Stability results for abstract evolution equations with intermittent time-delay feedback

0.17 MB·

by  Cristina Pignotti

Stability results for abstract evolution equations with
intermittent time-delay feedback
Cristina Pignotti
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica
7
1 Universit`a di L’Aquila
0
Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy
2

n
a
J Abstract

6
1 We consider abstract evolution equations with on–off time delay feedback. Without the
timedelayterm,themodelisdescribedbyanexponentiallystablesemigroup.Weshowthat,

] under appropriate conditions involving the delay term, the system remains asymptotically
P
stable.Underadditionalassumptionsexponentialstabilityresultsarealsoobtained.Concrete
A
examples illustrating the abstract results are finally given.
.
h
t
a
m
[ 1 Introduction

1
v
4
In this paper we study the stability properties of abstract evolution equations in presence
5
3 of a time delay term.
4 In particular, we include into the model an on–off time delay feedback, i.e. the time delay
0
is intermittently present.
.
1
Let be a Hilbert space, with norm , and let : bea dissipative operator gen-
0 H k·k A H → H
7 erating a C0 semigroup (S(t))t≥0 exponentially stable, namely there are two positive constants
−
1 M and µ such that
v: S(t) Me−µt, t 0, (1.1)
L(H)
i k k ≤ ∀ ≥
X
where ( ) denotes the space of bounded linear operators from into itself.
r L H H
a We consider the following problem
U (t) = U(t)+ (t)U(t τ) t > 0,
t A B − (1.2)
U(0) = U ,
0
(cid:26)
where τ , the time delay, is a fixed positive constant, the initial datum U belongs to and, for
0
H
t > 0, (t) is a bounded operator from to .
B H H
In particular, we assume that there exists an increasing sequence of positive real numbers
t , with t = 0, such that
n n 0
{ }
1) (t) = 0 t I = [t ,t ),
2n 2n 2n+1
B ∀ ∈
2) (t) = t I = [t ,t ).
2n+1 2n+1 2n+1 2n+2
B B ∀ ∈
1
We denote B = , n IN. Moreover, denoted by T the length of the interval
2n+1 2n+1 L(H) n
kB k ∈
I , that is
n
T = t t , n IN, (1.3)
n n+1 n
− ∈
we assume
T τ, n IN. (1.4)
2n
≥ ∀ ∈
Time delay effects are frequently present in applications and concrete models and it is now
well–understood that even an arbitrarily small delay in the feedback may destabilize a system
which is uniformly stable in absence of delay (see e.g. [7, 8, 22, 30]).
We want to show that, under appropriate assumptions involving the delay feedback coef-
ficients, the size of the time intervals where the delay appears and the parameters M and µ
in (1.1), the considered model is asymptotically stable or exponentially stable, in spite of the
presence of the time delay term.
Stability results for second–order evolution equations with intermittent damping were first
studied by Haraux, Martinez and Vancostenoble [14], without any time delay term. They con-
sidered a model with intermittent on–off or with positive–negative damping and gave sufficient
conditions ensuring that the behavior of the system in the time intervals with the standard
dissipative damping, i.e. with positive coefficient, prevails over the bad behavior in remaining
intervals where the damping is no present or it is present with the negative sign, namely as
anti–damping. Therefore, asymptotic/exponential stability results were obtained.
More recently Nicaise and Pignotti [23, 24] considered second–order evolution equations
with intermittent delay feedback. These results have been improved and extended to some semi-
linear equations in [9]. In the studied models, when the delay term (which possess a destabiliz-
ing effect) is not present, a not–delayed damping acts. Under appropriate sufficient conditions,
stability results are then obtained. Related results for wave equations with intermittent delay
feedback have been obtained, in 1-dimension, in [12], [13] and [3] by using a different approach
based on the D’Alembert formula. However, this last approach furnishes stability results only
for particular choices of the time delay.
In the recent paper [28], the intermittent delay feedback is compensated by a viscoelastic
damping with exponentially decaying kernel.
Theasymptotic behavior of wave–type equations with infinitememory andtime delay feed-
back has been studied by Guesmia in [11] via a Lyapunov approach and by Alabau–Boussouira,
Nicaise and Pignotti [2] by combining multiplier identities and perturbative arguments.
