1 Introduction

In this paper, we consider the following system of a viscoelastic beam in a two-dimensional space with nonlinear tension:

(1)

which is subject to the boundary conditions

(2)

and the initial conditions

(3)

where x and t represent the spatial and time variables, respectively. w(x, t) and v(x, t) are the transverse and longitudinal displacements of the beam at the position x for the time t, respectively. ρ is the uniform mass per unit length of the beam. L is the length of the beam. D, EI, T₀, and EA are the Kelvin-Voigt damping coefficient, bending stiffness, tension, and stiffness of the beam, respectively. The function U(t) is the control force applied at x=L.

In Eq. (1), (h * f)(t) is defined by

The convolution term in Eq. (1) represents the viscoelastic damping, and describes the history, and the kernel involved there is a relaxation function^[1-3].

Remark 1 In Eqs. (1)-(3), the subscripts mean partial derivatives.

Remark 2 The equations are nonlinear partial differential equations, in which and are nonlinear.

Such problems have a wide application in mechanical engineering. The questions related to vibration suppression or reduction have attracted considerable attention in the last decades. Structural control is an interesting research subject. Many authors have focused on such problems, and lots of results concerning the stability and controllability of such structures have been established^[4-19].

Zhang et al.^[20] investigated the existence and stabilization of global weak solutions for a class of second-order hyperbolic hemivariational inequalities in elasticity with a discontinuous nonlinear multi-valued term. In Ref. [21], the global existence and uniqueness of the dissipative Kirchhoff equations

were considered with the nonlinear boundary damping by the Galerkin approximation benefited from the idea of Zhang and Miao^[22]. In Ref. [22], the existence, uniqueness, and uniform stability of strong and weak solutions of the nonlinear wave equation u_tt -Δ u+b(x)u_t +f(u)=0 were proven in the bounded domains with the nonlinear damped boundary condition

the existence by means of the Galerkin method was proven, and the asymptotic behavior was obtained by using the multiplier technique from the idea in Ref. [23]. In Ref. [24], the stabilization of heat flow with the boundary time-varying delay effect was investigated. Under some assumptions, the exponential stability of the solution applying a variable norm technique was proven, and the Lyapunov functional approach was modified. In Ref. [25], the decay properties for the solutions of a class of PDEs were studied with the memory by the Lyapunov functional method. Moreover, it was proven that when the kernels of the convolutions decayed exponentially (resp. polynomially), the energy of the solutions would decay exponentially (resp. polynomially).

Recently, Zhang et al.^[26] proved the existence, uniqueness, and uniform stability of strong and weak solutions for the nonlinear generalized Klein-Gordon equation

in bounded domains under the nonlinear damped boundary condition

with some restrictions on f(u), h(∇u), g(u_t), and b(x). The authors proved the existence and uniqueness by means of the nonlinear semigroup method, and obtained the uniform stabilization by using the multiplier technique.

It has been proven in Refs. [27] and [28] that, for a smooth monotone function h, the solutions to the viscoelastic equations go to zero as t goes to infinity. For the specific behavior of the relaxation function h, one may refer to Refs. [29]-[39] for the subsequent results, where the energy decays exponentially (resp. polynomially) if h decays exponentially (resp. polynomially).

Then, a natural question is raised: how does the energy as the kernel function not necessarily decay exponentially or polynomially? Messaoudi^[40-41] looked at

for b =0 or 1 and h satisfying

(4)

where ξ is a nonincreasing differentiable function. Such a condition is then employed in a series of papers^[42-44].

Guided by the work of Lasiecka and Tataru^[45], another step forward, which was done by considering the condition

(5)

where χ is a convex function satisfying some smoothness properties, was introduced and used^[46-53].

The main objective of this paper is to prove an arbitrary decay result of the solutions to Eqs. (1)-(3) under an appropriate boundary control U(t) applied at the right boundary of the beam.

The main contributions of this paper are as follows:

(ⅰ) A coupled nonlinear dynamic model of a viscoelastic beam for the reduction of transverse and longitudinal vibrations is analyzed.

(ⅱ) A boundary control and a memory term are designed to reduce the transverse and longitudinal vibrations of the viscoelastic beam.

(ⅲ) The Lyapunov direct method is used to obtain the exponential stability under free vibration conditions.

This paper is organized as follows. In Section 2, we present some notations and technical lemmas needed in the proof of our results. In Section 3, we prove the existence and uniqueness of the solutions by using the Faedo-Galerkin method. The decay result under a suitable feedback U(t), acting on the right endpoint of the beam, is proven in Section 4.

