Research themes and keywords

I work in control theory for PDE's. I am both interested by applied aspects and the elaboration of general methods to answer some important questions in the control of PDE's and ODE's. I work on nonlinear and memory stabilization for dissipative PDE's and ODE's, uniform discretization of nonlinear damped PDE's, insensitizing control, observability/controllability/stabilization of coupled systems by a reduced number of controls, control questions for degenerate parabolic and hyperbolic PDE's, finite-time stabilization and more recently on some aspects of inverse problems. I worked in the past on nonlinear analysis for coupled stationary systems of PDE's which couples a Poisson's equation to two convection-diffusion equations. These systems model the transport of charges in semiconductors and in biology. I worked on uniqueness and multiplicity of solutions, singular perturbations (the normalized Debye length is a small parameter in front of the laplacian in the Poisson's equation) and asymptotic expansions for such singularly perturbed models. In particular I was interested in asymptotic estimates for the difference between the solution and the first terms of the asymptotic expansions. The methods I have developed rely on different parts of applied mathematics: analysis of PDE's, applied functional analysis, nonlinear analysis, convexity properties, energy methods, comparison principle, interpolation properties, ... and in the past on maximum principle, generalized maximum principle, nonlinear analysis for elliptic type systems and singular perturbation analysis.

- Control of PDE's: exact controllability, observability, stabilization

- Nonlinear stabilization of PDE's

- Coupled systems, reduced number of control, indirect stabilization, indirect observability, indirect controllability

- Insensitizing control

- Wave equations, hyperbolic equations of first and second order
in time

- Integro-differential PDE's, viscoelasticity, memory
stabilization

- Semi-discretization in space and time and full discretization of dissipative evolution PDE's

- Optimal energy decay rates and asymptotic behavior for dissipative evolution PDE's

- Degenerate parabolic and hyperbolic PDE's

and in the past:

- Drift-diffusion models for semiconductors and biological
membranes

- Nonlinear analysis

- Uniqueness and multiplicity of solutions

- Singular perturbation analysis

- Travelling waves in Combustion theory

I have introduced the following methods:

**The optimal-weight convexity method**: I have elaborated this new method in**Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. and Optimization, 51, no. 1, 61--105 (2005)**. These results were first announced in**A general formula for decay rates of nonlinear dissipative systems. C. R. Acad. Sci. Paris Sér. I Math, 338, 35--40 (2004)**. This method establishes a general and one-step formula for the energy decay rates of solutions of nonlinearly damped evolution PDE's, with arbitrary growth of the damping function. This formula is sharp (optimality is proved in some examples in infinite dimensions, and improves several existing rates in the literature) and gives a general methodology for many applications on examples of PDE's and of general damping functions. It applies to wave equations under abstract form either locally damped or damped on a part of the boundary, to the usual wave equation, Petrowsky equations, Schrödinger equations, ... Our method relies on new arguments which involve the dominant energy method, convexity properties, and general weighted nonlinear integral inequalities. This method is presented in the advanced course I gave at the CIME course given in Cetraro (Italy) in 2010 and the notes I wrote in**On some recent advances on stabilization for hyperbolic equations. Lecture Note in Mathematics/C.I.M.E. Foundation Subseries Control of Partial Differential Equations, Springer Verlag, volume 2048, 101 pages (2012)**

I have simplified the general one-step formula using a classification of the nonlinear damping function in

In collaboration with Piermarco Cannarsa, I have extended the optimal-weight convexity method to the case of memory stabilization for general decaying convolution kernels for abstract wave type equations in

This work extended my previous work in collaboration with Piermarco Cannarsa and Daniela Sforza in

In collaboration with Kaïs Ammari in

I have further generalized this approach in collaboration with Yannick Privat and Emmanuel Trélat in

In collaboration with Piermarco Cannarsa and Günter Leugering, I also proved that the optimal-weight convexity method applies to the boundary nonlinearly damped degenerate wave equation in

