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Showing papers in "ESAIM: Control, Optimisation and Calculus of Variations in 2007"


Journal ArticleDOI
TL;DR: In this paper, the authors provide KKT and saddle point optimality conditions, duality theorem and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces.
Abstract: This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas' lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint qualifications and/or a certain closedness assumption.

117 citations


Journal ArticleDOI
TL;DR: In this article, the authors present an algorithm called COTCOT (Conditions of Order Two and COnjugate Times) for computing the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal.
Abstract: The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions of Order Two and COnjugate Times), available on the web, and apply it to the minimal time problem of orbit transfer, and to the attitude control problem of a rigid spacecraft. This algorithm involves both normal and abnormal cases.

117 citations


Journal ArticleDOI
TL;DR: In this paper, the Laplace operator in a thin tube of with a Dirichlet condition on its boundary was studied asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero.
Abstract: We consider the Laplace operator in a thin tube of with a Dirichlet condition on its boundary We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame The main argument is a Γ -convergence theorem for a suitable sequence of quadratic energies

82 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one-dimensional vibrating systems or of square plates.
Abstract: We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are uniformly exponentially stable. This result is then applied to obtain strongly convergent approximations of the solutions of the algebraic Riccati equations associated to an LQR optimal control problem. We next give an application to a non-homogeneous string equation. Finally we apply similar techniques for approximating the equations of a damped square plate.

64 citations


Journal ArticleDOI
TL;DR: Pardoux and Peng as mentioned in this paper studied the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions, where both the control and the noise act on the boundary.
Abstract: We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C 1 regularity of such a solution is then used to construct the optimal feedback for the control problem. In order to overcome the difficulties arising from the degeneracy of the second order operator and from the presence of unbounded terms we study the HJB equation by introducing a suitable forward-backward system of stochastic differential equations as in the appraoch proposed in [Fuhrman and Tessitore, Ann. Probab. 30 (2002) 1397-1465; Pardoux and Peng, Lect. Notes Control Inf. Sci. 176 (1992) 200-217] for finite dimensional and infinite dimensional semilinear parabolic equations respectively.

60 citations


Journal ArticleDOI
TL;DR: It is demonstrated that tree leaves have different shapes and venation patterns mainly because they have adopted different efficient transport systems, and that optimal transportation plays a key role in the formation of tree leaves.
Abstract: In this article, we build a mathematical model to understand the formation of a tree leaf. Our model is based on the idea that a leaf tends to maximize internal efficiency by developing an efficient transport system for transporting water and nutrients. The meaning of "the efficient transport system" may vary as the type of the tree leave varies. In this article, we will demonstrate that tree leaves have different shapes and venation patterns mainly because they have adopted different efficient transport systems. The efficient transport system of a tree leaf built here is a modified version of the optimal transport path, which was introduced by 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 the phenomenon of ramifying structures in mass transportation. In the present paper, the cost functional on transport systems is controlled by two meaningful parameters. The first parameter describes the economy of scale which comes with transporting large quantities together, while the second parameter discourages the direction of outgoing veins at each node from differing much from the direction of the incoming vein. Under the same initial condition, efficient transport systems modeled by different parameters will provide tree leaves with different shapes and different venation patterns. Based on this model, we also provide some computer visualization of tree leaves, which resemble many known leaves including the maple and mulberry leaf. It demonstrates that optimal transportation plays a key role in the formation of tree leaves.

44 citations


Journal ArticleDOI
TL;DR: In this paper, the Mumford-Shah functional approximation of Ambrosio-Tortorelli's approximation of Mumford and Shah functions has been studied in an open subset Ω of R N, and the lower semicontinuous envelope of such energies has been computed.
Abstract: We consider, in an open subset Ω of R N , energies depending on the perimeter of a subset E ⊂ Ω (or some equivalent surface integral) and on a function u which is defined only on Ω \ E.W e compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the "holes" E may collapse into a discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application, a new proof of convergence for an extension of Ambrosio-Tortorelli's approximation to the Mumford-Shah functional.

41 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the inversion problem related to the manipulation of quantum systems using laser-matter interactions, focusing on the identification of the field free Hamiltonian and/or the dipole moment of a quantum system.
Abstract: This paper considers the inversion problem related to the manipulation of quantum systems using laser-matter interactions. The focus is on the identification of the field free Hamiltonian and/or the dipole moment of a quantum system. The evolution of the system is given by the Schrodinger equation. The available data are observations as a function of time corresponding to dynamics generated by electric fields. The well-posedness of the problem is proved, mainly focusing on the uniqueness of the solution. A numerical approach is also introduced with an illustration of its efficiency on a test problem.

38 citations


Journal ArticleDOI
TL;DR: In this paper, a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2D domain is studied, where the damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero.
Abstract: This work is devoted to the analysis of a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2-D domain. The damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero as time goes to infinity. Using discrete multiplier techniques, we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy for the solutions of the numerical scheme. The numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept, which guarantees that the low frequencies are damped correctly. Numerical experiments are presented and confirm these theoretical results. These results extend those by Tcheugoue-Tebou and Zuazua [ Numer. Math. 95 , 563–598 (2003)] where the 1-D case was addressed as well the square domain in 2-D. The methods and results in this paper extend to smooth domains in any space dimension.

