Convex Hamiltonians Research Articles

Computing the Effective Hamiltonian Using a Variational Approach

A numerical method for homogenization of Hamilton--Jacobi equations is presented and implemented as an $L^\infty$ calculus of variations problem. Solutions are found by solving a nonlinear convex optimization problem. The numerical method is shown to be convergent, and error estimates are provided. One and two dimensional examples are worked in detail, comparing known results with the numerical ones and computing new examples. The cases of nonstrictly convex Hamiltonians and Hamiltonians for which the cell problem has no solution are treated.

An a Posteriori Error Estimate for a Semi-Lagrangian Scheme for Hamilton–Jacobi Equations

We present an a posteriori estimate for a first order semi-Lagrangian method for Hamilton–Jacobi equations. The result requires piecewise C1,1 regularity of the viscosity solution and is stated for the Bellman equation related to the infinite horizon problem, although it can be applied to more general Hamilton–Jacobi equations with convex Hamiltonians. This estimate suggests different numerical indicators that can be used to construct an adaptive algorithm for the approximation of the viscosity solution.

Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with convex Hamiltonians $H=H(Du)$ is established, provided the discontinuous initial value function $\varphi(x)$ is continuous outside a set $\Gamma$ of measure zero and satisfies <p align="center"> (*)$ \qquad\qquad \varphi(x)\ge\varphi_{\star \star}(x) \equiv \lim$inf$_{y\rightarrow x, y\in\mathbb R^d\backslash\Gamma}\varphi(y). <p align="left" class="times"> The regularity of discontinuous solutions to Hamilton-Jacobi equations with locally strictly convex Hamiltonians is proved: The discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time. The $L^1$-accessibility of initial data and a comparison principle for discontinuous solutions are shown. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, $L$-solutions, minimax solutions, and $L^\infty$-solutions is also clarified.

A geometric proof of the existence of the Green bundles

We give a new proof of the existence of the Green bundles. Let (M, g) be a compact Riemannian manifold, and denote by 9t its geodesic flow on the tangent bundle TM. Let -r : TM -* M be the canonical projection and for all 0 E TM let V(0) be the kernel of d-ro. Two points 01, 02 are said to be conjugate if 02 g9tO and dgtV(01) n V(02) & o0 It was proved by Hopf [5] that a two-dimensional torus without conjugate points is flat. Afterwards, Green [8] proved that the integral of the scalar curvature of a manifold without conjugate points is nonpositive and it vanishes if the metric is locally flat. A main ingredient was the existence, under the condition of no conjugate points, of the following bundles: (1) Es(0) lim dg_tV(gt(0)), t-> oo (2) E'(0) = lim dgtV(g_t(0)). t-* oo Hopf's result was generalized to higher dimensions in [2], but there are still new rigidity type results using these bundles; see for example [1]. These bundles have other applications: among other ideas they where used by Freire and Mane' [7] to obtain estimates of the topological entropy. Foulon [6] generalized this result to the case of Finsler metrics. The bundles were also used by Eberlain [4] who proved that these are transverse if and only if the geodesic flow is Anosov. This result was also generalized to the case of convex Hamiltonians without conjugate points; see [31. The purpose of this note is to give a new proof of the following Theorem. If the geodesic flow 9t of a compact manifold does not have conjugate points, then for every 0 in TM the limits (1) and (2) exist. We recall from [4] the definition of the connection map K: T0TM -> T7r(o)M. For ( on T0TM let Z: (-c, e) -* TM be a curve with initial velocity (. Define K(s) = Z'(O) to be the covariant derivative of Z along the curve ir o Z. The definition does not depend on the curve Z. Received by the editors February 8, 2001. 2000 Mathematics Subject Classification. Primary 37D40. The author was partially supported by CONACYT-Mexico grant #28489-E and EPSRCUnited Kingdom GR/M5610. ?)2002 American Mathematical Society

Semicontinuous viscosity solutions to mixed boundary value problems with degenerate convex Hamiltonians

Semicontinuous viscosity solutions to mixed boundary value problems with degenerate convex Hamiltonians

On representation formula for solutions of Hamilton‐Jacobi equation for some types of initial conditions

This article investigates the representation formula for the semiconcave solutions of the Cauchy problem for Hamilton‐Jacobi equation with the convex Hamiltonian and the unbounded lower semicontino...

