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SUBSOLUTIONS OF TIMEPERIODIC HAMILTONJACOBI EQUATIONS
, 2006
"... Abstract. We prove the existence of C 1 critical subsolutions of the HamiltonJacobi equation for a timeperiodic Hamiltonian system. We draw a consequence for the Minimal Action functional of the system. 1. ..."
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Abstract. We prove the existence of C 1 critical subsolutions of the HamiltonJacobi equation for a timeperiodic Hamiltonian system. We draw a consequence for the Minimal Action functional of the system. 1.
Some new links between the weak KAM and Monge problems
, 903
"... The weak KAM theory predicts the survivals of invariant measures of Hamiltonian systems under large perturbations. It is the subject of an extensive research in the last few decades. The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number ..."
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Cited by 3 (0 self)
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The weak KAM theory predicts the survivals of invariant measures of Hamiltonian systems under large perturbations. It is the subject of an extensive research in the last few decades. The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Recently, some interesting links where discovered between these two fields. Here we investigate a new, surprising link involving the metric Monge distance. As a special case we get for any pair of nonegative measures λ +, λ − of equal mass a generalization of the identity W1(λ − , λ +) = lim ε→0 ε −2 inf µ W2(µ + ελ − , µ + ελ +) where Wp is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.
CONTINUITY OF OPTIMAL CONTROL COSTS AND ITS APPLICATION TO WEAK KAM THEORY
, 909
"... Abstract. We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the AubryMather problems corresponding to these systems. 1. ..."
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Abstract. We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the AubryMather problems corresponding to these systems. 1.
FAST WEAKKAM INTEGRATORS by
, 2012
"... Abstract. — We consider a numerical scheme for HamiltonJacobi equations based on a direct discretization of the LaxOleinik semigroup. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we p ..."
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Abstract. — We consider a numerical scheme for HamiltonJacobi equations based on a direct discretization of the LaxOleinik semigroup. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weakKAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing minplus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way. 1.
Regularization of subsolutions in discrete weak KAM theory
, 2012
"... We expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory. They allow to prove the existence and the density of C 1,1 subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set. ..."
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We expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory. They allow to prove the existence and the density of C 1,1 subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.