Showing papers in "ESAIM: Control, Optimisation and Calculus of Variations in 2012"
TL;DR: In this paper, a survey of local and global Carleman estimates for elliptic and parabolic partial differential equations is presented, and the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function is pointed out.
Abstract: Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.
229 citations
TL;DR: In this paper, the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent disipation potential is considered.
Abstract: In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of “BV solutions” involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play crucial roles in the description of the associated jump trajectories. We shall prove general convergence results for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.
133 citations
TL;DR: In this paper, the continuity and differentiability of the forward operator with respect to the con- ductivity parameter in Lp-norms are proved, and the analytical results are applied to several popular regularization formulations, which incorporate apriori information of smoothness/sparsity on the inho- mogeneity through Tikhonov regularization, for both linearized and nonlinear models.
Abstract: This paper analyzes the continuum model/complete electrode model in the electrical impedance tomography inverse problem of determining the conductivity parameter from boundary measurements. The continuity and differentiability of the forward operator with respect to the con- ductivity parameter in Lp-norms are proved. These analytical results are applied to several popular regularization formulations, which incorporate ap rioriinformation of smoothness/sparsity on the inho- mogeneity through Tikhonov regularization, for both linearized and nonlinear models. Some important properties, e.g., existence, stability, consistency and convergence rates, are established. This provides some theoretical justifications of their practical usage.
119 citations
TL;DR: In this paper, the authors considered a two-player zero-sum game in a bounded open domain, where players I and II play an e-step tug-of-war game with probability α, and with probability β, respectively.
Abstract: We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an e -step tug-of-war game with probability α , and with probability β (α + β = 1), a random point in the ball of radius e centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F . We give a detailed proof of the fact that the value functions of this game satisfy the Dynamic Programming Principle \begin{equation*} u(x) = \frac{\alpha}{2} \left\{ \sup_{y\in \ol B_{\eps}(x)} u (y) + \inf_{ y \in \ol B_{\eps}(x)} u (y) \right\} + \beta \kint_{ B_{\eps}(x)} u(y) \ud y, \end{equation*} u ( x ) = α 2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ sup y ∈ B e ( x ) u ( y ) + inf y ∈ B e ( x ) u ( y ) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + β ∫ B e ( x ) u ( y ) d y, for x ∈ Ω with u (y ) = F (y ) when y ∉ Ω. This principle implies the existence of quasioptimal Markovian strategies.
73 citations
TL;DR: Alabau et al. as mentioned in this paper showed that the energy of smooth solutions of two wave-like equations decays polynomially at infinity, whereas it is known that exponential stability does not hold.
Abstract: We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not hold (see (F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127-150)). We give applications of our result to locally or boundary damped wave or plate systems. In any space dimension, we prove polynomial stability under geometric conditions on both the coupling and the damping regions. In one space dimension, the result holds for arbitrary non-empty open damping and coupling regions, and in particular when these two regions have an empty intersection. Hence, indirect polynomial stability holds even though the feedback is active in a region in which the coupling vanishes and vice versa.
64 citations
TL;DR: In this article, an estimator/predictor based state feedback law is proposed to stabilize a one-dimensional wave equation system for which the boundary observation suffers from an arbitrary long time delay.
Abstract: The stabilization with time delay in observation or control represents difficult mathemat- ical challenges in the control of distributed parameter systems. It is well-known that the stability of closed-loop system achieved by some stabilizing output feedback laws may be destroyed by what- ever small time delay there exists in observation. In this paper, we are concerned with a particularly interesting case: Boundary output feedback stabilization of a one-dimensional wave equation system for which the boundary observation suffers from an arbitrary long time delay. We use the observer and predictor to solve the problem: The state is estimated in the time span where the observation is available; and the state is predicted in the time interval where the observation is not available. It is shown that the estimator/predictor based state feedback law stabilizes the delay system asymptoti- cally or exponentially, respectively, relying on the initial data being non-smooth or smooth. Numerical simulations are presented to illustrate the effect of the stabilizing controller.
