NONLINEAR FEEDBACK STABILIZATION OF A

NONLINEAR FEEDBACK STABILIZATION OF A ROTATING BODY-BEAM SYSTEM VIA TORQUE AND MOMENT CONTROLS ONLY ` KAÏS AMMARI, AHMED BCHATNIA, AND BOUMEDIENE CHENTOUF

Abstract. This article is dedicated to the investigation of the stabilization problem of a flexible beam attached to the center of a rotating disk. Contrary to previous works on the system, we assume that the feedback law contains a nonlinear torque control applied on the disk and most importantly only one nonlinear moment control exerted on the beam. Thereafter, it is proved that the proposed controls guarantee the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk and standard assumptions on the nonlinear functions governing the controls. We used here the strategy of Alabau-Boussouira [1, 2].

Contents 1.

Introduction

1

2.

Preliminaries

4

3.

Uniform stability of a subsystem

5

4.

Examples

9

4.1.

Example 1

10

4.2.

Example 2

10

5.

Stability of the closed-loop system

10

6.

Conclusion

12

References

12

1. Introduction Over the last decades, the stabilization problem of coupled elastic and rigid parts systems has attracted a huge amount of interest and in particular the rotating disk-beam system which arises in the study of large-scale flexible space structures [7]. It consists of a flexible beam (B), which models a flexible robot arm, clamped at one end to the center of a disk (D) and free at the other end (see Figure 1). Additionally, it is assumed that the center of mass of the disk is fixed in an inertial frame and rotates in that frame with a 2010 Mathematics Subject Classification. 35B35, 35R20, 93D20, 93C25, 93D15. Key words and phrases. Nonlinear stabilization, Rotating body-beam system, Dissipative systems, Energy decay rates.

1

2

` KAÏS AMMARI, AHMED BCHATNIA, AND BOUMEDIENE CHENTOUF

nonuniform angular velocity. Under the above assumptions, the motion of the whole structure is governed by the following nonlinear hybrid system (see [7] for more details):

(1.1)

   ρytt (x, t) + EIyxxxx (x, t) = ρω 2 (t) y(x, t),       y(0, t) = yx (0, t) = 0,       yxxx (`, t) = F(t),     yxx (`, t) = M(t), !) ( Z `   d   ω(t) Id + ρy 2 (x, t) dx = T (t),    dt 0       y(x, 0) = y0 (x), yt (x, 0) = y1 (x),      ω(0) = ω ∈ R, 0

(x, t) ∈ (0, `) × (0, ∞), t > 0, t > 0, t > 0, t > 0, x ∈ (0, `),

in which y is the beam’s displacement in the rotating plane at time t with respect to the spatial variable x and ω is the angular velocity of the disk. Furthermore, ` is the length of the beam and the physical parameters EI, ρ and Id are respectively the flexural rigidity, the mass per unit length of the beam, and the disk’s moment of inertia. Lastly, M(t) and F(t) are respectively the moment and force control exerted at the free-end of the beam, while the torque control T (t) acts on the disk.

x

B

D

O

Figure 1. The disk-beam system

y

NONLINEAR FEEDBACK STABILIZATION

3

After the pioneer work of Baillieul & Levi [7], the stabilization problem of the rotating system has been the object of a considerable mathematical endeavor (see for instance [8, 27, 23, 24, 21, 19, 11], [10]-[18] and the references therein). Instead of surveying the vast literature on this subject, we are going to highlight only those closely related to the context of our problem. To the best of our knowledge, very few works have been devoted to nonlinear stabilization of the system (1.1). Indeed, the authors in [19] constructed a nonlinear feedback torque control law T (t) which globally asymptotically stabilizes the system. Later, exponential stabilization results have been established in [11] via a class of nonlinear boundary and torque controls defined below

(1.2)

