The internal and boundary exact null controllability of nonlinear convective heat equations with homogeneous Dirichlet boundary conditions are studied. The methods we use combine Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system, interpolation inequalities and some estimates in the theory of parabolic boundary value problems in Lk .

Turnpike theorems deal with the optimality of trajectories reaching asingular solution, in calculus of variations oroptimal control problems.For scalar calculus of variations problems in infinite horizon, linear withrespect to the derivative, we use the theory of viscosity solutions ofHamilton-Jacobi equations to obtain a unique characterization of the valuefunction.With this approach, we extend for the scalar case the classical result based onGreen theorem, when there is uniqueness of the singular solution.We provide a new necessary and sufficient condition for turnpikeoptimality, even in the presence of multiple singular solutions.

In this paper, we study the stabilization of a two-dimensional Burgers equation around a stationary solution by a nonlinear feedback boundary control. We are interested in Dirichlet and Neumann boundary controls. In the literature, it has already been shown that a linear control law, determined by stabilizing the linearized equation, locally stabilizes the two-dimensional Burgers equation. In this paper, we define a nonlinear control law which also provides a local exponential stabilization of the two-dimensional Burgers equation. We end this paper with a few numerical simulations, comparing the performance of the nonlinear law with the linear one.

We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical simulation leads to approximated optimal configurations. We include several such simulations in 2d and 3d.

This essentially numerical study, sets out to investigate various geometrical properties of exact boundary controllability of the waveequation when the control is applied on a part of the boundary. Relationships between the geometry of the domain, the geometry ofthe controlled boundary, the time needed to control and the energy of the control are dealt with. A new norm of the control and anenergetic cost factor are introduced. These quantities enable a detailed appraisal of the numerical solutions obtained and the detectionof trapped rays.

We examine shape optimization problems in the context of inexact sequential quadraticprogramming. Inexactness is a consequence of using adaptive finite element methods (AFEM)to approximate the state and adjoint equations (via the dual weightedresidual method), update the boundary, and compute the geometric functional. We present anovel algorithm that equidistributes the errors due to shape optimization anddiscretization, thereby leading to coarse resolution in the early stages and fineresolution upon convergence, and thus optimizing the computational effort. We discuss theability of the algorithm to detect whether or not geometric singularities such as cornersare genuine to the problem or simply due to lack of resolution – a new paradigm inadaptivity.

We consider the problem of minimizing \hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathit{\ell }}\sqrt{{\mathit{\xi }}^{\mathrm{2}}\mathrm{+}{\mathit{K}}^{\mathrm{2}}\mathrm{\left(}\mathit{s}\mathrm{\right)}}\hspace{0.17em}\mathrm{d}\mathit{s}$ for a planar curve having fixedinitial and final positions and directions. The total length ℓ is free. Heres is thearclength parameter, K(s) is the curvature of the curveand ξ > 0 is a fixed constant. This problem comes from a model of geometry ofvision due to Petitot, Citti and Sarti. We study existence of local and global minimizersfor this problem. We prove that if for a certain choice of boundary conditions there is noglobal minimizer, then there is neither a local minimizer nor a geodesic. We finally giveproperties of the set of boundary conditions for which there exists a solution to theproblem.

We consider one-dimensional affine control systems. We show that if such a system is stabilizable by means of a continuous, time-invariant feedback, then it can be made input-to-state stable with respect to measurement disturbances, using a continuous, periodic time-varying feedback. We provide counter-examples showing that the result does not generally hold if we want the feedback to be time-invariant or if the control system is not supposed affine.

We study the global approximate controllability of the one dimensional semilinearconvection-diffusion-reaction equation governed in a bounded domain via the coefficient (bilinear control) in the additive reaction term. Clearly, even in the linear case, due to the maximum principle, such system is not globally or locally controllable in any reasonable linear space. It is also well known that for the superlinear terms admitting a power growth at infinity the global approximate controllability by traditional additive controls of localized support is out of question. However, we will show that a system like that can be steered in L²(0,1) from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static (x-dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.