59 citations
TL;DR: In this article, the authors studied the limit as p ǫ→ ∞ of minimizers of the fractional W s,p -norms and proved that the limit satisfies a non-local and non-linear equation.
Abstract: In this paper we study the limit as p → ∞ of minimizers of the fractional W s,p -norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.
59 citations
TL;DR: In this paper, the shape optimization of mechanical devices is investigated in the context of large, geometrically strongly nonlinear deformations and nonlinear hyperelastic constitutive laws, and a weighted sum of the structure compliance, its weight, and its surface area are minimized.
Abstract: Shape optimization of mechanical devices is investigated in the context of large, geomet- rically strongly nonlinear deformations and nonlinear hyperelastic constitutive laws. A weighted sum of the structure compliance, its weight, and its surface area are minimized. The resulting nonlinear elastic optimization problem differs significantly from classical shape optimization in linearized elas- ticity. Indeed, there exist different definitions for the compliance: the change in potential energy of the surface load, the stored elastic deformation energy, and the dissipation associated with the defor- mation. Furthermore, elastically optimal deformations are no longer unique so that one has to choose the minimizing elastic deformation for which the cost functional should be minimized, and this com- plicates the mathematical analysis. Additionally, along with the non-uniqueness, buckling instabilities can appear, and the compliance functional may jump as the global equilibrium deformation switches between different bluckling modes. This is associated with a possible non-existence of optimal shapes in a worst-case scenario. In this paper the sharp-interface description of shapes is relaxed via an Allen-Cahn or Modica-Mortola type phase-field model, and soft material instead of void is considered outside the actual elastic object. An existence result for optimal shapes in the phase field as well as in the sharp-interface model is established, and the model behavior for decreasing phase-field interface width is investigated in terms of Γ-convergence. Computational results are based on a nested opti- mization with a trust-region method as the inner minimization for the equilibrium deformation and a quasi-Newton method as the outer minimization of the actual objective functional. Furthermore, a multi-scale relaxation approach with respect to the spatial resolution and the phase-field parameter is applied. Various computational studies underline the theoretical observations.
58 citations
TL;DR: In this paper, the authors considered the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points and obtained an asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of the points goes to infinity.
Abstract: We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.
46 citations
TL;DR: In this article, sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived, and these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first-order optimality system associated with the regularized problems.
Abstract: In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems as well. It is further shown that these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first order optimality system associated with the regularized problems.
43 citations
TL;DR: In this paper, an equilibrium problem with equilibrium constraints arising from the modeling of competition in an electricity spot market (under ISO regulation) is considered and M-stationarity conditions are derived.
Abstract: We consider an equilibrium problem with equilibrium constraints (EPEC) arising from the modeling of competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called M-stationarity conditions are derived. This first requires a structural analysis of the problem, e.g., verifying constraint qualifications. Second, the calmness property of a certain multifunction has to be verified in order to justify using M-stationarity conditions. Third, for stating the stationarity conditions, the coderivative of a normal cone mapping has to be calculated. Finally, the obtained necessary conditions are made fully explicit in terms of the problem data for one typical constellation. A simple two-settlement example serves as an illustration.
TL;DR: A novel algorithm is presented that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort.
Abstract: We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approx- imate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution - a new paradigm in adaptivity.
TL;DR: In this paper, the optimal control problem in which the controlled system is described by a fully coupled anticipated forward-backward stochastic differential delayed equation was studied and the maximum principle for this problem was obtained under the assumption that the diffusion coefficient does not contain the control variables.
Abstract: This paper deals with the optimal control problem in which the controlled system is described by a fully coupled anticipated forward-backward stochastic differential delayed equation. The maximum principle for this problem is obtained under the assumption that the diffusion coefficient does not contain the control variables and the control domain is not necessarily convex. Both the necessary and sufficient conditions of optimality are proved. As illustrating examples, two kinds of linear quadratic control problems are discussed and both optimal controls are derived explicitly.
TL;DR: In this article, the authors study a one dimensional model of ferromagnetic nano-wires of finite length and prove the existence of wall profiles, and stabilize them by the mean of an applied magnetic field.