37 citations


Journal ArticleDOI
TL;DR: In this article, the authors apply integral control to the series interconnection of an L 2 -stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity.
Abstract: This paper is concerned with integral control of systems with hysteresis. Using an input- output approach, it is shown that application of integral control to the series interconnection of ei- ther (a) a hysteretic input nonlinearity, an L 2 -stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L 2 -stable, time-invariant linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain constant determined by a posi- tivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system. Mathematics Subject Classification. 34G20, 47J40, 47N70, 93C23, 93C25, 93D10, 93D25.

35 citations


Journal ArticleDOI
TL;DR: In this paper, the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium was considered.
Abstract: We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the macroscopic scale and on the periodic microscopic scale. Denoting by e the period, the potential or zero-order term is scaled as e �2 and the drift or first-order term is scaled as e �1 . Under a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain with non-degenerate quadratic behavior, we prove an exponential localization at this minimum point. The homogenized problem features a diffusion equation with quadratic potential in the whole space.

Journal ArticleDOI
TL;DR: In this article, it was shown that there are trajectories bifurcating from the trivial branch σλ if the generalized Morse indices µ(σa )a nd µ (σb) are different.
Abstract: Given a one-parameter family {gλ : λ ∈ (a, b)} of semi Riemannian metrics on an n- dimensional manifold M , a family of time-dependent potentials {Vλ : λ ∈ (a, b)} and a family {σλ : λ ∈ (a, b)} of trajectories connecting two points of the mechanical system defined by (gλ ,V λ), we show that there are trajectories bifurcating from the trivial branch σλ if the generalized Morse indices µ(σa )a nd µ(σb) are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index in the case of locally symmetric spaces and a comparison principle of Morse Schoenberg type.

Journal ArticleDOI
TL;DR: In this article, uniform local energy estimates of solutions to the damped Schrodinger equation in exterior domains under the hypothesis of the exterior geometric control are derived from the resolvent properties.
Abstract: We prove uniform local energy estimates of solutions to the damped Schrodinger equation in exterior domains under the hypothesis of the Exterior Geometric Control. These estimates are derived from the resolvent properties.

Journal ArticleDOI
TL;DR: In this paper, a relaxation theorem in BV for non-coercive functional with linear growth is proved, where no continuity of the integrand with respect to the spatial variable is assumed.
Abstract: We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.

Journal ArticleDOI
TL;DR: In this paper, the linearized elasticity system in a multidomain of R 3 is considered, where the lateral boundary of the plate and the top of the beam are assumed to be clamped.
Abstract: We consider the linearized elasticity system in a multidomain of R 3 . This multidomain is the union of a horizontal plate with fixed cross section and small thickness e, and of a vertical beam with fixed height and small cross section of radius r e . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When e and r e tend to zero simultaneously, with r e e 2 , we identify the limit problem. This limit problem involves six junction conditions.

Journal ArticleDOI
TL;DR: In this paper, the authors considered 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 does not move too far from a given state.
Abstract: 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 control point 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)].

Journal ArticleDOI
TL;DR: In this paper, an open-loop system of a multidimensional wave equation with variable coefficients, par-tial boundary Dirichlet control and collocated observation is considered, and it is shown that the system is wellposed in the sense of D. Salamon and regular in the senses of G. Weiss.
Abstract: An open-loop system of a multidimensional wave equation with variable coefficients, par- tial boundary Dirichlet control and collocated observation is considered. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The Riemannian geometry method is used in the proof of regularity and the feedthrough operator is explicitly computed.

Journal ArticleDOI
TL;DR: In this article, it was shown that local minimizers of functionals of the form ∫ Ω [f(Du(Χ))+g( Χ, u, u) +W 1,p 0 (ΩQ) are locally Lipschitz continuous provided f is a convex function with p-q growth satisfying a condition of qualified convexity at infinity.
Abstract: We show that local minimizers of functionals of the form ∫ Ω [f(Du(Χ}} + g(Χ, u(Χ))] dΧ, u e u 0 + W 1,p 0 (Ω), ∫ Ω [∫(Du(Χ))+g(Χ, u(Χ))] dΧ, ueu o +W 1,p 0 (ΩQ), are locally Lipschitz continuous provided f is a convex function with p-q growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

