We study properties of Lipschitz truncations of Sobolev functions with constant and variable exponent.As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in[Frehse et al., SIAM J. Math. Anal34 (2003) 1064–1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.

In this paper, we prove a controllabilityresult for a fluid-structure interaction problem. In dimension two,a rigid structure moves into an incompressible fluid governed byNavier-Stokes equations. The control acts on a fixed subset of thefluid domain. We prove that, for small initial data, this system isnull controllable, that is, for a given T > 0, the system can bedriven at rest and the structure to its reference configuration attime T. To show this result, we first consider a linearizedsystem. Thanks to an observability inequality obtained from aCarleman inequality, we prove an optimal controllability result witha regular control. Next, with the help of Kakutani's fixed pointtheorem and a regularity result, we pass to the nonlinear problem.

We present a revisited form of a resultproved in [Boccardo, Murat and Puel, Portugaliae Math.41 (1982) 507–534] and thenwe adapt the new proof in orderto show the existence for solutionsof quasilinear elliptic problems alsoif the lower order term has quadratic dependence on the gradient and singular dependence on the solution.

One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node $\pi/q$ during a simultaneous pluck or bow. This notion wasmade precise by a model of Bamberger, Rauch and Taylor. Their 'touch' isa damper of magnitude b concentrated at $\pi/q$ . The 'correct touch' is that b for which the modes, that do not vanishat $\pi/q$ , are maximally damped. We here examine the associated spectralproblem. We find the spectrum to be periodic and determined by a polynomialof degree $q-1$ . We establish lower and upper bounds on the spectral abscissaand show that the set of associated root vectors constitutes a Riesz basisand so identify 'correct touch' with the b that minimizes the spectral abscissa.

The propagation of fractures in a solid undergoing cyclic loadingsis known as the fatigue phenomenon. In this paper, we present a time continuousmodel for fatigue, in the special situation of the debonding of thin layers,coming from a time discretized version recently proposed byJaubert and Marigo [C. R. Mecanique333 (2005) 550–556].Under very general assumptions on the surface energy density andon the applied displacement, we discuss the well-posedness of ourproblem and we give the main properties of the evolution process.

The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization/reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is well-established in many situations where the governing p.d.e. system is of elliptic type. This paper focuses on the derivation of such formulas for parabolic and hyperbolic problems. Different kinds of cost functionals are considered.

The goal of this paper is to study the so-called worst-case or robustoptimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst,compliance when the loads are subject to some unknown perturbations.We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm tocompute it. Then, in the framework of Hadamard method, wecompute the directional shape derivative of this criterion which isused in a numerical algorithm, based on the level set method,to find optimal shapes that minimize the worst-case compliance.Since this criterion is usually merely directionally differentiable,we introduce a semidefinite programming approach toselect the best descent direction at each step of agradient method. Numerical examples are given in 2-d and 3-d.

Among ${\mathbb R}^3$ -valued triples of random vectors (X,Y,Z) having fixed marginal probability laws, what is the best way to jointly draw (X,Y,Z) in such a way that the simplex generated by (X,Y,Z) has maximal average volume? Motivated by this simple question, we study optimal transportation problems with several marginals when the objective function is the determinant or its absolute value.


For optimal control problems with ordinarydifferential equations where the $L^\infty$ -norm of the control isminimized, often bang-bang principles hold. For systems that aregoverned by a hyperbolic partial differential equation, thesituation is different:even if a weak form of the bang-bang principle still holds for the wave equation,it implies no restriction on the form of the optimal control.To illustrate thatfor the Dirichlet boundary control of the wave equation in general not even feasible controlsof bang-bang type exist,we examine the states that can be reached by bang-bang-offcontrols, that is controls that are allowed to attain only threevalues: Their maximum and minimum values and the value zero.We show that for certain control times,the difference between the initial and the terminal state can onlyattain a finite number of values.For the problems of optimal exact and approximate boundary control of the wave equationwhere the $L^\infty$ -norm of the control is minimized, we introduce dual problems and present the weak form of a bang-bangprinciple, that states that the values of $L^\infty$ -norm minimalcontrols are constrained by the sign of the dual solutions. Sincethese dual solutions are in general given as measures, thisis no restriction on the form of the control function:the dual solution may have a finite support, and when the dual solution vanishes,the control is allowed to attain all values from the intervalbetween the two extremal control values.


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.

