NEW CRITERIA OF STABILITY FOR TRAJECTORIES [601266]

NEW CRITERIA OF STABILITY FOR TRAJECTORIES
OF PERIODIC EVOLUTION FAMILIES IN HILBERT
SPACES
DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
Abstract. Letqbe a positive real number and let A(
) be a
q-periodic linear operator valued function on a complex Hilbert
spaceH;and letDbe a dense linear subspace of H;which is
invariant for the evolution family fU(t;s) :ts0g;generated
byA(
):We prove that if, the solution of the following well-posed
inhomogeneous Cauchy Problem_u(t) =A(t)u(t) +eity; t> 0
u(0) = 0;
is bounded on R+;for everyy2D;and every2R;by the
positive constant Kkyk; Kbeing an absolute constant, and if, in
addition, for some x2D;the trajectory U(
;0)xsatises a Lips-
chitz condition on the interval (0 ;q);then
sup
z2C;jzj=1sup
n2Z+

nX
k=0zkU(q;0)kx

:=N(x)<1:
The latter discrete boundedness condition has a lot of consequences
concerning the stability of solutions of the abstract nonautonomous
system _u(t) =A(t)u(t):To our knowledge, these results are new.
In the special case, when D=Hand for every x2H, the map
U(
;0)xsatises a Lipschitz condition on the interval (0 ;q);the
evolution familyU=fU(t;s) :ts0g;generated by the family
fA(t)g;is uniformly exponentially stable. In the autonomous case,
(i.e. whenU(t;s) =U(ts;0) for every pair ( t;s) withts0),
the latter assumption is too restrictive. More exactly, in this case,
the semigroup T:=fU(t;0)gt0;is uniformly continuous.
1991 Mathematics Subject Classication. 47A05, 47A30, 47D06, 47A10, 35B15,
35B10.
Key words and phrases. periodic evolution families, uniform exponential sta-
bility, boundedness, strongly continuous semigroup, periodic and almost periodic
functions.
1

2 DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
1.Introduction
In his famous paper, [16], Jan Pr uss showed, concerning a strongly
continuous semigroup ( etA)t0acting o a Hilbert space H;that the
following two statements are equivalent:
1.For everyf2L1([0;1];H);the equation _ u(t) =Au(t) +f(t);
has a unique 1-periodic mild solution.
2.The resolvent set of Acontains 2iZand
sup
n2ZkR(2in;A )k=M <1:
Earlier, such result has been obtained by A. Haraux ([10]), who addi-
tionally had to assume that
sup
n2ZknR(2in;A )kC <1:
By contrast with the autonomous case, the spectral criterion does not
work in the nonautonomous one. See, for example, [17], for further
details and counterexamples.
One typical assumption, concerning uniform exponential stability re-
sults (exponential stability results) for strongly continuous semigroups
acting in Banach spaces, is the existence of a bounded and holomor-
phic continuation of the resolvent (or of the local resolvent) to the right
half-plane of the complex plane. See [3], [4], [12], [20], [21], [22], and
the references therein.
Going on the similar way, the boundedness assumption of our an-
nounced result, may by written as:
(1.1) sup
2Rsup
t0

Zt
0eisU(t;s)xds

:=M(x)<1;8x2H:
Boundedness conditions, like (1.1), with e(ts)Ainstead ofU(t;s); A
being a bounded linear operator acting on a Banach space X;seems to
go back to the work of M. G. Krein. A history of this problem may be
followed in [6] or [5].
This paper is motivated by a recent result in [7], where the same
assertion, referred to uniformly bounded evolution families, is obtained
under the following stronger assumption,
sup
2Rsup
t0

Zt
eisU(t;s)xds

:=M(x)<1;8x2X;
Xbeing a complex Banach space.
It is interesting to compare the results from the present paper with
those in [7].

