t) 2C(s)C(t) for all s, t E
;
(ii)
(iii)
C(O) I;
C(t)x is continuous in
Define the associated sine family
s(t)
,
t on N for each fixed x E X.
S(t), t E
by
f
x,
t
e
0
Assume the following conditions on A.
(HI) A is the infinitesimal generator of
,
a strongly continuous cosine family
of bounded linear operators from X into itself and the adjoint
operator A* is densely defined i.e., D(A*)= X* (see [2]).
The infinitesimal generator of a strongly continuous cosine family C(t), t E
is
the operator A: X--X defined by
C(t), t E
,
-d-yC(t)x
xE
D(A),
o
where
D(A)- {x E X’C(t)x is twice continuously differentiable in t).
Define
E- {x E X: C(t)x is
once continuously differentiable in
t}.
To establish our main theorem we need the following lemmas.
Lemma 1: [14] Let (H1) hold. Then,
there exist constants M > 1 and co >_ 0 such that
(i)
t*
I c(t) I
Ne
and
[I S(t) co(t*)II < N
for t,t* E ;
Controllability
of Second Order oCemilinear
Ordinary
Differential Systems
S(t)X a E and S(t)E a D(A) for t ;
AS(t)x for x E and G ;
C(t)x
d2
-t2C(t)x AC(t)x for x E D(A) and t
Lemma 2: [14] Let (H1) hold, let v’-+X be such that
ff,a6 a,
q() f ’oS(, )()d. Tn,
(ii)
(iii)
(iv)
267
.
q is twice continuously
q’(t)-
,
differentiable and for t
/ C(t-s)v(s)ds,
and q"(t)-
0
v is continuously
D(A),
q(t)
/ C(t-s)v’(s)ds + C(t)v(O)
0
=Aq(t)+v(t).
Lemma 3: (Schaefer Theorem [13]) Let S be a convex subset of a normed linear
space Y and assume 0 S. Let F’S-S be a completely continuous operator, and let
(F)
AFx for
{x S:x
some 0
< A < 1}.
Then either (F) is unbounded or F has a fixed point.
We make the following assumptions"
(H2) f(t,., .)’XXX is continuous for each t E J and the function
f(.,x,y)’JX is strongly measurable (see [12, p. 116]) for each x,y X.
(H3) For every positive constant k there exists c k G LI(j) such that
sup
I111, I111 <_
(H4)
I f(t,x, y)II
There exists a continuous function
m:J[0, oe)
rn(t)a( I
I f(t, u, v)II
_< k(t) for almost
all t G J.
such that
I + I v I ), t
J, u, v e X,
where a" (0, cxz)--,(O, oe)is a continuous nondecreasing function and
T
M(T +
1)I m(s)ds < i ftd(;)
c
0
with
c
(M + M*)II 0 I
/
(1 + T)M I y0 I
/
(1 + T)M IIB I T I w- ill
T
[11 Xl [I + M I[ Xo
I
+ M I[ Yo
I
+ MT
/ m(s)f( I x()II + I x’()II
0
sup{ l] C(t)II" e J}, and M*
(H5) Bu(t)is continuous.
M
(H6)
sup{ I AS(t)II"
The linear operator W" L2(J,U)X defined by
J}.
K. BALACHANDRAN and S. MARSHAL ANTHONI
268
T
/ S(T- s)Bu(s)ds
Wu
0
has a bounded invertible operator W 1. XL2(j, U)/kerW.
(HT) C(t), t > 0 is compact.
Then the system (1) has a mild solution of the form (see [9])
C(t)x o + S(t)y o +
x(t)
/ S(t- s)f(s, x(s), x’(s))ds
0
+
/ S(t-s)Bu(s)ds.
(2)
0
Definition 2:
D(A), Yo E
solution x(. of (1)
xo E
The system (1) is said to be controllable on J if for every
and x I X there exists a control u-L2(j,U)such that the
satisfies x(T) x 1.
3. Main Result
Theorem: Suppose (H1)-(HT) hold. Then system (1) is controllable on J.
Proof: Using (H6) for an arbitrary function x(. we define the control
T
tt(t)
W
l[x 1 C(T)x o S(T)y o
S(T s)f(s, x(s), x’(s))ds](t).