   F(t) = g (yt (1, t)) ,   M(t) = −f (ytx (1, t)),     T (t) = −γ(ω(t) − $),

where $ ∈ R and f, g and γ are nonlinear real functions satisfying some standard hypotheses (see [11] for more details). It is important to point out that the presence of the force control F(t) = g (yt (1, t)) was unavoidable to obtain the exponential stability in [11] and hence it remained an open question whether the result is still valid if no force control is present in the feedback law (1.2). It is worth mentioning that the boundary stabilization of elastic systems has been the subject of extensive works. For instance, the case of the wave equation with nonlinear boundary feedback has been considered in [20, 26] for the asymptotic stability properties (see also [29]). We also note that the question of the uniform stabilization or pointwise stabilization of the Euler-Bernoulli beams or serially connected beams has been studied by a number of authors (cf. F. Alabau-Boussouira [1, 2], K. Ammari and M. Tucsnak [4], K. Ammari and S. Nicaise [5], M. A. Ayadi, A. Bchatnia, M. Hamouda and S. Messaoudi [6], G. Chen, M. C. Delfour, A. M. Krall and G. Payre [9], J. L. Lions [22], K. Nagaya [25] etc.). The novelty of the present work is to provide a positive answer to such a question. To be more precise, the contribution of this paper lies in showing that despite the absence of the boundary force control in (1.2), the closed-loop system possesses the very desirable property of exponential stability as long as the angular velocity is bounded (see (2.5)) and the functions f and γ obey standard conditions. Now, let us briefly present an overview of this paper. In Section 2, we formulate the closed-loop system in an abstract first-order evolution equation in an appropriate state space. Section 3 deals with the uniform exponential stability of a subsystem associated to the closed-loop system. In section 4, some examples are provided. Section 5 is devoted to the proof of the stability of the closed-loop system. Finally, the paper closes with conclusions and discussions.


4

2. Preliminaries First, one can use an appropriate change of variables so that one may assume that ` = 1. Then, the closed loop system (1.1)-(1.2) can be written as follows:  2  (x, t) ∈ (0, 1) × (0, ∞),  ρytt + EIyxxxx = ρω (t)y,     y(0, t) = y (0, t) = y (1, t) = 0,  t > 0, x xxx       t > 0,  yxx (1, t) = −f (ytx (1, t)),  Z 1    y(t)yt (t)dx −γ(ω(t) − $) − 2ρω(t) (2.3) 0  , t > 0, ω(t) ˙ =  Z 1    2  y(t) dx I + ρ  d   0     y(x, 0) = y (x), y (x, 0) = y1 (x), x ∈ (0, 1),  0 t     ω(0) = ω0 ∈ R. Thereafter, let z(·, t) = yt (·, t) and consider the state variable (φ(t), ω(t)) = (y(·, t), z(·, t), ω(t)). Next, for n ∈ N, n ≥ 2, we denote by Hln = {u ∈ H n (0, 1); u(0) = ux (0) = 0}. Now, consider the state space X defined by X = Hl2 × L2 (0, 1) ×R = H × R, | {z } H

equipped with the following real inner product (the complex case is similar): h(y, z, ω), (˜ y , z˜, ω ˜ )iX (2.4)

= h(y, z), (˜ y , z˜)iH + ω ω ˜ Z 1 = EIyxx y˜xx − ρ$2 y y˜ + ρz z˜ dx + ω ω ˜. 0

Note that the norm induced by this scalar product is equivalent to the usual one of the Hilbert space H 2 (0, 1) × L2 (0, 1) × R provided that (see [11]) (2.5)

p | $ |< 3 EI/ρ.

Now, we define a nonlinear operator A by (2.6)

D(A) = (y, z) ∈ Hl4 × Hl2 ; yxxx (1) = 0, yxx (1) = −f (zx (1)) ,

and (2.7)

EI 2 A(y, z) = −z, yxxxx − $ y . ρ


5

Whereupon the system (2.3) can be formulated in X as follows ˙ (2.8) φ(t), ω(t) ˙ + (A + B) (φ(t), ω(t)) = 0, in which   A(φ, ω) = (Aφ, 0) , with D(A) = D(A) × R,    

(2.9)

γ (ω − $) + 2ρ ω(t) < y, z >L2 (0,1) 0, ($2 − ω 2 )y, Id + ρkyk2L2 (0,1)

     B(φ, ω) =

! .