NEW CRITERIA OF STABILITY 3
First of all, the assertions in Theorem 2.1 and Corollary 2.1 below,
refer only to one trajectory, hence they have an individual character,
while those in [7] have a global character. Moreover, even Theorem 2.2
below could be stated as an individual result.
Mention that the state space in [7] is a Banach space, while in the
present paper, all results are formulated in the framework of Hilbert
spaces. It seems that the results in [7] could be applied to partial dif-
ferential equations (cf. [7, Remark 5.2]) while the autonomous version
of the Corollary 2.4 below, cannot be applied to such equations (cf.
Remark 2.1 below).
However, Corollaries 2.1, 2.2 and 2.3 below could be applied to evo-
lution equations with (possible) unbounded coecients.
Another dierence is that the boundedness assumptions (2.3) and
(2.6) are written along the solutions u(
) satisfying the initial condition
u(0) = 0;while in [7], are considered all solutions which verify u(s) = 0
for everys0:
The assertions of Corollaries 2.1, 2.2 and 2.3 below, are stated in
terms of strong stability. In the end of this paper we provide an example
which shows that uniform exponential stability of a strongly continuous
and uniformly bounded semigroup acting on a Hilbert space, is not a
consequence of boundedness assumption (2.6) below. A famous result,
known as ABLV theorem, see [1], [13], provides a sucient condition
for strong stability of semigroups (acting on Banach spaces) in terms
of countability of the boundary spectrum (A)\iR:In particular, lack
of the spectrum of the innitesimal generator of an uniformly bounded
semigroup which acts on a Banach space on the imaginary axis implies
the strong stability of semigroup. However, it is not always easy to
manipulate these criteria in specic examples, while the assumption
(2.6) can be checked easily, as shown by the two examples presented in
the last section of the article.
An interesting question may be risen when we connect the above
nonautonomous problem with the similar one in the semigroup case.
This latter one, is completely solved by van Neerven ([20]) and Vu
Phong ([15]). In these papers, the assumption refereing to Lipschitz
condition did not appear explicitly, but, in this case, it is automatically
veried for all xin the domain of the innitesimal generator. Then,
a natural question in the nonautonomous case is: can we replace in
the Corollary 2.4 the assumption referred to the Lipschitz condition
for everyx2H;with the similar one, but only for xin a dense subset
D;ofH? We have not an answer to this question yet.
The proof of the main result consists by an estimation of the integral
in (1.1), with xreplaced by fx(s); fxbeing aH-valued and q-periodic

4 DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
function, which is smooth in some sense. The proof of Corollary 2.4
is completed by using a well-known discrete criterion for exponential
stability of periodic evolution families.
2.Notations and statement of the result
LetHbe a complex separable Hilbert space and let L(H) be the
Banach algebra of all bounded linear operators acting on H. The inner
product inHis denoted byh
;
i, while the norms in Hand inL(H) are
denoted by the same symbol, namely by k
k. As usual, (T) denotes
the spectrum of a linear operator T:WhenTis bounded its spectral
radius, denoted by r(T);is given by the Gelfand formula
r(T) := supfjzj:z2(T)g= lim
n!1kTnk1=n:
Recall that the set fukgk2Z+, withuk2H, is a basis in Hif the linear
span offuk:k2Z+gis dense in H. Such basis is called orthogonal if
huk;uji= 0;for every pair ( k;j) of dierent nonnegative integers. As
is well-known, the orthonormal set fukgk2Z+is a basis in Hif and only
if for eachf2Hone hasP
k2Z+jhf;ukij2=kfk2. In this case, every
elementf2Hmay be represented as
f=X
k2Z+hf;ukiuk:
As usual, by L2([0;q];C);we denote the Hilbert space consisting of
allC-valued square integrable functions dened on the interval [0 ;q];
endowed with the usual inner product and norm. The set of functionsn
e2i(n=q)

pqo
n2Zis an orthonormal basis in L2([0;q];C), so any function
z2L2([0;q];C);may be represented as:
(2.1)z(t) =1
qX
n2ZZq
0z(s)e2ins=qds
e2int=q; t2[0;q]:
ByH:=L2([0;q];H) we denote the set of all H-valued measurable
functionsfdened on [0 ;q] satisfying the condition
Zq
0kf(t)k2dt1
2
:=kfk2
L2([0;q];H)<1:
InHthe functions equal almost everywhere are identied. Endowed
with the inner product hf;giH=Rq
0hf(t);g(t)idt;Hbecomes a Hilbert
space. In as follows, by '
ukwe denote the tensor product between
the scalar-valued function 'dened on [0 ;q] and the vector uk2H,

