一类三参数超越方程稳定区域的界面位置

本文讨论一类三参数超越方程根的分布．Ｂｅｌｌｍａｎ和 Ｃｏｏｋｅ在文［３］中对部分情况ａ＞０，ｂ≥０给出了结果．但这个结果是分析的，不便于应用． 本文对这类方程在整个参数空间（ａ，ｂ，ｃ）中给出根在左半平面的条件，并给出稳定区域的大致形状及边界曲面的大致位置的图形．应用上较为方便，特别是在分支理论中．     

Stability analysis of impulsive fractional differential systems with delay

In this paper, a class of impulsive fractional differential systems with finite delay is considered. Some sufficient conditions for the finite-time stability of above systems are obtained by using generalized Bellman–Gronwall’s inequality, which extend some known results.     

Primal-dual methods for the computation of trading regions under proportional transaction costs

Portfolio optimization problems on a finite time horizon under proportional transaction costs are considered. The objective is to maximize the expected utility of the terminal wealth. The ensuing non-smooth time-dependent Hamilton–Jacobi–Bellman equation is solved by regularization and the application of a semi-smooth Newton method. Discretization in space is carried out by finite differences or finite elements. Computational results for one and two risky assets are provided.     

Max-Plus Stochastic Processes

This paper is concerned with processes which are max-plus counterparts of Markov diffusion processes governed by Ito sense stochastic differential equations. Concepts of max-plus martingale and max-plus stochastic differential equation are introduced. The max-plus counterparts of backward and forward PDEs for Markov diffusions turn out to be first-order PDEs of Hamilton–Jacobi–Bellman type. Max-plus additive integrals and a max-plus additive dynamic programming principle are considered. This leads to variational inequalities of Hamilton–Jacobi–Bellman type.     

Hamilton–Jacobi–Bellman equations on time scales

In this paper, we consider a class of optimal control problems on time scales without state constraints, target conditions or the fixed terminal time. We first present and show a time scale version of the Bellman optimality principle. On this basis, using a chain rule of multivariables on time scales, we will derive Hamilton–Jacobi–Bellman equations on a time scale for these kind of optimal control problems. Finally, the quantum time scale is considered as an example to illustrate our results.     

On the general inverse problem of optimal control theory

The general inverse problem of optimal control is considered from a dynamic programming point of view. Necessary and sufficient conditions are developed which two integral criteria must satisfy if they are to yield the same optimal feedback law, the dynamics being fixed. Specializing to the linear-quadratic case, it is shown how the general results given here recapture previously obtained results for quadratic criteria with linear dynamics.Dedicated to R. Bellman     

Optimal control of diffusion processes and hamilton–jacobi–bellman equations part 2 : viscosity solutions and uniqueness

We consider general optimal stochastic control problems and the associated Hamilton–Jacobi–Bellman equations. We develop a general notion of week solutions – called viscosity solutions – of the amilton–Jocobi–Bellman equations that is stable and we show that the optimal cost functions of the control problems are always solutions in that sense of the Hamilton–Jacobi–Bellman equations. We then prove general uniqueness results for viscosity solutions of the Hamilton–Jacobi–Bellman equations.     

Properties of the Bellman Function in Time-Optimal Control Problems

In this paper, time-optimal control problems with closed terminal sets are considered. We give conditions which guarantee the Bellman function to be Hölder and Lipschitz continuous. We then prove that the condition for Lipschitz continuity is also necessary.     

基于内部竞争的保险公司的风险管理

讨论了具有内部竞争的保险公司的风险管理问题,保险公司的目标是:保险公司用于(股东)分红的净收益的期望现值最大。根据B e llm an最优性原理,得出了分红情况下的B e llm an偏微分方程,通过对所得方程的分析给出了解析解和最有控制策略。     

Numerical solution of optimization problems in vibrational mechanics using the characteristics method

Optimal control problems with a terminal pay-off functional are considered. The dynamics of the control system consists of rapid oscillatory and slow non-linear motions. A numerical method for solving these problems using the characteristics of the Hamilton–Jacobi–Bellman equation is presented. Estimates of the accuracy of the method are obtained. A theorem is proved which enables one to determine the class of functions containing the optimal preset control to be obtained. The results of the numerical solution of a terminal optimization problem for a fast non-linear pendulum are presented.     

