In this paper we study the stability of a flexible cable that is clamped at one end and free at t... more In this paper we study the stability of a flexible cable that is clamped at one end and free at the other. To stabilize this system we apply a control force in position and velocity at the free end of the cable. We prove that the closed-loop system is well-posed and is exponentially stable. We then analyze the spectrum of the system. Using a method due to Shkalikov we prove that the spectrum determines the exponential decay rate of the energy under certain conditions.
Molecular Crystals and Liquid Crystals Science and Technology. Section A. Molecular Crystals and Liquid Crystals, 1994
Merkel cell carcinoma of the eyelid in association with chronic lymphocytic leukaemia Merkel cell... more Merkel cell carcinoma of the eyelid in association with chronic lymphocytic leukaemia Merkel cell carcinoma (MCC) is a rare skin neoplasm. Tang and Toker 1 first described MCC in 1978 and since then 19 cases in association with chronic lymphocytic leukaemia (CLL) have been reported. 2-5 To the best of our knowledge, involvement of the eyelid by MCC has never been reported in the literature in association with CLL.
In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a for... more In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.
In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with va... more In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with variable coefficients clamped at one end and is free at the other. In order to stabilize the system, we apply a linear boundary control force in rotation and velocity rotation. By adopting the Riesz basis approach, it is shown that the closedloop system is a Riesz spectral system. Consequently, the spectrum-determined growth condition and the exponential stability are obtained. AMS subject classification: 35B35, 35P20, 93D15.
We study a Euler-Bernoulli beam using a special boundary feedback at the free and. The closed-loo... more We study a Euler-Bernoulli beam using a special boundary feedback at the free and. The closed-loop system is shown to be non dissipative. This gives rise to difficulties in analyzing the well-posedness and the stability of the considered system using the traditional dissipativity based-method. The major difficulty in answering this question comes from its special boundary conditions: the physical variables and their conjugate variables are assigned simultaneously at the same boundary point. Despite the lack of the dissipativity we obtain the Riesz basis property. As consequences, both the spectrum-determined growth conditions and exponential stability are concluded. AMS subject classification: 93C20, 93D15, 35B35, 35P10.
In this work, we consider a flexible Euler-Bernoulli beam with variable coefficients clamped at o... more In this work, we consider a flexible Euler-Bernoulli beam with variable coefficients clamped at one end subjected to a force control in rotation and velocity rotation. From the weak formulation, we construct spaces in order to better define the weak solution. Using Faedo-Galerkin techniques as well as a compactness result on intermediate spaces (see [9]), we demonstrate the existence of a weak solution of the considered problem passing to the limit. Another result is given by demonstrating the uniqueness of the solution. We finally study the regularity of this solution.
This paper studies the basis property and the stability of a distributed system described by a no... more This paper studies the basis property and the stability of a distributed system described by a nonuniform Euler-Bernoulli beam equation under linear boundary feedback control. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. The asymptotic distribution of eigenvalues, the spectrumdetermined growth condition, and the exponential stability are concluded. The results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam.
In this paper we consider a long flexible Euler-Bernoulli beam with boundary conditions imposed a... more In this paper we consider a long flexible Euler-Bernoulli beam with boundary conditions imposed at the two ends, the resulting model being called hybrid system. The beam is hybrid in the sense that it holds both rigid and elastic motions. Our main result is to show the existence and uniqueness of the weak solution of close-loop system. The closed-loop system stability is shown through Lyapunov-based analysis.
In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a for... more In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.
In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with va... more In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with variable coefficients clamped at one end and is free at the other. In order to stabilize the system, we apply a linear boundary control force in rotation and velocity rotation. By adopting the Riesz basis approach, it is shown that the closed-loop system is a Riesz spectral system. Consequently, the spectrum-determined growth condition and the exponential stability are obtained.
In this paper we study the stability of a flexible cable that is clamped at one end and free at t... more In this paper we study the stability of a flexible cable that is clamped at one end and free at the other. To stabilize this system we apply a control force in position and velocity at the free end of the cable. We prove that the closed-loop system is well-posed and is exponentially stable. We then analyze the spectrum of the system. Using a method due to Shkalikov we prove that the spectrum determines the exponential decay rate of the energy under certain conditions.
Molecular Crystals and Liquid Crystals Science and Technology. Section A. Molecular Crystals and Liquid Crystals, 1994
Merkel cell carcinoma of the eyelid in association with chronic lymphocytic leukaemia Merkel cell... more Merkel cell carcinoma of the eyelid in association with chronic lymphocytic leukaemia Merkel cell carcinoma (MCC) is a rare skin neoplasm. Tang and Toker 1 first described MCC in 1978 and since then 19 cases in association with chronic lymphocytic leukaemia (CLL) have been reported. 2-5 To the best of our knowledge, involvement of the eyelid by MCC has never been reported in the literature in association with CLL.
In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a for... more In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.
In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with va... more In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with variable coefficients clamped at one end and is free at the other. In order to stabilize the system, we apply a linear boundary control force in rotation and velocity rotation. By adopting the Riesz basis approach, it is shown that the closedloop system is a Riesz spectral system. Consequently, the spectrum-determined growth condition and the exponential stability are obtained. AMS subject classification: 35B35, 35P20, 93D15.
We study a Euler-Bernoulli beam using a special boundary feedback at the free and. The closed-loo... more We study a Euler-Bernoulli beam using a special boundary feedback at the free and. The closed-loop system is shown to be non dissipative. This gives rise to difficulties in analyzing the well-posedness and the stability of the considered system using the traditional dissipativity based-method. The major difficulty in answering this question comes from its special boundary conditions: the physical variables and their conjugate variables are assigned simultaneously at the same boundary point. Despite the lack of the dissipativity we obtain the Riesz basis property. As consequences, both the spectrum-determined growth conditions and exponential stability are concluded. AMS subject classification: 93C20, 93D15, 35B35, 35P10.
In this work, we consider a flexible Euler-Bernoulli beam with variable coefficients clamped at o... more In this work, we consider a flexible Euler-Bernoulli beam with variable coefficients clamped at one end subjected to a force control in rotation and velocity rotation. From the weak formulation, we construct spaces in order to better define the weak solution. Using Faedo-Galerkin techniques as well as a compactness result on intermediate spaces (see [9]), we demonstrate the existence of a weak solution of the considered problem passing to the limit. Another result is given by demonstrating the uniqueness of the solution. We finally study the regularity of this solution.
This paper studies the basis property and the stability of a distributed system described by a no... more This paper studies the basis property and the stability of a distributed system described by a nonuniform Euler-Bernoulli beam equation under linear boundary feedback control. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. The asymptotic distribution of eigenvalues, the spectrumdetermined growth condition, and the exponential stability are concluded. The results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam.
In this paper we consider a long flexible Euler-Bernoulli beam with boundary conditions imposed a... more In this paper we consider a long flexible Euler-Bernoulli beam with boundary conditions imposed at the two ends, the resulting model being called hybrid system. The beam is hybrid in the sense that it holds both rigid and elastic motions. Our main result is to show the existence and uniqueness of the weak solution of close-loop system. The closed-loop system stability is shown through Lyapunov-based analysis.
In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a for... more In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.
In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with va... more In this article, we examine the exponential stability for a flexible Euler-Bernoulli beam with variable coefficients clamped at one end and is free at the other. In order to stabilize the system, we apply a linear boundary control force in rotation and velocity rotation. By adopting the Riesz basis approach, it is shown that the closed-loop system is a Riesz spectral system. Consequently, the spectrum-determined growth condition and the exponential stability are obtained.
Uploads
Papers by Touré Augustin