ON REPRESENTATION FORMULA FOR SOLUTIONS OF HAMILTON‐JACOBI EQUATION FOR SOME TYPES OF INITIAL CONDITIONS

This article investigates the representation formula for the semiconcave solutions of the Cauchy problem for Hamilton‐Jacobi equation with the convex Hamiltonian and the unbounded lower semicontinous initial function. The formula like Hopf ‘s formula is given by forming envelope of some fundamental solutions of the equation.

One Functional Metric Space and Extensions of Nonlinear Differential Expressions

Some complete metric spaces consisting of functions are introduced. An extensive class of nonlinear partial differential expressions has there closed extensions with common dense essential domain stable under perturbations. Solutions of corresponding equations coincide with viscosity (discontinuous) solutions. Some applications to Hamilton-Jacobi equations with convex hamiltonian (usual in Control Theory) are given.

$L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions

We study the \(L^1\)-stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let \(\epsilon\) denote the `small scale' of such approximations (– the viscosity amplitude \(\epsilon\), the spatial grad-size \(\Delta x\), etc.), then our \(L^1\)-error estimates are of \({\cal O}(\epsilon)\), and are sharper than the classical \(L^\infty\)-results of order one half, \({\cal O}(\sqrt{\epsilon})\). The main building blocks of our theory are the notions of the semi-concave stability condition and \(L^1\)-measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the \(Lip^\prime\)-stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain \(L^1\)-bounds on their associated truncation errors; \(L^1\)-convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our \(L^1\)-theory.

A generalization of a theorem of Barron and Jensen and a comparison theorem for lower semicontinuous viscosity solutions

We extend and clarify some of observations due to Barron and Jensen concerning the relation between subdifferentials and superdifferentials of a function and extend the comparison principle for semicontinuous solutions of Hamilton–Jacobi equations with convex Hamiltonians to that in infinite-dimensional Hilbert spaces.

Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts

This paper is concerned with first-order time-dependent Hamilton-Jacobi equations. Exploiting some ideas of Barron and Jensen [9], we derive lower-bound estimates for the gradient of a locally Lipschitz-continuous viscosity solution $u$ of equations with a convex Hamiltonian. Using these estimates in the context of the level-set approach to front propagation, we investigate the regularity properties of the propagating front of $u$, namely $\Gamma_t = \{ x \in \mathbb R^n : u(x,t)=0 \}$ for $t \geq 0$. We show that, contrary to the smooth case, such estimates do not guarantee, in general, any expected regularity for $\Gamma_t$ even if $u$ is semiconcave.

Some counterexamples on the asymptotic behavior of the solutions of Hamilton–Jacobi equations

In this Note, we provide two counterexamples about the behavior as t →∞ of the solutions of first-order Hamilton–Jacobi equations. The first one concerns the behavior of the Lax–Oleinik semigroup for equations set in non-compact domains. We show that even for smooth, strictly convex Hamiltonians, the convergence, as t →∞ , of solutions of such equations may fail, in contrast to what happens in the compact framework, were convergence results where proved recently by Fathi, Namah and Roquejoffre and the authors. The second counterexample concerns the behavior of space periodic solutions in the case of space-time periodic Hamiltonians. Recently Fathi and Mather showed, using dynamical systems types arguments, that the convergence to a space-time periodic solution is not true in general. Here we provide very simple explicit counterexamples of this fact. Dans cette Note, nous décrivons deux types de contre-exemples sur le comportement quand t →∞ des solutions d'équations de Hamilton–Jacobi. Le premier concerne le comportement en temps grands du semi-groupe de Lax–Oleinik pour des équations posées dans un domaine non compact ; nous montrons que même pour des Hamiltoniens réguliers et strictement convexes, la convergence des solutions pour de telles équations peut être fausse contrairement à ce qui se passe dans des domaines compacts où des résultats de convergence ont été obtenus récemment par Fathi, Namah et Roquejoffre et les auteurs. Le second type de contre-exemple concerne le comportement de solutions périodiques en espace dans le cas d'Hamiltoniens qui sont périodiques à la fois en temps et en espace. Récemment, Fathi et Mather ont montré, par des arguments de systèmes dynamiques, que la convergence de telles solutions vers des solutions périodiques en temps et en espace est fausse en général. Nous fournissons, dans ce cas, des contre-exemples explicites très simples.

