Stabilization of a nonlinear Euler–Bernoulli beam

In this work, we study the vibration control of a flexible mechanical system. The dynamic of the problem is modeled as a viscoelastic nonlinear Euler–Bernoulli beam. To suppress the undesirable transversal vibrations of the beam, we adopt a control at the right boundary of the beam. This control law is simple to implement. We prove uniform stability of the system using a viscoelastic material, the multiplier method and some ideas introduced in [20]. It is shown that a large range of rates of decay of the energy can be achieved through a determined class of kernels. Unlike most of the existing classes in the market, ours are not necessarily strictly decreasing.


Introduction
Flexible systems exert an increasing influence in different industries and fields. For instance, we may cite flexible manipulators, flexible robot arm and marine risers for oil and gas transportation. The vibration problem of flexible systems has become a crucial topic of research. It is a widespread phenomena in engineering. The origin of these vibrations and their nature might be different. They can cause numerous harmful effects on the production process, including the damage of the equipment with significant financial consequences. There are many approaches to deal with vibration and stabilize flexible systems. Boundary control is the most practical and efficient one. In reference [5], active boundary controls to reduce vibration of an Euler-Bernoulli beam systems in one dimension are considered. In [12], nonlinear vibrations and stability issues are studied. In [4], boundary controllers are used to reduce the vibration of a coupled nonlinear flexible marine riser. Reference [11] considered an adaptive boundary control for an axially moving belt system to eliminate the vibration. In [8] using the direct method of Lyapunov, the exponential stability of a closed-loop system is proven with the help of boundary controls. Kelleche and Tatar [9] designed a nonlinear boundary control for a viscoelastic flexible system. Park et al. [14] studied the Euler-Bernoulli beam equation with memory, they proved the existence and the exponential stability of solutions for the problem under the boundary and initial conditions where They supposed that the kernel k verifies for some c i > 0, i = 1, ..., 3. Furthermore, a similar result in [15] was established under a boundary control Seghour et al. [17] investigated the following system where the positives coefficients d s , M s represents the vessel damping and the mass of the surface vessel. They showed an exponential decay result for solutions with the following conditions on the kernel: for some positive function ζ(t) and Moreover, in [18], the authors considered a similar problem under u(t) = d s v t (L , t), for kernels k verifying for some function η(t). Later, the authors in [1] established uniform stability of the same problem for kernels satisfying In [3], the authors considered the vibrating flexible beam system They imposed a linear control force at the boundary to achieve the exponential stability of the system. Motivated by this work [3], the objective of the present paper is to consider the nonlinear viscoelastic Euler-Bernoulli beam equation under the boundary conditions The initial conditions are where E I : the flexural rigidity ρ A: the mass per unit length E A: the axial stiffness v(x, t): the transverse displacement and P 0 : the tension force The variance length envisaged with the tension force will be assumed to be weak compared to the overall length of the beam. We show an arbitrary decay result for problem (1)-(3) with weaker hypotheses on the relaxation function ψ than the existing ones for similar problems. Namely, we do not limit ourselves to polynomially or exponentially decaying functions only. Relaxation functions that can have zero derivatives on certain subsets of (0, ∞) are considered, see [20][21][22]. We assume that the zone where the kernel is flat and is small. Consequently, a wide range of materials with various viscoelastic properties can be used in modern engineering. The rest of our paper is arranged as follows: In Section 2, we give some useful lemmas needed for our result. The arbitrary decay of the energy result is shown in Section 3.

Notation and main results
We introduce the following notation For the kernel ψ we assume: (H1) ψ : R + → R + is a differentiable function satisfying (H2) ψ (t) ≤ 0 for almost all t ≥ 0.
We denote and (., .), . the inner product and the norm of the space L 2 (0, L), respectively. The existence result for our problem (1)-(3) can be proved by Faedo-Galerkin method, the reader may consult [14].
We define the (classical) energy of problem (1)-(3) by Then, the time derivative of energy is equal to It is easy to see that Then, we consider the modified energy By differentiation, we obtain If our non-negative relaxation function satisfies ψ ≤ 0, it follows that e(t) is nonincreasing and uniformly bounded above by e(0) = E(0). Next, we introduce the functionals where ζ is a positive constant to be determined later, and θ (t) is specified below. We define the second modified functional by for λ i > 0, i = 1, 2, 3 to be specified later. Our first result shows that this functional is an appropriate one to consider.
Proof It is easy to see, from the above definitions, that where and for some constant q i > 0 and λ 1 such that The next result [21] gives a better estimate for

Asymptotic behavior
In this section we state and show our result. To this end we require some notation. For every measurable set A ⊂ R + , we define the probability measure where k = ∞ 0 ψ(s)ds. The flatness set and the flatness rate of ψ are (respectively) defined by F ψ = s ∈ R + , ψ(s) > 0 and ψ (s) = 0 and Let t > 0 and t 0 ψ(s)ds = ψ > 0.
Theorem 4 Let us suppose that ψ and θ satisfy the hypotheses (H1)-(H3) and R ψ < 1 4 . Then, there exist positive constants C and ν such that , with respect to t along the solution of (1)-(4), gives We decompose the first integral into Clearly and Moreover, that is and Using Young and Cauchy-Schwartz inequality we estimate the integral Next, Lemma 2 yields Then and Taking into account (14)- (18), we have The boundary control gives us Moreover, by Young's inequality and, therefore, Notice that After substitution of −ρ Av tt from (1) and integrating by part, we obtain Again utilizing Young's inequality, we get For the second and the third term in (21), we have Hence, For δ 2 > 0, we can write Now we proceed to estimate J 3 . For all measurable sets A and F such that A = R + \F, we see that We denote Q t = Q∩ [0, t]. Using Lemma 2, we obtain for δ 4 > 0 where ψ is defined in (11). We end up with For δ 5 > 0, we have Taking into account (19)- (22) and the above estimations of J 1 , J 2 , J 3 , J 4 , we obtain Further, a differentiation of ϕ 2 (t) yields Regarding ϕ 3 (t) it appears that Collecting the estimations (7), (23)-(25), we find for t ≥ t For n ∈ N, we introduce the sets [16] A n = s ∈ R + : nψ (s) + ψ (s) ≤ 0 .

Notice that
where N ψ is the null set in which ψ is not defined. The complement of A n in R + is denoted by F n = R + \A n . It appears that lim ψ ( Choosing we may write we choose ζ = 1 + 2k so that ⎧ ⎪ ⎨ ⎪ ⎩ Also, we need K θ (0) ≤ min 1 − k + 1 k , 4 and δ 1 = ψ + 1 + 2k 2 .
We also need λ 1 so small that As a consequence of the above consideration, for some positive constants c i i = 1, ..., 5. For λ 1 even smaller if necessary, we get where C 1 is some positive constant. As u(t) is nonincreasing, we have u(t) ≤ u(0) for all t ≥ t . Then (27) becomes By Proposition 1, we obtain for some positive constant C 2 . Integrating (28) over [t , t] yields Then using inequality (9) of Proposition 1, we find The continuity of E(t) over the interval [0, t ] makes it possible to deduce e(t) ≤ C θ (t) ν , t ≥ 0 for some positive constants C and ν.