Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (= Schrodinger Ginzburg-Landau) equation.

In the recent years, the stabilization of many PDEs was obtained by the backstepping approach. By considering a transformation mapping the
equation to stabilize to a stable equation, one reduces the stabilization problem to proving that the transformation is invertible. This strategy is applied on a linear Schrödinger equation with a distributed control. The rapid stabilization of this equation is then proved thanks to the introduction of a uniqueness condition.

In this talk, we consider the controllability properties of a strongly degenerate parabolic equation with Dirichlet boundary control. Despite the
strong degeneracy, we prove a result of well-posedness and null controllability with a control that acts on the degenerate part of the boundary. Then, we study the uniform controllability in the vanishing viscosity limit and prove that the cost of the control explodes exponentially fast in small time and converges exponentially fast in large time in some adapted weighted norm. The main tools used are a spectral decomposition involving Bessel functions and their zeros, some usual results on admissibility of scalar controls for diagonal semigroups, and the moment method of Fattorini and Russell.

This talk will review the biological phenomenon, and formulate a theoretical framework leading to a general theorem characterizing scale invariant behavior by equivariant actions on sets of vector fields that satisfy appropriate Lie-algebraic nondegeneracy conditions. The theorem allows one to make experimentally testable predictions, and the presentation will discuss the validation of these predictions using genetically engineered bacteria and microfluidic devices, as well their use as a "dynamical phenotype" for model invalidation. If there is time, the talk will close with some very speculative remarks about the role of the shape of transient responses in immune system self/other recognition and in current cancer immunotherapy.

I am interested in observers the design of which does not exploit the full a priori (= models) information and/or a posteriori (= measurements) information. Such observers compensate their design weakness by time evolution and are consequently dynamical systems with the estimate as output. In this context, the issue of estimate convergence comes out.
In the finite dimensional continuous time case, several properties are necessary to obtain convergence:
a) detectability always,
b) instantaneous observability when convergence speed is arbitrary
c) weak differential detectability given by a covariant tensor field of order 2, with non negative and symmetric values
when the observer output function is injective.
When strengthened these conditions become sufficient
a) observability (instead of detectability) is sufficient for the existence of a convergent nonlinear Luenberger observer
b) differential observability of some order (instead of instantaneous observability) is sufficient for the existence of a (practically) convergent high gain observer
c) strong (instead of weak) differential observability is sufficient for the existence of a locally convergent observer exploiting a Riemannian metric. The convergence is (semi-)global when a convexity property holds.

Noncooperative differential games with linear dynamics and quadratic costs have been widely studied. The existence of explicit solutions, determined by a finite set of algebraic equations, makes them a very attractive modeling tool. In applications, linear-quadratic games can be regarded as a natural approximation to more realistic, nonlinear models. In an attempt to justify this approximation, one is led to the following question:
Starting from a linear-quadratic differential game for two players, suppose that some small, nonlinear perturbations are added to the dynamics of the system and to the cost functions. Does the perturbed game still have a Nash equilibrium solution in  feedback form, close to the original one?
The talk will present results in this direction, in a finite as well in an infinite time horizon.

Difference equations are a useful tool in the analysis of hyperbolic systems of PDEs, providing a handy representation for some simplified dynamics. We start the talk by briefly presenting some recent results on the stability analysis of difference equations with time-varying parameters and providing an application to a system of wave equations on a network with time-varying damping terms at the exterior nodes. We then discuss relative, exact, and approximate controllability of difference equations. A necessary and sufficient condition for relative controllability is provided, together with an upper bound on the minimal relative controllability time, generalizing the usual Kalman controllability criterion. Exact and approximate controllability in $L^2$ are characterized in the case of two-dimensional difference equations with two delays and one control, highlighting the fact that even such simple-looking, low-dimensional case presents some non-trivial features.