We note that we will adapt here the approach of Alabau-Boussouira in [1, 2]. Hence, we suppose that the functions f and γ satisfy the following conditions: H.I: f : R → R is a continuous non-decreasing function such that f (0) = 0. H.II: There exists a continuous strictly increasing odd function f0 ∈ C([0, +∞)), continuously differentiable in a neighborhood of 0 such that   f (|s|) ≤ |f (s)| ≤ f −1 (|s|), 0 0  c |s| ≤ |f (s)| ≤ c |s|, 1

2

for all

|s| ≤ 1,

for all

|s| ≥ 1,

where ci > 0 for i = 1, 2. H.III: The function γ is Lipschitz on each bounded subset of R and for some K > 0, γ(x)x ≥ 0, | γ(x) | ≥ K | x |, ∀ x ∈ R. Moreover, we define a function H by (2.10)

H(x) =

√

√ xf0 ( x), ∀ x ≥ 0.

Thanks to the assumptions H.I-H.II, the function H is of class C 1 and is strictly convex on (0, r2 ], where r > 0 is a sufficiently small number. 3. Uniform stability of a subsystem In this section, we shall deal with a subsystem    ρytt + EIyxxxx − ρ$2 y = 0, 0 < x < 1, t > 0,      y(0, t) = yx (0, t) = yxxx (1, t) = 0, t > 0, (3.11)   yxx (1, t) = − f (ytx (1, t)), t > 0,      y(x, 0) = y (x), y (x, 0) = y (x), 0 < x < 1, 0 t 1 or equivalently (3.12)

  φ(t) ˙ + Aφ(t) = 0,  φ(0) = φ , 0

where φ = (y, z), z = yt and A is defined by (2.6)-(2.7).


6

We have the following proposition whose proof can be found in [11, Lemma 5 and Proposition 3] Proposition 3.1. Assume that the condition (2.5) holds, that is, | $ |< 3

p EI/ρ. Then, for any function

f satisfying solely the assumption H.I, the operator A defined by (2.6)-(2.7) is m-accretive in H with dense domain. Additionally, we have: (1) For any initial data φ0 = (y0 , y1 ) ∈ D(A), the system (3.12) admits a unique solution φ(t) = (y(t), z(t)) ∈ D(A) such that (y, yt ) ∈ L∞ (R+ ; D(A)), and

d (y, yt ) ∈ L∞ (R+ ; H). dt

The solution φ = (y, z) is given by φ(t) = e−At φ0 , for all t ≥ 0 where e−At

t≥0

is the semigroup

generated by −A on D(A) = H. Moreover, the function t 7−→ kAφ(t)kH is decreasing. (2) For any initial data φ0 = (y0 , y1 ) ∈ D(A) = H, the equation (3.12) admits a unique mild solution φ(t) = e−At φ0 which is bounded on R+ by ||φ0 ||H and φ = (y, yt ) ∈ C 0 (R+ ; H). (3) The semigroup e−At

t≥0

is asymptotically stable in H.

The first main result of this article is: Theorem 3.1. Let us suppose that the conditions (2.5) and H.I-H.II hold. Then there exist positive constants k1 , k2 , k3 and ε0 such that the modified energy E0 (t) associated to (3.11) satisfies for all (y0 , y1 ) ∈ D(A) E0 (t) (3.13)

Z

1

:=

1 2

≤

k3 H1−1 (k1 t + k2 ) E0 (0), ∀ t ≥ 0,

2 2 2 ρyt2 (x, t) + EIyxx (x, t) − ρ$2 y 2 + ρytt (x, t) + EIytxx (x, t) − ρ$2 yt2 dx

0

where Z

1

1 ds, H2 (t) = t H 0 (ε0 t). t H2 (s) Here H1 is a strictly decreasing and convex function on (0, 1], with lim H1 (t) = +∞. H1 (t) =

t→0

Hereby, the remaining task is to provide a proof of the above theorem. To fulfill this objective, we shall establish several lemmas. First, recall that the energy associated to the system (3.11) is defined by Z 1 1 1 2 (3.14) E(t) := k(y(t), yt (t))k2H = ρyt2 (x, t) + EIyxx (x, t) − ρ$2 y 2 dx. 2 2 0 We have Lemma 3.1. Let y be a solution of the system (3.11). Then, the functional E satisfies (3.15)

E 0 (t) = −EIytx (1, t)f (ytx (1, t)) ≤ 0.


