Section: New Results

Observality, controllability and stabilzation in the time domain

In [17] we consider $N$ Euler-Bernoulli beams and $N$ 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 $u$ taking values in $[0,1]$ and we prove, when $u$ 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.