We consider a linear coupled system of quasi-electrostatic equations which govern the evolution of a 3-D layered piezoelectric body. Assuming that a dissipative effect is effective at the boundary, we study the uniform stabilization problem. We prove that this is indeed the case, provided some geometric conditions on the region and the interfaces hold. We also assume a monotonicity condition on the coefficients. As an application, we deduce exact controllability of the system with boundary control via a classical result due to Russell.

The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.

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.

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.

We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value function of the generalized minimum time problem with discontinuous running cost.

In this paper, we consider the boundary stabilization of asandwich beam which consists of two outer stiff layers and acompliant middle layer. Using Riesz basis approach, we show thatthere is a sequence of generalized eigenfunctions, which forms aRiesz basis in the state space. As a consequence, thespectrum-determined growth condition as well as the exponentialstability of the closed-loop system are concluded. Finally, thewell-posedness and regularity in the sense of Salamon-Weiss classas well as the exact controllability are also addressed.

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.

We show that the set of nonnegative equilibrium-like states, namely, like $ (y_d, 0) $ of the semilinear vibrating string that can be reached from any non-zero initial state $ (y_0, y_1) \in H^1_0 (0,1) \times L^2 (0,1)$ , by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $ L^2 (0,1) \times \{0\} $ of $ L^2 (0,1) \times H^{-1} (0,1)$ . Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $ \; y \; $ and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the Γ-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.

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.

For a one-dimensional nonlocal nonconvex singular perturbation problemwith a noncoercive periodic well potential,we prove a Γ-convergence theorem and show compactnessup to translationin all Lp and the optimal Orlicz space for sequences of boundedenergy. This generalizes work of Alberti, Bouchitté and Seppecher(1994) for the coercive two-well case.The theorem has applications to a certain thin-film limit ofthe micromagnetic energy.

Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T 4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T 4-configurations in $\mathbb{R}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of $(\mathbb{R}^{2\times2})^{4}$ . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T 4-configurations.

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.

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 lower semicontinuous functional of the form $I_f(u)=\int_\Omega f(u){\rm d}x$ where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex functions.

In this work we study the multivalued complementarity problem onthe non-negative orthant. This is carried out by describingthe asymptotic behavior of the sequence of approximatesolutions to its multivalued variational inequality formulation.By introducing new classes of multifunctions we provide severalexistence (possibly allowing unbounded solution set), stability as well assensitivity results which extend andgeneralize most of the existing ones in the literature.We also present some kind of robustness results regarding existenceof solution with respect to certain perturbations.Topological properties of the solution-set multifunction areestablished and some notions of approximable multifunctions arealso discussed. In addition,some estimates for the solution set and its asymptotic cone are derived, as well as the existence of solutions forperturbed problems is studied.

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.