We refer also to Day and Yang [6] for the same kind of problem in the case of finite
memory. In these papers the authors prove exponential stability results if the coefficient of the
delay damping is sufficiently small. These stability results could be easily extended to a variable
coefficient b() L∞(0,+ ) under a suitable smallness assumption on the L∞ norm of b().
· ∈ ∞ − ·
In [28], instead, asymptotic stability results are obtained without smallness conditions re-
lated to the L∞ norm of the delay coefficient. On the other hand, the analysis is restricted
−
to intermittent delay feedback. Asymptotic stability is proved when the coefficient of the de-
lay feedback belongs to L1(0,+ ) and the length of the time intervals where the delay is not
∞
present is sufficiently large. The same paper considers also problems with on–off anti–damping
instead of a time delay feedback. Stability results are obtained even in this case under analogous
assumptions.
The idea is here to generalize the results of [28] by considering abstract evolution equations
for which, without considering the intermittent delay term, the associated operator generates
an exponentially stable C semigroup.
0
−
2
For such a class of evolution equations we already know that, under a suitable smalness
condition on the delay feedback coefficient, an exponential stability result holds true (see [25]).
We want to show that stability results are avalaible also under a condition on the L1 norm of
−
the delay coefficient, without restriction on the pointwise L∞ norm.
−
The paper is organized as follows. In section 2 we give a well–posedness result. In sections
3 and 4 we prove asymptotic and exponential stability results, respectively, for the abstract
model under appropriate conditions. Stability results are established also for a problem with
intermittent anti–damping instead of delay feedback in section 5. Finally, in section 6, we give
some concrete applications of the abtract results.
2 Well-posedness
In this section we illustrate a well-posedness results for problem (1.2).
Theorem 2.1 ForanyinitialdatumU thereexistsaunique(mild)solutionU C([0, ); )
0
∈ H ∈ ∞ H
of problem (1.2). Moreover,
t
U(t) = S(t)U + S(t s) (s)U(s τ)ds. (2.1)
0
− B −
Z0
Proof. Weprovetheexistence anduniquenessresultontheinterval[0,t ];thentheglobalresult
2
follows by translation (cfr. [23]). In the time interval [0,t ], since (t) = 0 t [0,t ), then
1 1
B ∀ ∈
there exists a unique solution U C([0,τ], ) satisfying (2.1). The situation is different in the
∈ H
time interval [t ,t ] where the delay feedback is present. In this case, we decompose the interval
1 2
[t ,t ] into the successive intervals [t +jτ,t +(j+1)τ), for j = 0,...,N 1, where N is such
1 2 1 1
−
that t +(N+1)τ t . The last interval is then [t +Nτ,t ]. Now, first we look at the problem
1 2 1 2
≥
on the interval [t ,t +τ]. Here U(t τ) can be considered as a known function. Indeed, for
1 1
−
t [t ,t +τ], then t τ [0,t ], and we know the solution U on [0,t ] by the first step. Thus,
1 1 1 1
∈ − ∈
problem (1.2) may be reformulated on [t ,t +τ] as
1 1
U (t) = U(t)+g (t) t (τ,2τ),
t A 0 ∈ (2.2)
U(τ) = U(τ ),
−
(cid:26)
where g (t) = (t)U(t τ). This problem has a unique solution U C([τ,2τ], ) (see e.g. Th.
0
B − ∈ H
1.2, Ch. 6 of [27]) given by
t
U(t) = S(t τ)U(τ )+ S(t s)g (s)ds, t [τ,2τ].
− 0
− − ∀ ∈
Zτ
Proceedings analogously in the successive time intervals [t + jτ,t + (j + 1)τ), we obtain a
1 1
solution on [0,t ].
2
3 Asymptotic stability results
Let T∗ be defined as
3
1
T∗ := lnM, (3.1)
µ
where M and µ are the constants in (1.1), that is T∗ is the time for which Me−µT∗ = 1.
We can state a first estimate on the intervals I where the delay feedback is not present.
2n
Proposition 3.1 Assume T > T∗. Then, there exists a constant c (0,1) such that
2n n
∈
U(t ) 2 c U(t ) 2, (3.2)
2n+1 n 2n
k k ≤ k k
for every solution of problem (1.2).
