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.

and in the past:

I have introduced the following methods:

I have simplified the general one-step formula using a classification of the nonlinear damping function in A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems. J. of Differential Equations 248, pp. pp. 1473–1517 (2010). I also prove that the one-step formula is optimal in finite dimension. The proof relies on a (new) energy comparison principle for finite dimensional dissipative systems, and  comparison arguments between time pointwise estimates and energy type estimates. This approach is  also valid for energy decay rates (nonuniform with respect to the mesh size) for the semi-discretized PDE's. I  have also proved lower energy estimates in the infinite dimensional framework, based on interpolation properties and energy comparison principles in the above 2010 JDE article but also in New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems. J. of Differential Equations 249, pp. 1145–1178 (2010) and Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations. Nonlinear Differential Equations and Applications 18, pp. 571—597 (2011)

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 A new method for proving sharp energy decay rates for memory-dissipative evolution equations for a quasi-optimal class of kernels. Comptes Rendus Mathématique, C. R. Acad. Sci. Paris, Sér. I 347, 867--872 (2009). In this case, the damping action is a convolution in the time variable, of the diffusion operator (acting on the solution) with a decaying kernel $k$. The damping mechanism is more complex to analyze. In particular the solution at time $t$ depends on all its states on $[0,t]$. As always, in this work I have been interested by proving sharp results through a general methodology, which relies on intrinsic properties and gives a simple, one-step and general energy decay formula. In particular, we prove that the energy of the solution decays at least as the convolution kernel at infinity for very general kernels. This requires to understand how nonlocal memory type stabilization acts for kernels which may have very different decaying behavior at infinity.

This work extended my previous work in collaboration with Piermarco Cannarsa and Daniela Sforza in Decay estimates for second order evolution equations with memory. J.F.A. 254, no 5, 1342--1372 (2008) which was devoted to the case of polynomially or exponentially decaying convolution kernels.

In collaboration with Kaïs Ammari in Sharp energy estimates for nonlinearly locally damped PDE’s via observability for the associated undamped system. J. of Functional Analysis 260, pp. 2424–2450 (2011) (the results were first announced in Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE’s. Comptes Rendus Mathématique 348, pp. 165-170 (2010)), I have proved that one can deduce the above one-step energy decay formulas for locally nonlinearly damped PDE's, under an observability inequality for the corresponding conservative system (with no damping). The result relies on the optimal-weight convexity method, comparison arguments between the conservative system, the linearly damped system and the nonlinearly damped system. It also establishes several interesting results in themselves, in particular how to link discrete energy estimates schemes and the corresponding continuous ones.
I have further generalized this approach in collaboration with Yannick Privat and Emmanuel Trélat in Nonlinear damped partial differential equations and their uniform discretizations, Preprint Hal hal-01162639 (2015) for general abstract first order nonlinearly damped systems. We also establish in this paper, space, time and full discretization schemes which are uniform with respect to the space discretization parameter. We also give several applications to important examples of PDE's.

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
Control and stabilization of degenerate wave equations, arXiv :1505.05720 (2015).

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).

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).