An optimal control problem forsemilinear parabolic partial differential equations is considered.The control variable appears in the leading term of the equation.Necessary conditions for optimal controls are established by themethod of homogenizing spike variation. Results for problems withstate constraints are also stated.

We prove pointwise gradient bounds for entire solutions of pde’s of the form

     ℒu(x) = ψ(x, u(x), ∇u(x)),

where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.

We study the stability of a flexible beam clamped at one end. A mass is attached at the other end, where a control moment is applied. The boundary control is proportional to the angular velocity at the end. By spectral analysis, we prove that the optimal decay rate of the energy is given by the spectrum of the generator of the semigroup associated to the system.

In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochasticdifferential games with reflection. We obtain an existence theorem and a characterizationtheorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games withnonlinear cost functionals defined by doubly controlled reflected backward stochasticdifferential equations.

We give a first contribution to the homogenization of many-body structures that are exposed to large deformations and obey the noninterpenetration constraint. The many-body structures considered here resemble cord-belts like they are used to reinforce pneumatic tires. We establish and analyze an idealized model for such many-body structures in which the subbodies are assumed to be hyperelastic with a polyconvex energy density and shall exhibit an initial brittle bond with their neighbors. Noninterpenetration of matter is taken into account by the Ciarlet-Nečas condition and we demand deformations to preserve the local orientation. By studying Γ-convergence of the corresponding total energies as the subbodies become smaller and smaller, we find that the homogenization limits allow for deformations of class special functions of bounded variation while the aforementioned kinematic constraints are conserved. Depending on the many-body structures’ geometries, the homogenization limits feature new mechanical effects ranging from anisotropy to additional kinematic constraints. Furthermore, we introduce the concept of predeformations in order to provide approximations for special functions of bounded variation while preserving the natural kinematic constraints of geometrically nonlinear solid mechanics.

The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: ${\displaystyle\inf_{{\displaystyle(u,v)\in {\cal U}_{ad}}}\int_{0}^{1} f\left(t, u(\theta_v(t)),u^{\prime}(t),v(t)\right){\rm d}t}$ , (1)where ${\cal U}_{ad} $ is a set of admissible controls and $\theta_v$ is the solution of the following equation: $\{ \frac{{\rm d}\theta(t)}{{\rm d}t}=g(t,\theta(t),v(t)), t\in [0,1]$ ; $\displaystyle\theta(0)=\theta_0, \theta(t)\in [0,1]\forall t$ . (2).The results are nonlocal and new.

We prove the existence of a principal eigenvalue associated to the∞-Laplacian plus lower order terms and the Neumann boundarycondition in a bounded smooth domain. As an application we getuniqueness and existence results for the Neumann problem and adecay estimate for viscosity solutions of the Neumann evolutionproblem.

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.

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}$

This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and two controls. Some of them are known to be flat, and this implies admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about “(x,u)-flatness” of these systems, with much more elementary techniques.

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).

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\dots SO\left(2\right)A_{N}$ where $A_1,A_2,\dots, A_{N}$ are matrices of non-zero determinant. Weestablish a sharp relation between the following two minimisationproblems in two dimensions. Firstly the N-well problem with surface energy. Let $p\in\left[1,2\right]$ , $\Omega\subset \mathbb{R}^2$ be a convex polytopal region. Define $$I^p_{\epsilon}\left(u\right)=\int_{\Omega} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2u\left(z\right)\right|^2 {\rm d}L^2 z$$ and let A F denote the subspace of functions in $W^{2,2}\left(\Omega\right)$ that satisfy the affine boundary conditionDu=F on $\partial \Omega$ (in the sense of trace), where $F\not\inK$ . We consider the scaling (with respect to ϵ) of $$m^p_{\epsilon}:=\inf_{u\in A_F} I^p_{\epsilon}\left(u\right).$$ Secondly the finite element approximation to the N-well problemwithout surface energy. We will show there exists a space of functions $\mathcal{D}_F^{h}$ whereeach function $v\in \mathcal{D}_F^{h}$ is piecewise affine on a regular(non-degenerate) h-triangulation and satisfies the affine boundarycondition v=lF on $\partial \Omega$ (where l F is affine with $Dl_F=F$ ) such that for $$\alpha_p\left(h\right):=\inf_{v\in \mathcal{D}_F^{h}}\int_{\Omega}d^p\left(Dv\left(z\right),K\right) {\rm d}L^2 z$$ there exists positive constants $\mathcal{C}_1<1<\mathcal{C}_2$ (depending on $A_1,\dots, A_{N}$ , p) for which the following holds true $$\mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\epsilon}\leq\mathcal{C}_2\alpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0.$$

