A new forecasting method of discrete dynamic system

Usually, a linear differential equation is used to represent continuous dynamic systems, but a linear difference equation is used to represent discrete dynamic systems. AGO is one of the most important characteristics of grey theory, and its main purpose is to reduce the random of data. A linear differential equation, instead of a linear difference equation, is used to replace the grey differential equation to analyze discrete systems in this paper. The k-order derivatives of 1-AGO data are calculated after cubic spline interpolation of them, and the model parameters are estimated by means of the deterministic convergence scheme. ARIMA models are used to analyze the leading indicator in advance, and Fourier series with suitably chosen values of parameters is used for fitting the leading indicator. The model presented in this paper is called Grey Dynamic Model GDM(1,1,1).     

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton–Jacobi–Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear–quadratic model.     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.     

Dynamical System Method for Solving Ill-Posed Operator Equations

Two dynamical system methods are studied for solving linear ill-posed problems with both operator and right-hand nonexact. The methods solve a Cauchy problem for a linear operator equation which possesses a global solution. The limit of the global solution at infinity solves the original linear equation. Moreover, we also present a convergent iterative process for solving the Cauchy problem.     

Smooth boundary integration method in a linear two-dimensional problem of incompressible fluid and solid

This paper presents a new application of a theoretical and computational method of smooth boundary integration which belongs to the methods of boundary integral equations. Smooth integration is not a method of approximation. In its final analytical form, a smooth-kernel integral equation is computerized easily and accurately.

Smooth integration is associated with a “pressure-vorticity” formulation which covers linear problems in elasticity and fluid mechanics. The solution presented herein is essentially the same as that reported in an earlier paper for regular elasticity. The constraint of incompressibility does not cause difficulties in the pressure-vorticity formulation.

The linear fluid mechanics problem formulated and solved in this paper covers Stokes' problem of a slow viscous flow, and has a wider interpretation. The translational inertia forces are incorporated in the linear problem, as in Euler's dynamic theory of inviscid flow. The centrifugal inertia forces are left for the non-linear problem. The linear problem is perceived as a step in solution of the non-linear problems.     

Projected equation methods for approximate solution of large linear systems

We consider linear systems of equations and solution approximations derived by projection on a low-dimensional subspace. We propose stochastic iterative algorithms, based on simulation, which converge to the approximate solution and are suitable for very large-dimensional problems. The algorithms are extensions of recent approximate dynamic programming methods, known as temporal difference methods, which solve a projected form of Bellman’s equation by using simulation-based approximations to this equation, or by using a projected value iteration method.     

IRREVERSIBLE INVESTMENT AND OPTIMAL FOREST EXPLOITATION

A forest harvest scheduling model which includes as activities the level of investment in harvest capacity and the accumulated harvest capacity in each period, is presented. The inclusion of these activities, in addition to the harvest activities, allows for the removal of harvest-flow constraints found in more typical Model II formulations of the harvest scheduling problem. The optimal harvest and investment policy can be determined by linear programming or quadratic programming methods, depending on whether prices are constant or supply-dependent. The new model better reflects economic reality than existing models, and provides a method for determining the optimal economic development of a forest industry, and the optimal draw-down of old growth forest. Numerical examples are given.     

The Cauchy Problem for the Kadomtsev–Petviashili II Equation with Nondecaying Data along a Line

The initial value problem for the Kadomstev–Petviashvili II (KPII) equation is considered with given data that are nondecaying along a line. The associated direct and inverse scattering of the two-dimensional heat equation is constructed. The direct problem is formulated in terms of a bounded Green's function. The inverse data are decomposed into scattering data along the line and     data from the decaying portion of the potential. The solution of the KPII equation is then obtained via coupled linear integral equations.     

Mathieu equation with application to analysis of dynamic characteristics of resonant inertial sensors

In this paper, Mathieu equation is applied to analyze the dynamic characteristics of resonant inertial sensors. Unlike previous work, Mathieu equation is not just a differential equation and analyzes the stability of the transition curves, but become an important method in analyzing parametric resonant characteristics and approximate output of resonant inertial sensors. It is demonstrated that the mathematical model of resonant inertial sensors is described by Mathieu equation. The relevant Mathieu equation theory and dynamic characteristics analysis methods were proposed, which include both stability and dynamic linear output. Finally, theoretical and experimental analysis show that the correlation of the theoretical curve and the experimental result coincide so perfectly, which means proposed analysis methods for Mathieu equation could be used to analyze the dynamic output characteristic of resonant inertial sensors. The theoretical analyzing approach of Mathieu equation and experimental results of resonant inertial sensors are obtained, which provide an application area for Mathieu equation and a reference for the robust design for resonant inertial sensors.     

Boundary conditions of the second-order differential equation and the Riccati equation

The conversion of a second-order linear ordinary differential equation with variable coefficients into a Riccati equation depends on whether the second-order problem is an initial-value or two-point boundary-value problem. The distinction is critical in determining the initial condition for the Riccati equation. If the second-order problem is an initial-value problem, the choice of the Riccati transformation depends on whether a zero initial condition for the function or its derivative is specified. If the problem is a two-point boundary-value problem, special methods must be introduced as described in the paper.     

