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Fatiha Alabau-Boussouira

Publications and source records attributed to Fatiha Alabau-Boussouira.

16 recordsLinked to original sources
arXiv2026

Internal controllability of first order quasilinear hyperbolic systems with a reduced number of controls

In this paper we investigate the exact controllability of $n \times n$ first order one-dimensional quasilinear hyperbolic systems by $m<n$ internal controls that are localized in space in some part of the domain. We distinguish two situations. The first one is when the equations of the system have the same speed. In this case, we can use the method of characteristics and obtain a simple and complete characterization for linear systems. Thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems. However, when the speed of the equations are not anymore the same, we see that we encounter the problem of loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system, we first use the notion of algebraic solvability due to M. Gromov. However, in contrast with this prior article where a standard fixed point argument could be used to treat the nonlinearities, we use here a fixed point theorem of Nash-Moser type due to M. Gromov in order to handle the problem of loss of derivatives.

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arXiv2023

Part I: Rebuttal to "Uniform stabilization for the Timoshenko beam by a locally distributed damping"

A paper, entitled "Uniform stabilization for the Timoshenko beam by a locally distributed damping" was published in 2003, in the journal Electronic Journal of Differential Equations. Its title concerns exclusively its Section 3, devoted to the case of equal speeds of propagation and to its main theorem, namely Theorem 3.1. It states that the solutions of the Timoshenko system (see (1.3) in [1]) decays exponentially when the damping coefficient b is locally distributed. The proof of Theorem 3.1 is crucially based on Lemma 3.6, which states the existence of a strict Lyapunov function along which the solutions of (1.3) decay when the speeds of propagation are equal. This rebuttal shows the major gap and flaws in the proof of Lemma 3.6, which invalidate the proofs of Lemma 3.6 and Theorem 3.1. Lemma 3.6 is stated at the top of page 12. The main part of its proof is given in the pages 12 and 13. In the last eight lines of page 13, eight inequalities are requested to hold together for the proof of Lemma 3.6. They don't appear in the statements of Lemma 3.6. The subsequent flaws come from the evidence that several of them are contradictory either between them or with claims in the title of the article. We also point in this rebuttal other flaws, or gaps in the proofs of Theorem 2.2 related to strong stability and non uniform stability for the case of distinct speeds of propagation. In [3], we correct and complete the proof of strong stability. We also correct, set up the missing functional frames, fill the gaps in the proof of non uniform stability in the cases of different speeds of propagation, and complete a missing argument in the proof of Theorem A in [4] (see Remark 4.3), the result of Theorem A being used in the paper [1] on which this rebuttal is mainly devoted.

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arXiv2023

Part II On strong and non uniform stability of locally damped Timoshenko beam: Mathematical corrections to the proof of Theorem 2.2 in the publication referenced as [1] in the bibliography

In part I of the rebuttal (see [2] to the article [1] entitled "Uniform stabilization for the Timoshenko beam by a locally distributed damping" published in 2003, in the journal Electronic Journal of Differential Equations, we prove that Lemma 3.6 and Theorem 3.1 are unproved due to major flaws (contradictory assumptions). We also show that Theorem 2.2 and its proofs of strong stability, and non uniform stability in the case of different speeds of propagation, contain several incorrect arguments and several gaps (including missing functional frames). In this part II, we give the precise missing functional frames, fill the gaps and correct several parts contained in the proof of Theorem 2.2 in [1]. We also complete a missing argument (see Remark 4.23 and Remark 3.2) in the proof of Theorem A in [5] used by [1]. For this we state and prove Proposition 4.4 (see also Proposition 4.6 for a general formulation in Banach spaces). We also give the correct formulations, and proofs of strong stability and non uniform stability (in case of different speeds of propagation) for Timoshenko beams.

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arXiv2023

Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation

We study the exact controllability of the evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$, which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of $A$, as initial data. The analogous problem (in a more general form) was addressed in our previous paper [Exact controllablity to eigensolutions for evolution equations of parabolic type via bilinear control, Alabau-Boussouira F., Cannarsa P. and Urbani C., Nonlinear Diff. Eq. Appl. (2022)] for a bounded operator $B$. The current extension to unbounded operators allows for many more applications, including the Fokker-Planck equation in one space dimension, and a larger class of control actions.

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arXiv2022

Rebuttal to "Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation"

A paper, entitled "Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation" was published in 2016, in the journal Boletim da Sociedade Paranaense de Matematica. The main results of that paper are stated in Theorem 1.3. Section 2 is devoted to stability estimates for the solutions of the Galerkin approximation (2.1)-(2.2) of (1.2). Section 3 is devoted to the proof of the well-posedness for the Kirchhoff equation (1.2). This rebuttal is written in view to show the major flaws, and gaps in the proofs of this publication.