High-Resolution Nonoscillatory Central Schemes for Hamilton--Jacobi Equations

In this paper, we construct second-order central schemes for multidimensional Hamilton--Jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle. Thus, these schemes provide the first examples of nonoscillatory second-order Godunov-type schemes based on global projection operators. Numerical experiments are performed; $L^1$/$L^{\infty}$-errors and convergence rates are calculated. For convex Hamiltonians, numerical evidence confirms that our central schemes converge with second-order rates, when measured in the L1 -norm advocated in our recent paper [Numer. Math, to appear]. The standard $L^{\infty}$-norm, however, fails to detect this second-order rate.

Convex Hamiltonians without conjugate points

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

Aubry-Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ1. It is well-known that its smooth invariant curves correspond to smooth Z2-periodic solutions of the PDE ut + H(x, t, u)x = 0. In this paper, we establish a connection between the Aubry-Mather theory of invariant sets of the Hamiltonian system and Z2-periodic weak solutions of this PDE by realizing the Aubry-Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry-Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry-Mather set, defined in (2.24). The graph itself is a backward-invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry-Mather theory into the characteristic fields of the above PDE. This is done by making use of one- and two-sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z2-periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two-sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.

Evolutions Determined by Control Problems and Hamilton-Jacoby-Bellman Equations

The functional spaces that are an analog of Sobolev spaces, but on the basis of another algebraic operations (min -plus), are constructed from a convex Hamiltonian. The non-stationary Hamilton-Jacoby-Bellman equations are considered in these spaces following the standard extension of corresponding differential operator. The Cauchy problem has a unique solution which belongs to the spaces constructed. The corresponding semigroup of transformations coincides with that one which discribes the evolutinary process in the Lagrange problem.

Comportement asymptotique des solutions d'équations de Hamilton-Jacobi monodimensionnelles

The long-time behaviour of one-dimensional Hamilton-Jacobi equations with convex Hamiltonians is studied. The solutions an shown to converge to steady solutions, modulo a linear growth in time.

Deterministic Exit Time Control Problems With Discontinuous Exit costs

We study deterministic exit time control problems with discontinuous exit costs. When the exit cost $\phi$ is upper semicontinuous and there is an outer field on the boundary, we show that all the value functions have the same lowersemicontinuous envelope which is the unique lower semicontinuous viscosity solution of the associated Dirichlet problem. We also prove uniqueness results for the generalized Dirichlet problem for first-order Hamilton--Jacobi equations with convex Hamiltonians and with discontinuous boundary conditions, under some nondegeneracy conditions on the Hamiltonians on the boundary.

On homogenization and scaling limit of some gradient perturbations of a massless free field

We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2-point correlation functions of these models.

Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations

This paper considers the questions of convergence of: (i) MUSCL type (i.e. second-order, TVD) finite-difference approximations towards the entropic weak solution of scalar, one-dimensional conservation laws with strictly convex flux and (ii) higher-order schemes (filtered to ``preserve'' an upper-bound on some weak second-order finite differences) towards the viscosity solution of scalar, multi-dimensional Hamilton-Jacobi equations with convex Hamiltonians.