Control Problem for a Nonlinear Distributed Equation

For a distributed second-order differential equation, we consider the problem of constructing a control law ensuring that the solution of this equation tracks the solution of a standard equation subjected to an unknown disturbance. A control design algorithm based on constructions of feedback control theory is proposed. The algorithm is stable under information noise and computational errors.     

On tracking solutions of parabolic equations

We consider a control problem for a parabolic equation. It consists in constructing an algorithm for finding a feedback control such that a solution of a given equation should track a solution of another equation generated by an unknown right-hand side. We propose two noise-resistant solution algorithms for the indicated problem. They are based on the method of extremal shift well-known in the guaranteed control theory. The first algorithm is applicable in the case of “continuous” measurements of phase states, whereas the second one implies discrete measurements.     

On a modification of the extremal shift method for a second-order differential equation in a Hilbert space

A problem of tracking a solution of a second-order differential equation in a Hilbert space by a solution of another equation is considered. It is assumed that the first (reference) equation is subject to the action of an unknown control, which is unbounded in time. In the case when the current states of both equations are observed with small errors, a solution algorithm stable with respect to informational noises and computational inaccuracies is designed. The algorithm is based on N.N. Krasovskii’s extremal shift method known in the theory of guaranteed control.     

Tracking of solutions to parabolic equations with memory in a general class of controls

We consider a control problem for a parabolic equation with memory. It consists in constructing an algorithm for finding a feedback control which enables one to track a solution of the given equation with an unknown right-hand side. For this problem we propose two noise-resistant solution algorithms based on the method of extremal shift. The first algorithm is applicable in the case of continuous measurements of phase states, whereas the second one presumes discrete measurements.     

含高次逆幂的矩阵方程对称解的双迭代算法

本文研究了在控制理论和随机滤波等领域中遇到的一类含高次逆幂的矩阵方程的等价矩阵方程对称解的数值计算问题.采用牛顿算法求等价矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立了求这类矩阵方程对称解的双迭代算法,数值算例验证了双迭代算法是有效的.     

On an input recovery problem in a linear delay system

We consider the problem of recovering the input of a linear differential equation with delay and propose a solution algorithm that is stable to perturbations. The algorithm is based on the extremal shift principle known in the theory of guaranteed control.     

Computation of the stabilizing solution of game theoretic Riccati equation arising in stochastic <Emphasis Type="Italic">H</Emphasis><Subscript> ∞ </Subscript> control problems

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H _∞ control problem for a class of stochastic systems affected by state dependent and control dependent white noise. The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.     

Approximation of optimal feedback control: a dynamic programming approach

We consider the general continuous time finite-dimensional deterministic system under a finite horizon cost functional. Our aim is to calculate approximate solutions to the optimal feedback control. First we apply the dynamic programming principle to obtain the evolutive Hamilton–Jacobi–Bellman (HJB) equation satisfied by the value function of the optimal control problem. We then propose two schemes to solve the equation numerically. One is in terms of the time difference approximation and the other the time-space approximation. For each scheme, we prove that (a) the algorithm is convergent, that is, the solution of the discrete scheme converges to the viscosity solution of the HJB equation, and (b) the optimal control of the discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the continuous counterpart. An example is presented for the time-space algorithm; the results illustrate that the scheme is effective.     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

Tracking the Solution to a Nonlinear Distributed Differential Equation by Feedback Laws

In this paper, a nonlinear distributed second order equation is considered. An algorithm for tracking a prescribed solution based on constructions of feedback control theory is designed. The algorithm is stable with respect to information noise and computation errors. It is designed for solutions in a sufficiently large time interval.     

On bi-pointwise control of a wave equation and algorithm

We are concerned with mathematical analysis related to the bi-pointwise control for a mixed type of wave equation. In particular, we are interested in the systematic build-up of the bi-pointwise control actuators; one at the boundary and the other at the interior point simultaneously. The main purpose is to examine Hilbert Uniqueness Method for the setting of bi-pointwise control actuators and to establish relevant algorithm based on our analysis. After discussing the weak solution for the state equation, we investigate bi-pointwise control mechanism and relevant mathematical analysis based on HUM. We then proceed to set up an algorithm based on the conjugate gradient method to establish bi-pointwise control actuators to halt the system.     