This talk is concerned with pointwise second-order necessary optimality conditions for optimal control problems. First we present a quite natural second-order extension of Pontryagin's maximum principle based on a second-order adjoint system that we will introduce. To further underline the links with Pontryagin's maximum principle we formulate second-order sensitivity relations along optimal trajectories that involve the second-order adjoint states. In a second part we show that using similar techniques as before but in the case when the system dynamics are differentiable with respect to controls, greatly generalized Jacobson necessary optimality conditions can be obtained.

This work is dedicated to the control of some classes of mechanical systems, not stabilizable by means of at least continuous state feedback laws. This is the case for non holonomic mechanical systems, an example being the unicycle robot, or for under-actuated mechanical systems, an example being the slider. The similarity between these two kinds of dynamical systems will be analyzed, with respect to controllability, stabilizability and flatness properties. The control design for slider stabilization could be viewed, to some extent, as an extension of the control law used for unicycle systems, but it is not so obvious if one wants to stabilize the slider at a chosen reference position, with a chosen fixed orientation. In fact, a new fixed point stabilization result will be given for the slider. Therefore, a control strategy will be proposed for both systems, based on differential flatness property to track moving non singular reference trajectories, and on periodic time-varying feedback laws to ensure final stabilization at rest equilibrium points.

I will explain new quantitative estimates on the space-time analyticity of solutions to parabolic initial value boundary problems with $L^2$ initial data. These are motivated by applications to the null-controllability of parabolic equations with controls supported on measurable sets with positive Lebesgue measure.

This talk is to show the applications of observability inequalities from measurable sets (which is discussed in the previous report by S. Montaner) in time-optimal control problems for parabolic equations: such as the existence and uniqueness of optimal controls, as well as the bang-bang property of optimal controls (which means that optimal controls are always on the boundary of the control constraint set).

The problem of stabilization has long played a central role in both the theory and application of feedback control. In the case of linear systems, key contributors include Hermite, Hurwitz, Maxwell, Routh, and Nyquist. Up until the 1950s nonlinear efforts were dominated by specific applications treated by method of Liapunov but in the 50s and 60s some success was had using a range of methods involving restricted classes of nonlinearities together with passivity properties of linear systems. At the same time, additional understanding of the complexity of differential equations was emerging, based in part on topological methods. Along with this additional understanding came the confidence to attack a wider classes of systems. The novelty that control theory brought to the picture was the idea of studying classes of differential equations which were only partially defined, having other parts that could be specified, subject to certain restrictions. This situation is discussed in detail in J.-M. Coron's recent book, Control and Nonlinearity. In this talk we will give some new results on such feedback stabilization questions and present a puzzle or two on the relationship between the existence of optimal controls, the existence of oscillations and feedback stabilization.

In this joint work with Remi Azouit and Alain Sarlette, we prove that a quantum harmonic oscillator governed  by a Lindblad master equation where the typical drive and loss channels are multi-photon processes instead of single-photon ones, converges to a protected subspace spanned by a finite set of coherent states with the same amplitude and uniformly distributed phases. We also show that this convergence features a finite set of invariant physical observables, such that the final state in the protected subspace can be directly predicted from the initial state. The proof
includes the full arguments towards the well-posedness of the corresponding dynamics in  proper Banach spaces of Hermitian trace-class operators equipped with adapted nuclear norms. It relies on Hille-Yosida theorem and Lyapunov convergence analysis.

In this talk I will present some recent results about the qualitative (symmetry and symmetry breaking) properties of extremal functions for functional inequalities, obtained in collaboration with J. Dolbeault and M. Loss. I will show that linear and nonlinear flows are a very good tool to do this in an optimal way. The main subject of my talk will be the symmetry of extremals for Caffarelli-Kohn-Nirenberg inequalities on the Euclidean espace. But I will also explain how this method can be used to derive similar inequalities and results on Riemanian manifolds.