NEW CRITERIA OF STABILITY 5
i.e. the map dened for t2[0;q], by: ('
uk)(t) ='(t)uk. The system
of vectors
B:=1pqe2i(n=q)

uk:n2Z;k2Z+
is an orthonormal basis in H. In fact, for k;p2Z+andm;n2Z, one
has: 1pqe2in
=q
uk;1pqe2im
=q
up
H
=1
qZq
0he2int=quk;e2imt=qupidt=kp
qZq
0e2i(nm)t=qdt
=
1; k=pandn=m
0; k6=porn6=m:
Moreover,
kfk2
H=Rq
0*
P
k2Z+hf(t);ukiuk;P
p2Z+hf(t);upiup+
dt
=P
k;p2Z+Rq
0hf(t);ukihf(t);upihuk;upidt
=P
k2Z+Rq
0hf(t);ukihf(t);ukidt:
In view of (2.1), for z(t) =hf(t);uki, obtain:
kfk2
H=1
q2X
k2Z+"X
n;m2ZZq
0hf(s);ukie2ins=qds


Zq
0hf(s);ukie2ims=qdsZq
0e2i(nm)t=qdt
=1
qX
(k;n)2Z+ZZq
0hf(s);e2ins=qukids
Zq
0hf(s);e2ins=qukids
=1
qX
(k;n)2Z+ZZq
0hf(t);e2int=qukidt2
=X
(k;n)2Z+Zhf;1pqe2i(n=q)

ukiH2
:
As consequence, every function f2H may be represented as
f(
) =1
qX
n2Ze2in
=qcn(f);

6 DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
wherecn(f)2H;thenthFourier coecient associated to f;is given
by
(2.2) cn(f) =X
k2Z+Zq
0hf(s);e2ins=qukids
uk:
A familyU=fU(t;s) :ts0gL (H) is called evolution family
onHifU(t;s)U(s;r) =U(t;r) for alltsr0 andU(t;t) =I
fort0:HereIis the identity operator of L(H). An evolution family
UonHis called strongly continuous if for eachx2H, the map
(t;s)7!U(t;s)x:f(t;s)2R2:ts0g!X
is continuous. We say that the evolution family Uhas exponential
growth if there exist the constants M1 and!2Rsuch that
kU(t;s)kMe!(ts), for allts:The evolution family Uisq-periodic,
for some positive q, ifU(t+q;s+q) =U(t;s) for all pairs ( t;s) with
ts0:Every strongly continuous and q-periodic evolution family
acting on a Banach space has an exponential growth, [8]. Let x2H
be xed.
The following assumptions, concerning the evolution family, and the
trajectoryU(
;0)x;are referred to several times. For this reason, we
state them separately.
(A(x)) : The map t7!ux(t) :=U(t;0)xsatises a Lipschitz
condition on the interval (0 ;q):
(HD) : The evolution family Uacts properly on the linear sub-
spaceD;i.e.U(t;s)(D)Dfor everyts0:
The result of this paper reads as follows.
Theorem 2.1. LetDbe a dense linear subspace of H;and let U=
fU(t;s)gts0be a strongly continuous and q-periodic evolution family
of bounded linear operators acting on Hwhich satisfy the assumption
(HD):If
(2.3) sup
2Rsup
t0

Zt
0eisU(t;s)yds

Kkyk<1;8y2D;
Kbeing a positive absolute constant, and if, in addition, for a given
x2D;the assumption (A(x))is fullled, then
(2.4) sup
z2C;jzj=1sup
n2Z+

nX
k=0zkU(q;0)kx

:=N(x)<1:

NEW CRITERIA OF STABILITY 7
In view of the density of DinHand by using the Theorem of Dom-
inated Convergence, the condition (2.3) is equivalent with the same
one, but with Hinstead ofD:
In the framework of semigroups, the assumptions of the previous
theorem may be relaxed as follows:
Theorem 2.2. LetT=fT(t)gt0be a strongly continuous semigroup
of bounded linear operators acting on H;and let (A;D(A))be its inn-
itesimal generator. If
sup
2Rsup
t0

