# Numerical Study of an Euler-Bernoulli Beam Stabilization Controlled in Displacement and Velocity

Authors: Yeo Brahima; Dakouri Narcisse; Adou Jérôme
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

---

## Abstract

Our model is based on an Euler-Bernoulli beam clamped at one end and subjected at its free end to a control in displacement and velocity. In this work, relative of the control parameters, we study the stabilization in displacement and then in energy of the model using a stable numerical scheme that we implement. This numerical scheme results from the Crank–Nicolson algorithm for the discretization in time and from the finite element method based on the approximation by Hermith cubic functions for the discretization in space. The study shows that, compared to the velocity control, the displacement control has an almost negligible effect on the stabilization of the beam. This result is confirmed later by a sensitivity study on the control parameters involved in our model.


---

## Body

## 1. Introduction

We consider an Euler-Bernoulli beam with variable coefficients, clamped at one end and free at the other end. We subject the free end of the beam whose length is assumed to be equal to unity, to a control resulting from the linear combination of its transverse displacement and its velocityvt,.. We note, respectively, mx and EIx the linear mass and the bending stiffness of the beam at any point of abscissa x. The model resulting from the study of the dynamics of the beam is written at any time t.(1)mx∂2ut,x∂t2+∂2∂x2EIx∂2ut,x∂x2=0;0<x<1;t>0,ut,0=∂ut,0∂x=∂2ut,1∂x2=0;∂∂xEI1∂2ut,1∂x2=αut,1+β∂ut,1∂t;t>0,u0,x=u0x;∂u0,x∂t=u1x,where α and β are positive constants.Previous studies [1, 2] have focused on the exponential stabilization of the Euler-Bernoulli beam, using the Riezs basis approach. This approach generally makes it possible to establish the spectrum of the operator associated with the original problem to analyze its properties. Authors My Driss and Abderrahman El, in [3], go further by proposing a numerical study of the operator spectrum to establish the influence of the control parameters on the speed of convergence of the energy associated with the model.In this work, we assume problem (1) is well posed in the sense of contraction semigroups according to [1]. Starting from this problem, we implemented an equivalent numerical model and we established its stability. This numerical scheme allows us to analyze graphically the influence of the control parameters on the stabilization of the beam and on the asymptotic behavior of the system energy. Results deduced from these graphical analysis are then confirmed by a study of the relative sensitivities of these control parameters to damping time of the vibrations of the beam.