**The higher order energy method:**I have announced the method in**Indirect boundary stabilization of weakly coupled systems (French title: Stabilisation frontière indirecte de systèmes faiblement couplés). C. R. Acad. Sci. Paris Sér. I Math, 328, 1015--1020 (1999)**and developed the methodology for coupled systems of two coupled abstract evolution equations with unbounded damping operators in**Indirect boundary stabilization of weakly coupled systems. Siam J. on Control and Optimization, 41, no. 2, 511--541 (2002)**and for two coupled abstract equations with bounded and coercive damping operators in collaboration with Piermarco Cannarsa, Vilmos Komornik in**Indirect internal damping of coupled systems. J. of Evolution Equations, 2, 127--150 (2002)**. This method gives a general approach for deriving polynomial stability for coupled systems with a reduced number of damping operators. It relies on the use of higher order energies, that is the usual energies of the time derivatives of the solution up to a suitable order, to counterbalance the lack of damping actions on some of the equations in the coupled system. For this, we have established a polynomial stability result for energies of semigroup solutions of abstract PDE's satisfying an integral inequality (which is not associated to a differential inequality) involving higher order energies of the solutions at the initial time. We give a very simple proof based on an induction argument together with a Fubini argument and semigroup property. I have further generalized this approach to partially coercive and bounded damping operators in collaboration with Matthieu Léautaud in**Indirect stabilization of locally coupled wave- type systems. ESAIM COCV 18, pp. 548—582 (2012)**and for hybrid boundary conditions in collaboration with Piermarco Cannarsa and Roberto Guglielmi relying on general interpolation results in**Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Mathematical Control and Related Fields 1, pp. 413—436 (2011)**

I have also studied relations between nonlinear and indirect
damping mechanisms for dissipative systems. More precisely, I have
also proved that it is possible to couple under certain conditions
the optimal-weight convexity method and the higher order energy
method in **Asymptotic behavior for Timoshenko beams subject to
a single nonlinear feedback control. NoDEA 14, no. 5-6, 643--669
(2007) **and in collaboration with Zhiqiang Wang and Lixin Yu
in **A one-step optimal energy decay formula for
indirectly nonlinearly damped hyperbolic systems coupled
by velocities, arXiv :1503.04126 (2015)**.

**The two-level energy method:**I have announced the method in**Indirect boundary observability of a weakly coupled wave system (French title: Observabilité et contrôlabilité frontière indirecte de deux équations des ondes couplées). C. R. Acad. Sci. Paris Sér. I Math, 333, 645--650 (2001)**and developed the general method in**A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems****Siam J. on Control and Optimization, 42, no. 3, 871--906 (2003)**for boundary damped symmetrically coupled systems. The general methodology relies on the use of different energy levels, the usual energy level for the observed component of the solution and a weakened energy for the unobserved energy component, a suitable sharp energy balance between the different energies which are involved, the conservation of the total energy of the system and suitable energy estimates. A key point is also a general uniform in time observability estimate for a forced abstract wave type equation. I have further extended this method in collaboration with Matthieu Léautaud to symmetrically coupled systems allowing partially coercive coupling operators in**Indirect controllability of locally coupled wave-type systems and applications. Journal de Mathématiques Pures et Appliquées 99, pp. 544--576 (2013)**and for bounded as well as unbounded observation opertors. This allowed us to derive the first positive observability and controllability properties for coupled systems for geometrical situations in which the control regions and coupling regions are at positive distance from each other.