Journal ArticleDOI
TL;DR: In this article, the authors consider probability measures μ and ν on a d-dimensional sphere in and cost functions of the form that generalize those arising in geometric optics where they prove that if μ and ǫ vanish on -rectifiable sets, if |l'(t)|>0, and is monotone then there exists a unique optimal map T o that transports μ onto where optimality is measured against c.
Abstract: In this paper, we consider probability measures μ and ν on a d -dimensional sphere in and cost functions of the form that generalize those arising in geometric optics where We prove that if μ and ν vanish on -rectifiable sets, if |l'(t)|>0, and is monotone then there exists a unique optimal map T o that transports μ onto where optimality is measured against c. Furthermore, Our approach is based on direct variational arguments. In the special case when existence of optimal maps on the sphere was obtained earlier in [Glimm and Oliker, J. Math. Sci. 117 (2003) 4096-4108] and [Wang, Calculus of Variations and PDE's 20 (2004) 329-341] under more restrictive assumptions. In these studies, it was assumed that either μ and ν are absolutely continuous with respect to the d -dimensional Haussdorff measure, or they have disjoint supports. Another aspect of interest in this work is that it is in contrast with the work in [Gangbo and McCann, Quart. Appl. Math. 58 (2000) 705-737] where it is proved that when l(t)=t then existence of an optimal map fails when μ and ν are supported by Jordan surfaces.

Journal ArticleDOI
TL;DR: In this paper, the authors derived a rigorous derivation of the Gross-Pitaevsky functional for vortex energy in a rotationally forced Bose-Einstein condensate.
Abstract: This paper gives a rigorous derivation of a functional proposed by Aftalion and Riviere [Phys. Rev. A 64 (2001) 043611] to characterize the energy of vortex filaments in a rotationally forced Bose-Einstein condensate. This functional is derived as a Γ -limit of scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence of large numbers of nontrivial local minimizers and we prove that, given any such local minimizer, the Gross-Pitaevsky functional has a local minimizer that is nearby (in a suitable sense) whenever a scaling parameter is sufficiently small.

Journal ArticleDOI
TL;DR: In this paper, an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot groups is studied, where the tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.
Abstract: We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups. The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.

Journal ArticleDOI
TL;DR: In this paper, the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence was studied, and it was shown that the Γ -limit is an integral functional on the matrix valued field.
Abstract: We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ -convergence. We prove that the Γ -limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γ -limit is also stable under volume constraint and various type of boundary conditions.

Journal ArticleDOI
TL;DR: In this article, a two-dimensional inverse problem in the field of tomography is solved by using the trace of the Riemann map as a fixed point equation and giving conditions for contraction.
Abstract: This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.

Journal ArticleDOI
TL;DR: In this paper, the homogenization process of ferromagnetic multilayers in the presence of surface energies was studied, where surface anisotropy was the dominant term and the magnitude of super-exchange interaction was inversely proportional to the interlayer distance.
Abstract: We study the homogenization process of ferromagnetic multilayers in the presence of surface energies: super-exchange, also called interlayer exchange coupling, and surface anisotropy. The two main difficulties are the non-linearity of the Landau-Lifshitz equation and the absence of a good sequence of extension operators for the multilayer geometry. First, we consider the case when surface anisotropy is the dominant term, then the case when the magnitude of the super-exchange interaction is inversely proportional to the interlayer distance. We establish the homogenized equation in these two situations.

Journal ArticleDOI
TL;DR: In this article, it was shown that every weak and strong local minimizer of the functional where, f grows like, g grows like and 1, is on an open subset of Ω such that.
Abstract: In this paper we prove that every weak and strong local minimizer of the functional where , f grows like , g grows like and 1 , is on an open subset of Ω such that . Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case is also treated for weak local minimizers.

Journal ArticleDOI
TL;DR: In this paper, partial regularity with optimal Holder exponent of vector-valued minimizers u of the quasiconvex variational integral was proved under polynomial growth.
Abstract: We prove partial regularity with optimal Holder exponent of vector-valued minimizers u of the quasiconvex variational integral $\int F( x,u,Du) \,{\rm d}x$ under polynomial growth. We employ the indirect method of the bilinear form.

Journal ArticleDOI
TL;DR: In this paper, a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form ut + H(x,Du )=0i n IR n × (0,T )w was proved.
Abstract: In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form ut + H(x,Du )=0i n IR n × (0,T )w here the Hamiltonian H may be noncoercive in the gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

Journal ArticleDOI
TL;DR: In this article, the optimal regularity for viscosity solutions of the pseudo infinity Laplacian was found and it was shown that the solutions are locally Lipschitz and showed an example that proves that this result is optimal.
Abstract: In this paper we find the optimal regularity for viscosity solutions of the pseudo infinity Laplacian. We prove that the solutions are locally Lipschitz and show an example that proves that this result is optimal. We also show existence and uniqueness for the Dirichlet problem.

Journal ArticleDOI
TL;DR: In this paper, the authors approximate free-discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls.
Abstract: We approximate, in the sense of Γ -convergence, free-discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls. The result extends to higher dimension what we already proved in the one-dimensional case.

Journal ArticleDOI
TL;DR: In this paper, a C k,α partial regularity result for local minimizers of variational integrals of the type I(u ) = � Ω f(D k u(x))dx, assuming that the integrand f satisfies (p,q) growth conditions was proved.
Abstract: We prove a C k,α partial regularity result for local minimizers of variational integrals of the type I(u )= � Ω f(D k u(x))dx, assuming that the integrand f satisfies (p,q) growth conditions.