A generalized cubic spline technique for identification of multivariable systems

The application of cubic splines to the identification of time-invariant systems is considered. The use of splines, initially proposed by Bellman in 1971, has been extended to the multidimensional case. In addition, the effects of noise on the identification procedure are considered and techniques are presented for improving the identification accuracy. A general spline technique, used in conjunction with a Kalman estimation procedure, has been developed for identifying physical systems described by a set of first-order differential equations. This method has been found to be superior to the exponential fitting technique proposed by Prony and to other finite-difference methods.     

Equations of attainable set dynamics,part 2: Partial differential equations

We derive three partial differential equations describing the attainable set dynamics from the local integral funnel equation. They can be considered as new partial differential equations for optimal control. The Bellman equation is a special case of one of them. Three examples are given.     

Solution of the Feedback Control Problem in the Mathematical Model of Leukaemia Therapy

A mathematical model of leukaemia therapy based on the Gompertzian law of cell growth is investigated. The effect of the medicine on the leukaemia and normal cells is described in terms of therapy functions. A feedback control problem with the purpose of minimizing the number of the leukaemia cells while retaining as much as possible the number of normal cells is considered. This problem is reduced to solving the nonlinear Hamilton–Jacobi–Bellman partial differential equation. The feedback control synthesis is obtained by constructing an exact analytical solution to the corresponding Hamilton–Jacobi–Bellman equation.     

Construction of a generalized solution to an equation that preserves the Bellman type in a given domain of the state space

A Cauchy problem is considered for a Hamilton-Jacobi equation that preserves the Bellman type in a spatially bounded strip. Sufficient conditions are obtained under which there exists a continuous generalized (minimax/viscosity) solution to this problem with a given structure in the strip. A construction of this solution is presented.     

The Bellman function of the dyadic maximal operator related to Kolmogorov’s inequality

We precisely evaluate the Bellman function of two variables of the dyadic maximal operator related to Kolmogorov’s inequality, thus giving an alternative proof of the results in [3]. Additionally, we characterize the sequences of functions that are extremal for this Bellman function. More precisely, we prove that they behave approximately like eigenfunctions of the dyadic maximal operator, for a specific eigenvalue.     

Asymmetric Wiener-Poisson control

A one-dimensional Wiener plus independent Poisson noise control problem, with asymmetric control bounds and integral discounted quadratic cost over an infinite horizon, is considered. The resultant Bellman equations are solved, allowing the optimal control to be expressed explicitly in closed-loop form.     

Optimal control problems with terminal functionals represented as the difference of two convex functions

Two control problems for a state-linear control system are considered: the minimization of a terminal functional representable as the difference of two convex functions (d.c. functions) and the minimization of a convex terminal functional with a d.c. terminal inequality contraint. Necessary and sufficient global optimality conditions are proved for problems in which the Pontryagin and Bellman maximum principles do not distinguish between locally and globally optimal processes. The efficiency of the approach is illustrated by examples.     

Periodic solutions of the hybrid system with small parameter

In this paper we investigate the existence and stability of the periodic solutions of a quasilinear differential equation with piecewise constant argument. The continuous and differentiable dependence of the solutions on the parameter and the initial value is considered. A new Gronwall–Bellman type lemma is proved. Appropriate examples are constructed.     

Some Results on Risk-Sensitive Control with Full Observation

The Bellman equation of the risk-sensitive control problem with full observation is considered. It appears as an example of a quasi-linear parabolic equation in the whole space, and fairly general growth assumptions with respect to the space variable x are permitted. The stochastic control problem is then solved, making use of the analytic results. The case of large deviation with small noises is then treated, and the limit corresponds to a differential game. Accepted 25 March 1996     

On the Rate of Convergence of Difference Approximations for Uniformly Nondegenerate Elliptic Bellman’s Equations

We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman’s equations is of order at least h ^2/3, where h is the mesh size. The equations are considered in smooth bounded domains.