The optimal control related to Riemannian manifolds and the viscosity solutions to Hamilton–Jacobi–Bellman equations

In this paper we study the optimal stochastic control problem for stochastic differential equations on Riemannian manifolds. The cost functional is specified by controlled backward stochastic differential equations in Euclidean space. Under some suitable assumptions, we conclude that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation which is a fully nonlinear parabolic partial differential equation on Riemannian manifolds.     

Non-uniqueness of solutions of the Hamilton–Jacobi–Bellman equation for time-average control

In control of diffusion processes a very useful instrument is the equation for optimal strategy and cost. For the version of infinite time horizon with time averaging this equation is much more complicated than for the version of finite time horizon, and even than for the version of infinite time horizon with discounting. In particular, the equation solution may be non-unique. This problem of non-uniqueness is researched in book of A. Arapostathis et al., 2012, for special models—near-monotone. The result received in the book is extended in the article to an important general case—models with restrictions in control which guarantee ergodicity of the process. Besides we correct the proofs from the book.     

Robust portfolio optimization via solution to the Hamilton–Jacobi–Bellman equation

We consider a problem of dynamic stochastic portfolio optimization modelled by a fully non-linear Hamilton–Jacobi–Bellman (HJB) equation. Using the Riccati transformation, the HJB equation is transformed to a simpler quasi-linear partial differential equation. An auxiliary quadratic programming problem is obtained, which involves a vector of expected asset returns and a covariance matrix of the returns as input parameters. Since this problem can be sensitive to the input data, we modify the problem from fixed input parameters to worst-case optimization over convex or discrete uncertainty sets both for asset mean returns and their covariance matrix. Qualitative as well as quantitative properties of the value function are analysed along with providing illustrative numerical examples. We show application to robust portfolio optimization for the German DAX30 Index.     

Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton–Jacobi–Bellman Equations

We analyse two practical aspects that arise in the numerical solution of Hamilton–Jacobi–Bellman equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes. This leads to computational difficulties not encountered there. In particular, we consider the overstepping of the domain boundary and analyse the accuracy and stability of stencil truncation. This truncation imposes a stricter CFL condition for explicit schemes in the vicinity of boundaries than in the interior, such that implicit schemes become attractive. We then study the use of geometric, algebraic and aggregation-based multigrid preconditioners to solve the resulting discretised systems from implicit time stepping schemes efficiently. Finally, we illustrate the performance of these techniques numerically for benchmark test cases from the literature.     

The use of a Legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic Hamilton–Jacobi equations

In this research, we study the problem of finding the approximate solution of a class of Hamilton–Jacobi equations, namely the Eikonal equation. We employ the Legendre pseudospectral viscosity method to solve this problem. This method basically consists of adding a spectral viscosity to the equation. This spectral viscosity, which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme. Several test problems are considered and the numerical results are given to show the efficiency of the proposed method.     

Hamilton–Jacobi–Bellman Quasi-Variational Inequality arising in an environmental problem and its numerical discretization

A Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) for a river environmental restoration problem with wise-use of sediment is formulated and its mathematical properties are analyzed. A finite difference scheme with a penalization technique is then established for solving the HJBQVI. The scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions. A demonstrative application example of the HJBQVI is finally presented.     

Higher-order Hamilton dynamics and Hamilton–Jacobi divergence PDE

This paper, using a non-standard Legendrian duality, investigates the Hamiltonian dynamics and formulates a Hamilton–Jacobi type divergence PDE governed by higher-order Lagrangians.     

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

In this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.     

New generalized method to construct new non-travelling wave solutions and travelling wave solutions of K–D equations

With the aid of computerized symbolic computation, we obtain new types of general solution of a first-order nonlinear ordinary differential equation with six degrees of freedom and devise a new generalized method and its algorithm, which can be used to construct more new exact solutions of general nonlinear differential equations. The (2+1)-dimensional K–D equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain non-travelling wave solutions and travelling wave solutions.     

Necessary condition for near optimal control of linear forward–backward stochastic differential equations

This paper investigates the near optimal control for a kind of linear stochastic control systems governed by the forward–backward stochastic differential equations, where both the drift and diffusion terms are allowed to depend on controls and the control domain is not assumed to be convex. In the previous work (Theorem 3.1) of the second and third authors, some problem of near optimal control with the control dependent diffusion is addressed and our current paper can be viewed as some direct response to it. The necessary condition of the near-optimality is established within the framework of optimality variational principle developed by Yong and obtained by the convergence technique to treat the optimal control of FBSDEs in unbounded control domains by Wu. Some new estimates are given here to handle the near optimality. In addition, an illustrating example is discussed as well.     

A numerical algorithm based on a variational iterative approximation for the discrete Hamilton–Jacobi–Bellman (HJB) equation

This paper presents a numerical algorithm based on a variational iterative approximation for the Hamilton–Jacobi–Bellman equation, and a domain decomposition technique based on this algorithm is also studied. The convergence theorems have been established. Numerical results indicate the efficiency and accuracy of the methods.     

A heuristic approach to optimal control†

Considerable difficulties have been encountered in the direct application of variational methods to control systems optimization because two-point boundary problems are involved which cannot normally be solved in real time. In this paper a method is described in which, by the use of heuristic computation techniques, this limitation is overcome.     

Homotopy solution of polynomial equations in H∞ optimal control

A method is described to solve the ‘standard’ H_∞ optimal control problem polynomially. The polynomial equations for equalizing solutions are of a homotopy type, so that the equations may be transformed accordingly to a set of non-linear ordinary differential equations.     

Jacobi wavelet collocation method for the modified Camassa–Holm and Degasperis–Procesi equations

Matrices representations of integrations of wavelets have a major role to obtain approximate solutions of integral, differential and integro-differential equations. In the present work, operational matrix representation of rth integration of Jacobi wavelets is introduced and to find these operational matrices, all details of the processes are demonstrated for the first time. Error analysis of offered method is also investigated in present study. In the planned method, approximate solutions are constructed with the truncated Jacobi wavelets series. Approximate solutions of the modified Camassa–Holm equation and Degasperis–Procesi equation linearized using quasilinearization technique are obtained by presented method. Applicability and accuracy of presented method is demonstrated by examples. The proposed method is also convergent even when a minor number of grid points. The numerical results obtained by offered technique are compatible with those in the literature.

     

Invariant imbedding and optimal feedback control†

The computational advantages of the invariant imbedding techniques for converting two-point boundary-value problems to initial-value problems are well known.—In this note an application of these techniques to the derivation of optimal feedback controllers is indicated.