The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem.

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

We prove Hölder regularity of the gradient, up to the boundary for solutions of somefully-nonlinear, degenerate elliptic equations, with degeneracy coming from thegradient.

A phase field approach for structural topology optimization which allows for topologychanges and multiple materials is analyzed. First order optimality conditions arerigorously derived and it is shown via formally matched asymptoticexpansions that these conditions converge to classical first order conditions obtained inthe context of shape calculus. We also discuss how to deal with triple junctions wheree.g. two materials and the void meet. Finally, we present severalnumerical results for mean compliance problems and a cost involving the least square errorto a target displacement.

We study the first eigenpair of a Dirichlet spectral problem for singularly perturbedconvection-diffusion operators with oscillating locally periodic coefficients. It followsfrom the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularlyperturbed operators with oscillating coefficients. Preprintwww.arxiv.org, arXiv:1206.3754] that thefirst eigenvalue remains bounded only if the integral curves of the so-called effectivedrift have a nonempty ω-limit set. Here we consider the case when theintegral curves can have both hyperbolic fixed points and hyperbolic limit cycles. One ofthe main goals of this work is to determine a fixed point or a limit cycle responsible forthe first eigenpair asymptotics. Here we focus on the case of limit cycles that was leftopen in [A. Piatnitski and V. Rybalko, Preprint.

We study properties of the functional

\begin{eqnarray} \mathscr{F}_{{\rm loc}}(u,\Omega):=\inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(\nabla u_{j})\udx\, \left| \!\!\begin{array}{rl} & (u_{j})\subset W_{{\rmloc}}^{1,r}\left(\Omega, \RN\right) \\ & u_{j}\tostar u\,\,\textrm{in}\BV\left(\Omega, \RN\right) \end{array} \right. \bigg\}, \end{eqnarray}$\begin{array}{ccc}{F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{:}\mathrm{=}\underset{\mathrm{\left(}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}}{\inf }\left\{\right.\underset{\mathit{j}\mathrm{\to }\mathrm{\infty }}{\lim \inf }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }{\mathit{u}}_{\mathit{j}}\mathrm{\right)}\mathrm{d}\mathit{x}\left|\begin{array}{c}\\ & \\ & \end{array}\right.\left\}\right.\mathit{,}& & \end{array}$

\hbox{$r\in[1,\frac{n}{n-1})$}$\mathit{r}\mathrm{\in }\mathrm{\left[}\mathrm{1}\mathit{,}\frac{\mathit{n}}{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

\begin{equation*} \scF_{{\rm loc}}(u,\Omega) \geq \int_{\Omega}f(\nabla u (x))\ud x + \int_{\Omega}\finf\bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|, \end{equation*}$F_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{\ge }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }\mathit{u}\mathrm{\right(}\mathit{x}\mathrm{\left)}\mathrm{\right)}\mathrm{d}\mathit{x}\mathrm{+}{\mathrm{\int }}_{\mathit{\Omega }}{\mathit{f}}^{\mathrm{\infty }}\left(\right.\frac{{\mathit{D}}^{\mathit{s}}\mathit{u}}{\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}}\left)\right.\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}\mathit{,}$

\hbox{$ f^{\infty}(\xi):= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$}${\mathit{f}}^{\mathrm{\infty }}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{:}\mathrm{=}{\overline{\lim }}_{\mathit{t}\mathrm{\to }\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{t\xi }\mathrm{\right)}\mathit{/}\mathit{t}$

We introduce the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external policy maker, and we propagate its effect for the number N of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This will be realized by considering cost functionals including L1-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of Γ-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.

In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of the calculus of variations are described. Necessary optimality conditions in the main problem of the calculus of variations and in the problem of Bolza in the nonsmooth case are derived. Examples comparing presented results with other approaches to nonsmooth problems of the calculus of variations are given.

This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier−Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-α equations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α → 0+ to a null control of the Navier−Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.

We show that we can reconstruct two coefficients of a wave equation by a single boundary measurement of the solution. The identification and reconstruction are based on Krein’s inverse spectral theory for the first coefficient and on the Gelfand−Levitan theory for the second. To do so we use spectral estimation to extract the first spectrum and then interpolation to map the second one. The control of the solution is also studied.

In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for scalar equations does not hold for vector-valued systems, we present an alternative approach based on an approximation result by Lipschitz functions due to Acerbi and Fusco combined with a Meyers Lp-estimate adapted to the functional ellipticity condition. The present framework includes in particular the elasticity case and the reinforcement by stiff thin fibers.

We prove the continuity and the Hölder equivalence w.r.t. an Euclidean distance of the value function associated with the L1 cost of the control-affine system q̇ = f0(q) + ∑j=1m uj fj(q), satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball–Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the L1 cost of time-dependent systems of the form q̇ = ∑j=1m uj fjt(q).