We consider spectral optimization problems of the form

$\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$

where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation

$\lambda_1(\Omega;D)=\min\lbrace\int_\Omega|\nabla u|^2dx+k\int_{\partial D}u^2d\mathcal{H}^{d-1}:$

$u\in H^1(D),u=0$ on $\partial\Omega\cap D,||u||_{L^2(\Omega)}=1\rbrace$

reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains. The case of Dirichlet condition on a $\textit{fixed}$ part and of Neumann condition on the $\textit{free}$ part of the boundary is also considered[-]

We study the small-time local controllability (STLC) for scalar input control affine systems, in finite dimension. It is known that the entire information about STLC is contained in the evaluation at zero of the Lie brackets of the vector fields. In the 80's, several authors formulated necessary conditions for controllability (obstructions), relying on particular 'bad' brackets. In this talk, I will present a unified approach to determine and prove obstructions to STLC, that allows to recover known obstructions and prove new ones, in a relatively systematic way. This approach relies on a recent Magnus-type representation of the state, a new Hall basis of the free Lie algebra over 2 generators and interpolation inequalities. This is a joint work with Frédéric Marbach and Jérémy Leborgne.[-]

We discuss properties of the viscous Hamilton-Jacobi equation$$\begin{cases}u_{t}-\Delta u=|D u|^{p} & \text { in }(0, \infty) \times \Omega, \\ u=0 & \text { in }(0, \infty) \times \partial \Omega, \\ u(0)=u_{0} & \text { in } \Omega,\end{cases}$$in the super-quadratic case $p>2$. Here $\Omega$ is a bounded domain in $\mathbf{R}^{\mathbf{N}}$. In the super-quadratic regime, solutions may be continuous but with a gradient blow up; in this case the second order equation exhibits very peculiar phenomena. Some properties are similar to first order problems, such as loss of boundary conditions and appearance of singularities, but the presence of diffusion let singularities appear and disappear, in a very unusual way. In the talk I will present results obtained in collaboration with Philippe Souplet which describe the qualitative behavior of the solution, starting from smooth initial data. This includes the analysis of blow-up rates, blow-up profiles, life after blow-up, loss and recovery of boundary conditions.[-]

In this lecture we shall present some recent results in collaboration with B. Geshkovski (MIT) on the design of optimal sensors and actuators for control systems. We shall mainly focus in the finite-dimensional case, using the Brunovsky normal form. This allows to reformulate the problem in a purely matricial context, which permits rewriting the problem as a minimization problem of the norm of the inverse of a change of basis matrix, and allows us to stipulate the existence of minimizers, as well as non-uniqueness, due to an invariance of the cost with respect to orthogonal transformations. We will present several numerical experiments to both visualize these artifacts and also point out towards further directions and open problems, in particular in the context of PDE infinite-dimensional models.[-]

In this talk we present a inequality obtained with Jérôme Le Rousseau, for sum of eigenfunctions for bi-Laplace operator with clamped boundary condition. These boundary conditions do not allow to reduce the problem for a Laplacian with adapted boundary condition. The proof follow the strategy used for Laplacian, namely we consider a problem with an extra variable and we prove Carleman estimates for this new problem. The main difficulty is to obtain a Carleman estimate up to the boundary.[-]

We consider the problem of lagrangian controllability for two models of fluids. The lagrangian controllability consists in the possibility of prescribing the motion of a set of particle from one place to another in a given time. The two models under view are the Euler equation for incompressible inviscid fluids, and the quasistatic Stokes equation for incompressible viscous fluids. These results were obtained in collaboration with Thierry Horsin (Conservatoire National des Arts et Métiers, Paris)[-]

We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We show the strong convergence of the approximation and then discussed several examples for $N = 1$ and $N = 2$. The reconstruction of both the state and the source term is also discussed, as well as the boundary case. The parabolic case - more delicate as it requires the use of appropriate weights - will be also addressed. Joint works with Nicolae Cîndea and Diego Araujo de Souza.[-]

We start by presenting some results on the stabilization, rapid or in finite time, of control systems modeled by means of ordinary differential equations. We study the interest and the limitation of the damping method for the stabilization of control systems. We then describe methods to transform a given linear control system into new ones for which the rapid stabilization is easy to get. As an application of these methods we show how to get rapid stabilization for Korteweg-de Vries equations and how to stabilize in finite time $1-D$ parabolic linear equations by means of periodic time-varying feedback laws.[-]

The importance of ultrasound is well established in the imaging of human tissue. In order to enhance image quality by exploiting nonlinear effects, recently techniques such as harmonic imaging and nonlinearity parameter tomography have been put forward. These lead to a coefficient identification problem for a quasilinear wave equation. Another characteristic property of ultrasound propagating in human tissue is frequency power law attenuation leading to fractional derivative damping models in time domain. In this talk we will first of all dwell on modeling of nonlinearity on one hand and fractional damping on the other hand. Then we will discuss the linear inverse problem of photoacoustic tomography with fractional damping. Finally some first results on nonlinearity parameter imaging are shown.[-]