In the framework of the linear fracture theory, a commonly-used toolto describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-calledenergy release rate defined as the variation of the mechanicalenergy with respect to the crack dimension. Precisely, thewell-known Griffith's criterion postulates the evolution of thecrack if this rate reaches a critical value. In this work, in the anti-plane scalar case, weconsider the shape design problem which consists in optimizing thedistribution of two materials with different conductivities in Ω in order to reducethis rate. Since this kind of problem is usually ill-posed, wefirst derive a relaxation by using the classical non-convexvariational method. The computation of the quasi-convex envelope ofthe cost is performed by using div-curl Young measures, leads to anexplicit relaxed formulation of the original problem, and exhibits fine microstructure in the form offirst order laminates. Finally, numerical simulations suggest thatthe optimal distribution permits to reduce significantly the value of the energy release rate.

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps $\{u_k\}_{k\in{\mathbb N}} \subset L^p(\Omega;{\mathbb R}^m)$ satisfying a linear differential constraint ${\mathcal A}u_k=0$ . Applications to sequential weak lower semicontinuity of integral functionals on ${\mathcal A}$ -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det $\nabla\varphi_k\stackrel{*}{\rightharpoonup}{\rm det}\nabla\varphi$ in measures on the closure of $\Omega\subset{\mathbb R}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^{1,n}(\Omega;{\mathbb R}^n)$ . This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemmaprecisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_{\Omega\setminus\Omega_j} v(u(x))\,{\rm d}x$ along $\{u_k\}\subset L^p(\Omega;{\mathbb R}^m)\cap{\rm ker}\ {\mathcal A}$ . Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”.

We study the potential which minimizes the fundamental gap of theSchrödinger operator under the total mass constraint. We considerthe relaxed potential and prove a regularity result for the optimalone, we also give a description of it. A consequence of this resultis the existence of an optimal potential under L 1 constraints.

We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploitsthe action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, withStokes equations governing the surrounding fluid.The action of thecilia is described by a set of controlled velocity fields on the surface of the organism. The first contribution of the paper is the proof that such a systemis generically controllable when the space of controlled velocity fields is at least three-dimensional. We also provide a complete characterization of controllable systems in the case in whichthe organism has a spherical shape. Finally, we offer a complete picture of controllable and non-controllable systems under the additional hypothesis thatthe organism and the fluid have densities of the same order of magnitude.

This article is devoted to the optimal control of state equations with memory of the form:

$\dot{x}(t)=F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) {\rm d}s), \; t>0,$

with initial conditions $x(0)=x, \; x(-s)=z(s), s>0$ .

Denoting by $y_{x, z, u}$ the solution of the previous Cauchy problem and:

$v(x,z):=\inf_{u\in V} \lbrace \int_0^{+\infty} {\rm e}^{-\lambda s } L(y_{x,z,u}(s), u(s)){\rm d}s\rbrace$

where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form:

$\lambda v(x,z)+H(x,z,\nabla_x v(x,z))+\langle D_z v(x,z), \dot{z} \rangle=0$

in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

Newton's problem of the body of minimal aerodynamic resistance is traditionallystated in the class of convex axially symmetric bodies withfixed length and width. We state and solve the minimal resistanceproblem in the wider class of axially symmetric but generallynonconvex bodies. The infimum in this problem is not attained. Weconstruct a sequence of bodies minimizing the resistance. Thissequence approximates a convex body with smooth front surface, whilethe surface of approximating bodies becomes more and morecomplicated. The shape of the resulting convex body and the value ofminimal resistance are compared with the corresponding results forNewton's problem and for the problem in the intermediate class ofaxisymmetric bodies satisfying the single impact assumption[Comte and Lachand-Robert, J. Anal. Math.83 (2001) 313–335]. In particular, the minimal resistance in our class issmaller than in Newton's problem; the ratio goes to 1/2 as(length)/(width of the body) → 0, and to 1/4 as(length)/(width) → +∞.

We study a class of hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C 0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.

We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693–702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.

Being a unique phenomenon in hybrid systems, mode switchis of fundamental importance in dynamic and control analysis. Inthis paper, we focus on global long-time switching and stabilityproperties of conewise linear systems (CLSs), which are a class oflinear hybrid systems subject to state-triggered switchingsrecently introduced for modeling piecewise linear systems. Byexploiting the conic subdivision structure, the “simple switchingbehavior” of the CLSs is proved. The infinite-time mode switchingbehavior of the CLSs is shown to be critically dependent on twoattracting cones associated with each mode; fundamental propertiesof such cones are investigated. Verifiable necessary andsufficient conditions are derived for the CLSs with infinite modeswitches. Switch-free CLSs are also characterized by exploringthe polyhedral structure and the global dynamical properties. Theequivalence of asymptotic and exponential stability of the CLSs isestablished via the uniform asymptotic stability of the CLSs thatin turn is proved by the continuous solution dependence on initialconditions. Finally, necessary and sufficient stability conditionsare obtained for switch-free CLSs.

We recall here some theoretical results of Helffer et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) doi:10.1016/j.anihpc.2007.07.004] about minimal partitions and propose numerical computations to check some of their published or unpublished conjectures and exhibit new ones.

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties.The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schrödinger equation.