## 2. Numerical Approximations of the Model

### 2.1. Implementation of the Numerical Scheme

Sincevt,.=∂ut,./∂t, the first equation of system (1) can still be written as follows:(2)∂v∂t=−1mx∂2∂x2EIx∂2ut,.∂x2;0<x<1;t>0.If we setYt=ut,.;vt,., then we introduce the functional space(3)H=HE20;1×L20;1,where(4)HE20;1=φ∈H20;1|φ0=∂φ0∂x=0,Then, the problem (1) is written as follows:(5)dYdt=A0Y;Y0=Y0=u0;v0,where A0 is a linear and unbounded operator defined on the following equation:(6)DA0=f,g∈H40;1∩HE20;1×HE20;1|f″1=0andEI1f″1′=αf1+βg1.For a fixed realT>0, we subdivide the interval 0;T into Nt intervals of the same length Δt. We then set at any point tn=nΔt0≤n≤Nt.(7)un=utn,x,vn=vtn,x,Yn=Ytn=un;vn.The time discretization of system (5) by the Crank–Nicolson scheme gives the following equations:(8)un+1−unΔt=vn+1+vn2;mxvn+1−vnΔt+12∂2∂x2EIx∂2∂x2un+1+un=0,withu0=u0andv0=v0.Forψ∈L20;1 and φ∈HE20;1, problem (8) is written in the following variational form:(9)∫01un+1−unΔtψxdx=∫01vn+1+vn2ψxdx,∫01mxvn+1−vnΔtφxdx+12∫01EIx∂2un+1∂x2+∂2un∂x2∂2φx∂x2dx+α·un+11+un12+β·vn+11+vn12φ1=0.We assume that the Euler-Bernoulli beam is the segment0;1 which we subdivide into N+1 fixed length intervals xi;xi+1i=0;…;N. For the discretization in the space of problem (9), relatively to study [4], we use the finite element method with the cubic Hermite polynomials φk1≤k≤4 as functions of references from which we build on 0;1 polynomials ϕ1ix and ϕ2ix defined for i=1,2,…,N−1 by the following equation:(10)ϕ1ix=φ2i−1x,ifx∈xi−1;xi,φ1ix,ifx∈xi;xi+1,ϕ2ix=φ4i−1x,ifx∈xi−1;xi,φ3ix,ifx∈xi;xi+1,then(11)ϕ1Nx=φ2N−1xifx∈xN−1;1withϕ2Nx=φ4N−1xifx∈xN−1;1,where(12)ϕ1i0=ϕ2i0=ϕ1i′0=ϕ2i′0=0.We set, respectively,uin and vin the approximate values of unxi and vnxi, and we set, for n∈ℕ∗ the following equation:(13)Un=u1n,…,uNn,∂u1n∂x,…,∂uNn∂xt,Vn=v1n,…,vNn,∂v1n∂x,…,∂vNn∂xt.Then, we get(14)Un=∑i=1Nϕ1ixuin+∑i=1Nϕ2ix∂uin∂x,Vn=∑i=1Nϕ1ixvin+∑i=1Nϕ2ix∂vin∂x,where the 2N− tuple ϕ11,ϕ21,ϕ12,ϕ22,ϕ13,ϕ23,…,ϕ1N,ϕ2N which we simply denote in the following Φi1≤i≤2N constitutes a basis of dimension 2N of the interpolation space,(15)Vh=v∈C00,1/v|xi;xi+1∈P3,1≤i≤N,v0=v′0=v″1=0.Equalities (14) are then written as follows:(16)Un=∑i=12NΦixuin+1,Vn=∑i=12NΦixvin+1,and variational problem (9) becomes, for all j=1,2,…,2N(17)∑i=12N2uin+1−Δtvin+1∫01ΦixΦjxdx=∑i=12N2uin+Δtvin∫01ΦixΦjxdx,∑i=12N∫01Δt.EIx∂xxΦix∂xxΦjxuin+1+2mxΦixΦjxvin+1dx+Φi1Φj1αuin+1+βvin+1∑i=12N∫01−Δt.EIx∂xxΦix∂xxΦjxuin+2mxΦixΦjxvindx+Φi1Φj1−αuin+βvin.


### 2.2. Stability of the Scheme

Assuming the linear massm=mx and the bending stiffness of the beam EI are constants, system (17) is written at any fixed point xj1≤j≤2N,(18)2Δt−112a/Δt+βcb+αcUjn+1Vjn+1=2Δt1−12a/Δt−βcb+αcUjnVjn,where(19)a=∫01∂xxϕjx2dx,b=∫01ϕjx2dx,c=ϕj21.We deduce(20)Ujn+1Vjn+1=1δδ2βcb+αc01+2/Δt2a/Δt−βcb+αcUjnVjn,with(21)δ=1+2/Δt2a/Δt+βcb+αc.To determine the associated Von-Neumann stability condition, we setUjn;Vjn=eikxjU^n;eikxjV^n. System (20) then becomes as follows:(22)U^n+1V^n+1=12βcb+αc+2/Δt2a/Δt+βc0b+αc+2/Δt2a/Δt−βcb+αc+2/Δt2a/Δt+βcU^nV^n.The eigenvalues λ1=1 and λ2=b+αc+2/Δt2a/Δt−βc/b+αc+2/Δt2a/Δt+βc of the amplification matrix associated with (22) satisfy the Von-Neumann stability condition,(23)λi≤1,i=1,2.We then deduce the stability of the numerical scheme (17).


