It has earlier been argued that there should exist a formulation of quantum mechanics which does not refer to a background spacetime. In this paper we propose that, for a relativistic particle, such a formulation is provided by a noncommutative generalisation of the Hamilton-Jacobi equation. If a certain form for the metric in the noncommuting coordinate system is assumed, along with a correspo...

An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the compl...

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 ...

We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of ...

We investigate the Cauchy problem for a nonlinear parabolic partial differential equation of Hamilton–Jacobi–Bellman type and prove some regularity results, such as Lipschitz continuity and semiconcavity, for its unique viscosity solution. Our method is based on the possibility of representing such a solution as the value function of the associated stochastic optimal control problem. The main f...

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...

Periodic solutions of the Hamilton-Jacobi equation determine invariant tori in phase space. The Fourier spectrum of a torus with respect to angular coordinates gives useful information about nonlinear resonances and their potential for causing instabilities. We describe a method to solve the Hamilton-Jacobi equation for an arbitrary accelerator lattice. The method works with Fourier modes of th...

In this paper, with the aim of estimating internal dynamics matrix of a gimbaled Inertial Navigation system (as a discrete Linear system), the discretetime Hamilton-Jacobi-Bellman (HJB) equation for optimal control has been extracted. Heuristic Dynamic Programming algorithm (HDP) for solving equation has been presented and then a neural network approximation for cost function and control input ...

This paper is a survey of the Hamilton-Jacobi partial differential equation. We begin with its origins in Hamilton’s formulation of classical mechanics. Next, we show how the equation can fail to have a proper solution. Setting this issue aside temporarily, we move to a problem of optimal control to show another area in which the equation arises naturally. In the final section, we present some ...