2 Hypothesis and preliminary results

In this section, we prepare the necessary material for the proof of our results. We give some notations, hypotheses, and lemmas, which will be needed later.

Let L²(0, L) be the Hilbert space equipped with the inner product (·, ·) and the norm ‖·‖.

(ⅰ) Hypotheses on memory kernel

Here, following Refs. [54] and [55], we shall consider the class of kernels h satisfying the following hypotheses:

(H1) is continuously differentiable, verifying h(0)>0 and

(H2) There is a positive constant γ satisfying

We define the energy functional of Eqs. (1)-(3) by

(6)

where E_k is the kinetic energy defined by

and E_p is the potential energy due to bending and axial deformation^[56] expressed by

(ⅱ) Control

In order to stabilize Eqs. (1)-(3), we propose the control force U(t) as follows:

(7)

where k is a positive constant.

Throughout this paper, we denote the binary operators by ◦ and ◊, e.g.,

(8)

(9)

where u∈ C([0, T]; L²(0, L)).

It is observed that

(10)

where t≥0.

Lemma 1 The energy E(t) given by Eq. (6) satisfies

(11)

Proof Multiply the first equation of Eq. (1) by w_t(x, t), similarly multiply the second equation of Eq. (1) by v_t(x, t), and integrate both equations over (0, L). Then, using the boundary conditions and Eq. (10), we get Eq. (11). This completes the proof of Lemma 1.

We define the modified energy functional of Eqs. (1)-(3) by

(12)

We can prove that the system (1)-(3) is dissipative as stated below.

Lemma 2 The energy (12) satisfies

(13)

Proof Multiply the first and second equations of Eq. (1) by w_t and v_t, respectively, multiply the second equation of Eq. (2) by v_t(L, t), and integrate by parts over (0, L). Then, taking the boundary conditions and hypotheses (H1) and (H2) into account, we get Eq. (13). This completes the proof of Lemma 2.

The assumption (H2) and control law (7) imply that is nonincreasing and

(14)

In the sequel, we prove some additional lemmas which will be very useful in the proof of our results.

Lemma 3 (see Ref. [57]) Let u be a function defined on , which satisfies

Then, the following inequalities hold:

Lemma 4 (see Ref. [56]) Let u ∈ C¹[0, L], which satisfies

Then, the following inequalities hold:

where ‖.‖_∞ is the norm of L^∞[0, L].

Lemma 5 (see Ref. [58]) For all a, b∈ , we have

where δ is a positive constant.

Lemma6 (see Ref. [54]) If w is a solution to Eqs. (1)-(3), assuming that h satisfies Assumptions (H1) and (H2), we have

(15)

(16)

Proof By applying Cauchy-Schwarz's inequality to and , we obtain Eqs. (15) and (16).

3 Well possedness

Now, we state the existence result of Eqs. (1)-(3), which will be proven by using the Faedo-Galerkin method. We define the Hilbert spaces

equipped with the norms

We also define the space W by

which is equipped with the norm

Here, H¹(0, L), H²(0, L), and H⁴(0, L) are usual Sobolev spaces.

Using Poincaré's inequality, we have that ‖u‖_K, ‖u‖_H, ‖u‖_V, and ‖u‖_W are equivalent to the standard norms of H¹(0, L), H²(0, L), H³(0, L), and H⁴(0, L), respectively.

Theorem 1 Let (w₀, v₀) ∈ W× H and (w₁, v₁) ∈ W× L²(0, L). Suppose that Assumptions (H1) and (H2) are satisfied. Then, the system (1)-(3) has a unique solution (w, v) in the sense that

Proof We first prove the existence and uniqueness of the regular solutions to Eqs. (1)-(3). Then, we extend the same result to weak solutions by using density arguments.

The variational problem of Eqs. (1) and (2) is given by: to find (w, v) ∈ V× K such that

(17)

(18)

where (φ, ϕ) ∈ V× K.

3.1 Approximate solutions

Galerkin's approximation is used to show that, for all (φ, ϕ) ∈ V× K, there exists (w, v) ∈ V× K such that Eqs. (17) and (18) hold.

Define (w_i, v_i)_{i≤ m} as the component of a complete orthogonal system W× H.

For each let

for which

We search for the functions

satisfying

(19)

(20)

where

The initial conditions are

3.2 A priori estimates

denote positive constants independent of m, and t∈ [0, T].

