Solution Of Hamilton-Jacobi Equation Research Articles

Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games

In the paper, constructions of the generalized method of characteristics are applied for calculating the generalized minimax (viscosity) solutions of Hamilton-Jacobi equations in dynamic bimatrix games. The structure of the game presumes interactions of two players in the framework of the evolutionary game model. Stochastic contacts between players occur according to the dynamic process, which can be interpreted as a system of Kolmogorov's differential equations with controls instead of probability parameters. It is assumed that control parameters are not fixed and can be constructed by the feedback principle. Two types of payoff functions are considered: short-term payoffs are determined in the current moments of time, and long-term payoffs are determined as limit functionals on the infinite time horizon. The notion of dynamic Nash equilibrium in the class of controlled feedbacks is considered for the long-term payoffs. In the framework of constructions of dynamic equilibrium, the solutions are designed on the basis of maximization of guaranteed payoffs. Such guaranteeing strategies are built in the framework of the theory of minimax (viscosity) solutions of Hamilton-Jacobi equations. The analytical formulas are obtained for the value functions in the cases of different orientations for the “zigzags” (broken lines) of acceptable situations in the static game. The equilibrium trajectories generated by the minimax solutions shift the system in the direction of cooperative Pareto points. The proposed approach provides new qualitative properties of the equilibrium trajectories in the dynamic bimatrix games which guarantee better results of payoffs for both players than static Nash equilibria. As an example, interactions of two firms on the market of innovative electronic devices are examined within the proposed approach for treating dynamic bimatrix games.

Fast Marching Method Aplication for Forward Modelling of Seismic Wave Propagation

One of the classical problem in seismology is to determine time travel and ray path of seismic wave betweentwo points at a given heterogeneous media. This problem is expressed by eikonal equation and can be seen as a propagation of a wavefront and interface evolution. One of methods to solve this problem is Fast Marching Method abbreviated as FMM. This method is used to produce entropy-satisfying viscosity solution of eikonal equation. FMM combines viscosity solution of Hamilton-Jacobi equation and Huygen's Principle that centered on min-heap data structure to determine the minimum value at every loop. In this study, FMM is applied to determine time travel and raypath of seismic wave. FMM also is used to determine the location of wavesource using simple algorithm. From our forward modeling schemes, it is found that FMM is an accurate, robust, and effcient method to simulate seismic wave propagation. For further study, FMM also can be used to be a part of passive seismic inverse scheme to locate hypocenter location.

A kind of non-conservative Hamilton system solved by the Hamilton-Jacobi method

The Hamilton-Jacobi equation is an important nonlinear partial differential equation. In particular, the classical Hamilton-Jacobi method is generally considered to be an important means to solve the holonomic conservative dynamics problems in classical dynamics. According to the classical Hamilton-Jacobi theory, the classical Hamilton-Jacobi equation corresponds to the canonical Hamilton equations of the holonomic conservative dynamics system. If the complete solution of the classical Hamilton-Jacobi equation can be found, the solution of the canonical Hamilton equations can be found by the algebraic method. From the point of geometry view, the essential of the Hamilton-Jacobi method is that the Hamilton-Jacobi equation promotes the vector field on the cotangent bundle T* M to a constraint submanifold of the manifold T* M R, and if the integral curve of the promoted vector field can be found, the projection of the integral curve in the cotangent bundle T* M is the solution of the Hamilton equations. According to the geometric theory of the first order partial differential equations, the Hamilton-Jacobi method may be regarded as the study of the characteristic curves which generate the integral manifolds of the Hamilton 2-form . This means that there is a duality relationship between the Hamilton-Jacobi equation and the canonical Hamilton equations. So if an action field, defined on UI (U is an open set of the configuration manifold M, IR), is a solution of the Hamilton-Jacobi equation, then there will exist a differentiable map from MR to T* MR which defines an integral submanifold for the Hamilton 2-form . Conversely, if * =0 and H1(UI)=0 (H1(UI) is the first de Rham group of U I), there will exist an action field S satisfying the Hamilton-Jacobi equation. Obviously, the above mentioned geometric theory can not only be applicable to the classical Hamilton-Jacobi equation, but also to the general Hamilton-Jacobi equation, in which some first order partial differential equations correspond to the non-conservative Hamiltonian systems. The geometry theory of the Hamilton-Jacobi method is applied to some special non-conservative Hamiltonian systems, and a new Hamilton-Jacobi method is established. The Hamilton canonical equations of the non-conservative Hamiltonian systems which are applied with non-conservative force Fi = (t)pi can be solved with the new method. If a complete solution of the corresponding Hamilton-Jacobi equation can be found, all the first integrals of the non-conservative Hamiltonian system will be found. The classical Hamilton-Jacobi method is a special case of the new Hamilton-Jacobi method. Some examples are constructed to illustrate the proposed method.