## 2.1. Implementation of the Numerical Scheme

Sincevt,.=∂ut,./∂t, the first equation of system (1) can still be written as follows:(2)∂v∂t=−1mx∂2∂x2EIx∂2ut,.∂x2;0<x<1;t>0.If we setYt=ut,.;vt,., then we introduce the functional space(3)H=HE20;1×L20;1,where(4)HE20;1=φ∈H20;1|φ0=∂φ0∂x=0,Then, the problem (1) is written as follows:(5)dYdt=A0Y;Y0=Y0=u0;v0,where A0 is a linear and unbounded operator defined on the following equation:(6)DA0=f,g∈H40;1∩HE20;1×HE20;1|f″1=0andEI1f″1′=αf1+βg1.For a fixed realT>0, we subdivide the interval 0;T into Nt intervals of the same length Δt. We then set at any point tn=nΔt0≤n≤Nt.(7)un=utn,x,vn=vtn,x,Yn=Ytn=un;vn.The time discretization of system (5) by the Crank–Nicolson scheme gives the following equations:(8)un+1−unΔt=vn+1+vn2;mxvn+1−vnΔt+12∂2∂x2EIx∂2∂x2un+1+un=0,withu0=u0andv0=v0.Forψ∈L20;1 and φ∈HE20;1, problem (8) is written in the following variational form:(9)∫01un+1−unΔtψxdx=∫01vn+1+vn2ψxdx,∫01mxvn+1−vnΔtφxdx+12∫01EIx∂2un+1∂x2+∂2un∂x2∂2φx∂x2dx+α·un+11+un12+β·vn+11+vn12φ1=0.We assume that the Euler-Bernoulli beam is the segment0;1 which we subdivide into N+1 fixed length intervals xi;xi+1i=0;…;N. For the discretization in the space of problem (9), relatively to study [4], we use the finite element method with the cubic Hermite polynomials φk1≤k≤4 as functions of references from which we build on 0;1 polynomials ϕ1ix and ϕ2ix defined for i=1,2,…,N−1 by the following equation:(10)ϕ1ix=φ2i−1x,ifx∈xi−1;xi,φ1ix,ifx∈xi;xi+1,ϕ2ix=φ4i−1x,ifx∈xi−1;xi,φ3ix,ifx∈xi;xi+1,then(11)ϕ1Nx=φ2N−1xifx∈xN−1;1withϕ2Nx=φ4N−1xifx∈xN−1;1,where(12)ϕ1i0=ϕ2i0=ϕ1i′0=ϕ2i′0=0.We set, respectively,uin and vin the approximate values of unxi and vnxi, and we set, for n∈ℕ∗ the following equation:(13)Un=u1n,…,uNn,∂u1n∂x,…,∂uNn∂xt,Vn=v1n,…,vNn,∂v1n∂x,…,∂vNn∂xt.Then, we get(14)Un=∑i=1Nϕ1ixuin+∑i=1Nϕ2ix∂uin∂x,Vn=∑i=1Nϕ1ixvin+∑i=1Nϕ2ix∂vin∂x,where the 2N− tuple ϕ11,ϕ21,ϕ12,ϕ22,ϕ13,ϕ23,…,ϕ1N,ϕ2N which we simply denote in the following Φi1≤i≤2N constitutes a basis of dimension 2N of the interpolation space,(15)Vh=v∈C00,1/v|xi;xi+1∈P3,1≤i≤N,v0=v′0=v″1=0.Equalities (14) are then written as follows:(16)Un=∑i=12NΦixuin+1,Vn=∑i=12NΦixvin+1,and variational problem (9) becomes, for all j=1,2,…,2N(17)∑i=12N2uin+1−Δtvin+1∫01ΦixΦjxdx=∑i=12N2uin+Δtvin∫01ΦixΦjxdx,∑i=12N∫01Δt.EIx∂xxΦix∂xxΦjxuin+1+2mxΦixΦjxvin+1dx+Φi1Φj1αuin+1+βvin+1∑i=12N∫01−Δt.EIx∂xxΦix∂xxΦjxuin+2mxΦixΦjxvindx+Φi1Φj1−αuin+βvin.


## 2.2. Stability of the Scheme

Assuming the linear massm=mx and the bending stiffness of the beam EI are constants, system (17) is written at any fixed point xj1≤j≤2N,(18)2Δt−112a/Δt+βcb+αcUjn+1Vjn+1=2Δt1−12a/Δt−βcb+αcUjnVjn,where(19)a=∫01∂xxϕjx2dx,b=∫01ϕjx2dx,c=ϕj21.We deduce(20)Ujn+1Vjn+1=1δδ2βcb+αc01+2/Δt2a/Δt−βcb+αcUjnVjn,with(21)δ=1+2/Δt2a/Δt+βcb+αc.To determine the associated Von-Neumann stability condition, we setUjn;Vjn=eikxjU^n;eikxjV^n. System (20) then becomes as follows:(22)U^n+1V^n+1=12βcb+αc+2/Δt2a/Δt+βc0b+αc+2/Δt2a/Δt−βcb+αc+2/Δt2a/Δt+βcU^nV^n.The eigenvalues λ1=1 and λ2=b+αc+2/Δt2a/Δt−βc/b+αc+2/Δt2a/Δt+βc of the amplification matrix associated with (22) satisfy the Von-Neumann stability condition,(23)λi≤1,i=1,2.We then deduce the stability of the numerical scheme (17).