Estimate 1 Upper bounds of

For the solutions w^m and v^m, using the same techniques as in Section 2 and Assumption (H2), from (13), we have

(21)

where is the modified energy functional defined by Eq. (12).

Integrating Eq. (21) over (0, t), we have

(22)

Since the initial conditions are sufficiently smooth, there exists a nonnegative constant M₁ which is independent of m and t such that

(23)

Hence, Eq. (23) ensures that

(24)

Estimate 2 Upper bounds of ‖w_tt^m(0)‖² and ‖v_tt^m(0)‖² in the L²-norm

Fix t=0, and take φ=w_tt^m(0) and

in Eqs. (19) and (20), respectively. Then, integrating by parts and applying the compatibility condition EAv_x^m(L, t)=-kv_t^m(L, t) and the boundary conditions, we have

(25)

(26)

From Eqs. (25) and (26), we have

(27)

Estimate 3 Upper bounds of ‖w_tt^m‖² and ‖v_tt^m‖² in the L²-norm

Let us fix t and ξ > 0 such that ξ < T-t. Take the difference of Eqs. (19) and (20) with t:=t+ξ and t=t. Simultaneously replace φ and ϕ with w_t^m(t+ξ)-w_t^m(t) and v_t^m(t+ξ)-v_t^m(t), respectively. Then, from the equality, we have

(28)

where

(29)

(30)

(31)

Integrate R₂ by parts. Note that w_x^m(0, t)=0 and w_x^m(L, t)=0. Then, taking the first estimate into account, from w_t^m(x, t), v_t^m(x, t) and Lemmas 3, 4 and 5, we have

(32)

(33)

(34)

Substitute Eqs. (32), (33), and (34) into Eq. (28), divide both sides by ξ², and take the limit ξ → 0. Then, we have

(35)

(36)

Integrate Eq. (35) over (0, t), take Eq. (36) under consideration, and estimate Eq. (44). Note that the initial values w₀, w₁, v₀, and v₁ are sufficiently smooth. Choose δ =D. Then, we have

Applying Gronwall's lemma to the last inequality, we have

3.3 Passage to the limit

From Eqs. (23) and (24), we have

(37)

(38)

Therefore, we can get some subsequences of (w^m) and (v^m), which we still denoted by (w^m) and (v^m), respectively, such that

(39)

and

3.4 Analysis of the nonlinear terms

Applying the Aubin-Lions compactness^[59] and Eq. (39), we have

(40)

From the last result, Eq. (39), and Lions lemma^[60], we have

We can now pass to the limit in Eqs. (19) and (20) to get a weak solution to Eqs. (1)–(3) (see Refs. [60] and [61]).

Proof of uniqueness Let us define (w₁, v₁) and (w₂, v₂) as two different sets of solutions to the system (1)–(3). The differences between the two sets are w=w₁-w₂ and v=v₁-v₂. It can be seen that w(x, 0)=v(x, 0)=0 and w_t(x, 0)=v_t(x, 0)=0. Therefore,

(41)

First, we start with the second equation in Eq. (41). Take φ =v_t, and integrate over (0, t). Then, using the fact that

we have

(42)

Now, let us go back to the first equation of Eq. (41). Take ϕ =w_t into Eq. (41), and integrate on (0, t). Then, we have

(43)

For the nonlinear term of the right-hand side of Eq. (43), applying Young's inequality and the fact that the function V× K→ L²(0, L) is locally Lipschitz continuous (see Ref. [62]), we have

(44)

We complete with the last term of the right-hand side of Eq. (43), since

(45)

Combining the estimates (44) and (45) with Eq. (43), we have

(46)

Adding Eqs. (43) and (46), then using Gronwall's Lemma and the facts that w(x, 0)=v(x, 0)=0 and w_t(x, 0)=v_t(x, 0)=0, we have

(47)

The proof of Theorem 1 is now completed.

4 Asymptotic behavior

In this section, our goal is to state and prove our decay result under the control force U(t) defined in Eq. (7). To this end, we need to construct a Lyapunov functional satisfying

for some positive constant α. This inequality gives the arbitrary decay of . To pass to E(t), we will need some "equivalence" between and . We define

(48)

where β, β₁, and β₂ are positive constants, is the modified energy given by Eq. (12), and

(49)

(50)

(51)

Under Assumption (H1), we can easily see that

The following proposition shows that E(t) and are equivalent.