Abstract: In this paper we study a one dimensional model of ferromagnetic nano-wires of finite length. First we justify the model by Γ-convergence arguments. Furthermore we prove the existence of wall profiles. These walls being unstable, we stabilize them by the mean of an applied magnetic field.
TL;DR: In this article, the authors prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure time for any given population size κ 1,...,κ n, where each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost.
Abstract: Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i -th group consists of κ i drivers, sharing the same departure and arrival costs ϕ i (t ),ψ i (t ). For any given population sizes κ 1 ,... ,κ n , we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure time. The possible non-uniqueness, and a characterization of this Nash equilibrium solution, are also discussed.
TL;DR: In this article, the authors considered quasilinear optimal control problems involving a thick two-level junction and a large number of thin cylinders with the cross-section of order O(e 2 ).
Abstract: We consider quasilinear optimal control problems involving a thick two-level junction Ωe which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order O(e 2 ). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary conditions depend on parameters e, α, β and the thin cylinders from each level are e-periodically alternated. Using the Buttazzo-Dal Maso abstract scheme for varia- tional convergence of constrained minimization problems, the asymptotic analysis (as e → 0) of these problems are made for different values of α and β and different kinds of controls. We have showed that there are three qualitatively different cases. Application for an optimal control problem involving a thick one-level junction with cascade controls is presented as well.
TL;DR: In this article, the authors consider the back-and-forward NN for 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers' equation), and prove that the convergence rate is always exponential in time.
Abstract: In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers' equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers' equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered.
TL;DR: In this paper, the authors studied a stochastic recursive optimal control problem for the systems described by SDDEs with delay, where not only the dynamics of the system but also the recursive utility depend on the past path segment of the state process in a general form.
Abstract: In this paper, we study one kind of stochastic recursive optimal control problem for the systems described by stochastic differential equations with delay (SDDE). In our framework, not only the dynamics of the systems but also the recursive utility depend on the past path segment of the state process in a general form. We give the dynamic programming principle for this kind of optimal control problems and show that the value function is the viscosity solution of the corresponding infinite dimensional Hamilton-Jacobi-Bellman partial differential equation.
TL;DR: In this paper, the authors considered linear Hamiltonian differential systems without the controllability (or normality) assumption and proved the Rayleigh principle for these systems with Dirichlet boundary conditions, which provided a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem.
Abstract: In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools are the extended Picone formula, which is proven here for this general setting, results on piecewise constant kernels for conjoined bases of the Hamiltonian system, and the oscillation theorem relating the number of proper focal points of conjoined bases with the number of finite eigenvalues. As applications we obtain the expansion theorem in the space of admissible functions without controllability and a result on coercivity of the corresponding quadratic functional.
TL;DR: In this article, an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on Hadamard manifolds was proposed, and the convergence of the iterations given by the method was shown to converge to a generalized critical point.
Abstract: In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on Hadamard manifolds. To reach this goal, we initially extend the concepts of regular and generalized subgradient from Euclidean spaces to Hadamard manifolds and prove that, in the convex case, these concepts coincide with the classical one. For the minimization problem, assuming that the function is bounded from below, in the quasiconvex and lower semicontinuous case, we prove the convergence of the iterations given by the method. Furthermore, under the assumptions that the sequence of proximal parameters is bounded and the function is continuous, we obtain the convergence to a generalized critical point. In particular, our work extends the applications of the proximal point methods for solving constrained minimization problems with nonconvex objective functions in Euclidean spaces when the objective function is convex or quasiconvex on the manifold.
TL;DR: In this paper, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpinski gasket is proved under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term.
Abstract: Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpinski gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpinski fractal. The abstract results are illustrated by explicit examples. Mathematics Subject Classification. 35J20, 28A80, 35J25, 35J60, 47J30, 49J52.
TL;DR: In this paper, a continuous stationary or discontinuous feedback strategy is proposed to ensure the asymptotic stability even in finite time for some variables, while other variables do not converge, and not necessarily toward equilibrium.
