Displaying similar documents to “Viscosity solutions methods for converse KAM theory”

Optimal control problems with upper semicontinuous Hamiltonians

Arkadiusz Misztela (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we give examples of value functions in Bolza problem that are not bilateral or viscosity solutions and an example of a smooth value function that is even not a classic solution (in particular, it can be neither the viscosity nor the bilateral solution) of Hamilton-Jacobi-Bellman equation with upper semicontinuous Hamiltonian. Good properties of value functions motivate us to introduce approximate solutions of equations with such type Hamiltonians. We show that the value...

Invariance of global solutions of the Hamilton-Jacobi equation

Ezequiel Maderna (2002)

Bulletin de la Société Mathématique de France

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We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.

Homogenization of Hamilton-Jacobi equations in Carnot Groups

Bianca Stroffolini (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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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.

A Hamilton-Jacobi approach to junction problems and application to traffic flows

Cyril Imbert, Régis Monneau, Hasnaa Zidani (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to...

Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations

Alessandra Cutrì, Francesca Da Lio (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form u t + H ( x , D u ) = 0 in I R n × ( 0 , T ) where the Hamiltonian may be noncoercive in the gradient As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

Convergence rates of symplectic Pontryagin approximations in optimal control theory

Mattias Sandberg, Anders Szepessy (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

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Many inverse problems for differential equations can be formulated as optimal control problems. It is well known that inverse problems often need to be regularized to obtain good approximations. This work presents a systematic method to regularize and to establish error estimates for approximations to some control problems in high dimension, based on symplectic approximation of the Hamiltonian system for the control problem. In particular the work derives error estimates and constructs...

Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls

Hong Qiu, Jiongmin Yong (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of...