We give sufficient conditions which allow the study of the exponentialstability of systems closely related to the linear thermoelasticity systemsby a decoupling technique. Our approach is based on the multiplierstechnique and our result generalizes (from the exponential stability pointof view) the earlier one obtained by Henry et al.

We consider a distributed system in which the state q is governed by a parabolic equation and a pair of controls v = (h,k) where h and k play two different roles: the control k is of controllability type while h expresses that the state q does not move too far from a given state. Therefore, it is natural to introduce the controlpoint of view. In fact, there are several ways to state and solve optimal control problems with a pair of controls h and k, in particular the Least Squares method with only one criteria for the pair (h,k) or the Pareto Optimal Control for multicriteria problems. We propose here to use the notion of Hierarchic Control. This notion assumes that we have two controls h, k where h will be the leader while k will be the follower. The main tool used to solve the null-controllability problem with constraints on the follower is an observability inequality of Carleman type which is “adapted” to the constraints. The obtained results are applied to the sentinels theory of Lions [Masson (1992)].

We give the definitions of exact and approximate controllability forlinear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” resultfor evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllabilityfor the linear Schrödinger equation and distributed additive control,and we show that the Hartree equation of quantum chemistry with bilinearcontrol $(E(t)\cdot x) u$ is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, andwe give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.

In this article, we build a mathematical model to understand theformation of a tree leaf. Our model is based on the idea that a leaftends to maximize internal efficiency by developing an efficienttransport system for transporting water and nutrients. The meaningof “the efficient transport system” may vary as the type of thetree leave varies. In this article, we will demonstrate that treeleaves have different shapes and venation patterns mainly becausethey have adopted different efficient transport systems.The efficient transport system of a tree leaf built here is amodified version of the optimal transport path, which was introducedby the author in [Comm. Cont. Math.5 (2003) 251–279; Calc. Var. Partial Differ. Equ.20 (2004) 283–299; Boundary regularity of optimal transport paths,Preprint] to study thephenomenon of ramifying structures in mass transportation. In thepresent paper, the cost functional on transport systems iscontrolled by two meaningful parameters. The first parameterdescribes the economy of scale which comes with transporting large quantities together, while thesecond parameter discourages the direction of outgoing veins at eachnode from differing much from the direction of the incoming vein.Under the same initial condition, efficient transport systemsmodeled by different parameters will provide tree leaves withdifferent shapes and different venation patterns.Based on this model, we also provide some computer visualization oftree leaves, which resemble many known leaves including the mapleand mulberry leaf. It demonstrates that optimal transportation playsa key role in the formation of tree leaves.

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures.The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases.In the particular situation of constant unloading response,the result contained in [G. Dal Maso and C. Zanini,Proc. Roy. Soc. Edinburgh Sect. A137 (2007) 253–279] is recovered.In this case, the convergence of the discrete time approximationsis improved.

In this paper we study a one dimensional model of ferromagnetic nano-wires of finitelength. First we justify the model by Γ-convergence arguments.Furthermore we prove the existence of wall profiles. These walls being unstable, westabilize them by the mean of an applied magnetic field.

In this paper, we consider a Borel measurable function on the space of $\scriptstyle m\times n$ matrices $\scriptstyle f: M^{m\times n}\rightarrow \bar{\mathbb{R}}$ taking the value $ \scriptstyle +\infty$ , such that its rank-one-convex envelope $\scriptstyle Rf$ is finite and satisfies for some fixed $\scriptstyle p>1$ : $$\scriptstyle -c_0\leq Rf(F)\leq c(1+\Vert F\Vert^p)\ \hbox{for all}\F\in M^{m\times n},$$ where $\scriptstyle c,c_0>0$ . Let $\scriptstyle\O$ be a given regular bounded open domain of $\scriptstyle \mathbb{R}^n$ . We define on $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ the functional $$\scriptstyle I(u)=\int_{\O}f(\nabla u(x))\ dx.$$ Then, under some technical restrictions on $\scriptstyle f$ , we show that the relaxed functional $\scriptstyle\bar I$ for the weak topology of $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ has the integral representation: $$\scriptstyle \bar I(u)=\int_{\O}Q[Rf](\nabla u(x))\ dx,$$ where for a given function $\scriptstyle g$ , $\scriptstyle Qg$ denotes its quasiconvex envelope.

