Lax-Oleinik Semigroup Research Articles

Homogenization for sub-riemannian Lagrangians

We consider Lagrangians that are defined only on the horizontal distribution of a sub-riemannian manifold. The associated Hamiltonian is neither strictly convex nor coercive. We prove a result on homogenization of the Hamilton–Jacobi equation following the approach in Contreras et al (2015 Calc. Var. Partial Differ. Equ. 52 237–52). To establish the result, we need to extend weak KAM and Aubry–Mather theories to the sub-riemannian setting. We obtain a Tonelli theorem to dispend of the Lagrangian dynamics tools used in standard weak KAM theory. In the way towards our main result we observe that, the long time convergence of the Lax-Oleinik semigroup, and the vanishing discount convergence of the discounted value function hold.

On the $$C^1$$ and $$C^2$$-Convergence to Weak K.A.M. Solutions

We introduce a notion of upper Green regular solutions to the Lax-Oleinik semi-group that is defined on the set of \(C^0\) functions of a closed manifold via a Tonelli Lagrangian. Then we prove some weak \(C^2\) convergence results to such a solution for a large class of approximated solutions as (1) the discounted solution (see [DFIZ16]); (2) the image of a \(C^0\) function by the Lax-Oleinik semi-group; (3) the weak K.A.M. solutions for perturbed cohomology class. This kind of convergence implies the convergence in measure of the second derivatives. Moreover, we provide an example that is not upper Green regular and to which we have \(C^1\) convergence but not convergence in measure of the second derivatives.

Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem

We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_xu)=h$. We prove that these solutions are fixed points of the associated Lax-Oleinik semigroup. The presented results can also be viewed as a new application of Marchal's theorem, whose main use in recent literature has been to prove the existence of periodic orbits.

Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation

We study the Cauchy problem for the first-order evolutionary Hamilton–Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax–Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.

The Lax-Oleinik semigroup on graphs

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. It follows that weak KAM solutions are viscosity solutions of the Hamilton-Jacobi equation [ 3 , 4 ], and in the case of Hamiltonians called of eikonal type in [ 3 ], we prove that the converse holds.

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.

A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on Rn

This paper is devoted to the study of the convergence of the Lax–Oleinik semigroup associated with reversible Hamiltonians H(x,p) on Rn. We provide a necessary and sufficient condition for the convergence of the semigroup. We also give an example to show that for irreversible Hamiltonians on Rn, even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax–Oleinik semigroup may not converge.

An infinite-dimensional weak KAM theory via random variables

We develop several aspects of the infinite-dimensional Weak KAM theory using a random variables' approach. We prove that the infinite-dimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup. Furthermore, we show the existence of invariant minimizing measures and calibrated curves defined on $\mathbb{R}$.

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup.

Fast weak–KAM integrators for separable Hamiltonian systems

We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. 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 {\em geometric integrator} satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus 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.

弱KAM理论和Hamilton-Jacobi方程

Mather theory studies the existence of various kinds of the action minimizing sets and the connecting orbits between these sets in higher-dimensional positive Lagrangian system. The work about the connecting orbits plays an important role in the study of Arnold diffusion. By studying the dynamical behavior of the action-minimizing curves for Tonelli Lagrangian systems, weak KAM theory founded by A. Fathi bridges Mather theory and the PDE methods concerning the associated Hamilton-Jacobi equation. However, because the convergence of the Lax-Oleinik semigroup which is critical in KAM theory does not hold in time-periodic Lagrangian system, the preliminary work about weak KAM theory focused on autonomous system. By introducing a new kind of Lax-Oleinik type operator, it is possible for us to build weak KAM theory in the time-periodic Lagrangian system and generalize the theory to more general Hamilton-Jacobi equation. In this paper, we introduce the basic knowledge of weak KAM theory and its latest development.

The convergence of Lax-Oleinik semigroup for time-periodic Lagrangian

In this article, the convergence of so-called Lax-Oleinik semigroup is studied for time-periodic Lagrangian systems when the degree of freedom is greater than 2. Under certain conditions, we show that the Lax-Oleinik semigroup converges if the rotation vector is completely irrational. Removing such conditions, we will give another kind of convergence of the sequence Fc((x, s), (x′, s′ +Tn)), the convergence of which is closely related to the Lax-Oleinik semigroup.

Translation invariance of weak KAM solutions of the Newtonian $N$-body problem

We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c \geq 0, of the classical N-body problem in an Euclidean space E of dimension k \geq 2. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant (c = 0) also called weak KAM solutions. We show that all these solutions are invariant under the action of E by translations on the space of configurations. We deduce the existence of non-invariant solutions for the super-critical equations (c > 0).

The rate of convergence of new Lax–Oleinik type operators for time-periodic positive definite Lagrangian systems

Assume that the Aubry set of the time-periodic positive definite Lagrangian L consists of one hyperbolic 1-periodic orbit. We provide an upper bound estimate of the rate of convergence of the family of new Lax–Oleinik type operators associated with L introduced by the authors in Wang and Yan (2012 Commun. Math. Phys. 309 663–91). In addition, we construct an example where the Aubry set of a time-independent positive definite Lagrangian system consists of one hyperbolic periodic orbit and the rate of convergence of the Lax–Oleinik semigroup cannot be better than O(1/t) as t → +∞.

A New Kind of Lax-Oleinik Type Operator with Parameters for Time-Periodic Positive Definite Lagrangian Systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.

On weak KAM theory for N-body problems

AbstractWe consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

The rate of convergence of the Lax-Oleinik semigroup-degenerate fixed point case

For the standard Lagrangian in classical mechanics, which is defined as the kinetic energy of the system minus its potential energy, we study the rate of convergence of the corresponding Lax-Oleinik semigroup. Under the assumption that the unique global minimum point of the Lagrangian is a degenerate fixed point, we provide an upper bound estimate of the rate of convergence of the semigroup.

Hyperbolicity and exponential convergence of the Lax–Oleinik semigroup

For a convex superlinear Lagrangian L : T M → R on a compact manifold M it is known that there is a unique number c such that the Lax–Oleinik semigroup L t + c t : C ( M , R ) → C ( M , R ) has a fixed point. Moreover for any u ∈ C ( M , R ) the uniform limit u ˜ = lim t → ∞ L t u + c t exists. In this paper we assume that the Aubry set consists in a finite number of periodic orbits or critical points and study the relation of the hyperbolicity of the Aubry set to the exponential rate of convergence of the Lax–Oleinik semigroup.

Weak kam theorem on non compact manifolds

In this paper, we consider a time independent C2 Hamiltonian, satisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.

The Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems: Basic Properties and Convergence Analysis

We introduce a max-plus analogue of the Petrov–Galerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation. We show that the error in the sup-norm can be bounded from the difference between the value function and its projections on max-plus and min-plus semimodules when the max-plus analogue of the stiffness matrix is exactly known. In general, the stiffness matrix must be approximated: this requires approximating the operation of the Lax–Oleinik semigroup on finite elements. We consider two approximations relying on the Hamiltonian. We derive a convergence result, in arbitrary dimension, showing that for a class of problems, the error estimate is of order $\delta+\Delta x(\delta)^{-1}$ or $\sqrt{\delta}+\Delta x(\delta)^{-1}$, depending on the choice of the approximation, where $\delta$ and $\Delta x$ are, respectively, the time and space discretization steps. We compare our method with another max-plus based discretization method previously introduced by Fleming and McEneaney. We give numerical examples in dimensions 1 and 2.