Dans ce travail, nous donnons une estimation logarithmique desdonnées de la solution u, d'un problème hyperboliqueavec condition aux limites de type Neumann, par la trace de urestreinte à un ouvert du bord, pendant un temps suffisammentgrand qui nous permet d'estimer la fonction de coût de ceproblème.

By means of a direct and constructive method based on the theory of semi-global C 1 solution, the local exact boundary observability is established for one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the exact boundary controllability and the exact boundary observability is then shown in the quasilinear case.

A general framework for calculating shape derivatives foroptimization problems with partial differential equations asconstraints is presented. The proposed technique allows to obtainthe shape derivative of the cost without the necessity to involvethe shape derivative of the state variable. In fact, the statevariable is only required to be Lipschitz continuous with respectto the geometry perturbations. Applications to inverse interfaceproblems, and shape optimization for elliptic systems and theNavier-Stokes equations are given.

The gradient based topological optimization tools introduced during thelast ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remainunchanged, at least during the topological phase of the optimizationalgorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not forimproving the cost function. Then, two numerical examples are presented:the first one compares topological optimization with standard shape optimization, and the second one, issued from a lake oxygenation problem, illustrates the use of the new asymptotic expansion.


We present an a posteriori error analysis of adaptive finiteelement approximations of distributed control problems for secondorder elliptic boundary value problems under bound constraints onthe control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and elementresiduals. Since we do not assume any regularity of the data ofthe problem, the error analysis further invokes data oscillations.We prove reliability and efficiency of the error estimator andprovide a bulk criterion for mesh refinement that also takes intoaccount data oscillations and is realized by a greedy algorithm. Adetailed documentation of numerical results for selected testproblems illustrates the convergence of the adaptive finiteelement method.

In this paper, we solve an optimal control problem using thecalculus of variation. The system under consideration is aswitched autonomous delay system that undergoes jumps at theswitching times. The control variables are the instants when theswitches occur, and a set of scalars which determine the jumpamplitudes. Optimality conditions involving analytic expressionsfor the partial derivatives of a given cost function with respectto the control variables are derived using the calculus ofvariation. A locally optimal impulsive control strategy can thenbe found using a numerical gradient descent algorithm.

We compare a general controlled diffusion process with a deterministic systemwhere a second controller drives the disturbance against the firstcontroller. We show that the two models are equivalent withrespect to two properties: the viability (or controlledinvariance, or weak invariance) of closed smooth sets, and theexistence of a smooth control Lyapunov function ensuring thestabilizability of the system at an equilibrium.


For a class of 2-D elastic energies we show that a radial equilibrium solutionis the unique global minimizer in a subclass of all admissible maps. Theboundary constraint is a double cover of S 1; the minimizer u is C 1and is such that $\det\nabla u$ vanishes at one point.


First, we consider a semilinear hyperbolic equation with a locally distributed damping in a boundeddomain. The damping is located on a neighborhood of a suitable portion of theboundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl.71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.

We study variational problems with volume constraints, i.e., with level sets of prescribed measure. We introduce a numerical method to approximate local minimizers and illustrate it with some two-dimensional examples. We demonstrate numerically nonexistence results which had been obtained analytically in previous work. Moreover, we show the existence of discontinuous dependence of global minimizers from the data by using a Γ-limit argument and illustrate this with numerical computations. Finally we construct explicitly local and global minimizers for problems with two volume constraints.

It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.

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.

The goal of this paper is to prove the first and second orderoptimality conditions for some control problems governed bysemilinear elliptic equations with pointwise control constraintsand finitely many equality and inequality pointwise stateconstraints. To carry out the analysis we formulate a regularityassumption which is equivalent to the first order optimalityconditions. Though the presence of pointwise state constraintsleads to a discontinuous adjoint state, we prove that the optimalcontrol is Lipschitz in the whole domain. Necessary and sufficientsecond order conditions are proved with a minimal gap betweenthem.

Let W be a function from the real m×n-matrices to the real numbers. If W is quasiconvex in the sense of the calculus of variations, then we show that W can be approximated locally uniformly by quasiconvex polynomials.

Here we present an approximation method for a rather broad class of first ordervariational problems in spaces of piece-wise constant functions overtriangulations of the base domain. The convergence of the method is based on aninequality involving $W^{-1, p}$ norms obtained by Nečas and on the generalframework of Γ-convergence theory.

In this paper, we are concerned with periodic, quasi-periodic (q.p.) and almost periodic (a.p.) Optimal Control problems. After defining these problems and setting them in an abstract setting by using Abstract Harmonic Analysis, we give some structure results of the set of solutions, and study the relations between periodic and a.p. problems. We prove for instance that for an autonomous concave problem, the a.p. problem has a solution if and only if all problems (periodic with fixed or variable period, q.p. or a.p.) have a constant solution. After that, we give some first order necessary conditions (weak Pontryagin) in the space of Harmonic Synthesis and we also give in this space an existence result.

The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control.In presence of (nonessential) touch points,the arc structure of the trajectory is not stable.Under some reasonable assumptions,we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. Assuming a weak second-order optimality condition (equivalent to uniform quadratic growth), stability and sensitivity results are given. The main tools are the study of a quadratic tangent problem and the notion of strong regularity.Those results enable us to design a new continuation algorithm,presented at the end of the paper, that handles automatically changes in the structure of the trajectory.

Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time $T>2\sqrt{2}$ ; this time, if the observation is made in $(-T/2,T/2)$ , is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I338 (2004) 413–418].Our proof follows an Ingham type approach.

We study the dynamic behavior and stability of two connectedRayleigh beams that are subject to, in addition to two sensors andtwo actuators applied at the joint point, one of the actuators alsospecially distributed along the beams. We show that with thedistributed control employed, there is a set of generalizedeigenfunctions of the closed-loop system, which forms a Riesz basiswith parenthesis for the state space. Then both thespectrum-determined growth condition and exponential stability areconcluded for the system. Moreover, we show that the exponentialstability is independent of the location of the joint. The range ofthe feedback gains that guarantee the system to be exponentiallystable is identified.

The irrigation problem is the problem of finding an efficient way to transport a measure μ+onto a measure μ-. By efficient, we mean that a structure that achieves the transport (which, following [Bernot, Caselles and Morel, Publ. Mat.49 (2005) 417–451], we call traffic plan)is better if it carries the mass in a grouped way rather than in a separate way.This is formalized by considering costs functionals that favorize this property.The aim of this paper is to introduce a dynamical cost functional on traffic plans that we argue to be more realistic.The existence of minimizers is proved in two ways: in some cases, we can deduce it from a classical semicontinuity argument; the other cases are treated by studying the link between our cost and the one introduced in [Bernot, Caselles and Morel, Publ. Mat.49 (2005) 417–451].Finally, we discuss the stability of minimizers with respect to specific variations of the cost functional.