0
Using this control we will show that the operator defined by
C(t)x o + S(t)y o +
(Fx)(t)
/ S(t- s)f(s, x(s), x’(s))ds
0
+
]" S(t
s)BW l[x
C(T)x o S(T)y o
0
T
S(T O)f(O, x(O), x’(O))dO](s)ds, rE J,
0
has a fixed point. This fixed point is then a solution of Equation (2).
Clearly, (Fx)(T) Xl, which means that the control u steers the system from the
initial state x 0 to x in time T, provided we obtain a fixed point of the nonlinear
operator F. Consider the space Z CI(J, X) with norm
In order to study the controllability problem for the system (1)
we apply
Lemma 3 to
Controllability
of Second Order Sernilinear
Ordinary
Differential Systems 269
the following system
x"(t)
Let x be
AAx(t) + Af(t,x(t),x’(t)) + Bu(t),
a mild solution of system
x(t)
(3).
J,
(0, 1).
Then from
A(C(t)x o + S(t)yo) + A
f
S(t- s)f(s, x(s), x’(s))ds
0
} (t
+
8)BW l[x 1 C(T)x 0 S(T)y o
0
T
/ S(T
O)f(O, x(O), x’(O))dO](s)ds
0
we h&ve
I x(t) I <- M I Xo I + MT I Yo I] + MT
f
rn(s)( [I x(s) ]] + I[ x’(s) I )ds
0
+ MT2 I B I W-1 I
I + M I Xo [[ + MT I[ Yo [[
x
T
+
MT} m(s)( I x(s)II + ]1 x’(s)]1 )ds]
+ } rn(s)( I x(s) I + I x’(s) I] )ds.
0
=_
K1
MT
-
0
Denoting by
v(t) the right-hand side of the above inequality we have
v(O)
tt’l,
I X(t) I
and
v’(t) + MTrn(t)a( I (t)II +
I
’(t)II ), t e J.
On the other hand,
x’(t)
[AS(t)x o + C(t)yo] +
/ C(t
s)f(s, x(s), x’(s))ds
0
2r- /
/ C(t
8)BW l[x
C(T)x 0 S(T)y o
0
T
} S(T
0
O)f(O,x(O),x’(O))dO](s)ds.
(3)
K. BALACHANDRAN and S. MARSHAL ANTHONI
270
Thus we have
I x’(t)II -< M* I Xo I + M I Yo I] +
M/m(s)f( I x(s)II + I x’(s)II )ds
0
+ MT I] B ]1 ]1 W- 111[]
x1
I + M I Xo I + MT ]l Yo I
T
/ m(s)f( I x(s)]1 + I x’(s)II )ds]
+ M/ m(sDa( I x(s)II + I x’(s)II )ds.
+ MT
0
K2
0
Denoting by
r(t)
the right-hand side of the above inequality we have
I ,’(t)II
K,
,-(o)
_< r(t)
and
r’(t)
Mm(t)f( ]l x(t) I + I x’(t) I ),
t E J.
Let
w(t)
v(t) + (t),
t
e J.
Then
(o)
v(O)+ r(o)
c,
and
w’(t)-v’(t)+r’(t)
<_ MTm(t)a(w(t)) + Mm(t)a(w(t))
M(T + 1)m(t)a(w(t)),
J.
This implies
w(t)
Z
f(s) <- M(T + 1)
(o)
m(s)ds <
0
c
a(s)’
The last inequality implies that there is a constant K such that
(t)- (t)+ r(t)</’,
Then
e J.
J.
Controllability
of Second Order Semilinear
I1 (t)II _< (t),
Ordinary
Differential Systems
271
,,
t
and hence
I x [[*
’
max{ I a II,
I I}
K,
where K depends only on T and on the functions m and ft.
We shall now prove that the operator F" Z-Z defined by
-
(r)(t)
c(t)o + s(t)vo +
/ s(t- )I(, (), x’())
0
/ S(t
s)BW l[x
C(T)x o S(T)y o
0
T
/ S(T
O)f(O, x(O), x’(O))dO](s)ds, rE J,
0
is a completely continuous operator.
