Showing papers in "ESAIM: Control, Optimisation and Calculus of Variations in 2003"
TL;DR: In this article, a closed loop parametrical identification procedure for continuous-time constant linear systems is introduced, which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus.
Abstract: A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.
510 citations
TL;DR: An approximation of the stiffest structure subject to a pressure load or a given field of internal forces is proposed in the framework of Γ-convergence, based on an approximation of the authors' three phases by a smooth phase-field.
Abstract: We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset S of a reference domain, and the complement of S is made of two other “phases”, the “void” and a fictitious “liquid” that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure S, which is the total work of the pressure and internal forces at the equilibrium displacement. In order to prevent from homogenization we add a penalization on the perimeter of S. We propose an approximation of our problem in the framework of Γ-convergence, based on an approximation of our three phases by a smooth phase-field. We detail the numerical implementation of the approximate energies and show a few experiments.
322 citations
TL;DR: This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems and Corresponding canonical forms are presented and sucient conditions which permit the design of constant and high gain observers for these systems.
Abstract: This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. In the single output case, it is known (4{6) that systems which are observable independently of the inputs, admit an observable canonical form. These systems are called uniformly observable systems. Moreover, a high gain observer for these systems can be designed on the basis of this canonical form. In this paper, we extend the above results to multi-output uniformly observable systems. Corresponding canonical forms are presented and sucient conditions which permit the design of constant and high gain observers for these systems are given. Mathematics Subject Classication. 37N35, 93Bxx.
120 citations
TL;DR: In this paper, it was shown that the system of equations is a well-posed linear system with input u and output y, where u is the energy identity and y is the state space.
Abstract: Let A0 be a possibly unbounded positive operator on the Hilbert space H ,w hich is boundedly invertible. Let C0 be a bounded operator from D A 1 2 0 to another Hilbert space U.W e prove that the system of equations ¨(t )+ A0z(t )+ 1 C 0C0 _ z(t )= C 0u(t); y(t )= C0 _ z(t )+ u(t); determines a well-posed linear system with input u and output y. The state of this system is x(t )= z(t) _ z(t) 2D A 1 2 0 H = X; where X is the state space. Moreover, we have the energy identity
105 citations
TL;DR: In this article, the free boundary problem for a nonlinear parabolic partial dif-ferential equation with a quadratic nonlinearity is considered and a series solution is proposed.
Abstract: In this paper we consider a free boundary problem for a nonlinear parabolic partial dif- ferential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary ap rioriand would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simu- lation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large. Mathematics Subject Classication. 93C20, 80A22, 80A23.
98 citations
TL;DR: In this article, the existence of bounded Palais-Smale sequences of strongly indefinite functional sequences was proved for the semilinear Schrodinger equation, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V ; K>0.
Abstract: In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrodinger equation , where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V ; K>0 . If for an appropriate constant c , we show that this equation has a nontrivial solution.
94 citations
TL;DR: In this article, the authors discuss the approximate reconstruction of inhomogeneities of small volume using boundary integrals of the (observed) electromagnetic fields, based on highly accurate asymptotic formulae.
Abstract: In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.
85 citations
TL;DR: In this article, it was shown that the Navier-Stokes equations with internal con- trollers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation.
Abstract: One proves that the steady-state solutions to Navier{Stokes equations with internal con- trollers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation. yt(x;t) y(x;t )+( y r)y(x;t )= m(x)u(x;t )+ f0(x )+ rp(x;t); (x;t) 2 Q ( ry)(x;t )=0 ; 8 (x;t) 2 Q =(0;1) y =0 ; on = @(0;1) y(x; 0) =y0(x) ;x 2 :
82 citations
TL;DR: The tree of shapes of an image as mentioned in this paper is a mix of the component trees of upper and lower level sets, and its existence under fairly weak assumptions and its completeness are proven.
Abstract: This chapter presents the tree of shapes of an image, a mix of the component trees of upper and lower level sets. Its existence under fairly weak assumptions and its completeness are proven. Ignoring the small details of the image, we show the essentially finite nature of the tree. Finally, we illustrate these theoretical results with a direct application to gray level quantization.
69 citations
TL;DR: In this paper, the authors conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-basis-property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams.
Abstract: We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.
64 citations
TL;DR: In this paper, an existence and regularity theorem is proved for integral equations of convolution type which contain hysteresis nonlinearities, and frequency-domain stability criteria are derived for feedback systems with a linear infinite-dimensional system in the forward path and a hystereis non-linearity in the feedback path.
Abstract: An existence and regularity theorem is proved for integral equations of convolution type which contain hysteresis nonlinearities. On the basis of this result, frequency-domain stability criteria are derived for feedback systems with a linear infinite-dimensional system in the forward path and a hysteresis nonlinearity in the feedback path. These stability criteria are reminiscent of the classical circle criterion which applies to static sector-bounded nonlinearities. The class of hysteresis operators under consideration contains many standard hysteresis nonlinearities which are important in control engineering such as backlash (or play), plastic-elastic (or stop) and Prandtl operators. Whilst the main results are developed in the context of integral equations of convolution type, applications to well-posed state space systems are also considered.
