A unified framework for analyzing the existence of ground states in wideclasses of elastic complex bodies is presented here. The approach makes useof classical semicontinuity results, Sobolev mappings and Cartesiancurrents. Weak diffeomorphisms are used to represent macroscopicdeformations. Sobolev maps and Cartesian currents describe the innersubstructure of the material elements. Balance equations for irregularminimizers are derived. A contribution to the debate about the role of thebalance of configurational actions follows. After describing a list ofpossible applications of the general results collected here, a concretediscussion of the existence of ground states in thermodynamically stablequasicrystals is presented at the end.

An alternative approach for the analysis and the numericalapproximation of ODEs, using a variational framework, ispresented. It is based on the natural and elementary idea of minimizing the residual of the differential equation measured in a usual Lp norm.Typical existence results for Cauchy problems can thus berecovered, and finer sets of assumptions for existence are made explicit. We treat, in particular, the cases of an explicit ODE and a differential inclusion. This approach also allows for a whole strategy to approximate numerically the solution. It is briefly indicated here as it will be pursued systematically and in a much more broad fashion in a subsequent paper.

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.

We study minimal surfaces in sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated with the canonical horizontal area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces in the horizontal spherical bundle over the base manifold. The singularities of minimal surfaces turn out to be the singularities of this projection, and we give a complete local classification of them. We illustrate our results by examples in the Heisenberg group and the group of roto-translations. 


For a general time-varying system, we prove that existence of an “Output Robust Control Lyapunov Function” implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result due to Coron and Rosier [J. Math. Syst. Estim. Control4 (1994) 67–84] concerning stabilization of autonomous systems by means of time-varying periodic feedback.

In this paper we study asymptotic behaviour of distributed parameter systems governedby partial differential equations (abbreviated to PDE). We first review some recently developed resultson the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionalswe prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDEsystems. Then we apply the result to establish exponential stability of various chemical engineeringprocesses and, in particular, exponential stability of heat exchangers. Through concrete examples weshow how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE.Meanwhile we explain how the method is adapted to the framework of Banach spaces Lp , $1<p\leq \infty$ . 


We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here. In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand, Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including “mixed” boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.

The purpose of this paper is to derive and study a new asymptoticmodel for the equilibrium state of a thin anisotropicpiezoelectric plate in frictional contact with a rigid obstacle.In the asymptotic process, the thickness of the piezoelectricplate is driven to zero and the convergence of the unknowns isstudied. This leads to two-dimensional Kirchhoff-Love plateequations, in which mechanical displacement and electric potentialare partly decoupled. Based on this model numerical examples arepresented that illustrate the mutual interaction between themechanical displacement and the electric potential. We observethat, compared to purely elastic materials, piezoelectric bodiesyield a significantly different contact behavior.

Optimal control problems for the heat equation with pointwisebilateral control-state constraints are considered. A locallysuperlinearly convergent numerical solution algorithm is proposedand its mesh independence is established. Further, for theefficient numerical solution reduced space and Schur complementbased preconditioners are proposed which take into account theactive and inactive set structure of the problem. The paper endsby numerical tests illustrating our theoretical findings andcomparing the efficiency of the proposed preconditioners.

In this work we study the structure of approximatesolutions of autonomous variational problems with a lowersemicontinuous strictly convex integrand f : Rn ×Rn $\to$ R 1 $\cup$ $\{\infty\}$ , where Rn is the n-dimensional Euclideanspace. We obtain a full description of the structure of theapproximate solutions which is independent of the length of theinterval, for all sufficiently large intervals.

We study Hamiltonian systems which generate extremal flows of regularvariational problems on smooth manifolds and demonstrate thatnegativity of the generalized curvature of such a system impliesthe existence of a global smooth optimal synthesis for the infinitehorizon problem.We also show that in the Euclidean case negativity of the generalized curvature is a consequence ofthe convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems.

In this paper we study Lavrentiev-type regularization concepts forlinear-quadratic parabolic control problems with pointwise state constraints. Inthe first part, we apply classical Lavrentiev regularization to a problem withdistributed control, whereas in the second part, a Lavrentiev-typeregularization method based on the adjoint operator is applied to boundarycontrol problems with state constraints in the whole domain. The analysis forboth classes of control problems is investigated and numerical tests areconducted. Moreover the method is compared with other numerical techniques.

In this work we consider the magnetic NLS equation $$ ( \frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox{ in } \mathbb{R}^N\qquad\qquad(0.1)$$ where $N \geq 3$ , $A \colon \mathbb{R}^N \to \mathbb{R}^N$ is a magnetic potential,possibly unbounded, $V \colon \mathbb{R}^N \to \mathbb{R}$ is a multi-well electricpotential, which can vanish somewhere, f is a subcriticalnonlinear term. We prove the existence of a semiclassical multi-peaksolution $u\colon \mathbb{R}^N \to \mathbb{C}$ to (0.1), under conditionson the nonlinearity which are nearly optimal.

We present below a new series of conjectures and openproblems in the fields of (global) Optimization and Matrix analysis, in thesame spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAMReview49 (2007) 255–273]. With each problem come a succinct presentation, a list of specificreferences, and a view on the state of the art of the subject.

A general framework for computing robust controllable sets ofconstrained nonlinear uncertain discrete-time systems as well ascontrolling such complex systems based on the computed robustcontrollable sets is introduced in this paper. The addressedone-step control approach turns out to be a robust model predictivecontrol scheme with feasible unit control horizon and contractiveconstraint. The solver of 1-dimensional quantified set inversion inmodal interval analysis is extended to 2-dimensional cases forcomputing robust controllable sets off-line with a clear semanticinterpretation, where both universal and existential quantifiers areconcerned simultaneously. An interval-based solver of constrainedminimax optimization is also proposed to compute one-step controlinputs online in a reliable way, which guarantee to drive the systemstate contractively along the computed robust controllable sets to aselected terminal robust control invariant set.

We consider continuous reformulations of the Euclidean travelling salesperson problem (TSP), based on certain clustering problemformulations. These reformulations allow us to apply a generalisation with perturbations of the Weiszfeld algorithm in an attempt tofind local approximate solutions to the Euclidean TSP.

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization insidethe magnetic sample is of particular relevance. We here investigate mathematicallythis problem by considering the full partial differential model of Landau-Lifschitzequations triggered by a uniform (in space) external magnetic field.

A direct construction of a stabilizing hybrid feedback that is robust togeneral measurement error is given for a general nonlinear control system that is asymptotically controllable to a compact set.

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 consider a mesoscopic model for phase transitions in a periodic mediumand we construct multibump solutions.The rational perturbative case is dealt with by explicitasymptotics.