HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS

In this paper,we give the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian(RCH)system and its regular reduced systems,which are called the Type I and Type II Hamilton-Jacobi equations.First,we prove two types of Hamilton-Jacobi theorems for an RCH system on the cotangent bundle of a configuration manifold by using the canonical symplectic form and its dynamical vector field.Second,we generalize the above results for a regular reducible RCH system with symmetry and a momentum map,and derive precisely two types of Hamilton-Jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH system.Third,we prove that the RCH-equivalence for the RCH system,and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries,leave the solutions of corresponding Hamilton-Jacobi equations invariant.Finally,as an application of the theoretical results,we show the Type I and Type II Hamilton-Jacobi equations for the Rp-reduced controlled rigid body-rotor system and the Rp-reduced controlled heavy top-rotor system on the generalizations of the rotation group SO(3)and the Euclidean group SE(3),respectively.This work reveals the deeply internal relationships of the geometrical structures of phase spaces,the dynamical vector fields and the controls of the RCH system.

THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS

We use Hopf-Lax formula to study local regularity of solution to Hamilton- Jacobi （HJ） equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T 〉 0, which depends only on the Hamiltonian and initial datum, for t 〉 T the solution of the IVP （1.1） is smooth except for ~ smooth n-dimensional hypersurface, across which Du（x, t） is discontinuous. And we show that the hypersurface 1 tends asymptotically to a given hypersurface with rate t-1/4.

A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS

The gradient blowup of the equation ut = △u ＋ a（x）｜△u｜p ＋ h（x）, where p 〉 2, is studied. It is shown that the gradient blowup rate will never match that of the self-similar variables. The exact blowup rate for radial solutions is established under the assumptions on the initial data so that the solution is monotonically increasing in time.

Arc Length-Based WENO Scheme for Hamilton-Jacobi Equations

In this article,novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations.These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil.The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and give a high-resolution numerical solution.Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.

UNIQUENESS OF VISCOSITY SOLUTIONS OF STOCHASTIC HAMILTON-JACOBI EQUATIONS

This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.

HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR HAMILTON-JACOBI EQUATIONS

In this paper, we use Hermite weighted essentially non-oscillatory （HWENO） schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes （HWENO-RK） of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.

AN EFFICIENT ADER DISCONTINUOUS GALERKIN SCHEME FOR DIRECTLY SOLVING HAMILTON-JACOBI EQUATION

This paper proposes an efficient ADER(Arbitrary DERivatives in space and time)discontinuous Galerkin(DG)scheme to directly solve the Hamilton-Jacobi equation.Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG(RKDG)schemes,the ADER scheme is one-stage in time discretization,which is desirable in many applications.The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time.In such predictor step,a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime.The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution.The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written.The explicit formulae of the scheme at third order is provided on two-dimensional structured meshes.The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme.Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

Convergence of the viscosity solution of non-autonomous Hamilton-Jacobi equations

In this paper,we investigate the non-autonomous Hamilton-Jacobi equation{ə_(t)u+H(t,x,ə_(x)=0,u(x,t)_(0))=φ(x),x∈M where H is 1-periodic with respect to t and M is a compact Riemannian manifold without boundary.We obtain the viscosity solution denoted by T_(t_(0))^(t)φ(x)and show T_(t_(0))^(t)φ(x)converges uniformly to a time-periodic viscosity solution u^(*)(x,t)ofə_(t)u+H(t,x,ə_(x)u,u)=0.

Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions

For a nonlinear finite time optimal control problem,a systematic numerical algorithm to solve the Hamilton-Jacobi equation for a generating function is proposed in this paper.This algorithm allows one to obtain the Taylor series expansion of the generating function up to any prescribed order by solving a sequence of first order ordinary differential equations recursively.Furthermore,the coefficients of the Taylor series expansion of the generating function can be computed exactly under a certain technical condition.Once a generating function is found,it can be used to generate a family of optimal control for different boundary conditions.Since the generating function is computed off-line,the on-demand computational effort for different boundary conditions decreases a lot compared with the conventional method.It is useful to online optimal trajectory generation problems.Numerical examples illustrate the effectiveness of the proposed algorithm.

