A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation

We estimate the variance of the value function for a random optimal control problem. The value function is the solution $w^\epsilon$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,\omega) = K(p) - V(x/\epsilon,\omega)$ in dimension $d \geq 2$. It is known that homogenization occurs as $\epsilon \to 0$, but little is known about the statistical fluctuations of $w^\epsilon$. Our main result shows that the variance of the solution $w^\epsilon$ is bounded by $O(\epsilon/|\log \epsilon|)$. The proof relies on a modified Poincar\'e inequality of Talagrand

Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates

We consider homogenization for weakly coupled systems of Hamilton--Jacobi equations with fast switching rates. The fast switching rate terms force the solutions converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.Comment: final version, to appear in Arch. Ration. Mech. Ana

A Limiting Case for Velocity Averaging

We complete the theory of velocity averaging lemmas for transport equations by studying the limiting case of a full space derivative in the source term. Although the compactness of averages does not hold any longer, a specific estimate remains, which shows compactness of averages in more general situations than those previously known. Our method is based on Calderon-Zygmund theory. R'esum'e Nous compl'etons les lemmes de moyenne pour les 'equations de transport en 'etudiant le cas limite d'une d'eriv'ee en espace dans le terme source. La compacit'e des moyennes ne peut etre obtenue, mais nous d'emontrons une estimation sp'ecifique qui permet de montrer la compacit 'e en moyenne dans les situations les plus g'en'erales connues actuellement. Notre m'ethode s'appuie sur la th'eorie de Calderon-Zygmund. I. Introduction. We consider the regularity properties of averaged quantities like (I:1) ae(t; x) = Z f(t; x; v)'(v)dv; where ' is a given function and f : IR \Theta IR d \Theta IR ..

Optimal Control Using Bisimulations: Implementation

We consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem which is to synthesize optimal enabling conditions for switching between locations in which the control is constant. We provide a singlepass algorithm to solve the dynamic programming problem that arises, with added constraints to ensure non-Zeno trajectories