Zt
0eisT(ts)yds

:=R(y)<1;8y2D(A);
then, for every x2D(A);one has:
sup
z2C;jzj=1sup
n2Z+

nX
k=0zkT(q)kx

:=N(x)<1:
In order to present some consequences of the Theorem 2.1, we state
and prove the following useful Lemma.
Lemma 2.1. LetTbe a bounded linear operator acting on Hand let
M1and!2Rsuch thatkTnkMe!nfor alln2Z+:If for some
x2H;one has
(2.5) sup
z2C;jzj=1sup
n2Z+

nX
k=0Tkx
zk+1

:=N(x)<1;
then limk!1Tkx= 0:
Proof. The proof is modeled after [[20], Theorem 4]. Obviously, the
mapz7!Rn;x(z) :=Pn
k=0Tkx
zk+1is holomorphic on Cnf0g:Letz02C
such thatjz0j=e!:By assumption, the sequence of functions ( Rn;x(
));
is uniformly bounded on the unit circle. On the other hand, for jzj
jz0j+ 1 we have
kRn;x(z)knX
k=0Mjz0jkkxk
(jz0j+ 1)k+1M(1 +jz0j)kxk:
Thus, by the Phragmen-Lindel of theorem, the sequence ( Rn;x(
));is
uniformly bounded on the circular crown 1 jzjjz0j+ 1;and then
it is uniformly bounded on the set fz2C:jzj1g;as well.
Since, forjzj>jz0j;one have that Rn;x(z)!R(z;T)x;asn!1;
the Vitali theorem, [[11], Theorem 3.14.1] assures us that the limit

8 DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
limn!1Rn;x(z);exists for all zin the circular crown 1 jzjjz0j+ 1:
This yields, lim k!1Tkx= 0:

Corollary 2.1. LetUbe a strongly continuous and q-periodic evolu-
tion family acting on H;such that all assumptions in Theorem 2.1 are
fullled. Then the trajectory U(
;0)xis strongly stable, i.e.
lim
t!1U(t;0)x= 0:
Corollary 2.2. LetUbe a strongly continuous and q-periodic evolution
family acting on H:Assume the following:
1.The condition (2.3) is fullled.
2.The mapU(
;0)ysatises a Lipschitz condition for every y2D:
3.The evolution family Uveries the assumption (HD):
Under these assumptions, the trajectory t7!U(t;s)x: [s;1)!H;is
strongly stable for every s0and everyx2D:
Proof. Letts0 andNbe any positive integer such that t
Nqs:SuchNexists fortlarge enough. Then U(t;s)x=U(t
Nq;0)U(Nq;s )x:Sinceys:=U(Nq;s )x2D;we may apply the previ-
ous Corollary to nish the proof.

Corollary 2.3. LetT=fT(t)gt0be a uniformly bounded and strongly
continuous semigroup acting on a Hilbert space H;and letD(A)the
maximal domain of its innitesimal generator. If
(2.6) sup
2Rsup
t0

Zt
0eisT(ts)yds

:=K(y)<1;8y2D(A);
then, the semigroup Tis strongly stable.
Proof. Since, for every x2D(A);the mapT(
)xis dierentiable, it
satises a Lipschitz condition on (0 ;q):Then, by applying the previous
Corollary for U(t;s) =T(ts);it follows that T(
)xis strongly stable
for everyx2D(A):Let now,y2Handxn2D(A) such that xn!y,
asn!1;in the norm of H:Then,
kT(t)ykkT(t)(yxn)k+kT(t)xnk
supfkT(t)k:t0gkyxnk+kT(t)xnk! 0;ast;n!1:

Corollary 2.4. If an evolution family U=fU(t;s)gts0;as in The-
orem 2.1, verify (2.3) for everyy2H;and also (A(v));for every

NEW CRITERIA OF STABILITY 9
v2H;then it is uniformly exponentially stable, i.e. there are two
positive constants Nandsuch that
kU(t;s)kNe(ts);for allts0:
Proof. Follows by applying Theorem 2.1 and Lemma 2.4, below.