Proof. Observe that in the time interval I = [t ,t ] the delay feedback is not present
2n 2n 2n+1
since (t) 0. Thus, (3.2) easily follows from (1.1) with √c = Me−µT2n < Me−µT∗ = 1.
n
B ≡
Let us now introduce the Lyapunov functional
1 1 t
F(t) = F(U,t) := U(t) 2 + (s+τ) U(s) 2ds. (3.3)
L(H)
2k k 2 kB k k k
Zt−τ
Proposition 3.2 Assume 1),2). Moreover, assume T τ, n IN. Then,
2n
≥ ∀ ∈
F′(t) B U(t) 2, t I = [t ,t ], n IN. (3.4)
2n+1 2n+1 2n+1 2n+2
≤ k k ∈ ∀ ∈
for any solution of problem (1.2).
Proof. By differentiating the energy F(), we have
·
1
F′(t) = U(t), U(t) + U(t), (t)U(t τ) + (t+τ) U(t) 2
L(H)
h A i h B − i 2kB k k k
1
(t) U(t τ) 2.
L(H)
−2kB k k − k
Then, since the operator is dissipative, one can estimate
A
1
F′(t) (t) U(t) U(t τ) + (t+τ) U(t) 2
L(H) L(H)
≤ kB k k kk − k 2kB k k k (3.5)
1
(t) U(t τ) 2.
L(H)
−2kB k k − k
Therefore, from Cauchy–Schwarz inequality,
1 1
F′(s) (t) U(t) 2 + (t+τ) U(t) 2.
L(H) L(H)
≤ 2kB k k k 2kB k k k
Now,observethat,sinceT τ,foreveryn IN,iftbelongstoI thent+τ belongsto
2n 2n+1
≥ ∈
I or to I .Inthefirstcase (t) = B while,in thesecondcase (t) = 0.
2n+1 2n+2 L(H) 2n+1 L(H)
kB k kB k
Thus (3.4) is proved.
4
Theorem 3.3 Assume 1), 2) and T τ for all n IN. Moreover assume T > T∗, for all
2n 2n
≥ ∈
n IN, where T∗ is the time defined in (3.1). Then, if
∈
∞
ln e2B2n+1T2n+1(c +T B ) = , (3.6)
n 2n+1 2n+1
−∞
n=0
X (cid:2) (cid:3)
the equation (1.2) is asymptotically stable, namely any solution U of (1.2) satisfies U(t) 0
k k →
for t + .
→ ∞
Proof. Note that from (3.4) we obtain
F′(t) 2B F(t), t I = [t ,t ), n IN.
2n+1 2n+1 2n+1 2n+2
≤ ∈ ∈
Then, by integrating on the time interval I ,
2n+1
F(t ) e2B2n+1T2n+1F(t ), n IN. (3.7)
2n+2 2n+1
≤ ∀ ∈
From the definition of the Lyapunov functional F,
1 1 t2n+1
F(t )= U(t ) 2+ (s+τ) U(s) 2ds. (3.8)
2n+1 2n+1 L(H)
2k k 2 kB k k k
Zt2n+1−τ
Notethat,fort [t τ,t ),thent+τ [t ,t +τ)andtherefore,since I τ
2n+1 2n+1 2n+1 2n+1 2n+2
∈ − ∈ | | ≥
itresultst+τ I I .Now,ift+τ I ,then (t+τ)= 0.Otherwise,ift+τ I ,
2n+1 2n+2 2n+2 2n+1
∈ ∪ ∈ B ∈
then (t+τ) = B . Then, from (3.8) we deduce
2n+1
kB k
1 1 min(t2n+2−τ,t2n+1)
F(t ) = U(t ) 2 + B U(s) 2ds, (3.9)
2n+1 2n+1 2n+1
2k k 2 k k
Zt2n+1−τ
since if t > t τ, then (t)= 0 for all t [t ,t +τ) [t ,t ).
2n+1 2n+2 2n+2 2n+1 2n+2 2n+3
− B ∈ ⊂
Then, since U() is decreasing in the intervals I (the operator is dissipative and
2n
k · k A
(t) 0), we deduce
B ≡
1 1
F(t ) U(t ) 2 + T B U(t τ) 2
2n+1 2n+1 2n+1 2n+1 2n+1
≤ 2k k 2 k − k (3.10)
1 1
U(t ) 2+ T B U(t ) 2.
