This article is dedicated to the investigation of the stabilization problem of a flexible beam at... more This article is dedicated to the investigation of the stabilization problem of a flexible beam attached to the center of a rotating disk. Contrary to previous works on the system, we assume that the feedback law contains a nonlinear torque control applied on the disk and most importantly only one nonlinear moment control exerted on the beam. Thereafter, it is proved that the proposed controls guarantee the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk and standard assumptions on the nonlinear functions governing the controls.
In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger e... more In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger equation i∂ty = −∂ xy+ γ∂ xy on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the left endpoint, where the parameter γ < 0. We prove that this system is exactly controllable in time T > 0, if and only if, the parameter γ does not belong to a critical countable set of negative real numbers. The analysis in this work is based on spectral analysis together with the nonharmonic Fourier series method. 2010 Mathematics Subject Classification. 35P05, 35G05, 81Q10, 81Q93, 93C15, 93D15.
In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger... more In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the left endpoint. We prove that this control system is exactly controllable at any time T > 0. The proofs are based on a detailed spectral analysis and on the use of nonharmonic Fourier series.
We consider the damped Schrödinger semigroup e −it d 2 dx2 on the tadpole graph R. We first give ... more We consider the damped Schrödinger semigroup e −it d 2 dx2 on the tadpole graph R. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of L2(R), we prove that the corresponding energy decay exponentially.
We survey some of our recent results on inverse problems for evolution equations. The goal is to ... more We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities and a special choice of initial conditions, we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their original versions.
Abstract: We study the global existence and the large time behavior of the system governing the n... more Abstract: We study the global existence and the large time behavior of the system governing the non-linear vibrations of a Timoshenko beam. For small initial data we prove global existence of strong solutions and exponential decay of the energy. 1
This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of o... more This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of order four. First, we prove the well-posedness of the system and give some regularity results. Then, we show that the zero solution of the system exponentially converges to zero when the time tends to infinity provided that the time-delay is small and the damping term satisfies reasonable conditions. Lastly, an intensive numerical study is put forward and numerical illustrations of the stability result are provided.
In this work, we study the bilinear optimal stabilization of a non-homogeneous Fokker-Planck equa... more In this work, we study the bilinear optimal stabilization of a non-homogeneous Fokker-Planck equation. We first study the problem of optimal control in a finite-time interval and then focus on the case of the infinite time horizon. We further show that the obtained optimal control guarantees the strong stability of the system at hand. An illustrating numerical example is given.
This paper is devoted to the analysis of the problem of stabilization of fractional (in time) par... more This paper is devoted to the analysis of the problem of stabilization of fractional (in time) partial differential equations. We consider the following equation $$ \partial^{\alpha,\eta}_{t} u(t)=\mathcal{A}u(t)-\frac{\eta}{\Gamma (1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha} \, e^{-\eta(t-s)}u(s)\, ds,\; t > 0, $$ with the initial data $u(0)=u^{0}$, where $\mathcal{A}$ is a unbounded operator in Hilbert space and $\partial_{t}^{\alpha,\eta}$ stands for the fractional derivative. We provide two main results concerning the behavior of the solutions when $t\longrightarrow+\infty$. We look first to the case $\eta>0$ where we prove that the solution of this problem is exponential stable then we consider the case $\eta=0$ when we prove under some consideration on the resolvent that the energy of the solution goes to $0$ as $t$ goes to the infinity as $1/t^\alpha$.
This article is dedicated to the investigation of the stabilisation problem of a flexible beam at... more This article is dedicated to the investigation of the stabilisation problem of a flexible beam attached to the centre of a rotating disk. We assume that the feedback law contains a nonlinear torque...
This article is dedicated to the investigation of the stabilization problem of a flexible beam at... more This article is dedicated to the investigation of the stabilization problem of a flexible beam attached to the center of a rotating disk. Contrary to previous works on the system, we assume that the feedback law contains a nonlinear torque control applied on the disk and most importantly only one nonlinear moment control exerted on the beam. Thereafter, it is proved that the proposed controls guarantee the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk and standard assumptions on the nonlinear functions governing the controls.
In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger e... more In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger equation i∂ty = −∂ xy+ γ∂ xy on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the left endpoint, where the parameter γ < 0. We prove that this system is exactly controllable in time T > 0, if and only if, the parameter γ does not belong to a critical countable set of negative real numbers. The analysis in this work is based on spectral analysis together with the nonharmonic Fourier series method. 2010 Mathematics Subject Classification. 35P05, 35G05, 81Q10, 81Q93, 93C15, 93D15.
In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger... more In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the left endpoint. We prove that this control system is exactly controllable at any time T > 0. The proofs are based on a detailed spectral analysis and on the use of nonharmonic Fourier series.
We consider the damped Schrödinger semigroup e −it d 2 dx2 on the tadpole graph R. We first give ... more We consider the damped Schrödinger semigroup e −it d 2 dx2 on the tadpole graph R. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of L2(R), we prove that the corresponding energy decay exponentially.
We survey some of our recent results on inverse problems for evolution equations. The goal is to ... more We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities and a special choice of initial conditions, we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their original versions.
Abstract: We study the global existence and the large time behavior of the system governing the n... more Abstract: We study the global existence and the large time behavior of the system governing the non-linear vibrations of a Timoshenko beam. For small initial data we prove global existence of strong solutions and exponential decay of the energy. 1
This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of o... more This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of order four. First, we prove the well-posedness of the system and give some regularity results. Then, we show that the zero solution of the system exponentially converges to zero when the time tends to infinity provided that the time-delay is small and the damping term satisfies reasonable conditions. Lastly, an intensive numerical study is put forward and numerical illustrations of the stability result are provided.
In this work, we study the bilinear optimal stabilization of a non-homogeneous Fokker-Planck equa... more In this work, we study the bilinear optimal stabilization of a non-homogeneous Fokker-Planck equation. We first study the problem of optimal control in a finite-time interval and then focus on the case of the infinite time horizon. We further show that the obtained optimal control guarantees the strong stability of the system at hand. An illustrating numerical example is given.
This paper is devoted to the analysis of the problem of stabilization of fractional (in time) par... more This paper is devoted to the analysis of the problem of stabilization of fractional (in time) partial differential equations. We consider the following equation $$ \partial^{\alpha,\eta}_{t} u(t)=\mathcal{A}u(t)-\frac{\eta}{\Gamma (1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha} \, e^{-\eta(t-s)}u(s)\, ds,\; t > 0, $$ with the initial data $u(0)=u^{0}$, where $\mathcal{A}$ is a unbounded operator in Hilbert space and $\partial_{t}^{\alpha,\eta}$ stands for the fractional derivative. We provide two main results concerning the behavior of the solutions when $t\longrightarrow+\infty$. We look first to the case $\eta>0$ where we prove that the solution of this problem is exponential stable then we consider the case $\eta=0$ when we prove under some consideration on the resolvent that the energy of the solution goes to $0$ as $t$ goes to the infinity as $1/t^\alpha$.
This article is dedicated to the investigation of the stabilisation problem of a flexible beam at... more This article is dedicated to the investigation of the stabilisation problem of a flexible beam attached to the centre of a rotating disk. We assume that the feedback law contains a nonlinear torque...
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