Hamilton-Jacobi equation

We construct exact solutions to the paraxial wave equation in free space characterized by stable caustics. First, we show that any solution of the paraxial wave equation can be written as the superposition of plane waves determined by both the Hamilton-Jacobi and Laplace equations in free space. Then using the five elementary stable catastrophes, we construct solutions of the Hamilton-Jacobi and Laplace equations, and the corresponding exact solutions of the paraxial wave equation. Therefore, the evolution of the intensity patterns is governed by the paraxial wave equation and that of the corresponding caustic by the Hamilton-Jacobi equation.

An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions

By employing the terminology of functional differential calculus, Hamilton-Jacobi theory is extended to apply to classical field equations. It is shown that an asymptotic solution to the Hamilton-Jacobi functional differential equation provides an asymptotic general solution to the associated nonlinear classical field equations.

This paper presents the hierarchical optimal multi-control algorithm for the partially unknown finite-horizon linear systems. The key to the multi-control problem is to find the solution to the coupled leader-follower Hamilton-Jacobi (HJ) equations. The coupled leader-follower HJ equations are derived and we define the Stackelberg equilibrium of the games. To solve the HJ equations for the follower, an integral reinforcement learning (IRL) approach is given on the basis of the PI algorithm, which uses partial knowledge of the dynamic systems. The Finite-Neural-Network (FNN) structure is used to approximate the value functions in the proposed algorithm. Simulation results are shown to verify the effectiveness of the scheme.

This paper presents a data-driven method for designing optimal controllers and robust controllers for unknown nonlinear systems. Mathematical models for the realization of the control are difficult to develop owing to a lack of knowledge regarding such systems. The proposed multidisciplinary method, based on optimal control theory and machine learning with kernel functions, facilitates designing appropriate controllers using a data set. Kernel-based system models are useful for representing nonlinear systems. An optimal and an H-infinity controller can be designed by solving Hamilton-Jacobi (HJ) equations, which unfortunately, are difficult to solve owing to the nonlinearity and complexity of the kernel-based models. The objective of this study consists of overcoming two challenges. The first challenge is to derive exact solutions to the HJ equations for a class of kernel-based system models. A key technique in overcoming this challenge is to reduce the HJ equations to easily solvable algebraic matrix equations, from which optimal and H-infinity controllers are designed. The second challenge is to control an unknown system using the obtained controllers, wherein the system is identified as a kernel-based model. Additionally, this study analyzes probabilistic stability of the feedback system with the proposed controllers. Numerical simulations demonstrate control performances of both the derived optimal and H-infinity controllers and stability of the feedback system.

We are interested in high-order numerical approximations for the solution of multi-dimensional Hamilton-Jacobi (HJ) equations of the form $$ {{\phi }_{t}} + H(\nabla \phi ) = 0,\quad x = ({{x}_{1}}, \ldots {{x}_{d}}) \in {{\mathbb{R}}^{d}}. $$ Here, H is the Hamiltonian, which we assume depends on ∇ϕ and possibly on x and t. The main difficulty in approximating the solutions of HJ equations is the discontinuous derivatives that develop in the solution in finite time even when the initial data is smooth. From an analytical point of view, solutions of HJ equations past the discontinuity are treated via the machinery of viscosity solutions (see [3, 4, 11] and the references therein). As far as the numerics is concerned, the first significant result were the converging first-order approximations by Souganidis [18]. High-order upwind methods, that are based on the ENO reconstruction of Harten et al [6], were introduced by Osher, Sethian and Shu [16, 17]. Jiang and Peng combined the Weighted ENO (WENO) interpolant [8, 14] with a monotone fiux obtaining the upwind-WENO schemes for HJ equations in [7].

Networked traffic state estimation involving mixed fixed-mobile sensor data using Hamilton-Jacobi equations

Hamilton–Jacobi theory, symmetries and coisotropic reduction

In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a "weighted-residual" or "stabilization-based" formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for Hamilton-Jacobi equations. The L 2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.

: This paper is the second in a series by the authors concerned with the theory of viscosity solutions Hamilton-Jacobi equations in infinite dimensional spaces. The first paper introduced a notion of viscosity solution appropriate for the study of Hamilton-Jacobi equations in spaces with the so-called Radon-Nikodym property and obtained uniqueness theorems under assumptions paralleling the finite dimensional theory. The main results of the current paper concern existence of solutions of stationary and time-dependent Hamilton-Jacobi equations. In order to establish these results it is necessary to overcome the difficulties associated with the fact that bounded sets are not precompact in infinite dimensions and this is done by sharp constructive estimates coupled with the use of differential games to solve regularized problems. Interest in this subject arises on the abstract side from the desire to contribute to the theory of linear partial differential equations in infinite dimensional spaces to treat natural questions raised by the finite dimensional theory. Interest also arises from potential applications to the theory of control of partial differential equations. However, the results herein do not apply directly to problems of the form arising in the control of partial differential equations, a question which wil be treated in the next paper of the series. Additional keywords: Banach spaces, Existence theory. (Author)

In the paper we investigate the regularity of the value function representing Hamilton- Jacobi equation: �Ut + H(t,x,U,�Ux) = 0 with a final condition: U(T,x )= g(x). Hamilton-Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of variation. In the paper we prove that absolutely continuous functions that are solutions of Mayer problem satisfy the Lipschitz condition. Using this fact we show that the value function is a bilateral solution of Hamilton-Jacobi equation. Moreover, we prove that continuity or the local Lipschitz condition of the function of final cost g is inherited by the value function. Our results allow to state the theorem about existence and uniqueness of bilateral solutions in the class of functions that are bounded from below and satisfy the local Lipschitz condition. In proving the main results we use recently derived necessary optimality conditions of Loewen-Rockafellar (P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 32 (1994) 442-470; P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 35 (1997) 2050-2069).

It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different. Quanta 2023; 12: 22–26.

Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations

Global subanalytic solutions of Hamilton–Jacobi type equations

We investigate the large-time behavior of viscosity solutions of Hamilton–Jacobi equations with noncoercive Hamiltonian in a multi-dimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka (Phys. D, 237 (2008), no. 22, 2845–2855). We prove that the average growth rate of a solution is constant only in a subset, which will be called effective domain, of the whole domain and give the asymptotic profile in the subset. This means that the large-time behavior for noncoercive problems may depend on the space variable in general, which is different from the usual results under the coercivity condition. Moreover, on the boundary of the effective domain, the gradient with respect to the $x$-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary Hamilton–Jacobi equations. We establish the existence and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain.