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Transformation of optimal control problems of descriptor systems into problems with state-space systems

Jovan Stefanovski (2012)

Kybernetika

We show how we can transform the ℋ ∞ and ℋ 2 control problems of descriptor systems with invariant zeros on the extended imaginary into problems with state-space systems without such zeros. Then we present necessary and sufficient conditions for existence of solutions of the original problems. Numerical algorithm for ℋ ∞ control is given, based on the Nevanlinna-Pick theorem. Also, we present an explicit formula for the optimal ℋ 2 controller.

Transport optimal et courbure de Ricci

Cédric Villani (2005/2006)

Séminaire Équations aux dérivées partielles

Des liens inattendus ont été récemment mis à jour entre le transport optimal de Monge–Kantorovich et certains problèmes de géométrie riemannienne, en liaison avec la courbure de Ricci. Une des retombées de ces interactions est la naissance d’une théorie “synthétique” des espaces métriques mesurés à courbure de Ricci minorée, venant compléter la théorie classique des espaces métriqes à courbure sectionnelle minorée. Dans ce texte (également fourni aux actes du Séminaire de Théorie Spectrale et Géométrie...

Transport problems and disintegration maps

Luca Granieri, Francesco Maddalena (2013)

ESAIM: Control, Optimisation and Calculus of Variations

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a...

Transportation flow problems with Radon measure variables

Marcus Wagner (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

For a multidimensional control problem ( P ) K involving controls u ∈ L ∞ , we construct a dual problem ( D ) K in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of ( L ∞ ) * . For this purpose, we add to ( P ) K a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

Turnpike theorems by a value function approach

Alain Rapaport, Pierre Cartigny (2004)

ESAIM: Control, Optimisation and Calculus of Variations

Turnpike theorems deal with the optimality of trajectories reaching a singular solution, in calculus of variations or optimal control problems. For scalar calculus of variations problems in infinite horizon, linear with respect to the derivative, we use the theory of viscosity solutions of Hamilton-Jacobi equations to obtain a unique characterization of the value function. With this approach, we extend for the scalar case the classical result based on Green theorem, when there is uniqueness of the...

Turnpike theorems by a value function approach

Alain Rapaport, Pierre Cartigny (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Turnpike theorems deal with the optimality of trajectories reaching a singular solution, in calculus of variations or optimal control problems. For scalar calculus of variations problems in infinite horizon, linear with respect to the derivative, we use the theory of viscosity solutions of Hamilton-Jacobi equations to obtain a unique characterization of the value function. With this approach, we extend for the scalar case the classical result based on Green theorem, when there is uniqueness of...

Two dimensional optimal transportation problem for a distance cost with a convex constraint

Ping Chen, Feida Jiang, Xiaoping Yang (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We first prove existence and uniqueness of optimal transportation maps for the Monge’s problem associated to a cost function with a strictly convex constraint in the Euclidean plane ℝ2. The cost function coincides with the Euclidean distance if the displacement y − x belongs to a given strictly convex set, and it is infinite otherwise. Secondly, we give a sufficient condition for existence and uniqueness of optimal transportation maps for the original Monge’s problem in ℝ2. Finally, we get existence...

Two Hartree-Fock models for the vacuum polarization

Philippe Gravejat, Christian Hainzl, Mathieu Lewin, Éric Séré (2012)

Journées Équations aux dérivées partielles

We review recent results about the derivation and the analysis of two Hartree-Fock-type models for the polarization of vacuum. We pay particular attention to the variational construction of a self-consistent polarized vacuum, and to the physical agreement between our non-perturbative construction and the perturbative description provided by Quantum Electrodynamics.

Two Numerical Methods for the elliptic Monge-Ampère equation

Jean-David Benamou, Brittany D. Froese, Adam M. Oberman (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The numerical solution of the elliptic Monge-Ampère Partial Differential Equation has been a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures (2009) 155–192; Oliker and Prussner, Numer. Math.54 (1988) 271–293; Oberman, Discrete Contin. Dyn. Syst. Ser. B10 (2008) 221–238; Dean and Glowinski, in Partial differential equations, Comput. Methods Appl. Sci. 16 (2008) 43–63; Glowinski et al., Japan...

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