A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations

It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m-1) for m>=3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

A Hamilton-Jacobi approach for front propagation in kinetic equations

In this paper we use the theory of viscosity solutions for Hamilton-Jacobi equations to study propagation phenomena in kinetic equations. We perform the hydrodynamic limit of some kinetic models thanks to an adapted WKB ansatz. Our models describe particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The scattering operator is supposed to satisfy a maximum principle. When the velocity space is bounded, we show, under suitable hypotheses, that the phase converges towards the viscosity solution of some constrained Hamilton-Jacobi equation which effective Hamiltonian is obtained solving a suitable eigenvalue problem in the velocity space. In the case of unbounded velocities, the non-solvability of the spectral problem can lead to different behavior. In particular, a front acceleration phenomena can occur. Nevertheless, we expect that when the spectral problem is solvable one can extend the convergence result.

安定多様体法におけるHamilton-Jacobi方程式の高速数値解法

安定多様体法におけるHamilton-Jacobi方程式の高速数値解法

Symplectic microgeometry II: generating functions

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an application, we describe hamiltonian flows as special symplectic micromorphisms whose local generating functions are the solutions of Hamilton-Jacobi equations. We obtain a purely categorical formulation of the temporal evolution in classical mechanics.

Time Remotely Almost Periodic Viscosity Solutions of Hamilton-Jacobi Equations

We study some properties of the remotely almost periodic functions. This paper studies viscosity solutions of general Hamilton-Jacobi equations in the time remotely almost periodic case. Existence and uniqueness results are presented under usual hypotheses.

Homogenization of generalized characteristics associated to solutions of Hamilton–Jacobi equations

It was proved in Arch. Ration. Mech. Anal. 162 (2002), 1-23 that singularities of solutions for Hamilton-Jacobi equa- tions will propagate along generalized characteristics. In this paper, we consider the homogenization of generalized character- istics. It is natural to think that (∗) generalized characteristics associated to viscosity solutions of oscillatory Hamilton-Jacobi equations will converge to generalized characteristics associated to solutions of the effective equation. We show that this is indeed correct if the spatial dimension is 1. For high dimensions, we need to add some extra assumptions of singularities near generalized characteristics. We provide a counterexample that (∗) fails without those assumptions. Some issues related to the weak KAM theory are also discussed.

A notion of extremal solutions for time periodic Hamilton-Jacobi equations

This paper is concerned with time periodic solutions ofHamilton-Jacobi equations in which the hamiltonian is increasingwrt to the unknown variable. When the uniqueness of the periodicsolution is not guaranteed, we define a notion of extremalsolution and propose two different ways to attain it, togetherwith the corresponding numerical simulations. In the course of theanalysis, the ode case, where we show that things are ratherexplicit, is also visited.

Birkhoff’s theorem and viscosity solutions of Hamilton-Jacobi equations

We obtain a partial generalization of Birkhoff’s theorem of invariant curve to higher dimesional case in the context of viscosity solutions of Hamilton-Jacobi equations, or weak KAM theory. This is a new approach after Herman’s proof.

