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.
In this paper we study the best decay rate of the solutions of a damped plate equation in a squar... more In this paper we study the best decay rate of the solutions of a damped plate equation in a square and with a homogeneous Dirichlet boundary conditions. We show that the fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup, if the damping coefficient is in L ∞ (Ω). Moreover, we give some numerical illustrations by spectral computation of the spectrum associated to the damped plate equation. The numerical results obtained for various cases of damping are in a good agreement with theoretical ones. Computation of the spectrum and energy of discrete solution of damped plate show that the best decay rate is given by spectral abscissa of numerical solution.
We study the problem of stabilization for the acoustic system with a spatially distributed dampin... more We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
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
In this work, we study the controllability of the bilinear Schrödinger equation on infinite graph... more In this work, we study the controllability of the bilinear Schrödinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schrödinger equation (BSE) i∂tψ = −∆ψ + u(t)Bψ in the Hilbert space L 2 p composed by functions defined on an infinite graph G verifying periodic boundary conditions on the infinite edges. The Laplacian −∆ is equipped with specific boundary conditions, B is a bounded symmetric operator and u ∈ L 2 ((0, T), R) with T > 0. We present the well-posedness of the (BSE) in suitable subspaces of D(|∆| 3/2). In such spaces, we study the global exact controllability and we provide examples involving tadpole graphs and star graphs with infinite spokes.
In this paper we consider a linear hybrid system which composed by two nonhomogeneous rods connec... more In this paper we consider a linear hybrid system which composed by two nonhomogeneous rods connected by a point mass and generated by the equations ρ 1 (x)u t = (σ 1 (x)u x) x − q 1 (x)u, x ∈ (−1, 0), t > 0, ρ 2 (x)v t = (σ 2 (x)v x) x − q 2 (x)v, x ∈ (0, 1), t > 0, u(0, t) = v(0, t) = z(t), t > 0, M z t (t) = σ 2 (0)v x (0, t) − σ 1 (0)u x (0, t), t > 0, with Dirichlet boundary condition on the left end x = −1 and a boundary control acts on the right end x = 1. We prove that this system is null controllable with Dirichlet or Neumann boundary controls. Our approach is mainly based on a detailed spectral analysis together with the moment method. In particular, we show that the associated spectral gap in both cases (Dirichlet or Neumann boundary controls) are positive without further conditions on the coefficients ρ i , σ i and q i (i = 1, 2) other than the regularities.
Journal of Dynamics and Differential Equations, 2015
We show that the best decay rate can be estimated by the observability (or controllability) cost ... more We show that the best decay rate can be estimated by the observability (or controllability) cost and open-loop admissibility cost. Moreover, we propose a numerical strategy to give an estimation for the best decay rate for a large class of evolution systems. Some examples are given to illustrate this new method.
Mathematics of Control, Signals, and Systems (MCSS), 2002
We consider the Rayleigh beam equation and the Euler-Bernoulli beam equation with pointwise feedb... more We consider the Rayleigh beam equation and the Euler-Bernoulli beam equation with pointwise feedback shear force and bending moment at the position x in a bounded domain ð0; pÞ with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if x=p is a rational number with coprime factorization x=p ¼ p=q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler-Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.
ESAIM: Control, Optimisation and Calculus of Variations, 2001
In this paper we consider second order evolution equations with unbounded feedbacks. Under a regu... more In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.
We study the problem of stabilization for a class of evolution systems with fractional-damping. A... more We study the problem of stabilization for a class of evolution systems with fractional-damping. After writing the equations as an augmented system we prove in this article first that the problem is well posed. Second, using the LaSalle's invariance principle we show that the system is strongly stable. Then, based on a resolvent approach we show a luck of uniform stabilization. Next, using multiplier techniques combined with the frequency domain method, we shall give a polynomial stabilization result under some consideration on the stabilization of an auxiliary dissipating system. Finally, we give some applications to the wave equation.
In this paper we consider some stabilization problems for the wave equation with switching. We pr... more In this paper we consider some stabilization problems for the wave equation with switching. We prove exponential stability results for appropriate damping coefficients. The proof of the main results is based on D'Alembert formula and some energy estimates.
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 paper, we prove the exponential stability of the solution of the nonlinear dissipative Sc... more In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network and where the damping is localized on one branch and at the infinity. Contents 1. Introduction 1 2. Well-posedness 3 3. Exponential stability 8 Appendix 12 References 13
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.
