4.1. Generation of a -Semigroup
If , then, under hypotheses (H1)–(H3), the linear operator defined by (3.3) is the infinitesimal generator of strongly continuous semigroup given by
then the -semigoup is exponentially stable, that is, there exist two positives constants such that
In order to apply the Proposition 2.3, we observe that can be written as follows:
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
We have also
From (4.10)-(4.11) into (4.7) we obtain
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.
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).
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.
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
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
Applying Lemma 4.3 and taking into account (4.21) we get with the same manner that for every
and or every
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
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 .
System (3.9) is approximately controllable on if and only if
Now, we are ready to formulate the third main result of this work.
If the following condition
is satisfied; then, under hypotheses (H1)–(H5), for all and all open subset system (3.9) is approximately controllable on .
The proof of this theorem relies on the Criteria 1 and the following lemma.
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
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.