## Section:
New Results2>
### Observality, controllability and stabilzation in the time domain3>

In [17] we consider Euler-Bernoulli beams and strings alternatively connected to one another and forming a chain beginning with a string. We study the strong and polynomial stabilities of this system on this network and the spectrum of the corresponding conservative system.

In [37] we study the asymptotic behavior of the solution of the non-homogeneous elastic systems with voids and a thermal effect. Our main results concern strong and polynomial stabilities (since this system suffers of exponential stability).

In [12] , we consider the approximation of two coupled wave equations with internal damping. Our goal is to damp the spurious high frequency modes by introducing a numerical viscosity term in the approximation schemes and prove the exponential or polynomial decay of the discrete scheme.

In [13] , we show similar results as in [12] for an abstract second order evolution equations.

In [44] we consider a class of infinite dimensional systems involving a control function taking values in and we prove, when is given in an appropriate feedback form and the system satisfies appropriate observability assumptions, that the system is weakly stable. The main example concerns the analysis and stabilization of a model of Boost converter connected to a load via a transmission line.

In [46] we present a course on stabilization of hyperbolic equations given at a CIME session on Control of PDE's in Italy in July 2010, including well-known results, together with recent ones including nonlinear stabilization, memory-damping and stabilization of coupled systems by a reduced number of controls. In particular, we present the optimal-weight convexity method (Alabau-Boussouira 2005, 2010) in both the finite dimensional and infinite dimensional framework and give applications to semi-discretization of hyperbolic PDE's.

In [41] , we consider stabilization of coupled systems of wave-type, with localized couplings and either localized internal closed loop controls or boundary control. We establish polynomial decay rates for coupling and damping regions which do not intersect in the one-dimensional case. We also derive results in the multi-dimensional case, under multiplier type conditions for both the coupling and damping regions. The novelty and difficulty is to consider localized couplings.

In [15] , we give a constructive proof of Gibson's stability theorem, some extension and further positive and negative applications of this result.

In [36] we prove that the boundary controls for the heat equation have the bang-bang property, at least in rectangular domains. This result is proved by combining methods from traditionally distinct fields: the Lebeau-Robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems, on one side, and a Remez-type inequality for Muntz spaces and a generalization of Turan's inequality, on the other side.

In [16] we prove exact controllability for symmetric coupled wave equations by a single control in the case of coupling and control regions which do not intersect. For this, we use and extend the two-level energy method introduced by Alabau-Boussouira (2001, 2003). Using transmutation, we derive null controllability results for coupled parabolic and Schrödinger equations. This is the first positive quantitative result, in a multi-dimensional framework with control and coupling regions with empty intersection.

In [14] , we prove controllability results for abstract systems of weakly coupled N evolution equations in cascade by a reduced number of boundary or locally distributed controls ranging from a single up to N-1 controls. We give applications to cascade coupled systems of N multi-dimensional hyperbolic, parabolic and diffusive (Schrödinger) equations. The results are valid for control and coupling regions which do not necessarily intersect.

In [22] , we study two notions of controllability, called respectively radial controllability and directional controllability. We prove that for families of linear vector fields, the two notions are actually equivalent.

In [24] we solve an optimization problem in convex geometry which, despite its seeming simplicity, offers a nice variety of solutions, some of them being unexpectable.

The paper [28] is devoted to prove that the union of two identical balls minimizes a non linear eigenvalue (related to the generalized Wirtinger inequality) among sets of given volume.

In [33] is considered a problem in population dynamics where we investigate the question of optimal location of the zone of control.

In [26] , we give a rigorous proof, valid also for unbounded operators, of the widely used “rotating wave approximations” for bilinear Schrödinger equations.

In [42] , we exploit the results of [26] on standard examples of bilinear quantum systems.