Chemoxis is a biological process in which a population moves due to a chemical stimulus in the enviroment. In this talk, we present some new results concerning the controllability of some models of chemotaxis. In particular, with these new results, we are able to avoid some blow up phenomena.

We study a free boundary problem modelling the motion of a piston in a viscous gas. The gas-piston system fills a cylinder with fixed extremities, which possibly allow gas from the exterior to penetrate inside the cylinder. The gas is modeled by the 1D compressible Navier-Stokes system and the piston motion is described by the second Newton's law. We prove the existence and uniqueness of global in time strong solutions. The main novelty brought in by our results is that they include the case of nonhomogeneous boundary conditions which, as far as we know, have not been studied in this context. Moreover, even for homogeneous boundary conditions, our results require less regularity of the initial data than those obtained in previous works. This is a joint work with Marius Tucsnak and Takeo Takahashi.

Our main goal in this lecture is to discuss the numerical solution of the system of ordinary and partial differential equations modelling the motion of rigid solid particles in a region filled with an incompressible viscoelastic liquid. The above system couples the unsteady Navier-Stokes equations modelling the flow of the liquid, the Newton-Euler equations modelling the motion of the particles, and the differential equation modelling the evolution of the elastic component of the stress tensor. The viscoelastic models to be considered are of the Oldroyd-B and FENE-CR (for Chilcott & Rallison) types. The key ingredients of the methodology we employ to achieve the numerical simulation of the above mixtures are:

The results of numerical experiments will be presented: they concern the settling of rigid solid particles subjected to gravity in an infinitely long vertical cylinder filled with an incompressible visco-elastic liquid. These experiments show, as expected, particle chaining phenomena taking place in the cylinder, with Oldroyd-B and FENE-CR fluids having significantly different chaining behaviors.

I will present a system of equations describing the effect of including a line (the "road") with a specific diffusion on biological invasions in a plane (the "field"). Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. One can determine the asymptotic shape of propagation in every direction influenced by the road. I will discuss several further effects such as transport, reaction on the road or the influence of various parameters. I will also describe some variants of this system. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.

In this talk we are concerned with the Vlasov-Navier-Stokes system, a nonlinear system describing the interaction of a large cloud of particles immersed in a viscous incompressible fluid. We achieve an internal null-controllability result for this system in the bidimensional torus, which allows to modify both the distribution of particles and the velocity field of the fluid in order to steer the system from small initial data to the zero equilibrium. The strategy of this result is obtained by means of Coron's return method, combined with a Leray-Schauder fixed-point scheme.

This talk is related to the theory of nonlocal stabilization by starting control for the equations of normal type. We consider the semilinear parabolic equation of normal type connected with the 3D Helmholtz equation with periodic boundary condition. Normal parabolic equation is derived from 3D Helmholtz equation by replacing the nonlinear term by its orthogonal projection B(v) on the vector v of velocity.
We will discuss the problem of stabilization to zero of the solution for NPE with arbitrary initial condition y0 by starting control in the form v(x) = λu(x), where λ is some constant depending on y0, and u is a universal function, depending only on a given arbitrary subset Ω ⊂ T^3, which contains the support of control u. This problem is reduced to three inequalities connected with starting control.
Semilinear normal parabolic equations have been derived in order to get better understanding of the structure of the solutions of the three-dimensional Helmholtz and Navier-Stokes system and other similar equations. In future we hope to use the obtained results to prove stabilization for Navier-Stokes and Helmholtz equations.

This lecture deals with various recent developments concerning the old and very classical concept of topological degree for continuous maps from the circle into itself (also called winding number or index). I will first explain how it can be extended beyond the class of continuous maps. This led to the "accidental"discovery of a simple, but intriguing  formula connecting the degree of a map  to its Fourier coefficients. The relation is easily justified when the map is smooth. However, the situation turns out to be extremely delicate if one assumes only continuity, or even Holder continuity. This "marriage" is more difficult than expected and there are many difficulties in this couple such as the following question I raised: "Can you hear the degree of a map from the circle into itself?"
I will also present new estimates for the degree leading to the question: "How much energy do you need to produce a map of given degree?".
Many simple looking problems remain open. The initial motivation for this research came from the analysis of the Ginzburg-Landau model in Physics.

