4.1. Generation of a
-Semigroup
Theorem 4.1.
If
, then, under hypotheses (H1)–(H3), the linear operator
defined by (3.3) is the infinitesimal generator of strongly continuous semigroup
given by
where
Moreover, if
then the
-semigoup
is exponentially stable, that is, there exist two positives constants
such that
Proof.
In order to apply the Proposition 2.3, we observe that
can be written as follows:
where
Therefore,
and 
Now, we have to verify condition (a) of the Proposition 2.3. We shall suppose that
Then, there exists a set
of complementary projections on
such that
If
is the matrix passage from the canonical basis of
to the basis composed with the eigenvectors of
, then
Hence,
We have also
From (4.10)-(4.11) into (4.7) we obtain
where
As
we get
As
as
then this implies the existence of a positive number
and a real number
such that
for every
Therefore
is a strongly continious semigroup
given by (4.1). We can even estimate the constants
and
as follows.
-
(i)
If
As
, then there exist constants
hence, if we put
we easily obtain
If
. If we put
then we find that
Therefore, the linear operator
generates a strongly continuous semigroup
on
given by expression (4.1).
Finally, if
we have already proved (4.20). Using (4.20) into (4.1) we get that the
-semigoup
is exponentially stable. The expression (4.5) is verfied with
and
is defined by (4.19).
Theorem 4.2.
If
then, under the hypotheses (H1)–(H3), the linear operator
defined by (3.3) is the infinitesimal generator of strongly continuous semigroup exponentially stable
defined by (4.1). Specially, there exist two positives constants
such that
To prove this result, we need the following lemma.
Lemma 4.3.
For every two real positives constants
and
, one has for every 
and for every 
Proof of Lemma 4.3.
It is easy to verify that for every
, for all
.
Let
and
, then we get
Hence, we get (4.23).
Also, it is easy to verify that for every
, for all
. Let
and
, then we get
Hence, from
(4.26) we get
for all
and
, which gives (4.24).
With the same manner we can prove that for every
and every
we have
and consequently, for every two real positives constants
and
and every
we have
Now, we are ready to prove Theorem 4.2.
Proof of Theorem 4.2.
By applying Proposition 2.3 we start from formula (4.12) and we put
where
To estimate
we have in taking into account 
and applying the Lemma 4.3
we get
for all
and
. But we have
, for all
Then we get for every
that
From (4.31)-(4.33) we get
where
and 
Applying Lemma 4.3 and taking into account (4.21) we get with the same manner that for every 
where
and or every 
where
From (4.34)-(4.40) into (4.12) we get
where
is defined by (4.17) and
Using (4.41) into (4.1) we get that the
-semigoup
generated by
is exponentially stable. Expression (4.22) is verfied with
and
is defined by (4.42).
4.2. Approximate Controllability
Befor giving the definition of the approximate controllabiliy for the sytem (3.9), we have the following known result: for all
and
the initial value problem (3.9) admits a unique mild solution given by
This solution is denoted by 
Definition 4.4.
System (3.9) is said to be approximately controllable at time
whenever the set
is densely embedded in
; that is,
The following criteria for approximate controllability can be found in [6].
Criteria 1.
System (3.9) is approximately controllable on
if and only if
Now, we are ready to formulate the third main result of this work.
Theorem 4.5.
If the following condition
is satisfied; then, under hypotheses (H1)–(H5), for all
and all open subset
system (3.9) is approximately controllable on
.
Proof.
The proof of this theorem relies on the Criteria 1 and the following lemma.
Lemma 4.6.
Let
and
be sequences of real numbers such that
,
and
, for all
, then for any
one has
Proof of Lemma 4.6.
By analyticity we get
and from this we get
. Under the assumptions of the lemma we get
as
and so
If
, we divide
by
and we pass
we get
. If
we divide
by
and we pass
and get
. If
, we divide
by
and we pass
and get
But in this we case we can integrate under the symbol of sommation over the intervall
and we get
. Hence
. Continuing this way we see that
for all 
We are now ready to prove Theorem 4.5. For this purpose, we observe that
where
is the
-semigroup generated by
.
Without lose of generality, we suppose that
Hence
where 
Now, suppose for
that
, for all
Then
If (4.46) is satisfied, then (4.50) take the form
Then, from lemma 4.6 we obtain that for
and all 
Since
we get that all 
On the other hand, from Theorem 2.4 we know that
are analytic functions, which implies the analticity of
and
Then we can conclude that for
and all 
Hence
for all
which implies that
This completes the proof of Theorem 4.5.