2n+1 2n+1 2n+1 2n
≤ 2k k 2 k k
Using this last estimate in (3.7), we obtain
U(t ) 2 2F(t ) e2B2n+1T2n+1(c +T B ) U(t ) 2, n IN, (3.11)
2n+2 2n+2 n 2n+1 2n+1 2n
k k ≤ ≤ k k ∀ ∈
where we have used also the estimate (3.2). By iterating this argument we arrive at
U(t ) 2 Πn e2B2k+1T2k+1(c +T B ) U 2, n IN. (3.12)
k 2n+2 k ≤ k=0 k 2k+1 2k+1 k 0k ∀ ∈
Now observe that U(t) is not decreasing in the whole (0,+ ). However, it is decreasing
k k ∞
for t [t ,t ), n IN, where the destabilizing delay feedback does not act and so
2n 2n+1
∈ ∈
U(t) U(t ) , t [t ,t ). (3.13)
2n 2n 2n+1
k k ≤ k k ∀ ∈
5
Moreover, from (3.10), for t [t ,t ) we have
2n+1 2n+2
∈
U(t) 2 2F(t) e2B2n+1T2n+1(c +B T ) U(t ) 2, (3.14)
n 2n+1 2n+1 2n
k k ≤ ≤ k k
where in the second inequality we have used (3.2).
Then, we have asymptotic stability if
Πn e2B2k+1T2k+1(c +T B ) 0, for n ,
k=0 k 2k+1 2k+1 −→ → ∞
or equivalently
ln Πn e2B2k+1T2k+1(c +T B ) , for n ,
k=0 k 2k+1 2k+1 −→ −∞ → ∞
h i
namely under the assumption (3.6). This concludes the proof.
Remark 3.4 In particular, (3.6) is verified if the following conditions are satisfied:
∞ ∞
B T < + and lnc = . (3.15)
2n+1 2n+1 n
∞ −∞
n=0 n=0
X X
Indeed, it is easy to see that (3.15) is equivalent to
∞ ∞
B T < + and ln(c +B T ) = (3.16)
2n+1 2n+1 n 2n+1 2n+1
∞ −∞
n=0 n=0
X X
and that (3.16) implies (3.6).
Therefore,from(3.15),wehave stability if (t) L1(0,+ ) and,forinstance,thelength
kB k ∈ ∞
of the good intervals I is greater than a fixed time T¯, T¯ > T∗ and T¯ τ, namely
2n
≥
T T¯, n IN.
2n
≥ ∀ ∈
Indeed, in this case there exists c¯ (0,1) such that 0 < c < c¯.
n
∈
If we assume that the length of the delay intervals, namely the time intervals where the
delay feedback is present, is lower than the time delay τ, that is
T τ, n IN. (3.17)
2n+1
≤ ∀ ∈
we can prove another asymptotic stability result which is, in some sense, complementary to the
previous one.
In this case we can directly work with U(t) instead of passing trough the function F().
k k ·
We can give the following preliminary estimates on the time intervals I , n IN.
2n+1
∈
Proposition 3.5 Assume 1), 2). Moreover assume T τ and T τ , n IN. Then,
2n+1 2n
≤ ≥ ∀ ∈
for t I ,
2n+1
∈ d
U(t) 2 B U(t) 2 +B U(t ) 2. (3.18)
2n+1 2n+1 2n
dtk k ≤ k k k k
6
Proof: By differentiating U(t) 2 we get
k k
d
U(t) 2 =2 U(t), U(t) +2 U(t), (t)U(t τ) .
dtk k h A i h B − i
Then, by using the dissipativness of the operator ,
A
d
U(t) 2 2 U(t), (t)U(t τ) .
dtk k ≤ h B − i
Hence, from 2),
d
U(t) 2 B U(t) 2 +B U(t τ) 2.
2n+1 2n+1
dtk k ≤ k k k − k
We can now conclude observing that since T τ and T τ , then for t I it is
2n+1 2n 2n+1
≤ ≥ ∈
t τ I . Then, since U(t) is decreasing in I , the estimate in the statement is proved.
2n 2n
− ∈ k k
The stability result follows.
