In this section we state and prove our main result. For this purpose we establish several lemmas.
Lemma 3.1 Let be the solution for system (7a)-(7d). Then
(11)
(12)
(13)
Proof See the Appendix. □
Now, we give a property of the energy function E.
Proposition 3.1 The time-derivative of the energy function E in equation (10), along the solution of system (7a)-(7d) satisfies
(14)
for all .
Proof Differentiating the energy function (10) with respect to t, we get
(15)
According to equation (11) and boundary control (7c), we get
(16)
Substituting equation (16) into equation (15) and observing (7a), we obtain
(17)
for all . We obtain equation (14). □
Remark 3.1 From Proposition 3.1, we obtain the energy identity for system (7a)-(7d),
Therefore, the energy E is a decreasing function of time.
During the subsequent stability analysis, we utilize the following inequality.
Lemma 3.2 Let be the solution for system (7a)-(7d). Then
(18)
for all .
Proof Applying the Cauchy-Schwarz inequality, we get
(19)
for all . On the other hand, the definition of energy function (10) implies
for all . It follows from the above inequality that
(20)
Together with (19) and (20), we get equation (18). Hence we complete the proof of Lemma 3.2. □
Now, we present a Gronwall-type lemma (see Komornik [22], pp.124), which will play an essential role when establishing the stabilization result.
Lemma 3.3 Let be a non-increasing function. Assume that there exists a constant such that
Then the following estimation is true, for all ,
We give a priori estimation for the energy function , which was established in [15]. For the sake of completeness, we give the proof here.
Lemma 3.4 The energy function E in equation (10), along the solution of system (7a)-(7d), satisfies
(21)
for all .
Proof We multiply equation (7a) by and do integration over , with respect to x. We obtain
(22)
using equations (13) and (12) in Lemma 3.1. It follows from (10) that
(23)
Since , according to boundary control (7c), we have
(24)
Hence, substituting equation (23) into equation (22) and using equation (24), one has
Since , we complete the proof of Lemma 3.4. □
Lemma 3.5 For any constant , the energy function E along the solution of system (7a)-(7d) satisfies the following estimation, for all ,
(25)
where .
Proof According to inequality (21) in Lemma 3.4, we have, for all ,
(26)
Moreover, using integration by parts, we get
Hence, inequality (26) becomes
(27)
where
Firstly, we estimate and ,
(28)
where the last inequality follows from the fact is a decreasing function. On the other hand,
(29)
Finally, inserting the two inequalities, (28) and (29), in (27), we get inequality (25). This completes the proof of Lemma 3.5. □
We now state the main stabilization result for system (7a)-(7d).
Theorem 3.1 Assume that assumption (H) holds. Then there exist three constants such that, for all ,
(30)
Remark 3.2 It is worth to mention that Theorem 2.4 in [13] can be viewed as a special cases of Theorem 3.1. Indeed, in the linear control case (6), the exponential stability in Theorem 3.1 coincides with the result in [13].
Proof of Theorem 3.1 We distinguish two cases related to the parameter r to establish the energy decay rate.
Case (I): ;
Case (II): .
In Case (I), we choose . According to hypothesis (H), we know that
Hence, from inequality (25) and equation (14), we deduce that, for all ,
(31)
where and is given in Lemma 3.5.
Now we deal with Case (II). In this case, we choose . We first admit the following fact (the proof is given in the Appendix).
Claim 1 For any , we have the following estimates, for all ,
(32)
(33)
Now, inserting inequalities (32) and (33) into (25), we obtain, for all ,
(34)
where , and is given in Lemma 3.5. Now we choose . Then it is obvious that . Hence, inequality (34) becomes
Recalling , we get . Hence, the above inequality is rewritten as
(35)
where the last inequality follows from Remark 3.1. Finally, by letting in (31), (35) and using Lemma 3.3 with , we complete the proof of Theorem 3.1. □
Remark 3.3 According to the proof of Theorem 3.1, it is easy to see that the constants σ, and in Theorem 3.1 can be chosen as, respectively, , , and with , . This means that the coefficients of the exponential or polynomial decay rate are exactly determined only by the initial tension a, the initial energy and the feedback control u. However, in the polynomial decay case, the order of decay rate is determined only by the feedback control u.
Finally, it is shown that the boundary control u stabilizes the nonlinear Kirchhoff string.
Theorem 3.2 Assume that assumption (H) holds. Then there exist two constants such that for all and ,
(36)
where , and , , σ are given in Theorem 3.1.
Proof According to the fact that , for all , we get
(37)
for all and . By combining (37) with (30) in Theorem 3.1, we complete the proof of Theorem 3.2. □