Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.

In this paper we prove a unique continuationresult for a cascade system of parabolic equations, in which the solution of the firstequation is (partially) used as a forcing term for the second equation. As aconsequence we prove the existence of ε-insensitizing controls for someparabolic equations when the control region and the observability region do not intersect.

We consider some discrete and continuous dynamics in a Banach spaceinvolving a non expansive operator J and a corresponding family ofstrictly contracting operators Φ (λ, x): = λJ( $\frac{1-\lambda}{\lambda}$ x) for λ ∈ ] 0,1] . Our motivationcomes from the study of two-player zero-sum repeated games, wherethe value of the n-stage game (resp. the value of theλ-discounted game) satisfies the relationv n = Φ( $\frac{1}{n}$ , $v_{n-1}$ ) (resp.  $v_\lambda$ = Φ(λ, $v_\lambda$ )) where J is the Shapleyoperator of the game. We study the evolution equationu'(t) = J(u(t))- u(t) as well as associated Eulerian schemes,establishing a new exponential formula and a Kobayashi-likeinequality for such trajectories. We prove that the solution of thenon-autonomous evolution equationu'(t) = Φ(λ(t), u(t))- u(t) has the same asymptoticbehavior (even when it diverges) as the sequence v n (resp. as thefamily $v_\lambda$ ) when λ(t) = 1/t (resp. whenλ(t) converges slowly enough to 0).

Sensitivity analysis (with respect to the regularization parameter)of the solution of a class of regularized state constrainedoptimal control problems is performed. The theoretical results arethen used to establish an extrapolation-based numerical scheme forsolving the regularized problem for vanishing regularizationparameter. In this context, the extrapolation technique providesexcellent initializations along the sequence of reducingregularization parameters. Finally, the favorable numericalbehavior of the new method is demonstrated and a comparison toclassical continuation methods is provided.

Homogenization of integral functionals is studiedunder the constraint that admissible maps have to take their valuesinto a given smooth manifold. The notion of tangentialhomogenization is defined by analogy with the tangentialquasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, aΓ-convergence result is established in Sobolev spaces, thehomogenization problem in the space of functions of boundedvariation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq.36 (2009) 7–47].

For external magnetic field h ex ≤Cε–α , we provethat a Meissner state solution for the Chern-Simons-Higgs functional exists. Furthermore, if the solutionis stable among all vortexless solutions, then it is unique.

This paper deals with topology optimization of domains subject to a pointwise constraint on the gradient of the state. To realize this constraint, a class of penalty functionals is introduced and the expression of the corresponding topological derivative is obtained for the Laplace equation in two space dimensions. An algorithm based on these concepts is proposed. It is illustrated by some numerical applications.

Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $\eta,\bar\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost

$J[\gamma]=\int_0^T \left({\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))+K^2_{\gamma(t)}{\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) \right) ~{\rm d}t$

along all smooth curves starting from x with direction η and ending in $\bar{x}$ with direction $\bar\eta$ . Here g is the standard Riemannian metric on S 2 and $K_\gamma$ is the corresponding geodesic curvature.The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1).We compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.

In this paper we construct upper bounds for families offunctionals of the form

$$E_\varepsilon(\phi):=\int_\Omega\Big(\varepsilon |\nabla\phi|^2+\frac{1}{\varepsilon }W(\phi)\Big){\rm d}x+\frac{1}{\varepsilon }\int_{{\mathbb{R}}^N}|\nabla \bar H_{F(\phi)}|^2{\rm d}x$$

where Δ $\bar H_u$ = div { $\chi_\Omega$ u}. Particular cases of such functionals arise inMicromagnetics. We also use our technique to construct upper boundsfor functionals that appear in a variational formulation ofthe method of vanishing viscosity for conservation laws.

