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.

This paper investigates the problem of exponential stability for a damped Euler-Bernoulli beam with variable coefficients clamped at one end and subjected to a force control in rotation and velocity rotation. We adopt the Riesz basis approach for show that the closed-loop system is a Riesz spectral system. Therefore, the exponential stability and the spectrumdetermined growth condition are obtained.

In this paper, we prove the existence and uniqueness of the weak solution of a flexible beam that is clamped at one end and free at the other; a mass is also attached to the free end of the beam. Also, we construct a finite element method, based on piecewise cubic Hermitian shape functions. Next, we derive error estimates for the semi-discrete Galerkin approximations. The results are derived from \cite{BS}. Finally, we implement the results of numerical schemes developed.

In this paper, we prove the existence and uniqueness of the weak solution of a system of nonlinear equations involved in the mathematical modeling of cancer tumor growth with a non homogeneous divergence condition. We also present  a new concept of generalized differentiation of non linear operators : C-differentiability. Through this notion, we also prove the uniqueness and the C-differentiability of the solution when the system is perturbed by a certain number of parameters. Two results have been established. In the first one, differentiability is according to Fréchet. The proof is given uses the theorem of reciprocal functions in Banach spaces. First of all, we give the proof of strict differentiability of a direct mapping, according to Fréchet. In the second result, differentiability is understood in a weaker sense than that of Fréchet. For the proof we use Hadamard's theorem of small perturbations of Banach isomorphism of spaces as well as the notion of strict differentiability.

The main goal of this paper is to establish the first order necessary optimality conditions for a tumor growth model that evolves due to cancer cell proliferation. The phenomenon is modeled by a system of three-dimensional partial differential equations. We prove the existence and uniqueness of optimal control and necessary conditions of optimality are established by using the variational formulation.