Proposition 1 Let and be the functionals defined by Eqs. (48) and (12), respectively. Then, for β, β₁, and β₂ small enough, we have

(52)

where α₁ and α₂ are some positive constants.

Proof Using Young's inequality and Lemma 3, ∀t≥ 0, we have

(53)

Clearly,

(54)

Applying Holder's inequality, Lemma 4, and Eq. (14) to the last term of Eq. (54), we have

(55)

where

The relations (53) and (55) give

(56)

where C₂=ρ L²/2+(ρ L²+Lk) C₁.

For Φ₁, using Young's inequality and Lemma 6, we have

(57)

For Φ₂, applying Young's inequality and Eq. (55), we have

(58)

According to the bounds (56), (57), and (58), we have

where

We take β, β₁, and β₂ so small that λ < 1. This completes the proof.

Lemma 7 Let be the functional defined by Eq. (12). Then, for t≥ 0, the time derivative can be upper bounded by

(59)

Proof Using the control law (7) into the Eq. (13), we obtain Eq. (59).

Lemma 8 Let Φ(t) be the functional given by Eq. (49). Then, for t≥ 0, the time derivative of Φ(t) can be upper bounded by

(60)

Proof A differentiation of Φ(t), using the governing equations of Eq. (1), yields

(61)

where

Integrating A₁ by parts twice and using the boundary conditions, we have

Adding and subtracting the term in the above equality and using Lemma 6, we have

(62)

Using the boundary conditions and integrating the terms A₂ and A₃ by parts, we have

(63)

(64)

Now, substituting Eqs.(62)-(64) into Eq.(61), using Eq.(7), and choosing , we arrive at Eq.(60).

Lemma 9 Let Φ₁(t) be the functional given by Eq. (50). Then, for t≥ 0, the time derivative of Φ₁(t) can be upper bounded by

(65)

where δ_i (i=1, 2, 3) are some positive constants, and

Proof The derivative of Φ₁(t) with respect to the time t gives

Substitute the first governing equation (1) in the third term of the right-hand side of the above equality, we have

(66)

where

Let us estimate these terms for B₁. Similar to Eq. (57), according to Lemma 6, we have

(67)

Integrating the terms B₂ and B₃ by parts, and using the boundary conditions, Young's inequality, and Lemma 6, we have

(68)

(69)

For B₄, integrating by parts, and using the boundary conditions, Young and Holder's inequalities, and Lemma 6, we have

(70)

where C₄ is a positive constant.

Collecting Eqs.(67)–(70), we get Eq.(65).

Lemma 10 Let Φ₂(t) be the functional given by Eq. (51). Then, for t≥ 0, the time derivative of Φ₂(t) can be upper bounded by

(71)

where C₁ is a positive constant.

Proof A differentiation of Φ₂(t) yields

Substitute the second governing equation of Eq. (1) into the last term of the right-hand side of the above equality, and integrate by parts. Then, we have

(72)

(73)

(74)

Combining Eqs.(72)–(74) and Eq.(7), we obtain Eq.(71).

Theorem 2 For the system dynamics described by Eqs. (1)-(3), under Assumptions (H1) and (H2) and the control law (7), it is given that (w₀, v₀) ∈ W× H and (w₁, v₁) ∈ W× L²(0, L), satisfying

(75)

where C₁ is given in Eq. (71). Then,

(76)

where A and α are positive constants.

Proof Take the estimates (13), (60), (65), and (71). For all t≥ t₀ > 0, we have

(77)

where

Substitute Assumption (H1) into the first term of the right-hand side of Eq. (77), and combine the obtained result with the second term. Then, we have

where

Next, we choose δ_i (i=1, 2, 3) so small that

(78)

If β, β₁, and β₂ are small enough and satisfy Eq. (78), we have

Combine the above inequality and Eq. (52), and integrate this differential inequality over (t₀, t). Then, we have

(79)

Now, we estimate on [0, t₀]. For 0≤ t≤ t₀, we have

(80)

(81)

From Eqs. (79)–(81), we find Eq. (76) with and

5 Conclusions

The reduction of the transverse vibrations to a viscoelastic beam in a two-dimensional space with nonlinear tension is studied. A controller mechanism is attached at the right boundary to control the undesirable vibrations, and the multiplier method is used to design a boundary control law ensuring an exponential stabilization result.

Acknowledgements The authors would like to express their gratitude to the anonymous referees for giving constructive and fruitful suggestions.