The paper deals with connections between optimality and passivity-like properties in discrete time. The problem is set in the framework of differential/difference representations of discrete-time dynamics. The Hamilton–Jacobi–Bellman equality associated with a given cost and the corresponding optimal control solution are characterized. On these bases the connection with u-average passivity is c...

We consider two theorems formulated in the derivation of the Quantum Hamilton–Jacobi Equation from the EP. The first one concerns the proof that the cocycle condition uniquely defines the Schwarzian derivative. This is equivalent to show that the infinitesimal variation of the stress tensor “exponentiates” to the Schwarzian derivative. The cocycle condition naturally defines the higher dimensio...

We establish a rate of convergence for a semi-discrete operator splitting method applied to Hamilton-Jacobi equations with source terms. The method is based on sequentially solving a Hamilton-Jacobi equation and an ordinary diierential equation. The Hamilton-Jacobi equation is solved exactly while the ordinary diierential equation is solved exactly or by an explicit Euler method. We prove that ...

We introduce a Monte Carlo scheme for the approximation of the solutions of fully non–linear parabolic non–local PDEs. The method is the generalization of the method proposed by [Fahim-Touzi-Warin, 2011] for fully non–linear parabolic PDEs. As an independent result, we also introduce a Monte Carlo Quadrature method to approximate the integral with respect to Lévy measure which may appear inside...

We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of diffusion processes. Under suitable ergodicity assumptions on the underlying diffusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense.

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the reduction of the vector field and of the Poisson tensors. We show explicitly that, after the reduction on each one of the symplectic leaves, the vector field of the Lagrange top is separable in the sense of Hamilton–Jacobi.

This paper studies the H 1 control problem for a nonlinear, unbounded, innnite dimensional system with state constraints. We characterize the solvability of the problem by means of a Hamilton-Jacobi-Isaacs equation, proving that the H 1 problem can be solved if and only if the HJI equation has a nonnegative viscosity supersolution, vanishing at the origin.

We discuss an extension of the Hamilton–Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholonomic analogue of the Hamilton–Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton–Jacobi theory for unconstrained systems. The proof also helps...