On the Value Function of a Differential Game with Simple Dynamics and Piecewise Linear Data

A differential game with simple motions is considered under assumptions that the Hamiltonian and the cost terminal function are piecewise linear and positively homogeneous. The structure of the value function of the differential game is investigated in the framework of the theory of minimax (or/and viscosity) solutions for Hamilton-Jacobi equations. Inequalities are provided to estimate the value function. Cases of explicit formulas for the value function are pointed out.

HYPERBOLIC DOMAINS OF DETERMINACY AND HAMILTON–JACOBI EQUATIONS

If L(t, x, ∂t, ∂x) is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets Ω0⊂ {t = 0}. The frozen constant coefficient operators [Formula: see text] determine local convex propagation cones, [Formula: see text]. Influence curves are curves whose tangent always lies in these cones. We prove that the set of points Ω which cannot be reached by influence curves beginning in the exterior of Ω0is a domain of determinacy in the sense that solutions of Lu = 0 whose Cauchy data vanish in Ω0must vanish in Ω. We prove that Ω is swept out by continuous spacelike deformations of Ω0and is also the set described by maximal solutions of a natural Hamilton–Jacobi equation (HJE). The HJE provides a method for computing approximate domains and is also the bridge from the raylike description using influence curves to that depending on spacelike deformations. The deformations are obtained from level surfaces of mollified solutions of HJEs.

Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with convex Hamiltonians $H=H(Du)$ is established, provided the discontinuous initial value function $\varphi(x)$ is continuous outside a set $\Gamma$ of measure zero and satisfies <p align="center"> (*)$ \qquad\qquad \varphi(x)\ge\varphi_{\star \star}(x) \equiv \lim$inf$_{y\rightarrow x, y\in\mathbb R^d\backslash\Gamma}\varphi(y). <p align="left" class="times"> The regularity of discontinuous solutions to Hamilton-Jacobi equations with locally strictly convex Hamiltonians is proved: The discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time. The $L^1$-accessibility of initial data and a comparison principle for discontinuous solutions are shown. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, $L$-solutions, minimax solutions, and $L^\infty$-solutions is also clarified.

ニューラルネットによるHamilton-Jacobi方程式の解法と非線形システムの最適フィードバック制御則

ニューラルネットによるHamilton-Jacobi方程式の解法と非線形システムの最適フィードバック制御則

Ergodic problem for the Hamilton-Jacobi-Bellman equation. II

We study the ergodic problem for the first-order Hamilton-Jacobi-Equations (HJBs), from the view point of controllabilities of underlying controlled deterministic systems. We shall give sufficient conditions for the ergodicity by the estimates of controllabilities.Next, we shall give some results on the Abelian-Tauberian problem for the solutions of HJBs. Our solutions of HJBs satisfy the equations in the sense of viscosity solutions.

Comparison results for hamilton-jacobi equations without grwoth condition on solutions from above

We prove comparison results for solutions of Hamilton-Jacobi equations, which adapt and extend a recent result by 0. Alvarez [A] to the Cauchy problem. Our results can be applied even to Hamilton-Jacob1 equations withnonconvex Hamiltonians. We examine the comparison results with examples treated recently by M.Bar& and F. DaLio [BD] and by W. M. McEneaney [M].

Moser-type inequalities for solutions of linear elliptic equations with lower-order terms

Moser-type estimates for solutions of linear elliptic equations with lower-order terms are given. The results are proved under various assumptions on the coefficients and on the known term in the equations. Also solutions of Hamilton-Jacobi equations are considered.

Semicontinuous solutions for Hamilton-Jacobi equations and theL ?control problem

We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation forL ∞-control problems. This result is then used to prove uniqueness for lsc solutions of Hamilton-Jacobi equations of the form −u t +H(t, x, u, −Du)=0, whereH(t, x, r, p) is convex inp. The remaining assumptions onH in the variablesr andp extend the currently known results.

Local behavior of solutions of Hamilton-Jacobi equations

Local behavior of solutions of Hamilton-Jacobi equations