Theorem 3.6 Assume 1), 2), T τ and T τ , n IN. Moreover assume T > T∗,
2n+1 2n 2n
≤ ≥ ∀ ∈
for all n IN, where T∗ is the time defined in (3.1). If
∈
∞
ln eB2n+1T2n+1(c +1 e−B2n+1T2n+1) = , (3.19)
n
− −∞
n=0
X (cid:2) (cid:3)
then every solution U of (1.2) satisfies U(t) 0 for t + .
k k → → ∞
Proof. For t I = [t ,t ), from estimate (3.18) we have
2n+1 2n+1 2n+2
∈
t
U(t) 2 eB2n+1(t−t2n+1) U(t ) 2 +B U(t ) 2e−B2n+1(s−t2n+1)ds .
2n+1 2n+1 2n
k k ≤ k k k k
n Zt2n+1 o
Then we deduce
U(t) 2 eB2n+1T2n+1 U(t ) 2 +eB2n+1(t−t2n+1) U(t ) 2 1 e−B2n+1(t−t2n+1) ,
2n+1 2n
k k ≤ k k k k −
h i
and therefore
U(t) 2 eB2n+1T2n+1 U(t ) 2 +eB2n+1T2n+1 U(t ) 2 U(t ) 2,
2n+1 2n 2n
k k ≤ k k k k −k k
for t I = [t ,t ), n IN.
2n+1 2n+1 2n+2
∈ ∈
Now we use the estimate (3.2) obtaining
U(t ) 2 eB2n+1T2n+1 c +1 e−B2n+1T2n+1 U(t ) 2, n IN.
2n+2 n 2n
k k ≤ − k k ∈
Thus, (cid:0) (cid:1)
1
U(t ) Πn eB2k+1T2k+1(c +1 e−B2k+1T2k+1) 2 U . (3.20)
k 2n+2 k ≤ k=0 k − k 0k
Then the asymptotic stability hresult follows if i
Πn eB2k+1T2k+1 c +1 e−B2k+1T2k+1 0, for n ,
k=0 k − → → ∞
namely if (cid:0) (cid:1)
∞
ln eB2n+1T2n+1(c +1 e−B2n+1T2n+1) , for n .
n
− → −∞ → ∞
nX=0 h i
7
Remark 3.7 Observe that, when the odd intervals I have length lower or equal than the
2n+1
time delay τ , the assumption (3.19) is a bit less restrictive than (3.6). Indeed,
eB2n+1T2n+1(c +1 e−B2n+1T2n+1) < e2b2n+1T2n+1(c +B T ), n IN.
n n 2n+1 2n+1
− ∀ ∈
Remark 3.8 Arguing as in Remark 3.4 we can show that condition (3.19) is verified, in par-
ticular, if (3.15) holds true.
4 Exponential stability
Under additional assumptions on the coefficients T ,B ,c , exponential stability results
n 2n+1 n
hold true.
Theorem 4.1 Assume 1), 2). Moreover, assume
T = T0 n IN, (4.1)
2n
∀ ∈
with T0 τ and T0 > T∗, where T∗ is the constant defined in (3.1),
≥
T = T˜ n IN (4.2)
2n+1
∀ ∈
and
sup e2B2n+1T˜(c+B T˜)= d < 1, (4.3)
2n+1
n∈IN
where c = Me−µT0. Then, there exist two positive constants C,α such that
U(t) Ce−αt U , t > 0, (4.4)
0
k k ≤ k k
for any solution of problem (1.2).
Proof. Note that, from the definition of the constant c, estimate (3.2) holds with c = c, n
n
∀ ∈
IN. Thus, from (4.3) and (3.11) we obtain
U(T0+T˜) d21 U0 ,
k k ≤ k k
and then,
U(n(T0+T˜)) dn2 U0 , n IN.
k k ≤ k k ∀ ∈
Therefore, U(t) satisfies an exponential estimate like (4.4) (see Lemma 1 of [12]).
k k
Concerning the case of small delay intervals, namely I τ, n IN, one can state
2n+1
| | ≤ ∀ ∈
the following asymptotic stability result.
Theorem 4.2 Assume 1), 2). Moreover assume
T = T0 n IN,
2n
∀ ∈
with T0 τ and T0 > T∗, where the time T∗ is defined in (3.1),
≥
T = T˜, with T˜ τ n IN (4.5)
2n+1
≤ ∀ ∈
8
and
sup eB2n+1T˜(c+1 e−B2n+1T˜) = d< 1, (4.6)
−
n∈IN
where c = Me−µT0. Then, there exist two positive constants C,α such that
U(t) Ce−αt U , t > 0, (4.7)
0
k k ≤ k k
for any solution of (1.2).