Remark 2.1. The assumption that (A(v))fulls for every v2H;
is too restrictive. For example, in the particular case when U(t;s) =
T(ts)for everyts0;wherefT(t)gt0is a strongly contin-
uous semigroup acting on H;the familyUverify the above Lipschitz
assumption if and only if the semigroup is uniformly continuous, i.e.
if and only if there exists a bounded linear operator acting on Hsuch
thatT(t) =etAfor allt0:
Proof. Indeed, under the assumption ( A(x));the mapt7!ux(t) :=
T(t)xbelongs to W1;1((0;q);H);thereforeux(
) is dierentiable a.e.
on (0;q);ku0
x(t)kLx(Lxbeing the constant of Lipschtitz of ux(
))
andux(t) belongs in the domain D(A) of the innitesimal generator a.e.
on (0;q):Then, there exists a sequence ( tn) of positive real numbers,
withtn!0 asn!1 such thatux(tn)2D(A);ux(tn) =T(tn)x!x;
asn! 1 and the sequence ( u0
x(tn)) = (AT(tn)) converges in the
weak topology of H:This latter fact is an obvious consequence of the
Banach-Steinhaus theorem by using the classical fact that the adjoint
A;ofA;is the innitesimal generator of the semigroup fT(t)gt0:
Since,Ais closed in the weak topology of H;it followsx2D(A):
ThenD(A) =XandAis bounded. 
The following lemmas are useful in the proof of Theorem 2.1.
Lemma 2.2. [2, Lemma 2.2] Let us consider the functions h1,h2:
[0;q]!C, dened by:
h1(s) =
s ; s2[0;q
2)
qs ; s2[q
2;q]andh2(s) =s(qs); s2[0;q]:
DenoteH1() :=Rq
0h1(s)eisdsandH2() :=Rq
0h2(s)eisds. Then,
jH1()j+jH2()j6= 0 for all2R:
The continuation by periodicity on the real axis of the function hj,
forj2f1;2g, will be denoted by the same symbol.
Lemma 2.3. Letx2Xsuch that the map s7!U(s;0)xsatises
a Lipschitz condition on (0;q). For each j2f1;2g;let consider the
q-periodic function fj:R!H, given on [0;q];by:
fj(t) :=hj(t)U(t;0)x:

10 DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
The following two statements hold true:
1.Each function fjsatises a Lipschitz condition on R.
2.The Fourier series associated to fjis absolutely and uniformly
convergent on R.
Lemma 2.4. [6, Lemma 1] If
(2.7) sup
n1

nX
k=0eikU(q;0)k

:=M()<1
theneibelongs to the resolvent set of U(q;0). Moreover, if (2.7) holds
for every2R;thenr(U(q;0))<1;i.e. the familyUis uniformly
exponentially stable.
3.Proof of the Theorem 2.1
Proof of Lemma 2.3
1.Letx2H; Kx:= sups2[0;q]jjU(s;0)xjjandLx>0 such that
kU(t;0)xU(s;0)xkLxjtsjfor allt;s2(0;q):One has
kfj(t)fj(s)k=khj(t)U(t;0)xhj(s)U(s;0)xk
jhj(t)jkU(t;0)xU(s;0)xk+jhj(t)hj(s)jkU(s;0)xk
(maxfq
2;q2
4gLx+Kx
maxf1;qg)jtsj:
Using the continuity of the map U(
;0)x;the previous inequality may
be extended rst for t;s2[0;q] and then, by using the periodicity of
the mapfj;to the entire axis.
2.An argument of this type for scalar valued functions may be found
in [18, Exercise 16, pp. 92-93]. For sake of completeness we present the
details.
For eacht2Rand each positive number , which will be chosen
later, denote gj(t) :=fj(t+)fj(t). Using (2.2), we get
cn(gj) =X
k2Z+Zq
0hgj(s);e2ins=qukids
uk
=X
k2Z+Zq
0hfj(s+);e2ins=qukidsZq
0hfj(s);e2ins=qukids
uk
=X
k2Z+Zq+
hfj();e2in()=qukidZq
hfj();e2in(+)=qukid
uk

NEW CRITERIA OF STABILITY 11
=X
k2Z+Zq
0
e2in=qe2in=q

hfj();e2in=qukid
uk
=X
k2Z+Zq
02isin(2n=q )hfj();e2in=qukid
uk
= 2isin(2n=q )cn(fj):
In view of the Bessel inequality and taking into account that the
functionfjsatises a Lipschitz condition on Rwith a constant L(x);
follows
4q2L2(x)Zq
0kgj(t)k2dtX
n2Zkcn(gj)k2
=X
n2Z4kcn(fj)k2jsin(2n=q )j2:
Letpbe a positive integer and :=q
2p+2.
SetAp:=fn2Zj2p1<jnj2pg. Obviously,jApj= 2pandp
2
2<
jsin(2n=q )jfor eachn2Ap. Furthermore,S
p1Ap=Znf1;0;1g.
Using the Schwartz inequality, we get