_
Let B k-{xZ, Ilxll*_<k} for some k>_l. We first show that F maps B k
into an equicontinuous family. Let x B k and tl, t 2 J. Then if 0 < t I < 2 T,
IIC(tl)-C(ta)l[ Ilxoll + IlS(tl)-S(t)ll [lyoll
II(Fx)(tl)-(Fx)(t)ll
2
112:1
C(T)x 0 S(T)y 0
1
T
/ S(T
O)f(O, x(O), x’(O))dO](s)ds I
0
_<
I C(tl) C(t2)II I x0 I
/
I S(tl)
S(t2)II
I y0 l[
K. BALACHANDRAN and S. MARSHAL ANTHONI
272
/o
/
I S(tl
s)- s(t 2 s)[I Cek(S)ds
t2
1
1
I S(tl
s) s(t 2 s)II I B I I W- 111111 x I I + M I o I + MT I yo I
T
+
MT/ ak(O)dO]ds
0
T
+ MT
/ ak(O)dO]ds,
0
and similarly,
<- I c’(tl)
[I (Fx)’(tl) (Fx)’(t2)[I
c’(t2)II I[ Xo I + I s’(tl) s’(t2)I[ I yo I
1
+I
/ [C(tl s) C(t s)]f(s,x(s),x’(s))ds I
t2
II/tl C(t s)f(s, x(s), x’(s))ds I
2
0
-
/
1
I
f [C(tl
s) C(t 2 s)]BW l[x 1 C(Z)x 0 S(T)y 0
0
T
/ S(T
O)f(O, z(O), x’(O))dO](s)ds [I
0
2
+I
/
c(t )BW- l[x
J
C(T)x o S(T)y o
T
S(T O)f(O, x(O), x’(O))dO](s)ds I
0
<- I A(S(tl)- S(t2))II I o I + I C(tl)- C(t2)[I I 0 I
I C(tl
s)- c(t 2 s)II
(4)
Controllability
of Second Order Semilinear Ordinary Differential Systems 273
2
I c(t- s)II ()d
I
T
+
MT/ ck(O)dO]ds
0
2
I c(t2- )II I B I W- 11111
x1
I + M I x0 I + MT I yo I
T
+ MT
/ ck(O)dO]ds.
(5)
0
The right-hand side of (4) and (5) are independent of y E B/C and tends to zero as
t 2 -tl+0 since C(t) and S(t) are uniformly continuous for t E d. The compactness
of C(t) and co(t) for t > 0 imply the continuity in the uniform operator topology (or
norm topology, see [12, p. 182]). Furthermore, the compactness of co(t) follows from
that of C(t) (see [15]).
Thus F maps B/C into an equicontinuous family of functions. It is easy to see
that the family FB/C is uniformly bounded.
Next we show that FB/C is compact. Since we have shown that FB/C is an equicontinuous collection, it suffices by the Arzela-Ascoli theorem to show that F maps
B/C into a precompact set in X.
Let 0<t_<T be fixed and let e be a real number satisfying 0<e<t. For
x G B/c we define
(F)(t) C(t)o + S(t)vo +
/
S(t- )f(, (), ’())
0
--[-
/
S(t- s)BW- 112:1 c()o s()o
o
T
S(T O)f(O, 2:(0), 2:’(O))dO](s)ds.
0
Since C(t) and S(t) are compact operators, the set Y e(t)compact in X for every e, 0 < e < t. Moreover, for every x
I (Fx)(t)- (Fex)(t)II
/
Bk,
we have that
/ I s(t- s)f(s,x(s),x’(s))II
/ I s(t- )BW- l[x -C(T)x
t-e_
{(Fx)(t)’x B/C}
0
S(T)y 0
d
is pre-
K. BALACHANDRAN and S. MARSHAL ANTHONI
274
T
/ S(T-O)f(O,x(O),x’(O))dO](s)II
ds.
0
T
+ MT
/ ak(O))dO]ds
0
and
I (F)’(t)- (F)’(t)II
+
fI
J
JI
C(t- )f(s, re(s), m’(8))II d
c(t s)BW e[x 1 C(T)x o S(T)y o
T
S(T- O)f(O,x(O),x’(O))dO](s)II d.
0
_<
+
/ I c(t- )II k()
d
/ IIC(t-s)[[ IIBII [[W-11111 Ix111 +MI]zoll +MTllYoll
+ / ak(O)dO]ds.
T
MT
0
Therefore, there are precompact sets arbitrarily close to the set {(Fx)(t):x E Bk}.
Hence the set {(Fx)(t)’x Bk} is precompact in X.
It remains to show that F" ZZ is continuous. Let {xn} C_ Z with xnx in Z.