We consider networks of open channels which can be monitored and controlled at the junctions. These control systems can be open loop unstable and subject to unmeasured boundary disturbances. We address the issue of feedback stabilization and disturbance rejection under proportional-integral control. When the dynamics are represented by Saint-Venant equations with non-uniform steady states, we show how an energy Lyapunov function can be used to analyze the closed-loop stability. The issue is presented through the specific case study of the control of navigable rivers with a particular focus on the Meuse river. An important challenge is to regulate the water level while mitigating the flow rate variations and rejecting disturbances induced by hydro-electric power plants.

For boundary inputs, ISS has heretofore been proved only in a restrictive sense - with respect to the disturbance’s derivative. I will present ISS results that are analogous to those for ODEs, establishing gains from the supremum of the disturbance in time to various spatial norms on the state. Such results are not obtainable using those same spatial norms of the state as Lyapunov functionals. Instead, we develop approaches that rely on alternative representations of the PDE. With ISS shown, I will illustrate the application of small-gain arguments on a model of thermoelasticity in which a feedback coupling of the thermal and structural dynamics calls for a small gain analysis in various spatial norms.

In this joint work with Jean-Michel Coron and Franck Sueur, we consider the Navier-Stokes equation within a 2D or 3D smooth bounded domain. We use controls only acting on a small section of the whole boundary (or supported within any small open subset). On the remaining parts of the boundary, the fluid obeys the Navier slip-with-friction homogeneous boundary condition. We prove small time global exact null
controllability. Our strategy is to deduce our result from the controllability of the corresponding Euler equations by means of a small-time-to-small-viscosity scaling. The main difficulty is to handle the boundary layers that appear near all the uncontrolled parts of the boundary.

This talk takes place in the context of the stabilization of partial differential equations. More precisely it is considered the Korteweg-de Vries equation, which models long waves of relatively shallow depth. After a short introduction on the KdV equation and on the design of a unconstrained distributed feedback law stabilizing the origin of the equation, the talk will deal with the case where a stabilizing feedback law has to be bounded by a saturation function. The sketch of the proof of the stability is given. Finally, the results are illustrated by simulations.

We will present the problem of bilinear controllability for Schrödinger equation in a bounded domain, i.e., the possibility of controlling the equation by an action on a potential, with the difficulties and the few available results and methods up to now. We will then present a new result of local bilinear controllability in dimension 3 in a neighborhood of an eigenfunction corresponding to a simple eigenvalue and with a non-degeneracy condition of the normal derivative on a part of the boundary satisfying the geometric control condition.

This talk deals with the analysis of Stackelberg techniques for the control of linear and nonlinear PDEs. Roughly speaking, the aim is to find a hierarchy of controls that can be used to drive the solution exactly to a trajectory at a final time and, also, minimize the distances of the controlled trajectory to one or several prescribed states at appropriate locations. First, we will consider the so called Stackelberg-Nash controllability strategy for parabolic equations or systems. Then, we will present other related problems and questions. In particular, we will consider similar formulations in the context of Navier-Stokes systems, Stackelberg-Pareto strategies, etc.

This talk is devoted  to the  partial null controllability issue of parabolic linear systems with n equations. This mathematical problem appears for example in mathematical descriptions of the growth of brain tumors (see Chakrabarty Hanson 2009).  Given a bounded  domain,  we will study the effect of m localized controls in a nonempty open subset  only controlling  p components of the solution (p, m <n).  The first main result  is a necessary and sufficient condition  when the coupling and control matrices are constant.  The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled 2x2 parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent. We will finish with numerical illustrations.