We are concerned with the asymptotic analysis of optimal controlproblems for 1-D partial differential equations defined on aperiodic planar graph, as the period of the graph tends to zero. Wefocus on optimal control problems for elliptic equations withdistributed and boundary controls. Using approaches of the theory ofhomogenization we show that the original problem on the periodicgraph tends to a standard linear quadratic optimal control problemfor a two-dimensional homogenized system, and its solution can beused as suboptimal controls for the original problem.

We establish two new formulations of the membrane problem by working in the space of $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -Young measures and $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -varifolds. The energyfunctional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. Theinterest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classicalformulation. The second formulation moreover accounts for concentration effects.

We consider in $\mathbb{R}^2$ all curvature equation $\frac{\partial u}{\partial t}=|Du|G({\rm curv}(u))$ where G is a nondecreasing function and curv(u) is the curvature of the level line passing by x. These equations are invariant with respectto any contrast change u → g(u), with g nondecreasing.Consider the contrast invariant operator $T_t: u_o\to u(t)$ .A Matheron theorem asserts that all contrast invariant operatorT can be put in a form $(Tu)({\bf x}) = \inf_{B\in {\cal B}}\sup_{{\bf y}\in B} u({\bf x}+{\bf y})$ .We show the asymptotic equivalence of both formulations.More precisely, we show that all curvature equations can be obtained asthe iteration of Matheron operators $T_h^n$ whereh → 0 and n → ∞ with nh=t.

This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is \hbox{$I_{\ep}(u)=\frac{1}{2}\int_{\Omega} \ep^{-1}\lt|1-\lt|Du\rt|^2\rt|^2+\ep\lt|D^2 u\rt|^2 {\rm d}z$}${\mathit{I}}_{\mathit{ϵ}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int }}}_{\mathit{\Omega }}{\mathit{ϵ}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{ϵ}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ where u belongs to the subset of functions in \hbox{$W^{2,2}_{0}(\Omega)$}${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202] Jabin et al. characterized a class of functions which includes all limits of sequences \hbox{$u_n\in W^{2,2}_0(\Omega)$}${\mathit{u}}_{\mathit{n}}\mathrm{\in }{\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ with Iϵn(un) → 0 as ϵn → 0. A corollary to their work is that if there exists such a sequence (un) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u: = limn → ∞un = dist(·,∂Ω). Recently [Lorent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted), http://arxiv.org/abs/0902.0154v1] we provided a quantitative generalization of this corollary over the space of convex domains using ‘compensated compactness’ inspired calculations of DeSimone et al. [Proc. Soc. Edinb. Sect. A 131 (2001) 833–844]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B1(0) without the requiring the trace condition on Du.

In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram’s lemma and semigroup theory. The validity of cutting-edge method is proved by spectral analysis approach. In particular, we give a detailed procedure of cutting-edge for the bush-type wave networks. The results show that if we impose feedback controllers, consisting of velocity and position terms, at all the boundary vertices and at most three velocity feedback controllers on the cycle, the system is asymptotically stabilized. Finally, some examples are given.

For a hybrid system composed of a cable with masses at both ends, we prove the existence of solutions for a class of nonlinear and nonmonotone feedback laws by means of a priori estimates. Assuming some local monotonicity, strong stabilization is obtained thanks to some Riemann's invariants technique and La Salle's principle.

This article aims at studying the controllability of a simplified fluid structure interaction model derived and developed in [C. Conca, J. Planchard and M. Vanninathan, RAM: Res. Appl. Math. John Wiley & Sons Ltd., Chichester (1995); J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. This interaction is modeled by a wave equation surrounding a harmonic oscillator. Our main result states that, in the radially symmetric case, this system can be controlled from the outer boundary. This improves previous results [J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. Our proof is based on a spherical harmonic decomposition of the solution and the so-called lateral propagation of the energy for 1d waves.