We prove convergence of stationary distributions for the randomly forced Burgers and Hamilton-Jacobi equations in a limit when viscosity tends to zero. It turns out that for all values of the viscosity ν there exists a unique (up to an additive constant) solution to the randomly forced Hamilton-Jacobi equation which is extendible for all times. The main result follow from the convergence of the...

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi’s method by applying it to the same equation studied by Musielak et al. with their own method [Musielak ZE, Roy D and Swift LD. Method to derive Lagrangian and Hamiltonian for a nonlinear dynami...

In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi e...

The paper proves a new uniqueness result for viscosity solutions of the Dirichlet problem for Hamilton-Jacobi equations of the form H(x; u; D x 0 u) = 0 in ; u = g on @; where is an open subset or R n and D x 0 u is the partial gradient of the scalar function u with respect to the rst n 0 variables (n 0 n). The main theorem states that there is a viscosity solution of the equation which is uniq...

Nondominated sorting is a fundamental combinatorial problem in multiobjective optimization and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science. In a previous work [SIAM J. Math. Anal., 46 (2014), pp. 603–638], we showed that nondominated sorting has a continuum limit that corresponds to solving a Hamilton–Jacobi equation. In...

The design of new optimal nonlinear voltage regulator for power systems is considered. In the proposed approach, the Hamilto-Jacobi-Belman (HJB), a partial differential equation, is solved using Taylor series expansion of its nonlinear terms. The performance of the proposed controller in a single-machine infinite-bus power system is simulated and the advantages of the nonlinear feedback control...

We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutati...

We propose a new formulation to compute effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations. Our formulation utilizes an observation made by BarronJensen [3] about viscosity supersolutions of Hamilton-Jacobi equations. The key idea is to link the effective Hamiltonian to a suitable effective equation. The main advantage of our formulation is that only one auxiliary...

An Adaptive Dynamic Programming algorithm for nonlinear systems with unknown dynamics is developed. The algorithm is initialized with a positive definite cost functional / stabilizing control law pair (V0, k0) (coupled via the Hamilton Jacobi Bellman Equation). Given (Vi, ki), one runs the system using control law ki recording the state and control trajectories, with these trajectories used to ...