Controlled Ordinary Differential Equations with Random Path-Dependent Coefficients and Stochastic Path-Dependent Hamilton-Jacobi Equations

An example of failure of stochastic homogenization for viscous Hamilton-Jacobi equations without convexity

We give an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with non-convex Hamiltonian.

Hamilton-Jacobi-Bellman equation of nonlinear optimal control problems with fractional discount rate

This paper derives the Hamilton-Jacobi-Bellman equation of nonlinear optimal control problems for cost functions with fractional discount rate from the Bellman's principle of optimality. The fractional discount rate is described by Mittag-Leffler function that can be considered as a generalized exponential function.

Controlled Continuous Time Random Walks and fractional Hamilton Jacobi Bellman equations

In this paper we study controlled continuous time random walks (CTRWs) and heuristically derive pay-off function dynamic programming (DP) equations which turn in the limit of standard scaling to fractional Hamilton Jacobi Bellman type equations. This paper aims to extend results from [1] in a controlled setting.

Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample

This paper provides a counterexample about the asymptotic behavior of the solutions of a discounted Hamilton-Jacobi equation, as the discount factor vanishes. The Hamiltonian of the equation is a 1-dimensional continuous and coercive Hamiltonian.

Reduction of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations

This article is devoted to the study of lower semicontinuous solutions of Hamilton-Jacobi equations with convex Hamiltonians in a gradient variable. Such Hamiltonians appear in the optimal control theory. We present a necessary and sufficient condition for a reduction of a Hamiltonian satisfying optimality conditions to the case when the Hamiltonian is positively homogeneous and also satisfies optimality conditions. It allows us to reduce some uniqueness problems of lower semicontinuous solutions to Barron-Jensen and Frankowska theorems. For Hamiltonians, which cannot be reduced in that way, w…

Twisted Lax--Oleinik formulas and weakly coupled systems of Hamilton--Jacobi equations

We show that viscosity solutions of evolutionary weakly coupled systems of Hamilton--Jacobi equations can be approximated by iterated twisted Lax--Oleinik like operators. We establish convergence to the solution of the iterated scheme and discuss further properties of the approximate solutions.

Slow periodic homogenization for Hamilton-Jacobi equations

Capuzzo-Dolcetta and Ishii proved that the rate of periodic homogenization for coercive Hamilton-Jacobi equations is $O(\varepsilon^{1/3})$. We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is $Ω(\varepsilon^{1/2})$.

Hamilton-Jacobi in metric spaces with a homological term

The Hamilton-Jacobi equation on metric spaces has been studied by several authors; following the approach of Gangbo and Swiech, we show that the final value problem for the Hamilton-Jacobi equation has a unique solution even if we add a homological term to the Hamiltonian. In metric measure spaces which satisfy the $RCD(K,\infty)$ condition one can define a Laplacian which shares many properties with the ordinary Laplacian on $\R^n$; in particular, it is possible to formulate a viscous Hamilton-Jacobi equation. We show that, if the homological term is sufficiently regular, the viscous Hamilton…

An initial condition reconstruction in Hamilton-Jacobi equations

We describe the family of initial conditions for Hamilton-Jacobi equations (HJE) corresponding to optimal control problems that can be retrieved by solving HJE backward in time.

Characteristic curves for set-valued Hamilton-Jacobi equations

The method of characteristics is extended to set-valued Hamilton-Jacobi equations. This problems arises from a calculus of variations' problem with a multicriteria Lagrangian function: through an embedding into a set-valued framework, a set-valued Hamilton-Jacobi equation is derived, where the Hamiltonian function is the Fenchel conjugate of the Lagrangian function. In this paper a method of characteristics is described and some results are given for the Fenchel conjugate.

A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks

Three definitions of viscosity solutions for Hamilton-Jacobi equations on networks recently appeared in literature ([1,4,6]). Being motivated by various applications, they appear to be considerably different. Aim of this note is to establish their equivalence.

带控制约束的Hamilton-Jacobi-Isaacs方程的策略迭代及其与Hamilton-Jacobi-Bellman方程的比较

Policy iteration for Hamilton-Jacobi-Isaacs equations with control constraints and comparison with Hamilton-Jacobi-Bellman equations

本文分析带控制约束的HJI方程双层策略迭代的收敛性，并比较一阶与二阶HJB方程在有无约束控制下的数值解。

Convergence to time-periodic solutions in Hamilton-Jacobi equations on the circle

The goal of this paper is to give a simple proof of the convergence to time-periodic states of the solutions of time-periodic Hamilton-Jacobi equations on the circle with convex Hamiltonian. Note that the period of the limiting solutions may be greater than the period of the Hamiltonian.

Some results on the large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations

Systems of Hamilton-Jacobi equations arise naturally when we study the optimal control problems with pathwise deterministic trajectories with random switching. In this work, we are interested in the large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. The large time behavior for systems of Hamilton-Jacobi equations have been obtained by Camilli-Loreti-Ley and the author (2012) and Mitake-Tran (2012) under quite strict conditions. In this work, we use a PDE approach to extend the convergence result proved by Barles-Souganidis (2000) in …

Optimal convergence rate for homogenization of convex Hamilton-Jacobi equations in the periodic spatial-temporal environment

We study the optimal convergence rate for homogenization problem of convex Hamilton-Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of [8], which means the optimal convergence rate is also $O(\varepsilon)$.

Weak KAM theorem for Hamilton-Jacobi equations

In this paper, we generalize weak KAM theorem from positive Lagrangian systems to "proper" Hamilton-Jacobi equations. We introduce an implicitly defined solution semigroup of evolutionary Hamilton-Jacobi equations. By exploring the properties of the solution semigroup, we prove the convergence of solution semigroup and existence of weak KAM solutions for stationary equations: \begin{equation*} H(x, u, d_x u)=0. \end{equation*}

A system of of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations

We introduce a class of systems of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e. tessellations of domains with respect to geodesic distances where generators and centroids coincide. Typical examples are given by geodesic centroidal Voronoi tessellations and geodesic centroidal power diagrams. An appropriate version of the Fast Marching method on unstructured grids allows computing the solution of the Hamilton-Jacobi system and therefore the associated tessellations. We propose various numerical examples to illustrate the features of the technique.