TL;DR: In this paper, the authors investigated the asymptotic behavior of a class of monotone nonlinear Neumann problems, with growth p -1 (p ∈]1, +∞[), on a bounded multidomain (N ≥ 2).
Abstract: We investigate the asymptotic behaviour, as e → 0, of a class of monotone nonlinear Neumann problems, with growth p -1 (p ∈]1, +∞[), on a bounded multidomain (N ≥ 2). The multidomain ΩE is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness hE in the x N direction, as e → 0. The second one is a “forest" of cylinders distributed with e -periodicity in the first N - 1 directions on the upper side of the plate. Each cylinder has a small cross section of size e and fixed height (for the case N=3 , see the figure). We identify the limit problem, under the assumption: . After rescaling the equation, with respect to hE , on the plate, we prove that, in the limit domain corresponding to the “forest" of cylinders, the limit problem identifies with a diffusion operator with respect to x N , coupled with an algebraic system. Moreover, the limit solution is independent of x N in the rescaled plate and meets a Dirichlet transmission condition between the limit domain of the “forest" of cylinders and the upper boundary of the plate.
TL;DR: In this article, the authors consider the Laplace equation in a smooth bounded domain and prove logarithmic estimates of solutions on a part of the boundary or of the domain without known boundary conditions.
Abstract: We consider the Laplace equation in a smooth bounded domain. We prove logarithmic estimates, in the sense of John [5] of solutions on a part of the boundary or of the domain without known boundary conditions. These results are established by employing Carleman estimates and techniques that we borrow from the works of Robbiano [8,11]. Also, we establish an estimate on the cost of an approximate control for an elliptic model equation.
TL;DR: The aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold and states that complexity and entropy are equivalent for generic paths.
Abstract: We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset A of a metric space is the minimum number of balls of a given radius ? needed to cover A. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-Riemannian manifold as the infimum of the lengths of all trajectories contained in an ?-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold. We construct first a norm || * ||? on the tangent space that depends on a parameter ? > 0. Our main result states then that the entropy of a path is equivalent to the integral of this ?-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen. © EDP Sciences, SMAI 2003.
TL;DR: In this paper, the diusion limit for general conservative Boltzmann equations with Oscillations has been investigated, where the coecients may depend on both slow and fast variables.
Abstract: We investigate the diusion limit for general conservative Boltzmann equations with os- cillating coecients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coecients may depend on both slow and fast variables. Passing to the limit, we are led to an eective drift-diusion equation. We also describe the diusive behaviour when the equilibrium function has a non-vanishing flux. Mathematics Subject Classication. 35Q35, 82C70, 76P05, 74Q99, 35B27.
TL;DR: In this article, a method for motion planning and boundary control for a class of linear PDEs with constant coefficients is presented, where transitions from rest to rest can be achieved in a prescribed finite time.
Abstract: Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived employing Mikusinski's operational calculus. The method is illustrated through an application to a model of a Timoshenko beam, which is clamped on a rotating disk and carries a load at its free end.
TL;DR: In this paper, the authors consider the stabilization of a set of equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver-Muller boundary condition.
Abstract: We consider the stabilization of Maxwell's equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver–Muller boundary condition. This is based on some stability estimates that are obtained using the “standard" identity with multiplier and appropriate properties of the feedback. We deduce an explicit decay rate of the energy, for instance exponential, polynomial or logarithmic decays are available for appropriate feedbacks.
TL;DR: In this article, the authors consider general nonlinear systems with observations, containing a (single) unknown function φ, and study the possibility to learn about this unknown function via the observations: if it is possible to determine the values of the unknown function from any experiment [on the set of states visited during the experiment], and for any arbitrary input function, on any time interval, the system is "identifiable".
Abstract: In this paper, we consider general nonlinear systems with observations, containing a (single) unknown function φ . We study the possibility to learn about this unknown function via the observations: if it is possible to determine the [values of the] unknown function from any experiment [on the set of states visited during the experiment], and for any arbitrary input function, on any time interval, we say that the system is “identifiable”. For systems without controls, we give a more or less complete picture of what happens for this identifiability property. This picture is very similar to the picture of the “observation theory” in [7]: Contrarily to the case of the observability property, in order to identify in practice, there is in general no hope to do something better than using “approximate differentiators”, as show very elementary examples. However, a practical methodology is proposed in some cases. It shows very reasonable performances. As an illustration of what may happen in controlled cases, we consider the equations of a biological reactor, [2,4], in which a population is fed by some substrate. The model heavily depends on a “growth function”, expressing the way the population grows in presence of the substrate. The problem is to identify this “growth function”. We give several identifiability results, and identification methods, adapted to this problem.
TL;DR: It is shown that the generalized smoothing spline obtained by solving an optimal control problem for a linear control system converges to a deterministic curve even when the data points are perturbed by random noise.