We consider an equilibrium problem with equilibrium constraints (EPEC) arising from themodeling of competition in an electricity spot market (under ISO regulation). For acharacterization of equilibrium solutions, so-called M-stationarity conditions arederived. This first requires a structural analysis of the problem, e.g.,verifying constraint qualifications. Second, the calmness property of a certainmultifunction has to be verified in order to justify using M-stationarity conditions.Third, for stating the stationarity conditions, the coderivative of a normal cone mappinghas to be calculated. Finally, the obtained necessary conditions are made fully explicitin terms of the problem data for one typical constellation. A simple two-settlementexample serves as an illustration.

We consider the Laplace equation in a smooth bounded domain. Weprove logarithmic estimates, in the sense of John [5] of solutions ona part of the boundary or of the domain without known boundary conditions.These results are established by employing Carleman estimates and techniquesthat we borrow from the works of Robbiano [8,11]. Also, we establishan estimate on the cost of an approximate control for an elliptic model equation.

Let W be a non-negative function of class C3 from $\xR^2$ to $\xR$ , which vanishes exactly at two points a and b. LetS 1(a, b) be the set of functions of a real variable which tend to a at -∞and to b at +∞ and whose one dimensional energy $$E_1(v)=\int_\xR\bigl[W(v)+\lvert v'\rvert^2/2\bigr]\,\xdif x$$ is finite.Assume that there exist two isolated minimizers z + and z - of the energy E 1over S 1(a, b). Under a mild coercivity condition on thepotential W and a generic spectral condition on the linearization of theone-dimensional Euler–Lagrange operator at z + and z - , it ispossible to prove that there exists a function ufrom $\xR^2$ to itself which satisfies the equation $$-\Delta u + \xDif W(u)^\mathsf{T}=0,$$ and the boundary conditions $$\lim_{x_2\to +\infty} u(x_1,x_2)=z_+(x_1-m_+),\phantom{\mathbf{a}}\lim_{x_2\to-\infty} u(x_1,x_2)=z_-(x_1-m_-),\lim_{x_1\to -\infty}u(x_1,x_2)=\mathbf{a},\phantom{z_+(x_1-m_+)} \lim_{x_1\to+\infty}u(x_1,x_2)=\mathbf{b}.$$ The above convergences are exponentially fast; the numbers m + and m - are unknowns of the problem.

We consider a hybrid, one-dimensional, linear system consisting in two flexible strings connected by a point mass. It is known that this system presents two interesting features. First, it is wellposed in an asymmetric space in which solutions have one more degree of regularity to one side of the point mass. Second, that the spectral gap vanishes asymptotically. We prove that the first property is a consequence of the second one. We also consider a system in which the point mass is replaced by a string of length 2ε and density 1/2ε. We show that, as ε → 0, thesolutions of this system converge to those of the original one. We also analyze the convergence of the spectrum and obtain the well-posedness of the limit system in the asymmetric space as a consequence of non-standard uniform bounds of solutions of the approximate problems. Finally we consider the controllability problem. It is well known that the limit system with L-controls on one end is exactly controllable in an asymmetric space. We show how this result can be obtained as the limit when ε → 0 of partial controllability results for the approximate systems inwhich the number of controlled frequencies converges to infinity as ε → 0. This is done by means of some new results on non-harmonic Fourier series.