P
n2Apkcn(fj)k!2
2pP
n2Apkcn(fj)k2
<2p+1P
n2Apkcn(fj)k2sin2(2n=q )
q3L2(x)
2p+3;
andX
n2Znf1;0;1gkcn(fj)k(p
2 + 1)qpqL(x)
23=2:=L1(x):
The restriction of the function fjto the interval [0 ;q] belongs to
L2([0;q];H) and, in addition, fj= (1=q)P
n2Ze2in
=qcn(fj):Set
sN;j(t) :=NX
n=Ne2int=qcn(fj):
Clearly, (Rq
0kfj(t)sN;j(t)k2dt)1
2);decays to 0 when N!1 . As is
already shown, the series (P
n2Ze2int=qcn(fj));is uniformly convergent
onR, hence there exists a continuous function sj:R!H;such that
sup
t2Rksj(t)sN;j(t)k! 0 whenN!1:
Since
ksN;jsjkL2([0;q];H)pqksN;jsjk1!0;

12 DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
sj=fjinL2([0;q];H). The functions sjandfjare continuous and
equal almost everywhere on [0 ;q], sofj(t) =sj(t) for eacht2[0;q].
Taking into account that both functions are q-periodic, they are equal
onR:To conclude, the Fourier series associated to fj;is absolutely and
uniformly convergent on Rtofj:
Proof of Theorem 2.1
For anyn2Z+and for every x2X;one has
Znq
0eisU(nq;s)fj(s)ds=n1X
k=0Z(k+1)q
kqeisU(nq;s)fj(s)ds
=n1X
k=0Zq
0ei(kq+)U(nq;kq +)fj(kq+)d
=Hj()n1X
k=0eikqU((nk)q;0)x:
LetAbe the set of all real for whichH1() = 0:In view of Lemma
2.2, we get:

n1X
k=0eikqU(q;0)nkx

=8
<
:1
jH1()j

Rnq
0eisU(nq;s)f1(s)ds

;  =2A
1
jH2()j

Rnq
0eisU(nq;s)f2(s)ds

; 2A:
On the other hand

Znq
0eisU(nq;s)fj(s)ds

=

1
qX
k2ZZnq
0ei(+2k=q)sU(nq;s)ck(fj)

K
qX
k2Zkck(fj)kK
q
0
@L1(x) +X
k2f1;0;1gkck(fj)k1
A:

Proof of Theorem 2.2 . Is the same with the proof of Theorem 2.1,
taking into account that in this case the Fourier coecients ck(fj);k2
Z;j2f1;2g;belongs toD(A):

NEW CRITERIA OF STABILITY 13
4.Some examples
Example 4.1. LetH:=L2([0;];C)endowed with the usual norm
and letfT(t)gt0be the semigroup dened on H;by
(T(t)x) () =2
1X
n=1etn2sinnZ
0x(s) sinnsds
; 2[0;];t0;
havingAas innitesimal generator. Further details about this semi-
group and its innitesimal generator may be nd, for example, in [23,
pp. 179 and 199] . We recall that the domain of Aconsists by all abso-
lutely continuous functions x(
)such thatx0(
)is absolutely continuous,
x00(
)2Handx(0) =x() = 0:
Also, consider the map a:R+!(0;1)verifying the following
conditions:
i)a(
)is a-periodic map.
ii)a(t)1for everyt0.
iii)There exist 2(0;1]andc>0such that
ja(t)a(s)jcjtsj;for allt;s0:
SetA(t) =a(t)A; t0. The familyfA(t)gt0is well-posed (i.e.
there exists a strongly continuous and periodic evolution family U=
fU(t;s)gts0such that any solution u(
)of the system _x(t) =A(t)x(t);
veriesu(t) =U(t;s)u(s)for allts0):See, [19],[17],[14], for
further details concerning well-posedness. In this case, the evolution
family is given by, [9, Example 2.9b] ,
U(t;s)x=TZt
sa(r)dr
x;8x2H;ts0:
Obviously, every continuous function f2Hveries the inequality
kfkL2([0;];C)pkfk1. Setx(
)2D(A):Taking into account that
^xn:=R
0x(s) sinnsds satises the estimation k^xnk kxk1;and
denoting by F(t)the primitive function of a(t), we get

Zt
0eiTZt
a(r)dr
x()d

=
=2


Zt
0ei 1X
n=1en2Rt
a(r)drsinn!
^xnd

2kxk11X
n=1Zt
0en2Rt
a(r)drd2kxk11X
n=1en2F(t)Zt
0a()en2F()d

14 DOREL BARBU, JO EL BLOT, CONSTANTIN BUS E AND OLIVIA SAIERLI
2kxk11X
n=11
n2
1en2(F(0)F(t))
2kxk11X
n=11
n2<1:
Thus, for each x=x(
)2D(A);have that

