The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire et al., J. Comput. Phys.194 (2004) 363–393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the ersatz material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.

In this paper we give an existence theorem for the equilibrium problem for nonlinear micropolar elastic body. We consider the problem in its minimization formulation and apply the direct methods of the calculus of variations. As the main step towards the existence theorem, under some conditions, we prove the equivalence of the sequential weak lower semicontinuity of the total energy and the quasiconvexity, in some variables, of the stored energy function.

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphismto a linear system on a Lie group or a homogeneous space if and only if the vectorfields of the system are complete and generate a finite dimensionalLie algebra.

A vector field on a connected Lie group is linear if its flow is a one parametergroup of automorphisms. An affine vector field is obtained by adding aleft invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine.

Affine vector fields on homogeneous spaces can be characterized by their Lie brackets withthe projections of right invariant vector fields.

A linear system on a homogeneous space is a system whose drift part isaffine and whose controlled part is invariant.

The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which has its own interest. The present proof makes use of geometric control theory arguments.

We prove global internal controllability in large time for the nonlinear Schrödinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2 . We also get a regularity result about the control if the data are assumed smoother.

An a priori Campanato type regularity condition is established for a class of W1X local minimisers $\overline{u}$ of the general variational integral $\int_{\Omega} F(\nabla u(x))\,{\rm d}x$ where $\Omega \subset \mathbb{R}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition $F(\xi)\leq c(1+|\xi|^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space $\mathcal{L}^{p,\mu} (\Omega,\mathbb{R}^{N\times n})$ , µ < n, Campanato space $\mathcal{L}^{p,n}(\Omega,\mathbb{R}^{N\times n})$ and the space of bounded mean oscillation $ {\rm BMO}\Omega,\mathbb{R}^{N\times n})$ . The admissible maps $u\colon \Omega \to \mathbb{R}^N$ are of Sobolev class W1,p , satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W1BMO local minimisers is extended from Lipschitz maps to this admissible class.

The paper studies optimal portfolio selection for discrete timemarket models in mean-variance and goal achieving setting. Theoptimal strategies are obtained for models with an observed processthat causes serial correlations of price changes. The optimalstrategies are found to be myopic for the goal-achieving problem andquasi-myopic for the mean variance portfolio.

We consider a class of variationalproblems for differential inclusions, related to thecontrol of wild fires. The area burned by the fire at time t> 0is modelled as the reachable set fora differential inclusion $\dot x$ ∈F(x), starting froman initial set R 0. To block the fire, a barrier can be constructedprogressively in time. For each t> 0, the portion of the wall constructedwithin time t is described by a rectifiable setγ(t) ⊂ $\mathbb{R}^2$ . In this paperwe show that the searchfor blocking strategies and for optimal strategies can be reduced toa problem involving one single admissible rectifiable set Γ⊂ $\mathbb{R}^2$ ,rather than the multifunction t $\mapsto$ γ(t) ⊂ $\mathbb{R}^2$ .Relying on this result, we then developa numerical algorithm for the computation ofoptimal strategies, minimizing the total area burned by the fire.

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.

We consider an optimal control problem for the three-dimensional non-linear Primitive Equations of the ocean in a vertically bounded and horizontally periodic domain. We aim to reconstruct the initial state of the ocean from Lagrangian observations. This inverse problem is formulated as an optimal control problem which consists in minimizing a cost function representing the least square error between Lagrangian observations and their model counterpart, plus a regularization term. This paper proves the existence of an optimal control for the regularized problem. To this end, we also prove new energy estimates for the Primitive Equations, thanks to well-chosen functional spaces, which distinguish the vertical dimension from the horizontal ones. We illustrate the result with a numerical experiment.

Motivated by the observationthat the gain-loss criterion, while offering economically meaningful prices of contingent claims,is sensitive to the reference measure governing the underlying stock price process (a situationreferred to as ambiguity of measure), we propose a gain-loss pricing model robust to shifts in the reference measure. Using a dual representation property of polyhedral risk measures we obtain a one-step, gain-loss criterion based theorem ofasset pricing under ambiguity of measure, and illustrate its use.

In this paper we prove a regularityresult for local minimizers of functionals of the Calculus of Variations of thetype

$$\int_{\Omega}f(x, Du)\ {\rm d}x$$

where Ω is a bounded open set in $\mathbb{R}^{n}$ , u∈ $W^{1,p}_{\rm loc}$ (Ω; $\mathbb{R}^{N}$ ), p> 1, n≥ 2 and N≥ 1.We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to givea bound on the Hausdorff dimension of the singular set of minimizers.

We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improvePólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extendHersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals.

We consider abstract second order evolution equations with unboundedfeedback with delay. Existence results are obtained under somerealistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.

The work focuses on the Γ-convergence problem and the convergence of minimizers for a functional defined in a periodic perforated medium andcombining the bulk (volume distributed) energy and the surfaceenergy distributed on the perforation boundary. It is assumed that the mean valueof surface energy at each level set of test function is equal tozero.Under natural coercivity and p-growth assumptions on the bulk energy, and the assumption that the surface energy satisfies p-growth upper bound, weshow that the studied functional has a nontrivial Γ-limit andthe corresponding variational problem admitshomogenization.


In this paper, we study thecontrol system associated with the incompressible 3D Euler system.We show that the velocity field and pressure of the fluid areexactly controllable in projections by the same finite-dimensionalcontrol. Moreover, the velocity is approximately controllable. We also prove that 3D Eulersystem is not exactly controllable by a finite-dimensionalexternal force.

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized.Lower and upper bounds on the first conjugate time are proved.The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

In this paper we consider the initialboundary value problem of a parabolic-elliptic system for imageinpainting, and establish the existence and uniqueness of weaksolutions to the system in dimension two.

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40], is due to Li and Nirenberg [Comm. Pure Appl. Math. 58 (2005) 85–146]. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a C 4-deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jumpset A or a controlled jump set C where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary, finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded.

We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in ${\mathbb R}^N$ . An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value $+\infty$ .