## 3. Influence of Control Parameters on the Beam Stabilization

### 3.1. Stabilization in Motion

If we set(24)Mm=∫01mΦixΦjxdx1≤i,j≤2N,S=Φi1,Φj11≤i,j≤2N,K=∫01EIx∂xxΦix∂xxΦjxdx1≤i,j≤2N, numerical scheme (17) is then written as follows:(25)PYn+1=QYn,Y0=Y0,where(26)P=2I2N−ΔtI2NΔtK+αS2Mm+βΔtS,Q=2I2NΔtI2N−ΔtK+αS2Mm−βΔtS,I2N denotes the unit matrix of order 2N.For the simulation, we set the following numerical values as follows:(27)EI=0,004725,m=0,78,and(28)N=10,T=1,5,Δt=0,001,for the model parameters. In addition, we note for the initial conditions as follows:(29)u0x=x3−3x2,v0x=0.We also choose different values for each parametersα and β involved in problem (1), in order to highlight their influence on the system. Thus, the following results are depicted on MATLAB [5].Figure 1
Influence of the parameterβ on the beam vibrations in x=0,5.Figure 2
Influence of the parameterα on the beam vibrations in x=0,5.


### 3.2. Stabilization in Energy

If we multiply the first equation of system (1) by the test function φ∈HE20;1, we get, after two integrations by parts, the following variational formulation:(30)∫01mx∂2ut,x∂t2φxdx+∫01EIx∂2ut,x∂x2∂2φx∂x2dx+αut,1+β∂ut,1∂tφ1=0.By replacing in (30) the test function φ by ∂ut,x/∂x, we get the following equation:(31)dEtdt+β∂ut,1∂t2=0,where the total energy of the system is written as follows:(32)Et=12∫01mx∂ut,x∂x2dx+12∫01EIx∂2ut,x∂x22dx+α2ut,12.Then, we deduce from relations (31) and (32) that(33)dEtdt≤0.Therefore, the energyEt defines a Lyapunov function and satisfies(34)0≤Et≤E0<∞,∀t∈ℝ+.Relative to the Crank–Nicolson scheme used in system (8), we set, for 0≤n≤Nt,(35)En=12∫01mxvn2dx+12∫01EIx∂2un∂x22dx+α2un12,the associate discrete energy. Then we get the following result.Proposition 1.
For any integern, there is a positive real λn such that(36)En+1−EnΔt+λn=0.Proof.
According to (34), we get(37)En+1−En=12∫01mxvn+12−vn2dx+12∫01EIx∂2un+1∂x22−∂2un∂x22dx+α2un+112−un12.
Forx∈0;1, we set in the first equation of the variational problem (9) the following equation:(38)ψx=mxvn+1−vn.
Then, we get(39)∫01mxun+1−unΔtvn+1−vndx=∫01mxvn+12−vn22dx.
Moreover, if we set(40)φx=un+1,as a test function in the second equation of problem (9), we get the following equation:(41)∫01mxvn+1−vnΔtun+1dx+12∫01EIx∂2un+1∂x2+∂2un∂x2∂2un+1∂x2dx+α2un+11+un1+β2vn+11+vn1un+11=0.
Analogously, if we choose as test functionφx=un, the second equation of system (8) is written as follows:(42)∫01mxvn+1−vnΔtundx+12∫01EIx∂2un+1∂x2+∂2un∂x2∂2un∂x2dx+α2un+11+un1+β2vn+11+vn1un1=0.
By differencing (41) and (42) and relative the fact that(43)un+1−un=Δt2vn+1+vn,we get(44)En+1−EnΔt+λn=0,forn∈ℕ,whereλn=βvn+11+vn122.
Consequently, we state the following result.Corollary 1.
For any integern≥1, we get(45)En+1≤En≤E0.
Finally, we notice that the energy of the discrete system decreases and is controlled by that of the initial data. For the numerical simulation of the energy, we deduce from Proposition1 the following equality:(46)En+1=En−βΔt4vn+11+vn12,0≤n≤Nt,where(47)E0=EI2∫01u0″x2dx+α2u012.
Under initial conditions (29) and relation (35), we set(48)Δt=0,001.
Curves above are depicted on MATLAB [5].Figure 3
Influence of the parameterα on the energy.Figure 4
Influence of the parameterβ on the energy.


