We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Hölder continuous with Hölder exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse Hölder inequality.

We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups. The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.

This paper is devoted to the computation of distance to set, called S, defined by polynomial equations. First we consider the case of quadratic systems. Then, application of results stated for quadratic systems to the quadratic equivalent of polynomial systems (see [5]), allows us to compute distance to semi-algebraic sets. Problem of computing distance can be viewed as non convex minimization problem: $ d(u,S) = \inf_{x \in S} \| x-u\|^2$, where u is in $\mathcal{R}^{n}$. To have, at least, lower approximation of distance, we consider the dual bound m(u) associated with the dual problem and give sufficient conditions to guarantee m(u) = d(u,S). The second part deal with numerical computation of m(u) using an interior point method in semidefinite programming. Last, various examples, namely from chemistry and robotic, are given.

In this paper necessary and sufficient conditions of L∞-controllability andapproximate L∞-controllability are obtained for the control systemwtt = wxx − q2w,w(0,t) = u(t),x > 0, t ∈ (0,T), whereq ≥ 0, T > 0,u ∈ L∞(0,T) is a control. This system isconsidered in the Sobolev spaces.

This paper is mainlyconcerned with a class of optimal control problems of systemsgoverned by the nonlinear dynamic systems on time scales.Introducing the reasonable weak solution of nonlinear dynamicsystems, the existence of the weak solution for the nonlineardynamic systems on time scales and its properties are presented.Discussing L1-strong-weak lower semicontinuity of integralfunctional, we give sufficient conditions for the existence ofoptimal controls. Using integration by parts formula and Hamiltonianfunction on time scales, the necessary conditions of optimality arederived respectively. Some examples on continuous optimal controlproblems, discrete optimal control problems, mathematicalprogramming and variational problems are also presented for demonstration.

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.

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.

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.

In this paper we prove that every weakand strong localminimizer $u\in{W^{1,2}(\Omega,\mathbb{R}^3)}$ of the functional $I(u)=\int_\Omega|Du|^2+f({\rm Adj}Du)+g({\rm det}Du),$ where $ u:\Omega\subset\mathbb{R}^3\to \mathbb{R}^3$ , f grows like $|{\rm Adj}Du|^p$ , g growslike $|{\rm det}Du|^q$ and1<q<p<2, is $C^{1,\alpha}$ on an opensubset $\Omega_0$ of Ω such that ${\it meas}(\Omega\setminus \Omega_0)=0$ . Suchfunctionals naturally arise from nonlinear elasticity problems. The keypoint in order to obtain the partial regularity result is toestablish an energy estimate of Caccioppoli type, which is based onan appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers.

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.

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of Γ-convergence of the elastic energy, as the thickness tends to zero.

New L 1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved,under a very weak dependence of the integrand with respect to the spatial variable x. Moreprecisely, only the lower semicontinuity in the sense of the 1-capacity is assumed inorder to obtain the lower semicontinuity of the functional.This condition is satisfied, for instance, by the lower approximate limit of the integrand, ifit is BV with respect to x. Under this further BV dependence,a representation formula for the relaxed functional is also obtained.

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

We are concerned with the Lipschitz modulus of the optimal set mappingassociated with canonically perturbed convex semi-infinite optimizationproblems. Specifically, the paper provides a lower and an upper bound forthis modulus, both of them given exclusively in terms of the problem's data.Moreover, the upper bound is shown to be the exact modulus when the numberof constraints is finite. In the particular case of linear problems theupper bound (or exact modulus) adopts a notably simplified expression. Ourapproach is based on variational techniques applied to certain difference ofconvex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First](which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554])constitute the starting point of the present work.

The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of Γ-convergence, in the framework of finite plasticity. Denoting by ε the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order ε2α−2, with α ≥ 3. According to the value of α, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized plate theory.

In the context of a variational model for the epitaxial growth of strained elastic films,we study the effects of the presence of anisotropic surface energies in the determinationof equilibrium configurations. We show that the threshold effect that describes thestability of flat morphologies in the isotropic case remains valid for weak anisotropies,but is no longer present in the case of highly anisotropic surface energies, where we showthat the flat configuration is always a local minimizer of the total energy. Following theapproach of [N. Fusco and M. Morini, Equilibrium configurations of epitaxially strainedelastic films: second order minimality conditions and qualitative properties of solutions.Preprint], we obtain these results by means of a minimality criterion based on thepositivity of the second variation.

We study the asymptotic behavior of $\lambda v_\lambda$ as $\lambda\rightarrow 0^+$ , where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) \[ \lambda v_\lambda + H(x,Dv_\lambda)=0,\] with \[H(x,p):=\min_{b\in B}\max_{a \in A} \{-f(x,a,b)\cdot p -l(x,a,b)\}.\] We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions,or in the closure of a bounded connected domain $\Omega\subset{\xR}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat boththe case of the Neumann boundary conditions(reflection on the boundary) andthe case of state constraints boundary conditions.Under the uniform approximate controllability assumption of one player, we extendthe uniform convergence result of the value function to a constant as $\lambda\rightarrow 0^+$ to differential games.As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous.

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.


Critical points of a variant of the Ambrosio-Tortorelli functional,for which non-zero Dirichlet boundary conditions replace thefidelity term, are investigated. They are shown to converge toparticular critical points of the corresponding variant of theMumford-Shah functional; those exhibit many symmetries. ThatDirichlet variant is the natural functional when addressing aproblem of brittle fracture in an elastic material.

The regularity of Lagrange multipliers for state-constrained optimal control problems belongs to the basic questions of controltheory. Here, we investigate bottleneck problems arising from optimal control problems for PDEs with certain mixed control-stateinequality constraints. We show how to obtain Lagrange multipliers in Lp spaces for linear problems and give an application to linearparabolic optimal control problems.