Several recent results in the area of robust asymptotic stability of hybrid systems show that the concept of a generalized solution to a hybrid system is suitable for the analysis and design of hybrid control systems. In this paper, we show that such generalized solutions are exactly the solutions that arise when measurement noise in the system is taken into account.

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\{\nabla u_k\}$ , bounded in $L^p(\Omega;{\mathbb R}^{m\times n})$ if p > 1 and $\Omega\subset{\mathbb R}^n$ is a bounded domain with the extension property in $W^{1,p}$ .Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Caseswhere no boundary conditions nor regularity of Ω arerequired and links with lower semicontinuity results by Meyers andby Acerbi and Fusco are also discussed.

We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given $\phi_{0}$ close enough to an eigenstate corresponding to the length l = 1 and $\phi_{f}$ close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T] \rightarrow \mathbb{R}^{*}_{+}$ with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from $\phi_{0}$ to $\phi_{f}$ in time T.In particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way.Our proof relies on local controllability results proved with moment theory, a Nash-Moser implicit function theorem and expansions to the second order.

We prove the interior null-controllability of one-dimensionalparabolic equations with time independent measurable coefficients.

The aim of this paper is to study problems of the form: $inf_{(u\in V)}J(u)$ with $J(u):=\int_0^1 L(s,y_u(s),u(s)){\rm d}s+g(y_u(1))$ where V is a set of admissible controls and y u is the solution of the Cauchy problem: $\dot{x}(t) =\langle f(.,x(.)), \nu_t \rangle + u(t), t \in (0,1)$ , $x(0) = x_{\rm 0}$ and each $\nu_t$ is a nonnegative measure with support in [0,t]. After studying the Cauchy problem, we establish existence of minimizers, optimality conditions (in particular in the form of a nonlocal version of the Pontryagin principle) and prove some regularity results. We also consider the more general case where the control also enters the dynamics in a nonlocal way.

This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in thedesign of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmostcharacteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precludingthe use of standard optimisation methods. Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations. In dealing with systems of time-delay type, we extend the use of this algorithm toinfinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms toefficiently and accurately compute the rightmost characteristic roots of such time-delay systems. Furthermore, the optimisation procedure is adapted, enabling it to perform a local stabilisation of a nonlinear time-delay system along a branch of steady state solutions. We illustrate the use of the algorithm by presenting results for some numerical examples.

Let $F^{\rm p} \in {\rm GL}(3)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form ${\rm Curl}[{F^{\rm p}}]\cdot (F^{\rm p})^T$ applied to rotations controls the gradient in the sense that pointwise $ \forall R \in C^1(\mathbb{R}^3, {\rm SO}(3)): \Arrowvert {\rm Curl}[R] \cdot R^T \Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}R\Arrowvert_{\mathbb{R}^{27}}^2$ .This result complements rigidity results[Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying that $ \forall A \in C^1(\mathbb{R}^3, \mathfrak{so}(3)): \Arrowvert {\rm Curl}[A]\Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}A\Arrowvert_{\mathbb{R}^{27}}^2 = \Arrowvert\nabla{\rm axl}[A]\Arrowvert_{\mathbb{R}^9}^2$ .

In this work we study the nonlinear complementarity problem on thenonnegative orthant. This is done by approximating its equivalentvariational-inequality-formulation by a sequence of variationalinequalities with nested compact domains. This approach yieldssimultaneously existence, sensitivity, and stability results. Byintroducing new classes of functions and a suitable metric forperforming the approximation, we provide bounds for the asymptoticset of the solution set and coercive existence results, which extendand generalize most of the existing ones from the literature. Suchresults are given in terms of some sets called coercive existencesets, which we also employ for obtaining new sensitivity andstability results. Topological properties of thesolution-set-mapping and bounds for it are also established.Finally, we deal with the piecewise affine case.

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as aweighted dissipation-energy functional with a weight decaying with a rate $1/\epsilon$ . The corresponding Euler-Lagrange equation leads to anelliptic regularization of the original evolutionary problem. The Γ-limit of these functionals for $\epsilon\to 0$ is highlydegenerate and provides limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.

This paper presents the role of vector relative degree in theformulation of stationarity conditions of optimal control problemsfor affine control systems. After translating the dynamics into anormal form, we study the Hamiltonian structure. Stationarityconditions are rewritten with a limited number of variables. Theapproach is demonstrated on two and three inputs systems, then, weprove a formal result in the general case. A mechanical systemexample serves as illustration.