7

Proof. By multiplying the first equation in (3.11), by yt , using the integration by parts with respect to x over (0, 1), the boundary conditions and the hypotheses H.I, we obtain (3.15).

The second lemma is Lemma 3.2. Let (y, yt ) be a solution of the system (3.11). Then, there exists θ > 0 such that the functional Z 1 (3.16) F (t) := 2 xyt (x, t)yx (x, t) dx, t ≥ 0, 0

verifies the following estimate F 0 (t) ≤

(3.17)

1 3

Z

1 2 ytxx (x, t) dx + (1 + θ)

0

EI 2 f (yxt (1, t)) ρ

− Kk(y(t), yt (t)k2H , a.e. t ≥ 0. Proof. We differentiate F with respect to t, we get (3.18)

F 0 (t)

EI 2 EI yx (1, t)f (yxt (1, t)) + yt2 (1, t) + f (yxt (1, t)) + $2 y 2 (1, t) ρ ρ Z 1 EI 2 yxx (x, t) + $2 y 2 (x, t) dx, a.e. t ≥ 0. yt2 (x, t) + 3 ρ 0 −2

= −

Using Young’s inequality, we obtain : (3.19)

−2

EI EI 2 EI 2 yx (1, t)f (yxt (1, t)) ≤ y (1, t) + θ f (yxt (1, t)), ρ ρθ x ρ

for any θ > 0. On the other hand, we have Z

1

yx (1, t) = yx (1, t) − yx (0, t) =

yxx (x, t) dx ≤ kyxx kL2 (0,1) . 0

Consequently, we obtain (3.20)

yx2 (1, t)

Z ≤

1 2 yxx (x, t) dx.

0

Similarly, we use the fact y(0, t) = 0 to obtain Z Z 1 1 2 1 1 2 (3.21) y 2 (1, t) ≤ yxx (x, t) dx and yt2 (1, t) ≤ y (x, t) dx. 3 0 3 0 txx Combining (3.16), (3.19), (3.20) and (3.21), we get Z 1 1 2 EI 2 ytxx dx + (1 + θ) (3.22) F 0 (t) ≤ f (yxt (1, t)) 3 0 ρ Z 1 1 1 ρ$2 2 + −ρyt2 (x, t) + + − 3 EIyxx (x, t) − ρ$2 y 2 (x, t) dx, a.e. t ≥ 0. ρ 0 θ 3EI Recalling the condition (2.5), one can choose θ such that

1 2θ

+

ρ$ 3EI

− 3 < 0 . Then, we deduce the existence

of a positive constant K such that the inequality (3.17) is verified. Now, we can are able to achieve the main objective of this section:


8

Proof of Theorem 3.1. First of all, recall that (see (3.13) and (3.14)) 1 E0 (t) := E(t) + k(yt , ytt )k2H , 2

(3.23) and (3.24)

E1 (t) := E0 (t) + ε F (t),

in which F (t) is defined by (3.16). It follows from (3.22) and (3.23) that E10 (t) ≤

(3.25)

+

2 2 −EIytx (1, t)f (ytx (1, t)) − EIyttx (1, t)f 0 (ytx (1, t)) +

ε(1 + θ)

ε 3

Z

1

ytxx 2 (x, t) dx

0

EI 2 f (yxt (1, t)) − εKk(y, yt k2H , a.e. t ≥ 0. ρ

On the other hand, thanks to the assumptions H.I-H.II, we obtain (1 + θ) +(1 + θ)

EI EI 2 f (yxt (1, t)) ≤ (1 + θ) c2 yxt (1, t)f (yxt (1, t))1{|ytx |≥1} ρ ρ

EI 2 EI 2 y (1, t) + f02 (yxt (1, t)) 1{|ytx |≤1} − (1 + θ) y (1, t)1{|ytx |≤1} ρ xt ρ xt