HIGH ORDER FINITE DIFFERENCE HERMITE WENO FAST SWEEPING METHODS FOR STATIC HAMILTON-JACOBI EQUATIONS

In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.

DISCONTINUOUS SOLUTIONS IN L~∞ FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions in L~∞ for HamiltonJacobi equations. This approach allows the initial data only in L~∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discoatinuous solutions in L~∞. The existence of global discontinuous solutions in L~∞ is established. These solutions in L~∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed to examine the L~∞ stability of our L~∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.

CONVERGENCE OF FINITE VOLUME SCHEMES FOR HAMILTON-JACOBI EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper, we use Daubechies scaling functions as test functions for the Galerkin method, and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations. It can be proved that the schemes are TVD schemes. Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations. Furthermore, they have high-order accuracy in smooth regions and good resolution of singularities.

A STOPPING CRITERION FOR HIGHER-ORDER SWEEPING SCHEMES FOR STATIC HAMILTON-JACOBI EQUATIONS

We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a fifth-order Weighted PowerENO sweeping scheme for static Hamilton-Jacobi equations with convex Hamiltonians and present numerical examples that validate the effectiveness of the new stopping criterion.

A PDE Approach to the Long-Time Asymptotic Solutions of Contact Hamilton-Jacobi Equations

We study the long-time asymptotic behaviour of viscosity solutions u(x,t)of the Hamilton-Jacobi equation u_(t)(x,t)+H(x,u(x,t),Du(x,t))=0 in T^(n)×(0,∞)with a PDE approach,where H=H(x,u,p)is coercive in p,non-decreasing in u and strictly convex in(u,p),and establish the uniform convergence of u(x,t)to an asymptotic solution u_(∞)(x)as t→∞.Moreover,u_(∞) is a viscosity solution of Hamilton-Jacobi equation H(x,u(x),Du(x))=0.

Exponential Convergence to Time-Periodic Viscosity Solutions in Time-Periodic Hamilton-Jacobi Equations

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold,where the Hamiltonian satisfies the condition:The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit.It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.

An h-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In[35,36],we presented an h-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws.A tree data structure(binary tree in one dimension and quadtree in two dimensions)is used to aid storage and neighbor finding.Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled"children".Extensive numerical tests indicate that the proposed h-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities.In this paper,we apply this h-adaptive method to solve Hamilton-Jacobi equations,with an objective of enhancing the resolution near the discontinuities of the solution derivatives.One-and two-dimensional numerical examples are shown to illustrate the capability of the method.

Symmetric Reduction and Hamilton-Jacobi Equations of the Controlled Underwater Vehicle-Rotor System

.As an application of the theoretical results,in this paper,we study the symmetric reduction and Hamilton-Jacobi theory for the underwater ve-hicle with two internal rotors as a regular point reducible RCH system,in the cases of coincident and non-coincident centers of the buoyancy and the gravity.At first,we give the regular point reduction and the two types of Hamilton-Jacobi equations for a regular controlled Hamiltonian(RCH)system with sym-metry and a momentum map on the generalization of a semidirect product Lie group.Next,we derive precisely the geometric constraint conditions of the reduced symplectic forms for the dynamical vector fields of the regular point reducible controlled underwater vehicle-rotor system,that is,the two types of Hamilton-Jacobi equations for the reduced controlled underwater vehicle-rotor system,by calculations in detail.These work reveal the deeply internal relationships of the geometrical structures of the phase spaces,the dynamical vector fields and the controls of the system.

High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations

In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function φ.They are now computed from the current value of φ and the previous spatial derivatives of φ. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear Hamiltonian, therefore any HamiltonJacobi equations with complex Hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameterε used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods.

Finite Volume Hermite WENO Schemes for Solving the Hamilton-Jacobi Equation

In this paper,we present a new type of Hermite weighted essentially nonoscillatory(HWENO)schemes for solving the Hamilton-Jacobi equations on the finite volume framework.The cell averages of the function and its first one(in one dimension)or two(in two dimensions)derivative values are together evolved via time approaching and used in the reconstructions.And the major advantages of the new HWENO schemes are their compactness in the spacial field,purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial reconstructions.Extensive numerical tests are performed to illustrate the capability of the methodologies.