The Beverton–Holt dynamic equation

The Cushing–Henson conjectures on time scales are presented and verified. The central part of these conjectures asserts that based on a model using the dynamic Beverton–Holt equation, a periodic environment is deleterious for the population. The proof technique is as follows. First, the Beverton–Holt equation is identified as a logistic dynamic equation. The usual substitution transforms this equation into a linear equation. Then the proof is completed using a recently established dynamic version of the generalized Jensen inequality.     

A dynamic boundary-value problem without initial conditions for almost linear parabolic equations

We study a dynamic boundary-value problem without initial conditions for linear and almost linear parabolic equations. First, we establish conditions for the existence of a unique solution of a problem without initial conditions for a certain abstract implicit evolution equation in the class of functions with exponential behavior as t → −∞. Then, using these results, we prove the existence of a unique solution of the original problem in the class of functions with exponential behavior at infinity.     

常系数线性常微分方程组的显式解

本文利用张量分析给出了常系数线性常微分方程组和n阶常系数线性常微分方程初值问题一般解的显式表示,包括特征根有重根时的情况.实际上本文给出了计算矩阵exp[At]的元素的一般公式.这种方法不仅在公式表示上简洁方便,而且更适用于计算机的程序设计,大大加快了运算速度.     

伊藤-泊松型随机微分方程的线性二次控制

本文研究伊藤-泊松型随机微分方程的线性二次控制问题,利用动态规划方法、伊藤公式等技巧,通过解HJB方程,我们得到了随机Riccati方程及另外两个微分方程,求出控制变量,解决了线性二次最优控制最优问题.     

The infinite-dimensional Riccati equation

The infinite-dimensional versions of the linear quadratic cost control problem and of the linear filtering problem lead to an infinite-dimensional Riccati equation with unbounded operators. Existence and uniqueness theorems for mild solutions of this Riccati equation are established using a sequential approach based on approximating controllers for the linear quadratic cost control problem. By using a semigroup and evolution equation approach, the results are valid for a large class of unbounded operators and, hence, are applicable to many linear infinite-dimensional control systems.     

Indefinite LQ optimal control with cross term for discrete‐time uncertain systems

Recently, there has been an increasing interest in the study on uncertain optimal control problems. In this paper, a linear quadratic (LQ) optimal control with cross term for discrete‐time uncertain systems is considered, whereas the weighting matrices in the cost function are allowed to be indefinite. Firstly, a recurrence equation for the problem is presented based on Bellman's principle of optimality in dynamic programming. Then, a necessary condition for the existence of an optimal linear state feedback control of the indefinite LQ problem is given by the recurrence equation. Moreover, a sufficient condition of well‐posedness for the indefinite LQ problem is presented by introducing a linear matrix inequality (LMI) condition. Furthermore, it is shown that the well‐posedness of the indefinite LQ problem, the solvability of the indefinite LQ problem, the LMI condition, and the solvability of the constrained difference equation are equivalent to each other. Finally, an example is presented to illustrate the results obtained.     

Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs

We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimisation problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a state constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general class of Lévy processes. Owing to the complexity of our investment-consumption model, it is not possible to derive closed form solutions for the value function. Hence, the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore, we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large class of numerical methods for the investment-consumption model in question.     

Existence for an abstract control problem in banach spaces

This paper is concerned with existence in L¹ for a control problem with a lower semicontinuous cost functional associated with a general linear process of causal type in Banach space.

The state system considered includes several standard infinite dimensional systems such as: linear evolution equation, wave equation, linear functional differentional equation. In particular, linear distributed parameter systems are included. The existence results are consequences of an intricate analysis regarding the lower semicontinuity and compactness of level sets of Bolza functional in L¹ spaces. All these results are given under minimal requirements and using sharp functional methods. Specialized to finite dimensional problems the results given here extend some recent results of Ioffe and Rockafellar, but the proofs are quite different.     

反五对角与拟反五对角方程组的追赶法

本文研究了反五对角和拟五对角线性方程组的求解问题.利用矩阵分解方法以及将系数矩阵A分解成三个简单矩阵的乘积A=LUD,获得了反五对角线性方程组以及拟反五对角线性方程组的追赶法,从而推广了对角型线性方程组追赶法.     

等式约束非线性控制系统的时程精细计算

针对等式约束非线性最优控制问题,通过一阶Taylor级数展开,得到线性化的动力学方程,进而在方程原变量的基础上,引入对偶向量(Lagrange乘子向量),将动力学方程从Lagrange体系引入到了Hamilton体系,在全状态下,从一个新的角度对等式约束非线性控制问题进行了描述,进一步基于时程精细积分理论,对其方程进行了有效的精细求解,并通过算例说明了文中方法的有效性。