On a certain problem on feedback control for a parabolic equation with memory

A game control problem for a parabolic differential equation with memory is considered. An algorithm for its solution based on Krasovskii’s method of extreme shift and the method of stable paths is proposed.     

The Policy Iteration Algorithm for Average Continuous Control of Piecewise Deterministic Markov Processes

The main goal of this paper is to apply the so-called policy iteration algorithm (PIA) for the long run average continuous control problem of piecewise deterministic Markov processes (PDMP’s) taking values in a general Borel space and with compact action space depending on the state variable. In order to do that we first derive some important properties for a pseudo-Poisson equation associated to the problem. In the sequence it is shown that the convergence of the PIA to a solution satisfying the optimality equation holds under some classical hypotheses and that this optimal solution yields to an optimal control strategy for the average control problem for the continuous-time PDMP in a feedback form.     

On identification in fuzzy systems and its applications in control problems

The problem of identification in fuzzy systems described by the use of fuzzy equation is considered. The identification method and its performance index is also presented. The formal procedure of the identification algorithm is illustrated by means of a numerical example. The possibility of using the proposed algorithm for the solution of a control problem is given as well.     

Method of controlled models in the problem of reconstructing a boundary input

The problem of dynamic reconstruction of boundary controls in a nonlinear parabolic equation is considered. In the case of a control concentrated in the Neumann boundary conditions, a solution algorithm is described, which is stable with respect to the information noise and calculation errors. The algorithm is based on the construction of feedback-controlled auxiliary models.     

Hopf bifurcation analysis for congestion control with heterogeneous delays

In this paper, a congestion control algorithm with heterogeneous delays is considered. Local stability of the equilibrium solution of this algorithm is investigated based on analyzing the corresponding transcendental characteristic equation. Especially, using one of the system parameters of congestion control algorithm as the bifurcation parameter, when the system parameter exceeds a critical value, the congestion control algorithm undergoes a supercritical Hopf bifurcation, and the explicit formulae determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al.’s approaches. Finally, some numerical simulations are performed to verify the theoretical results.     

AN OPTIMAL CONTROL MODEL FOR ANALYSIS OF TIMBER RESOURCE UTILIZATION IN SOUTHEAST ASIA

A discrete time optimal forestry model is built and a shooting method solution algorithm identified. The applicability of the model and algorithm to public policies that affect forestry resources is demonstrated in an application of the model to examine the development of wood processing capacity in Southeast Asia. The necessary conditions of the optimal control model are manipulated to identify a difference equation problem with initial and terminal conditions. The solution to this boundary value problem is identified using a search routine that repetitively, numerically evaluates (shoots) the difference equations. The solution is the trajectory that satisfies the initial and terminal conditions.     

Numerical method for a quadratic minimization problem with an ellipsoidal constraint and an a priori estimate for the solution norm

An algorithm for solving a quadratic minimization problem on an ellipsoidal set in a Hilbert space is proposed. The algorithm is stable to nonuniform perturbations of the operators. A key condition for its application is that we know an estimate for the norm of the exact solution. Applications to boundary control problems for the one-dimensional wave equation are considered. Numerical results are presented.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

A generalized structured doubling algorithm for the numerical solution of linear quadratic optimal control problems

We propose a generalization of the structured doubling algorithm to compute invariant subspaces of structured matrix pencils that arise in the context of solving linear quadratic optimal control problems. The new algorithm is designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because the stable and unstable invariant subspaces are not well separated (because of eigenvalues near or on the imaginary axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence of the method and compare the new method with the classical structured doubling algorithm as well as some structured QR methods. Copyright © 2012 John Wiley & Sons, Ltd.