### 3.3. Results and Discussion

By scrutinizing the curves depicted in Figures1 and 2, we notice that for a fixed value of β and for different values of α, the vibrations tend to stabilize at almost the same time. Conversely, if α is fixed, we notice that for larger values of β the beam stabilizes much more quickly.Furthermore, the analysis of Figures3 and 4 shows that contrary to the parameter α, for increasingly large values of β, the energy of the system decreases much faster from its initial value E0 towards 0.Finally, we deduce from the graphical results that the velocity control has more impact on the stabilization of the beam than the displacement control.


## 3.1. Stabilization in Motion

If we set(24)Mm=∫01mΦixΦjxdx1≤i,j≤2N,S=Φi1,Φj11≤i,j≤2N,K=∫01EIx∂xxΦix∂xxΦjxdx1≤i,j≤2N, numerical scheme (17) is then written as follows:(25)PYn+1=QYn,Y0=Y0,where(26)P=2I2N−ΔtI2NΔtK+αS2Mm+βΔtS,Q=2I2NΔtI2N−ΔtK+αS2Mm−βΔtS,I2N denotes the unit matrix of order 2N.For the simulation, we set the following numerical values as follows:(27)EI=0,004725,m=0,78,and(28)N=10,T=1,5,Δt=0,001,for the model parameters. In addition, we note for the initial conditions as follows:(29)u0x=x3−3x2,v0x=0.We also choose different values for each parametersα and β involved in problem (1), in order to highlight their influence on the system. Thus, the following results are depicted on MATLAB [5].Figure 1
Influence of the parameterβ on the beam vibrations in x=0,5.Figure 2
Influence of the parameterα on the beam vibrations in x=0,5.


## 3.2. Stabilization in Energy

If we multiply the first equation of system (1) by the test function φ∈HE20;1, we get, after two integrations by parts, the following variational formulation:(30)∫01mx∂2ut,x∂t2φxdx+∫01EIx∂2ut,x∂x2∂2φx∂x2dx+αut,1+β∂ut,1∂tφ1=0.By replacing in (30) the test function φ by ∂ut,x/∂x, we get the following equation:(31)dEtdt+β∂ut,1∂t2=0,where the total energy of the system is written as follows:(32)Et=12∫01mx∂ut,x∂x2dx+12∫01EIx∂2ut,x∂x22dx+α2ut,12.Then, we deduce from relations (31) and (32) that(33)dEtdt≤0.Therefore, the energyEt defines a Lyapunov function and satisfies(34)0≤Et≤E0<∞,∀t∈ℝ+.Relative to the Crank–Nicolson scheme used in system (8), we set, for 0≤n≤Nt,(35)En=12∫01mxvn2dx+12∫01EIx∂2un∂x22dx+α2un12,the associate discrete energy. Then we get the following result.Proposition 1.
For any integern, there is a positive real λn such that(36)En+1−EnΔt+λn=0.Proof.
According to (34), we get(37)En+1−En=12∫01mxvn+12−vn2dx+12∫01EIx∂2un+1∂x22−∂2un∂x22dx+α2un+112−un12.
Forx∈0;1, we set in the first equation of the variational problem (9) the following equation:(38)ψx=mxvn+1−vn.
Then, we get(39)∫01mxun+1−unΔtvn+1−vndx=∫01mxvn+12−vn22dx.
Moreover, if we set(40)φx=un+1,as a test function in the second equation of problem (9), we get the following equation:(41)∫01mxvn+1−vnΔtun+1dx+12∫01EIx∂2un+1∂x2+∂2un∂x2∂2un+1∂x2dx+α2un+11+un1+β2vn+11+vn1un+11=0.
Analogously, if we choose as test functionφx=un, the second equation of system (8) is written as follows:(42)∫01mxvn+1−vnΔtundx+12∫01EIx∂2un+1∂x2+∂2un∂x2∂2un∂x2dx+α2un+11+un1+β2vn+11+vn1un1=0.
By differencing (41) and (42) and relative the fact that(43)un+1−un=Δt2vn+1+vn,we get(44)En+1−EnΔt+λn=0,forn∈ℕ,whereλn=βvn+11+vn122.
Consequently, we state the following result.Corollary 1.
For any integern≥1, we get(45)En+1≤En≤E0.
Finally, we notice that the energy of the discrete system decreases and is controlled by that of the initial data. For the numerical simulation of the energy, we deduce from Proposition1 the following equality:(46)En+1=En−βΔt4vn+11+vn12,0≤n≤Nt,where(47)E0=EI2∫01u0″x2dx+α2u012.
Under initial conditions (29) and relation (35), we set(48)Δt=0,001.
Curves above are depicted on MATLAB [5].Figure 3
Influence of the parameterα on the energy.Figure 4
Influence of the parameterβ on the energy.