2EI −1 1 H (yxt (1, t)f0 (yxt (1, t))) 1{|ytx |≤1} . ≤ −(1 + θ) c2 E 0 (t) + (1 + θ) ρ ρ Furthermore, as E0 (t) and E1 (t) are equivalent, we get E00 (t) ≤ −εKE0 (t) + (1 + θ)

(3.26)

2EI −1 H (yxt (1, t)f0 (yxt (1, t))) 1{|ytx |≤1} , a.e. t ≥ 0. ρ

Thereafter, for ε0 < r2 , we have H 0 ≥ 0 and H 00 ≥ 0 over (0, r2 ]. Moreover, E 0 ≤ 0. Consequently, the functional R defined by R(t) := H

0

E0 (t) E0 (t) + δE(t), ε0 E0 (0)

is equivalent to E0 (t), for δ > 0. E 0 (t) E0 (t) In addition, we have ε0 E00(0) H 00 ε0 E E0 (t) ≤ 0. This, together with (3.26), allows us to conclude that 0 (0) E00 (t) 00 E0 (t) E0 (t) H ε0 E0 (t) + H 0 ε0 E00 (t) + δE 0 (t) E0 (0) E0 (0) E0 (0) E0 (t) 2EI 0 E0 (t) 0 −εKE0 (t)H ε0 + (1 + θ) H ε0 H −1 (yxt (1, t)f0 (yxt (1, t))) + δE 0 (t). E0 (0) ρ E0 (0)

R0 (t) = ε0 (3.27)

≤

Our objective now is to estimate the second term in the right-hand side of (3.27). For that purpose, we introduce, as in [1], the convex conjugate H ∗ of H defined by (3.28)

H ∗ (s) = s(H 0 )−1 (s) − H((H 0 )−1 )(s) for s ∈ (0, H 0 (r2 )),

and H ∗ satisfies the following Young inequality: (3.29)

AB ≤ H ∗ (A) + H(B) for A ∈ (0, H 0 (r2 )), B ∈ (0, r2 ).


9

E0 (t) Now, taking A = H 0 ε0 E and B = H −1 (yxt (1, t)f0 (yxt (1, t))), we obtain 0 (0)

2EI ∗ E0 (t) 0 R (t) ≤ −εKE0 (t)H + (1 + θ) H H ε0 ρ E0 (0) 2EI +(1 + θ) H H −1 (yxt (1, t)f0 (yxt (1, t)) + δE 0 (t) ρ 2EI E0 (t) 0 E0 (t) E0 (t) + (1 + θ) ε0 H ε0 ≤ −εKE0 (t)H 0 ε0 E0 (0) ρ E0 (0) E0 (0) 2EI E0 (t) 2EI −(1 + θ) H ε0 + (1 + θ) yxt (1, t)f0 (yxt (1, t)) + δE 0 (t) ρ E0 (0) ρ 2EI E0 (t) 0 E0 (t) E0 (t) 0 + (1 + θ) ε0 H ε0 ≤ −εKE0 (t)H ε0 E0 (0) ρ E0 (0) E0 (0) 2 0 −(1 + θ) E (t) + δE 0 (t). ρ 0

0

E0 (t) ε0 E0 (0)

With a suitable choice of ε0 and δ, we deduce from the last inequality that E0 (t) 0 E0 (t) E0 (t) 2EI 0 ε0 H ε0 ≤ −kH2 , (3.30) R (t) ≤ − εKE0 (0) − (1 + θ) ρ E0 (0) E0 (0) E0 (0) 0 where k = εKE0 (0) − (1 + θ) 2EI ρ ε0 > 0 and H2 (s) = sH (ε0 s).

Since E0 (t) and R(t) are equivalent, there exist positive constants a1 and a2 such that a1 R(t) ≤ E0 (t) ≤ a2 R(t). We set now S(t) =

a1 R(t) E0 (0) .

It is clear that S(t) ∼ E0 (t).

