We give the definitions of exact and approximate controllability forlinear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” resultfor evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllabilityfor the linear Schrödinger equation and distributed additive control,and we show that the Hartree equation of quantum chemistry with bilinearcontrol $(E(t)\cdot x) u$ is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, andwe give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.

The energy in a square membrane Ω subject to constant viscous dampingon a subset $\omega\subset \Omega$ decays exponentially in timeas soon as ωsatisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau(\omega)$ of this decay satisfies $\tau(\omega)= 2 \min( -\mu(\omega),g(\omega))$ (see Lebeau [Math.Phys. Stud.19 (1996) 73–109]). Here $\mu(\omega)$ denotes the spectral abscissa of thedamped wave equation operator and  $g(\omega)$ is a number called the geometrical quantity of ω and defined as follows.A ray in Ω is the trajectory generated by thefree motion of a mass-point in Ω subject to elastic reflections on theboundary. These reflections obey the law of geometrical optics.The geometrical quantity $g(\omega)$ is then defined as the upper limit (large timeasymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\omega)$ when ωis a finite union of squares.

We consider the differential game associated with robust control of asystem in a compact state domain, using Skorokhod dynamics on theboundary. A specific class of problems motivated by queueing network controlis considered. A constructive approach to the Hamilton-Jacobi-Isaacsequation is developed which is based on an appropriate family ofextremals, including boundary extremals for which the Skorokhoddynamics are active. A number of technical lemmas and a structuredverification theorem are formulated to support the use of thistechnique in simple examples. Two examples are considered whichillustrate the application of the results. This extends previous workby Ball, Day and others on such problems, but with a new emphasis onproblems for which the Skorokhod dynamics play a critical role. Connections with the viscosity-sense oblique derivative conditions ofLions and others are noted.

In this paper we analyze a typical shape optimization problem intwo-dimensional conductivity. We study relaxation for this problemitself. We also analyze the question of the approximation of thisproblem by the two-phase optimal design problems obtained when wefill out the holes that we want to design in the original problemby a very poor conductor, that we make to converge to zero.

We prove the periodicity of all H 2-local minimizers with low energyfor a one-dimensional higher order variational problem.The results extend and complement an earlier work of Stefan Müllerwhich concerns the structure of global minimizer.The energy functional studied in this work is motivated by theinvestigation of coherent solid phase transformations and thecompetition between theeffects from regularization and formation of small scale structures.With a special choice of a bilinear double well potential function, wemake use of explicit solution formulas to analyze the intricateinteractions between the phase boundaries. Our analysis can provideinsights for tackling the problem with general potential functions.

We consider the problem of placing a Dirichlet region made by n small balls of given radius in a given domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look for the Γ-limit of suitably scaled functionals, in order to get informations on the asymptotical distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.

In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu)$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop systemgenerates a C 0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors andgeneralized eigenvectors that forms a Riesz basis for the state Hilbert space.Furthermore, we show that when the weight $\mu>\frac{1}{2}$ , for any time delay,we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When $\mu=\frac{1}{2}$ , we show that the system is at most asymptotically stable. When $\mu<\frac{1}{2}$ , the system is always unstable.

An elliptic PDE is studied which is a perturbation of an autonomousequation. The existence of a nontrivial solution is proven viavariational methods. The domain of the equation is unbounded, whichimposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. Inan earlier paper with this assumption, a solution was obtained using asimple application of topological (Brouwer) degree. Here, a more subtledegree theory argument must be used.

Let $\Lambda $ be a Lagrangian submanifold of $T^{*}X$ for some closedmanifold X. Let $S(x,\xi )$ be a generating function for $\Lambda $ whichis quadratic at infinity, and let W(x) be the corresponding graph selectorfor $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that thereexists a subset $X_{0}\subset X$ of measure zero such that W is Lipschitzcontinuous on X, smooth on $X\backslash X_{0}$ and $(x,\partial W/\partialx(x))\in \Lambda $ for $X\backslash X_{0}.$ Let H(x,p)=0 for $(x,p)\in\Lambda$ . Then W is a classical solution to $H(x,\partial W/\partialx(x))=0$ on $X\backslash X_{0}$ and extends to a Lipschitz function on thewhole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions.We also prove that these conditions are satisfied in many cases, includingcertain commonly occuring cases where H(x,p) is not convex in p.