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arXiv2021

Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

In a separable Hilbert space $X$, we study the controlled evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A\geq-σI$ ($σ\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the $j$th eigensolution for any $j\geq1$. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.

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arXiv2019

Exact controllability to the ground state solution for evolution equations of parabolic type via bilinear control

In a separable Hilbert space $X$, we study the linear evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A$ is an accretive self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground state solution, that is, the solution of the free equation ($p\equiv0$) starting from the ground state of $A$. We also derive global controllability results in large time and discuss applications to parabolic equations in low space dimension.

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arXiv2019

Superexponential stabilizability of evolution equations of parabolic type via bilinear control

We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\geq0 \end{equation*} where the operator $-A$ is a self-adjoint accretive operator on a Hilbert space and $p(\cdot)$ is the control function. The proof is based on a linearization argument. We prove that the linearized system is exacly controllable and we apply the moment method to build a control $p(\cdot)$ that steers the solution to the ground state in finite time. Finally, we use such a control to bring the solution of the nonlinear equation arbitrarily close to the ground state solution with doubly exponential rate of convergence. We give several applications of our result to different kinds of parabolic equations.

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arXiv2015

Nonlinear damped partial differential equations and their uniform discretizations

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequalities with respect to the discretization parameters. We establish our results, first in the continuous setting, then for space semi-discrete models, and then for time semi-discrete models. The full discretization is inferred from the previous results. Our results cover, for instance, the Schrödinger equation with nonlinear damping, the nonlinear wave equation, the nonlinear plate equation, as well as certain classes of equations with nonlocal terms.

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arXiv2015

Control and stabilization of degenerate wave equations

We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter $μ_a>0$. We establish observability inequalities for weakly (when $μ_a \in [0,1[$) as well as strongly (when $μ_a \in [1,2[$) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when $μ_a>2$) and the blow-up of the observability time when $μ_a$ converges to $2$ from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degenerate control problem when $μ_a \in [0,2[$. We conclude the paper by studying the boundary stabilization of the degenerate linearly damped wave equation and show that a suitable boundary feedback stabilizes the system exponentially. We extend this stability analysis to the degenerate nonlinearly boundary damped wave equation, for an arbitrarily growing nonlinear feedback close to the origin. This analysis proves that the degeneracy does not affect the optimal energy decay rates at large time. We apply the optimal-weight convexity method of \cite{alaamo2005, alajde2010} together with the results of the previous section, to perform this stability analysis.

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arXiv2015

A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in \cite{AB01, ACK01, AB02, AL12} for the linear damping case, and in \cite{AB07} for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of \cite{AB05, AB10}.

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arXiv2014

Finite-time stabilization of a network of strings

We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term $αu_t$ with a coefficient $α$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $α$, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node.

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arXiv2014

A hierarchic multi-level energy method for the control of bi-diagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls

This work is concerned with the exact controllability/observability of abstract cascade hyperbolic systems by a reduced number of controls/observations. We prove that the observation of the last component of the vector state allows to recover the initial energies of all of its components in suitable functional spaces under a necessary and sufficient condition on the coupling operators for cascade bi-diagonal systems. The approach is based on a multi-level energy method which involves $n$-levels of weakened energies. We establish this result for the case of bounded as well as unbounded dual control operators and under the hypotheses of partial coercivity of the $n-1$ coupling operators on the sub-diagonal of the system. We further extend our observability result to mixed bi-diagonal and non bi-diagonal $n+p$-coupled cascade systems by $p+1$ observations. Applying the HUM method, we derive the corresponding exact controllability results for $n$-coupled bi-diagonal cascade and $n+p$-coupled mixed cascade systems. Using the transmutation method for the wave operator, we prove that the corresponding heat (resp. Schrödinger) multi-dimensional cascade systems are null-controllable for control regions and coupling regions which are disjoint from each other and for any positive time for $n \le 5$ for dimensions larger than $2$, and for any $n \ge 2$ in the one-dimensional case. The controls can be localized on a subdomain or on the boundary and in the one-dimensional case the coupling coefficients can be supported in any non-empty subset of the domain.

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arXiv2014

Insensitizing exact controls for the scalar wave equation and exact controllability of $2$-coupled cascade systems of PDE's by a single control

We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE's and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in \cite{sicon03, alaleau11} for symmetric coupled systems, to cascade systems which are examples of non symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos Lebeau and Rauch \cite{blr92}. By duality, this solves the exact controllability, by a single control, of $2$-coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger $2$-coupled cascade systems under (GCC) and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa \cite{DeT00}. Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by J.-L. Lions \cite{lions89} and later on by Dáger \cite{Dager06} and Tebou \cite{tebou08}.

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