Taking into consideration the fact that

H20 (t), H2 (t) > 0 over (0, 1] (this is a direct consequence of strict convexity of H on (0, r2 ]) we deduce from (3.30) that S 0 (t) ≤ −k1 H2 (S(t)),

for all t ∈ R+ ,

with k1 > 0. Integrating the last inequality, we obtain H1 (S(t)) ≥ H1 (S(0)) + k1 t. Finally, since H1−1 is decreasing (as for H1 ), we have S(t) ≤ H1−1 (k1 t + k2 ) ,

with k2 > 0.

Invoking once again the equivalence of E0 (t) and R(t), we deduce the desired result (3.13).

4. Examples The objective of this section is to apply the inequality (3.13) on some examples in order to show explicit stability results in term of asymptotic profiles in time. To proceed, we choose the function H strictly convex near zero.


10

4.1. Example 1. Let f be a function that satisfies 1

c3 min(|s|, |s|p ) ≤ |f (s)| ≤ c4 max(|s|, |s| p ), with some c3 , c4 > 0 and p ≥ 1. For f0 (s) = csp , hypothesis H.II is verified. Then H(s) = cs

p+1 2

.

Therefore, we distinguish the following two cases: and H1−1 (t) = exp(−ct). • If p = 1, we have f0 is linear and H2 (s) = cs, H1 (s) = − ln(s) c Applying (3.13) of Theorem 3.1, we conclude that E0 (t) ≤ k3 exp(−c(k1 t + k2 )). p−1

• If p > 1; this implies that f0 is nonlinear and we have H2 (s) = cs 2 and Z 1 p−1 1 − p−1 1 H1 (t) = s 2 ds = 0 (1 − t 2 ). c c t Therefore, 2

H1−1 (t) = (c0 t + 1)− p−1 . Using again (3.13), we obtain 2

E0 (t) ≤ H1−1 (k1 t + k2 ) = (c0 (k1 t + k2 ) + 1)− p−1 . 4.2. Example 2. Let f0 (s) =

1 s

exp(− s12 ). Following the same steps as in example 1, one can conclude

that the energy of (3.11) satisfies E0 (t) ≤ ε ln

k1 t + k2 + c exp( ε10 )

!!−1

c

.

Remark 4.1. We also know that the system (3.11) is exponentially stable if f is linear. This has been proved by H. Laousy and al. [21], They used the frequency method as it seems that the multipliers method does not lead to the result. 5. Stability of the closed-loop system Throughout this section, we shall assume without loss of generality that the physical parameters EI, ρ are unit. Additionally, we shall suppose that (2.5), H.I and H.III are fulfilled. Moreover, the hypothesis H.II is assumed to be satisfied with f0 (s) = cs (see Example 1) and hence by invoking Theorem 3.1, the subsystem (3.11) is uniformly exponentially stable. This implies that there exist positive constants M and η such that (5.31)

ke−At kL(H) ≤ M e−ηt , ∀t ≥ 0.


11

The aim of this section is to show the exponential stability of the closed-loop system (2.8). We shall use the same arguments as put forth in [11] but with a number of changes born out of necessity. First of all, thanks to the assumptions (2.5) and H.I-H.III we can claim that for any initial data (φ, ω) ∈ D(A), the system has a unique global solution on (0, ∞) [11]. Moreover, the functional Z 1 Z 1 Id 1 1 2 2 2 2 2 (5.32) yt + yxx − $2 y 2 dx y dx + (ω(t) − $) (ω(t) − $) + V(t) = 2 2 2 0 0 is a Lyapunov function and satisfies along the solutions of (2.8) d (5.33) V(φ, ω)(t) = −yxt (1, t) f yxt (1, t) − (ω(t) − $) γ ω(t) − $ ≤ 0 dt almost everywhere on [0, +∞) [11]. Secondly, we consider the subsystem   ψ(t) ˙ + Aψ(t) + B(t, ψ(t)) = 0, t > 0,  ψ(0) = ψ , 0

where ψ = (y, z), ψ0 ∈ D(A) and B(t, ψ) = ($2 − ω 2 (t))P (ψ),

(5.34)