In this paper we study the best decay rate of the solutions of a damped plate equation in a squar... more In this paper we study the best decay rate of the solutions of a damped plate equation in a square and with a homogeneous Dirichlet boundary conditions. We show that the fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup, if the damping coefficient is in L ∞ (Ω). Moreover, we give some numerical illustrations by spectral computation of the spectrum associated to the damped plate equation. The numerical results obtained for various cases of damping are in a good agreement with theoretical ones. Computation of the spectrum and energy of discrete solution of damped plate show that the best decay rate is given by spectral abscissa of numerical solution.
We study the problem of stabilization for the acoustic system with a spatially distributed dampin... more We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
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
In this work, we study the controllability of the bilinear Schrödinger equation on infinite graph... more In this work, we study the controllability of the bilinear Schrödinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schrödinger equation (BSE) i∂tψ = −∆ψ + u(t)Bψ in the Hilbert space L 2 p composed by functions defined on an infinite graph G verifying periodic boundary conditions on the infinite edges. The Laplacian −∆ is equipped with specific boundary conditions, B is a bounded symmetric operator and u ∈ L 2 ((0, T), R) with T > 0. We present the well-posedness of the (BSE) in suitable subspaces of D(|∆| 3/2). In such spaces, we study the global exact controllability and we provide examples involving tadpole graphs and star graphs with infinite spokes.
In this paper we consider a linear hybrid system which composed by two nonhomogeneous rods connec... more In this paper we consider a linear hybrid system which composed by two nonhomogeneous rods connected by a point mass and generated by the equations ρ 1 (x)u t = (σ 1 (x)u x) x − q 1 (x)u, x ∈ (−1, 0), t > 0, ρ 2 (x)v t = (σ 2 (x)v x) x − q 2 (x)v, x ∈ (0, 1), t > 0, u(0, t) = v(0, t) = z(t), t > 0, M z t (t) = σ 2 (0)v x (0, t) − σ 1 (0)u x (0, t), t > 0, with Dirichlet boundary condition on the left end x = −1 and a boundary control acts on the right end x = 1. We prove that this system is null controllable with Dirichlet or Neumann boundary controls. Our approach is mainly based on a detailed spectral analysis together with the moment method. In particular, we show that the associated spectral gap in both cases (Dirichlet or Neumann boundary controls) are positive without further conditions on the coefficients ρ i , σ i and q i (i = 1, 2) other than the regularities.
Journal of Dynamics and Differential Equations, 2015
We show that the best decay rate can be estimated by the observability (or controllability) cost ... more We show that the best decay rate can be estimated by the observability (or controllability) cost and open-loop admissibility cost. Moreover, we propose a numerical strategy to give an estimation for the best decay rate for a large class of evolution systems. Some examples are given to illustrate this new method.
Mathematics of Control, Signals, and Systems (MCSS), 2002
We consider the Rayleigh beam equation and the Euler-Bernoulli beam equation with pointwise feedb... more We consider the Rayleigh beam equation and the Euler-Bernoulli beam equation with pointwise feedback shear force and bending moment at the position x in a bounded domain ð0; pÞ with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if x=p is a rational number with coprime factorization x=p ¼ p=q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler-Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.
ESAIM: Control, Optimisation and Calculus of Variations, 2001
In this paper we consider second order evolution equations with unbounded feedbacks. Under a regu... more In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.
We study the problem of stabilization for a class of evolution systems with fractional-damping. A... more We study the problem of stabilization for a class of evolution systems with fractional-damping. After writing the equations as an augmented system we prove in this article first that the problem is well posed. Second, using the LaSalle's invariance principle we show that the system is strongly stable. Then, based on a resolvent approach we show a luck of uniform stabilization. Next, using multiplier techniques combined with the frequency domain method, we shall give a polynomial stabilization result under some consideration on the stabilization of an auxiliary dissipating system. Finally, we give some applications to the wave equation.
In this paper we consider some stabilization problems for the wave equation with switching. We pr... more In this paper we consider some stabilization problems for the wave equation with switching. We prove exponential stability results for appropriate damping coefficients. The proof of the main results is based on D'Alembert formula and some energy estimates.
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 paper, we prove the exponential stability of the solution of the nonlinear dissipative Sc... more In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network and where the damping is localized on one branch and at the infinity. Contents 1. Introduction 1 2. Well-posedness 3 3. Exponential stability 8 Appendix 12 References 13
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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