Abstract: We consider chained systems that model various systems of mechanical or biological origin. It is known according to Brockett that this class of systems, which are controllable, is not stabilizable by continuous stationary feedback (i.e. independent of time). Various approaches have been proposed to remedy this problem, especially instationary or discontinuous feedbacks. Here, we look at another stabilization strategy (by continuous stationary or discontinuous feedbacks) to ensure the asymptotic stability even in finite time for some variables, while other variables do converge, and not necessarily toward equilibrium. Furthermore, we build feedbacks that permit to vanish the two first components of the Brockett integrator in finite time, while ensuring the convergence of the last one. The considering feedbacks are continuous and discontinuous and regular outside zero. Mathematics Subject Classification. 93D15, 93C10, 93D09.
TL;DR: In this paper, the flatness of two-input driftless control systems is studied and the problems of describing all flat outputs and of calculating them are open and solved in the paper.
Abstract: We study the problem of flatness of two-input driftless control systems. Although a characterization of flat systems of that class is known, the problems of describing all flat outputs and of calculating them is open and we solve it in the paper. We show that all x-flat outputs are parameterized by an arbitrary function of three canonically defined variables. We also construct a system of 1st order PDE's whose solutions give all x-flat outputs of two-input driftless systems. We illustrate our results by describing all x-flat outputs of models of a nonholonomic car and the n-trailer system.
TL;DR: In this paper, the multiplicity of solutions for a class of non-cooperative p-Laplacian operator elliptic systems is studied and a sequence of solutions is obtained by using the limit index theory.
Abstract: In this paper, we study the multiplicity of solutions for a class of noncooperative p-Laplacian operator elliptic system. Under suitable assumptions, we obtain a sequence of solutions by using the limit index theory.
TL;DR: In this article, the role of concavity inequalities in shape optimization is discussed, and several counterexamples to the Blaschke-concavity of variational functionals, including capacity.
Abstract: Motivated by a long-standing conjecture of Polya and Szego about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application to isoperimetriclike inequalities. As a byproduct of this approach we also obtain a quantitative version of the Kneser-Suss inequality. Finally, for a large class of functionals involving Dirichlet energies and the surface measure, we perform a local analysis of strictly convex portions of the boundary via second order shape derivatives. This allows in particular to exclude the presence of smooth regions with positive Gauss curvature in an optimal shape for Polya-Szego problem.
TL;DR: In this article, two different Boltzmann integrals that represent the memory of materials are considered and the spectral properties for both cases are thoroughly analyzed, and it is shown that the spectrum of system determines completely the dynamic behavior of the vibration, which forms a Riesz basis for the state space.
Abstract: In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the latter case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded. The results of this paper expositorily demonstrate the proper modeling the elastic systems with Boltzmann damping.
TL;DR: In this article, the authors investigated analytic affine control systems where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type.
Abstract: In this paper we investigate analytic affine control systems = X + uY , u ∈ [ a,b ] , where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type. We construct normal forms for such systems and, among other things, we study the connection between the presence of the singular trajectory starting at q 0 on the boundary of the reachable set from q 0 with the minimal number of analytic functions needed for describing the reachable set from q 0 .
TL;DR: In this article, second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type.
Abstract: Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth of the so-called “violation function”. Together with corresponding necessary conditions they constitute a no-gap pair of conditions.
TL;DR: In this paper, an evolution equation similar to that introduced by Vese in [ Comm. Partial Diff. Eq. 1573-1591] and whose solution converges in large time to the convex envelope of the initial datum was considered.
Abstract: We consider an evolution equation similar to that introduced by Vese in [ Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
TL;DR: In this article, a linearization approach to the L ∞ -control problems is proposed, where the set of constraints appearing in the linearized formulation of (standard) control problems is shown to be semigroup-type.
Abstract: The aim of the paper is to provide a linearization approach to the L ∞ -control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the L p approach and the associated linear formula- tions. This seems to be the most appropriate tool for treating L ∞ problems in continuous and lower semicontinuous setting.