Zt
0eiTZt
a(r)dr
xd

L2([0;];C)2pkxk11X
n=11
n2<1:
On the other hand, the map
t7!U(t;0)x=TZt
0a(s)ds
x
is derivable for all x2D(A);and its derivative is bounded by constant
kAxk:Then, it satises a Lipschitz condition on the interval (0;q);and
the Corollaries 2.1 and 2.2 above can be applied in this particular case.
Mention that Corollary 2.4 above cannot be applied to the evolution
family in this example, because, in the special case when a(t) = 1 for ev-
eryt0;there exists at least one x(
)such that the trajectory U(
;0)x
does not satisfy any Lipschitz condition on the interval (0;q):Thus, our
theoretical results allow us to establish the strong stability of the peri-
odic evolution family fU(t;s)gts0rather than its uniform exponential
stability. As is well-known, the semigroup fT(t)gis uniformly exponen-
tially stable. Combining this with the inequalityRt
sa()d(ts)for
ts;is easily to see that the evolution family fT(Rt
sa()d)gts0is
uniformly exponentially stable as well.
The next example shows that the boundedness integral conditions
(1.1) and (2.6) are not equivalent. More exactly, there exist semigroups
which verify (2.6) and does not verify (1.1).
Example 4.2. LetZ+the set of all nonnegative integers and let H:=
l2(Z+;C)endowed with the usual norm denoted by k
k2:Letn:=1
n+
in;n2Z+;x= (xn)n2Z+be a sequence in Hand letT(t)x:= (entxn):
Obviously, the one parameter family T=fT(t)gt0is a strongly con-
tinuous and uniformly bounded semigroup having as innitesimal gen-
erator the "diagonal" operator dened by ((Ax)n) := (nxn):The max-
imal domain of Aconsists by all sequences (xn)2Hwhich verify the
conditionP1
n=0jnxnj2<1:Obviously, the semigroup Tis not uni-
formly exponentially stable. Indeed, supposing the contrary, there are
two positive constants Kandsuch that
e2
NtjxNjK2e2tkxk2
2

NEW CRITERIA OF STABILITY 15
holds true for every t0; N2Z+andx2H:This provide a contra-
diction when Nis large enough and xN6= 0:Then (1.1) is not fullled
(cf.[15]). In the following we prove that (2.6) is fullled.
Letx= (xn)inD(A)andbe any real number. Then,
0
@tZ
0eise(ts)Axds1
A(n) =eite1
n+int
1
n+i(n)xn:
Therefore,

tZ
0eise(ts)Ax

2
221X
n=1n2jxnj2<1;
i.e.(2.6) holds true.
Acknowledgement 1. The authors thank the reviewer for their help-
ful comments on the second draft of the manuscript. In particular,
these comments have helped us to eliminate a mistake in Lemma 2.1.
Also, the reviewer pointed us new elementary way by which one can
prove that the semigroup contained in Example 4.1 is uniformly expo-
nentially stable.
Acknowledgement 2. The authors thanks to the anonymous referees
for their useful suggestions on the preliminary version of this manu-
script, especially concerning the Remark 2.1.
Acknowledgement 3. An important part of this work was done while
the third named author was Visiting Professor of University Paris 1
Sorbonne-Pantheon. The support of the university is gratefully ac-
knowledged.
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NEW CRITERIA OF STABILITY 17
West University of Timisoara, Department of Mathematics, Bd. V.
Parvan No. 4, 300223-Timisoara, Rom ^ania
E-mail address :barbu@math.uvt.ro
Laboratoire SAMM EA 4543, University Paris 1 Sorbonne-Pantheon,
Centre P. M. F., 90 rue de Tolbiac, 75634 Paris Cedex 13, France
E-mail address :Joel.Blot@univ-paris1.fr
West University of Timisoara, Department of Mathematics, Bd. V.
Parvan No. 4, 300223-Timisoara, Rom ^ania
E-mail address :buse@math.uvt.ro
Tibiscus University of Timisoara, Department of Computer Science
and Applied Informatics, Str. Lasc ar Catargiu, No. 4-6,300559-Timisoara,
Rom^ania
E-mail address :saierli olivia@yahoo.com