We define and characterize weak and strong two-scale convergence in Lp ,C 0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness;in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions.We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.

We obtain an existence-uniqueness result fora second order Neumann boundary value problem including caseswhere the nonlinearity possibly crosses several points ofresonance. Optimal and Schauder fixed points methods are used toprove this kind of results.

Geometric control theory and Riemannian techniques are used to describe the reachable set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to estimate the minimal time needed to reach any point from identity. This method provides an effective way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum system with a drift from a given initial position to a given final position.The bounds include diameters of the flag manifolds; the latter arealso explicitly computed in the paper.

In this paper, we prove the global null controllability ofthe linear heat equation completed with linear Fourierboundary conditions of the form ${\partial y\over\partial n} + \beta\,y = 0$ . We consider distributed controls with support in a small set andnonregular coefficients $\beta=\beta(x,t)$ . For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heatequation with nonhomogeneous Neumann boundary conditions.

This paper is concerned with the global exact controllability ofthe semilinear heat equation (with nonlinear terms involving the state andthe gradient) completed with boundary conditions of the form ${\partialy\over\partial n} + f(y) = 0$ . We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzedin a previous first part of this work. In this second part we show that, when the nonlinear terms arelocally Lipschitz-continuous and slightly superlinear, one has exactcontrollability to the trajectories.

In this paper we deal with the local exact controllability of theNavier-Stokes system with nonlinear Navier-slip boundaryconditions and distributed controls supported in small sets. In afirst step, we prove a Carleman inequality for the linearizedNavier-Stokes system, which leads to null controllability of thissystem at any time T>0. Then, fixed point arguments lead to thededuction of a local result concerning the exact controllabilityto the trajectories of the Navier-Stokes system.

We consider a controllability problem for a beam, clamped at one boundary andfree at the other boundary, with an attached piezoelectric actuator. ByHilbert Uniqueness Method (HUM)and new results on diophantine approximations, weprove that the space of exactly initial controllable data depends on thelocation of the actuator. We also illustrate these results with numericalsimulations.

This paper addresses the Cauchy problem for thegradient flow equation in a Hilbert space  $\mathcal{H}$ \[\begin{cases}u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t)&\text{{\it a.e.}\ in }(0,T),u(0)=u_0, \end{cases}\] where $\phi: \mathcal{H} \to (-\infty,+\infty]$ is a proper,lower semicontinuous functional which is not supposed to be a(smooth perturbation of a) convex functional and $\partial_{\ell}\phi$ is(a suitable limiting version of) its subdifferential.We will present some new existence results for the solutions of theequation by exploiting a variational approximationtechnique, featuring some ideas from the theory of Minimizing Movementsand of Young measures.
Our analysisis also motivated by some models describing phase transitionsphenomena, leading tosystems of evolutionary PDEs which have a common underlying gradient flow structure:in particular, we will focus onquasistationary models, which exhibithighly non convex Lyapunov functionals.