Then there is an integer qsuch that IlXn(t) l <q, IIx’n(t) ll <qfr all n and tGJ,
so I (t)II _< q, I ’(t)II _< q and x,x’ e Z. By (H)
f (t, xn(t), x’n(t))---, f (t x(t), x’(t))
J and since
for each
I f(t, xn(t),xn(t))- f(t,x(t),x’(t)) I 2aq(t),
by the dominated convergence theorem that
we have
tl
n
F
I
sup
tEJ
I
J
0
s(t s)[f(s, xn(s), X’n(S)) f(s, x(s), x’(s))]ds
Controllability
of Second Order Semilinear Ordinary Differential Systems
f
/ S(T
0
S(t s)BW-
0
T
x
275
O)[f(O, xn(O), x’(O)) f(O, x(O), x’(O))]dOds I]
T
I S(t- s)[f(S, Xn(S),X’n(s))- f(s,x(s),x’(s))] I ds
T
+
fI
T
x
J
s(t- )BW-
0
S(T O)[f(O, x,(O), x’n(O)) f(O, x(O), x’(O))]dO I dsO
0
and
I (Fn)’- (F)’ I
up
tGJ
I
Z
C(t 8)[f(s, xn(8), x(8)) f(, x(8), x’(8))]d8
0
/ C(t- s)BW-1
o
T
x
J
0
S(T O)[f(O, xn(O), x’n(O)) f(O, x(O), x’(O))]dOds I
T
I c(t
s)[f(s, x,(s), x’n(s)) f(s, x(s), x’(s))] I d
T
nt-
o
T
x
j S(T
] I C(t- 8)BW-1
O)[f(O, xn(O), x’n(O)) f(O, x(O), x’(O))]dO I dsO.
0
,
Thus F is continuous. This completes the proof that F is completely continuous.
Finally, the set (F)= {x E Z:x ,Fx, e (0, 1)} is bounded as we proved it in
the first step. Consequently, by Schaefer’s theorem, the operator F has a fixed point
in Z. This means that any fixed point of F is a mild solution of (1) on J satisfying
(Fx)(t) x(t). Thus the system (1) is controllable on J.
4. Exaxnple
Consider the partial differential equation
z,(v, t)
z,,(v, t) + (t, z(v, t), zt(v, t)) + (, t),
z(O, t)
z(-, t)
o,
K. BALACHANDRAN and S. MARSHAL ANTHONI
276
z(y,O)-- Zo(y), zt(y ,0)- Zl(y), 0 < y < 7r, t e J-[0, T].
Now we have to show that there exists a control # which steers
initial state to the final state in a Banach space X.
Let X- L2[0, r] and let A: XX be defined by
from any specified
w", wED(A),
Aw
where
(6)
(6)
D(A)- {w X: w, w’ are absolutely continuous, w" X, w(0)- w(r)- 0}.
Then,
E
Aw
n2(w’ Wn)Wn’ w e D(A),
n--1
V//rsinns, n
1,2, 3,... is the orthogonal set of eigenvalues of A.
It can be easily shown that A is the infinitesimal generator of a strongly continuous cosine family C(t), t
in X given by
where
wn(s
,
c(t)-
cost(w, )w,
e x,
and that the associated sine family is given by
S(t)w-
E
lgsinnt(w’wn)Wn
wEX.
n=l
Let f: J x X x XX be define by
f(t, v, w)(y)
r(t, v(y), w(y)), v, w e X,
y
e [0, w],
r" J [0, r] [0, ’]---[0, ’] is continuous and strongly measurable.
Let u: J---.U C X be defined by
where
((t))()
,(, t),
e [0, ],
where #: [0, r] J-,[0, r] is continuous.
Assume that there exists a bounded invertible operator W -1
L(J, U)/kerW) such that
(with range
T
w
j" S(T- )()d.
0
Further, the function
(H)
r satisfies the following condition"
There exists a continuous function p" g--,[0, )such that
where
a: (0, oc)---(O, oc) is a continuous nondecreasing function and
T
0
where c is a known constant.
oc
c
Controllability
of Second Order Semilinear
Ordinary
Differential Systems
277
With this choice of A, f and B= I, (1) is an abstract formulation of (6).
Furthermore, all the conditions stated in the above theorem are satisfied. Hence,
system (6) is controllable on J.
Acknowledgement
The work of the first author is supported by CSIR, New Delhi, India
(Grant No:
25(89)97-EMR-II).
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[14]
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