Abstract: In this paper it is shown that the generalized smoothing spline obtained by solving an optimal control problem for a linear control system converges to a deterministic curve even when the data points are perturbed by random noise. We furthermore show that such a spline acts as a filter for white noise. Examples are constructed that support the practical usefulness of the method as well as gives some hints as to the speed of convergence.
TL;DR: In this paper, asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs are constructed on an arbitrary reflexive Banach space, where the existence of persistent inputs which make the system observable is discussed.
Abstract: On an arbitrary reflexive Banach space, we build asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs. An important part of this paper is devoted to various examples, where we discuss the existence of persistent inputs which make the system observable. These results make a wide generalization to a nonlinear framework of previous works on the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).
TL;DR: In this article, the authors prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient.
Abstract: We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is
TL;DR: In this article, the authors studied the sequence u n, which is solution of in Ω an open bounded set of R N and un = 0 on ∂Ω, when f n tends to a measure concentrated on a set of null Orlicz-capacity.
Abstract: We study the sequence u n , which is solution of in Ω an open bounded set of R N and un = 0 on ∂Ω, when f n tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the N -function Φ , and prove a non-existence result.
TL;DR: Lower semicontinuity results are obtained for multiple integrals of the kind, where μ is a given positive measure on, and the vector-valued function u belongs to the Sobolev space associated with μ as mentioned in this paper.
Abstract: Lower semicontinuity results are obtained for multiple integrals of the kind , where μ is a given positive measure on , and the vector-valued function u belongs to the Sobolev space associated with μ . The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ . More precisely, for fully general μ , a notion of quasiconvexity for f along the tangent bundle to μ , turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when μ belongs to a suitable class of rectifiable measures.
TL;DR: For systems with slowly varying parameters, the controllability behavior is studied and the relation to the control sets for the systems with frozen parameters is clarified in this article, where the relation between the control set and the control behavior is discussed.
Abstract: For systems with slowly varying parameters the controllability behavior is studied and the relation to the control sets for the systems with frozen parameters is clarified.
TL;DR: In this article, the authors considered parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes.
Abstract: The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period " of the holes goes to zero. Since standard conservation laws do not hold in this model, a rst diculty is to get ap rioriestimates of the solutions. We obtain them in a weighted space where the weight is the principal eigenfunction of an \adjoint" periodic time-dependent eigenvalue problem. This problem is not a classical one, and its investigation is an important part of this work. Then, by using the multiple scale method, we construct the leading terms of a formal expansion (with respect to ") of the solution and give the limit \homogenized" problem. An interesting peculiarity of the model is that, depending on the geometry of the holes, a large convection term may appear in the limit equation. Mathematics Subject Classication. 35B27, 74Q10, 76M50.
TL;DR: In this paper, the authors studied the lower semicontinuity problem for a supremal functional of the form F(u, Ω) = ess sup f(x, u(x),Du(x)) with respect to the strong convergence in L ∞ (Ω), furnishing a comparison with the analogoustheory developed by Serrin for integrals.
Abstract: In this paper we study the lower semicontinuity problem for a supremal functional of the form F(u,Ω) = ess sup f(x, u(x),Du(x)) with respect to the strong convergence in L ∞ (Ω), furnishing a comparison with the analogoustheory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.
TL;DR: In this article, the authors examined a stored free energy model for such crystals involving a (higher order) Landau/de Gennes type parameter" term and provided a proof for the existence of a minimizer.
Abstract: Controlling growth at crystalline surfaces requires a detailed and quantitative understand- ing of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal (\step-terrace") type surface (for a recent review see (4)). In the case of orthodox crystals one nds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical mechanical view of the meanderings as arising from atomic interchanges). The classical theory due to Herring, Mullins and others (5), coupled with advances in real-time experimental microscopy tech- niques, has proven very successful in the applied development of such crystalline materials. However in 1997 a series of experimental observations on vicinal defects of heavily boron-doped Silicon crystals revealed that these crystals were quite unorthodox in the sense that a lowering of the equilibrium temperature led to increased amplitude for the isothermal wanderings of a step edge (3). In addition, at low temperatures the step prole adopted a periodic saw-tooth structure rather than the straight prole predicted by the classical theories. This article examines a stored free energy model for such crystals involving a (higher order) Landau/de Gennes type \order parameter" term and provides a proof for the existence of a minimizer. Mathematics Subject Classication. 49J45, 49S05.
TL;DR: In this paper, the viscosity solution of a related Hamilton-Jacobi equation was shown to provide a minimizer for the integral functional in the form where the Borel function is assumed to be neither convex nor coercive.
Abstract: We consider minimization problems of the form where is a bounded convex open set, and the Borel function is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Ω and the zero level set of f , we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.
TL;DR: In this article, the authors studied the long-time effect of a nonlinear damping term, with special attention to the model case with α real, A>0, and characterized the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.
Abstract: We study the large-time behaviour of the nonlinear oscillator where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case with α real, A>0 . We characterize the existence and behaviour of fast orbits, i.e. , orbits that stop in finite time.
TL;DR: In this article, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations is minimized with respect to shape and the necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.
Abstract: We minimize, with respect to shape, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.