## 3.3. Results and Discussion

By scrutinizing the curves depicted in Figures1 and 2, we notice that for a fixed value of β and for different values of α, the vibrations tend to stabilize at almost the same time. Conversely, if α is fixed, we notice that for larger values of β the beam stabilizes much more quickly.Furthermore, the analysis of Figures3 and 4 shows that contrary to the parameter α, for increasingly large values of β, the energy of the system decreases much faster from its initial value E0 towards 0.Finally, we deduce from the graphical results that the velocity control has more impact on the stabilization of the beam than the displacement control.


## 4. Sensitivity of Parametersα and β on the Stability

### 4.1. Sensitivity Indexes Values ofα and β

In this part, we objectively study the influence of the parametersα and β on the stability of the Euler-Bernoulli beam. For this purpose, we note in the table below the damping time δτ of the beam vibrations, for different values of α and β.Table 1
Vibrations damping timesδτ of the beam (in seconds).
αβ0.50.851.02.53.55.0501.821.150.900.420.240.221001.851.120.980.400.200.202001.791.150.950.400.200.152501.821.050.900.350.250.203501.801.100.900.350.300.20The first-order Sobol sensitivity index as defined by McKay in [6] is written as follows:(49)Si=VarEδτ|YiVarδτ,where Varδτ and Eδτ|Yi denote, respectively, the variance of δτ and the conditional expectation of δτ obtained by fixing Yi. From the values of δτ contained in Table1, if we set, respectively, Yi=α and Yi=β in (49). Then we get the following indexes of sensitivity of α and β.(50)Sα≃0,00016,Sβ≃0,58450.


### 4.2. Discussion

The comparison of the sensitivity indexes obtained in (50) clearly shows that, compared to the parameter α, the parameter β has more impact on the stabilization time of the Euler-Bernoulli beam.Then, the conclusion of the analyses deduced from the graphical results obtained above is therefore confirmed.


## 4.1. Sensitivity Indexes Values ofα and β

In this part, we objectively study the influence of the parametersα and β on the stability of the Euler-Bernoulli beam. For this purpose, we note in the table below the damping time δτ of the beam vibrations, for different values of α and β.Table 1
Vibrations damping timesδτ of the beam (in seconds).
αβ0.50.851.02.53.55.0501.821.150.900.420.240.221001.851.120.980.400.200.202001.791.150.950.400.200.152501.821.050.900.350.250.203501.801.100.900.350.300.20The first-order Sobol sensitivity index as defined by McKay in [6] is written as follows:(49)Si=VarEδτ|YiVarδτ,where Varδτ and Eδτ|Yi denote, respectively, the variance of δτ and the conditional expectation of δτ obtained by fixing Yi. From the values of δτ contained in Table1, if we set, respectively, Yi=α and Yi=β in (49). Then we get the following indexes of sensitivity of α and β.(50)Sα≃0,00016,Sβ≃0,58450.


## 4.2. Discussion

The comparison of the sensitivity indexes obtained in (50) clearly shows that, compared to the parameter α, the parameter β has more impact on the stabilization time of the Euler-Bernoulli beam.Then, the conclusion of the analyses deduced from the graphical results obtained above is therefore confirmed.


## 5. Conclusion

In this work, we have studied the stabilization of the Euler-Bernoulli beam clamped at one extreme and subjected at the other extreme to a linear combination of displacement and velocity. To do this, we have implemented a numerical stable scheme. With this numerical scheme, we have studied the stabilization of our original model in displacement and in energy, respectively. The analysis of the graphical results allowed us to affirm that the control in velocity allows the beam to stabilize faster than when the control is done in displacement. This result, which we confirmed after a statistical study of the sensitivity of the control parameters, corroborates that obtained in the numerical study of the spectrum of the operator associated with the model by Aouragh and Yebari in [7] and later by Kouassi et al. in [2].



---

*Source: 7647568-2022-10-28.xml*