I have further shown that the two-level energy method also
applies to abstract cascade coupled systems (which are simpler to
study than symmetrically coupled systems) in **Insensitizing
exact controls for the scalar wave equation and exact
controllability of $2$-coupled cascade systems of PDE's by a
single control. Mathematics of Control, Signals, and Systems,
26, 1—46 (2014)**. I gave a general necessary and sufficient
condition for partially coercive coupling operators and for
bounded as well as unbounded observation operators. I also prove
general insensitizing controllability results for scalar abstract
wave equations for bounded as well as unbounded control operators.**
**

**Multi-level energy method:** I generalized the two-level
method to a multi-levels energy method for abstract cascade
systems of n equations subjected to a single control, where n is
any integer larger or equal to 2 in **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. Advances in Differential Equations, 18, 1005-1072
(2013)**. This extension is based on a sharp and subtile
induction argument on the number of equations of the cascade
system. Its proof relies on the use of weakened energies of order
0, -1 up to 2-n of the corresponding component of the solution.
The method is constructive and uses the property that one can
derive from the original system set in the natural energy space a
hierarchy of related systems similar to the original one, but set
in weakened energy spaces. The solutions of these hierarchic
systems are linked to each other and this rich structure allows us
to get positive controllability results. I also extended the
necessary and sufficient condition (derived for 2 equations) to n
equations for partially coercive coupling operators and for
bounded as well as unbounded observation operators. These results
apply to cascade wave equations, as well as cascade systems of
Petrowsky or plates equations or cascade systems of Schrödinger
equations. We also explain how these results can be used to derive
simultaneous control properties for systems of PDE's.

I have also studied the influence of the
coupling action in cascade systems when the coupling operator is
no longer assumed to be partially coercive (in practice this means
that the coupling coefficient in the coupled systems changes sign
within the spatial domain) in **On the influence of the coupling
on the dynamics of single-observed cascade systems of pde’s.
Mathematical Control and Related Fields, 5, 1—30 (2015)**. I
give general positive results as well as negative results for
examples of coupling coefficients, showing how the coupling may
interact with the modes of the solution at any frequency,
leading to non unique continuation results.

**Control/observability and stabilization of degenerate PDE's: **I
have studied a semilinear degenerate heat equation in
collaboration with Piermarco Cannarsa and Genni Fragnelli in **Carleman
estimates for degenerate parabolic operators with applications
to null controllability. J. of Evolution Equations, 6, 161****—204
(2006). **The proof relies as usual on an obervability
inequality for the adjoint equation, the main difficulty is the
generator of the underlying semigroup is no longer uniformly
elliptic, the diffusion coefficient degenerating at some part of
the boundary of the spatial domain. We prove Carleman estimates
with weights adapted to the degeneracy of the coefficients (in the
spirit of ideas which were introduced by Piermarco Cannarsa,
Judith Vancostenoble and Patrick Martinez), suitable Hardy's
inequalities. We then derive the desired observability inequality.
In collaboration with Piermarco Cannarsa and Günter Leugering, I
also studied control, observation and nonlinear stabilization for
degenerate wave equations in **Control and stabilization of
degenerate wave equations, arXiv :1505.05720 (2015)**.

**Boundary stabilization for the anisotropic elasticity
system: ****
**for this system, it was
conjectured that the natural boundary feedback was not
sufficient to stabilize the energy of the solutions. In
collaboration with Vilmos Komornik in **Boundary
observability, controllability and stabilization of
linear elastodynamic systems. Siam J. on Control and
Optimization, 37, no. 2, 521-542 (1998)**, I proved that the
conjecture is false for spatial domains close to balls in any
dimensions. Indeed the natural feedback is sufficient to
stabilize exponentially the energy of the solutions.**
**

I describe below my former contributions and works (in the past)
on the drift-diffusion model for the transport of charges in
semiconductors and biological membranes.