This paper is devoted to an analysis of vortex-nucleationfor a Ginzburg-Landau functional withdiscontinuous constraint. This functional has been proposedas a model for vortex-pinning, and usuallyaccounts for the energyresulting from the interface of two superconductors. Thecritical applied magnetic field for vortex nucleation is estimated inthe London singular limit,and as a by-product, results concerning vortex-pinning andboundary conditions on the interface are obtained.

The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math.102 (2006) 413–462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.

We study the integral representation of relaxed functionals in the multi-dimensional calculus of variations, for integrands which are finite in a convex bounded set with nonempty interior and infinite elsewhere.

We study a comparison principle and uniqueness of positive solutions forthe homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations withlower order terms. A model example is given by

$ -\Delta u+\lambda\frac{|\nabla u|^2}{u^r} = f(x), \qquad\lambda,r>0.$

The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right handside. Furthermore, they could be applied to obtain uniqueness results for nonlinear equations having the p-Laplacian operator as the principalpart. Our results improve those already known, even if the gradient term is not singular.

Optimal control problems for semilinear elliptic equationswith control constraints and pointwise state constraints arestudied. Several theoretical results are derived, which arenecessary to carry out a numerical analysis for this class ofcontrol problems. In particular, sufficient second-order optimalityconditions, some new regularity results on optimal controls and asufficient condition for the uniqueness of the Lagrange multiplierassociated with the state constraints are presented.

In this paper we study homogenization for a class of monotone systems of first-order time-dependent periodic Hamilton-Jacobi equations.We characterize the Hamiltonians of the limit problem by appropriate cell problems. Hence weshow the uniform convergence of the solution of the oscillating systems tothe boundeduniformly continuous solution of thehomogenized system.

In this work, we propose a methodology for the expression of necessary and sufficient Lyapunov-like conditions for the existence of stabilizing feedback laws. The methodology is an extension of the well-known Control Lyapunov Function (CLF) method and can be applied to very general nonlinear time-varying systems with disturbance and control inputs, including both finite and infinite-dimensional systems. The generality of the proposed methodology is also reflected upon by the fact that partial stability with respect to output variables is addressed. In addition, it is shown that the generalized CLF method can lead to a novel tool for the explicit design of robust nonlinear controllers for a class of time-delay nonlinear systems with a triangular structure.

In this paper we investigate the equivalence of the sequentialweak lower semicontinuity of the total energy functional and the quasiconvexity of thestored energy function of the nonlinear micropolar elasticity. Based on techniques of Acerbi and Fusco [Arch. Rational Mech. Anal.86 (1984) 125–145] we extend the result from Tambača and Velčić [ESAIM: COCV (2008) DOI: 10.1051/cocv:2008065] for energies thatsatisfy the growth of order p≥ 1. This result is the mainstep towards the general existence theorem for the nonlinear micropolarelasticity.

In terms of the normal cone and the coderivative,we provide some necessary and/or sufficient conditions of metric subregularity for(not necessarily closed) convex multifunctions in normed spaces. As applications, we present someerror bound results for (not necessarily lower semicontinuous) convex functions on normedspaces. These results improve and extend some existing error bound results.

This paper is concerned with the following periodic Hamiltonianelliptic system

$ \{ -\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\\varphi(x)\to 0\ \hbox{and }\psi(x)\to0\ \hbox{as }|x|\to\infty.$

Assuming the potential V is periodic and 0 lies in a gap of $\sigma(-\Delta+V)$ , $G(x,\eta)$ is periodic in x andasymptotically quadratic in $\eta=(\varphi,\psi)$ , existence andmultiplicity of solutions areobtained via variational approach.


In this paper, we study the stabilization of a two-dimensional Burgers equation around a stationary solution by a nonlinear feedback boundary control. We are interested in Dirichlet and Neumann boundary controls. In the literature, it has already been shown that a linear control law, determined by stabilizing the linearized equation, locally stabilizes the two-dimensional Burgers equation. In this paper, we define a nonlinear control law which also provides a local exponential stabilization of the two-dimensional Burgers equation. We end this paper with a few numerical simulations, comparing the performance of the nonlinear law with the linear one.