In this talk, we analyze the null controllability problem for some equations of Sobolev-Galpern's type. In general, this type of equations are not controllable when the control region is confined in the domain. Moreover, by allowing the controls to move, we show that it is possible to obtain  controllability results for such kind of equations.

In this talk, I would like to explain how a symplectic interpretation of some standard notions and facts like Lagrange multipliers, Pontryagin maximum principle, Legendre and Goh conditions, Jacobi equation e.t.c. leads to effective optimality conditions that survive deep singularities and complicated constraints typical for the optimal control problems. Main hero of the talk is a version of Maslov index for, maybe discontinuous, "monotone" curves in the Lagrange Grassmannian. All necessary information about Lagrange Grassmannians will be introduced in the talk.

The problem whether all subriemannian minimizers are smooth has been open since subriemannian structures  were introduced in the 1980s. We will discuss several results on this question,  and in particular the proof of the existence of points of smoothness in the real analytic case and for rank 2 distributions in the C-infinity case, and some more recent results about abnormal extremals.

How much does it cost...to knot a closed simple curve? To cover the sphere twice? to realize such or such  homotopy class? ... etc.
All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis.
In this talk we will concentrate on the operation consisting of everting the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem.
We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.

In this talk we study the stabilization and controllabibility properties of first-order linear integro-differential hyperbolic equations. For such equations we prove that these properties are both equivalent to the criterion of Fattorini. The proof relies on a Fredholm transformation that maps the original system into a finite-time stable target system. The criterion of Fattorini is used to prove the invertibility of such a transformation. Finally, we discuss several particular examples. This is a joint work with Jean-Michel Coron and Long Hu.

In this talk, we analyze, via a simple academic example from quantum control, the problem of dynamic output stabilization, when the point where we want to stabilize corresponds to a control value that makes the system unobservable. We give, for this example, a method to stabilize the system in an arbitrarily small neighborhood of the unobservable point. We also discuss our ongoing work about exact stabilization.

Control theory for ODE systems is now relatively mature. There exist a huge list of works on control theory for (deterministic) distributed parameter systems though it is still quite active; while the same can be said for control theory for stochastic systems in finite dimensions. In this
talk, I will give a short introduction to control theory of stochastic distributed parameter systems (governed by stochastic differential equations in infinite dimensions, typically by stochastic PDEs), which is, in my opinion, almost at its very beginning stage. I will mainly explain the new phenomenon and difficulties in the study of controllability and optimal control problems for these sort of equations. In particular, I will show by some examples that both the formulation of stochastic control problems and the tools to solve them may differ considerably from their deterministic / finite-dimensional counterparts. Interestingly enough, one has to develop new mathematical tools to solve some problems in this field. Meanwhile, in some sense, the stochastic distributed parameter control system is the most general control system in the framework of classical physics, and therefore the study of this field may provide some useful hints for that of quantum control systems.

In this lecture, based on recent joint work with Martin Lazar (Dubrovnik), we discuss the problem of controllability for  parameter-dependent linear finite-dimensional systems and PDEs. The goal is to identify the most distinguished realisations of those parameters to better approximate the whole range of parameter-dependent controls.  
With this aim we adapt recent results on greedy and weak greedy algorithms for parameter depending Partial Differential Equations (PDE) or, more generally, abstract equations in Banach spaces.  
Our results lead to optimal approximation procedures that, in particular, perform better than simply sampling the parameter-space to compute the control for each realisation of the parameter values. This is confirmed by numerical experiments. 
The number of weak-greedy samplings that are required is shown to be particularly low when dealing with heat-like equations,  because of the intrinsic dissipativity that the model introduces for high frequencies.
Following a recent joint work with M. Choulli (Metz), we also discuss the problem of the stability of the recovery of the coefficients of elliptic PDE in terms of their resolvents, which arises naturally when trying to develop greedy algorithms for robust control, independent of the data under consideration.