I proved the existence of multiple solutions for doping profiles
(an inhomogeneity which appears in the Poisson's equation and
which defines the type of semiconductor) which change sign three times
in space (case of thyristor) in **On the existence of multiple
steady-state solutions in the theory of electrodiffusion}. Part
I: the nonelectroneutral case. Part II: a constructive method
for the electroneutral case. Trans. of the A.M.S., 350, no. 12,
4709--4756 (1998). **This model
is complex : it is (strongly) nonlinear, it is a singularly
perturbed systems (the small parameter --the Debye length --
appears in front of the Laplacian in the Poisson's equation
for the electrostatic potential), it is well-known physically
that uniqueness or multiple solutions may exist depending on
the number of sign changes of the given doping profile (which
is the inhomogeneity appearing in the Poisson's equation)
which defines the type of semiconductor : p-n jonction (1
single sign change), a transistor (2 sign changes), a
thyristor (3 sign changes) , the voltage amplitudes and other
physical parameters.

For this,
instead of studying directly the voltage driven model, I
studied the current driven model. I proved existence and
uniqueness for this model in **Structural properties of the
one dimensional drift-diffusion models for semiconductors.
Trans. of the A. M. S., 348, 823--871 (1996) **and that
if multiple solutions may exist they give rise only to turning
points for the curve giving the total current in terms of the
applied voltage. As a
consequence, I showed that the saturation phenomenon which was
predicted by numerical simulations by M. S. Mock (Compel. 1
(1982), pp. 165--174), never occurs in the mathematical
drift-diffusion model.

To derive
the multiplicity results, I studied the linearized
current driven equations. I proved that the corresponding
linearized operator is invertible and I gave a uniform bound
of the norm of its inverse in with respect to the singular
perturbation parameter. This is a key result based on a
suitable choice of either the electron or the hole
convection-diffusion equation, this information being then
used in the Poisson's equation through a "maximum principle
type property". I proved that the singular asymptotic
expansion of the singularly perturbed systems is accurate at
any arbitrary order. This allows to deduce that if multiple
solution exist for the reduced model, then the original
singularly perturbed system has multiple solutions. I then
gave a constructive method based on a suitable normalization
process for high currents.

I also
proved several uniqueness results either close to electric
equilibrium (small voltages) or for large voltages in **A
method for proving uniqueness theorems for the stationary
semiconductor device and electrochemistry equations.
Nonlinear Anal., 18, 861--872 (1992)**, **in A
uniqueness theorem for reverse biased diodes. Applicable
Anal., 52, 261--276 (1994)**, **New uniqueness theorems
for the one dimensional drift-diffusion semiconductor device
equations. Siam J. on Math. Anal., 26, 715--737 (1995)**,
and **Results for the steady-state electrodiffusion
equations in case of monotonic potentials and multiple
junctions. Nonlinear Anal. 29, no. 8, 849--887 (1997)**.

I
also proved several results of uniform asymptotic estimates
for the singularly perturbed drift-diffusion model with
respect to the Debye length in **Uniform asymptotic error
estimates for semiconductor device and electrochemistry
equations. Nonlinear Anal., 14, 123--139 (1990)**, in
collaboration with Mohand Moussaoui in **Asymptotic
estimates for the multi-dimensional electro-diffusion
equations. Math. Models Methods Appl. Sci. 8, no. 3,
469--484 (1998) **and** **in
collaboration with Abdelghafour Jabir and Mohand Moussaoui in
**An asymptotic analysis of a unipolar junction model.
Applicable Analysis, 72, 127--153 (1999)**.

** **I also worked on the
asymptotic analysis of an intransient Vlasov-Poisson model in
collaboratin with Kamel Hamdache and Yue-Jun Peng in **Analyse
asymptotique du système de Vlasov-Poisson instationnaire dans le
cas d'une diode plane. Asymptotic Analysis, 16, no. 1, 25--48
(1998) **and in collaboration with Alessandra Lunardi on the
analysis of stability of travelling wave solutions for a model of
flame in combustion theory in **Behaviour near the travelling
wave solution of a free boundary system in combustion theory.
Dynamic Systems and Applications, 1, 391--418 (1992)**.** **
** **

** **
** **
** **
** **
** **
** **** **
** **** **
** **