in which P is the compact operator on H defined by P (y, z) = (0, y), ∀(y, z) ∈ H. Moreover, ω(t) is the second component of the solution (ψ(t), ω(t)) of the global system (2.8) with initial data (ψ0 , ω0 ) ∈ D(A). Accordingly, the solution ψ(t) can be written as follows Z t (5.35) ψ(t) = e−(t−T0 )A ψ(T0 ) + e−(t−ν)A (ω 2 (ν) − $2 )P ψ(ν) dν, T0

for any t ≥ T0 . Clearly, it follows from (5.34) that kB(t, ψ(t))kH ≤ |$2 − ω 2 (t)|kψ(t)kH . Furthermore, in the light of (5.32), (5.33) and the assumptions H.I-H.III, we have $ − ω(·) ∈ L2 ([0, ∞[; R) ∩ L∞ ([0, ∞[; R) and hence limt→+∞ ω(t) = $ [11]. Herewith, for any ε > 0, there exists T0 sufficiently large such that for each t ≥ T0 , we have (5.36)

|ω 2 (t) − $2 | < ε.

Combining (5.36) as well as (5.31) and (5.35) and applying Gronwall’s Lemma yields (5.37)

kψ(t)kH ≤ M e−(η−εM )(t−T0 ) kψ(T0 )kH ,

for any t ≥ T0 . Thereafter, one can choose ε so that η − εM > 0, and thus the solution ψ(t) of (2.8) is exponentially stable in H. Finally, amalgamating these properties and going back to (2.8) one can show the


12

exponential convergence of ω(t) − $ in R by means of H.III. The proof runs on much the same lines as that of [11, Theorem 2]. Indeed, we deduce from (2.8) that (5.38)

γ (ω(t) − $) kyk2L2 (0,1) − 2Id ω(t) < y, yt >L2 (0,1) 1 d − γ (ω(t) − $) . (ω(t) − $) = dt Id I I + kyk2 2 d

d

L (0,1)

Multiplying (5.38) by ω(t) − $ and using the assumption H.III, we get 1 d 2 (ω(t) − $) 2 dt (5.39)

≤

(ω(t) − $) γ (ω(t) − $) kyk2L2 (0,1) − 2Id ω(t) (ω(t) − $) < y, yt >L2 (0,1) Id Id + kyk2L2 (0,1)

−

K 2 γ (ω(t) − $) . Id

Lastly, solving the above differential inequality and using (5.37) as well as the boundness of solutions, one can obtain the exponential convergence of ω(t) towards $. To recapitulate, we have proved the second main result: Theorem 5.1. Assume that (2.5), H.I and H.III are fulfilled. We also suppose that H.II holds with f0 (s) = cs. Then, for each initial data (φ0 , ω0 ) ∈ D(A) the solution (φ(t), ω(t)) of the closed-loop system (2.8) tends exponentially to (0, $) in X as t → +∞. Remark 5.1. We note that (with slight modifications of the proof of Theorem 5.1) we obtain the polynomial stability if we suppose that H.II holds with f0 (s) = csp , p > 3. 6. Conclusion This note was concerned with the stabilization of the rotating disk-beam system. We proposed a feedback law which consists of a nonlinear torque control exerted on the disk and only a nonlinear boundary moment control applied on the beam. Then, it is shown that the closed-loop system is exponentially stable provided that the angular velocity of the disk does not exceed a well-defined value and the nonlinear functions involved in the feedback law obey reasonable conditions. This result provides a positive answer to the open problem left in [11]. References [1] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51 (2005), no. 1, 61–105. [2] F, Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems. J. Differ. Equ. 248, (2010), 1473–1517. [3] K, Ammari, A. Bchatnia and K. EL Mufti, Stabilization of the nonlinear damped wave equation via linear weak observability, NoDEA Nonlinear Differential Equations and Applications, 2 (2016), 23:6. [4] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler Beams by Means of a Point Feedback Force, SIAM Journal on Control and Optimization, 4 (2000), 1160–1181.


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UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia E-mail address: [email protected] áire et Ge óme ´trie, UR13ES32, Department of Mathematics, Faculty of Sciences of UR Analyse Non-Line Tunis, University of Tunis El Manar, Tunisia E-mail address: [email protected] Kuwait University, Faculty of Science, Department of Mathematics, Safat 13060, Kuwait E-mail address: [email protected]