Proof. The proof is analogous to the one of Theorem 4.1.
5 Intermittent anti–damping
With analogous technics we can also deal with an intermittent anti–damping term. More
precisely, let us consider the model
U (t)= U(t)+ (t)U(t) t > 0,
t A B (5.1)
U(0) = U ,
0
(cid:26)
where τ is the time delay, the initial datum U belongs to and, for t > 0, (t) is a bounded
0
H B
operator from such that
H
(t)U,U 0, U .
hB i ≥ ∀ ∈ H
Thus (t)U(t) is an anti–damping term (cfr. [14]). In particular we consider an intermittent
B
feedback, that is we assume that there exists an increasing sequence of positive real numbers
t , with t = 0, such that
n n 0
{ }
3) (t)= 0 t I = [t ,t ),
2n 2n 2n+1
B ∀ ∈
4) (t)= t I = [t ,t ).
2n+1 2n+1 2n+1 2n+2
B D ∀ ∈
We denote D = , n IN.
2n+1 2n+1 L(H)
kD k ∈
As before, denote by T the length of the interval I , that is
n n
T = t t , n IN.
n n+1 n
− ∈
Note that Proposition 3.1, which gives an observability estimate on the intervals I where
2n
the anti-damping is not present, still holds true. Concerning the time intervals I where the
2n+1
anti-damping acts one can obtain the following estimate.
Proposition 5.1 Assume 3) and 4). For every solution of problem (5.1),
d
U(t) 2 2D U(t) 2, t I = [t ,t ], n IN.
2n+1 2n+1 2n+1 2n+2
dtk k ≤ k k ∈ ∀ ∈
Proof. Being dissipative, the estimate follows immediately from 3).
A
From Proposition 5.1 we deduce an asymptotic stability result.
9
Theorem 5.2 Assume 3), 4). Moreover assume T > T∗, for all n IN, where T∗ is the time
2n
∈
defined in (3.1). If
∞
ln e2D2n+1T2n+1c = , (5.2)
n
−∞
n=0
X (cid:0) (cid:1)
thentheproblem(5.1)isasymptotically stable,thatisanysolutionU of(5.1)satisfies U(t) 0
k k →
for t + .
→ ∞
Proof. From Proposition 5.1 we have the estimate
d
U(t) 2 2D U(t) 2, t I = [t ,t ], n IN.
2n+1 2n+1 2n+1 2n+2
dtk k ≤ k k ∈ ∀ ∈
This implies
U(t ) 2 e2D2n+1T2n+1 U(t ) 2, n IN. (5.3)
2n+2 2n+1
k k ≤ k k ∀ ∈
Then, from estimate (3.2) which is always valid of course in the time intervals without damping,
U(t ) 2 e2D2n+1T2n+1c U(t ) 2, n IN. (5.4)
2n+2 n 2n
k k ≤ k k ∀ ∈
By repeating this argument we obtain
U(t ) 2 Πn e2D2k+1T2k+1c U 2, n IN. (5.5)
k 2n+2 k ≤ k=0 kk 0k ∀ ∈
Therefore, asymptotic stability is ensured if
Πn e2D2k+1T2k+1c 0, for n ,
k=0 k −→ → ∞
or equivalently
ln Πn e2D2k+1T2k+1c , for n .
k=0 k −→ −∞ → ∞
This concludes. (cid:16) (cid:17)
Remark 5.3 In particular (5.2) is verified under the following assumptions:
∞ ∞
D T < + and lnc = . (5.6)
2n+1 2n+1 n
∞ −∞
n=0 n=0
X X
Under additional assumptions on the problem coefficients T ,D ,c , an exponential
n 2n+1 n
stability result holds.
Theorem 5.4 Assume 3), 4) and
T = T0 n IN, (5.7)
2n
∀ ∈
with T0 > T∗, where the time T∗ is defined in (3.1). Assume also that
T = T˜ n IN (5.8)
2n+1
∀ ∈
and
sup e2D2n+1T˜c = d < 1, (5.9)
n∈IN
where, c = Me−µT0. Then, there exist two positive constants C,α such that
U(t) Ce−αt U , t > 0, (5.10)
0
k k ≤ k k
for any solution of problem (5.1).
10