A control operator and some of its applications
type: Item
instance of: scholarly article
title: A control operator and some of its applications
zbMATH Open document ID: 0553.49022
DOI: 10.1016/0022-247X(84)90154-9
author: M. A. Ibiejugba
author: P. Onumanyi
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: This paper develops a conjugate gradient method algorithm for solving the optimal control problem of the linear system \(x'(t)=Cx(t)+Du(t)\), \(0\leq t\leq \sigma\) with an integral quadratic cost \[ \int^{\sigma}_{0}(x^ T(t)Qu(t)+u^ T(t)Ru(t)\,dt, \] where \(x\in {\mathbb R}^ n\), \(u\in {\mathbb R}^ q\), \(C\in L({\mathbb R}^ n)\), \(D\in L({\mathbb R}^ q,{\mathbb R}^ n)\) and \(Q\) and \(R\) are symmetric, positive definite, square matrices of dimensions \(n\) and \(q\), respectively. The control operator, proposed in this paper, is defined by \(A\in L(K)\), \(K=H^ 1([0,\sigma];{\mathbb R}^ n)\times L^ 2([0,\sigma];{\mathbb R}^ q)\) and \[ \langle z,Az\rangle_ K=J(x,u,\mu)=\int^{\sigma}_{0}(x^ T(t)Qx(t)+u^ T(t)Ru(t))\,dt+\mu \int^{\sigma}_{0}\| x'(t)-Cx(t)- Du(t)\|^ 2\,dt, \] where \(\mu >0\) is the penalty constant. If \(A\) can be determined explicitly, then the conjugate gradient method algorithm is applicable. The construction of the control operator \(A\) is stated by several equalities. Thus, an extended conjugate gradient method algorithm can be working. Some further applications of the control operator are discussed, too.
Mathematics Subject Classification ID: 90C52
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 90C20
zbMATH DE Number: 3881369
zbMATH Keywords: conjugate gradient method algorithm
zbMATH Keywords: integral quadratic cost
MaRDI profile type: Publication
cites work: Methods of conjugate gradients for solving linear systems
cites work: Iterative methods for solving linear equations
cites work: Q5832645
cites work: Function minimization by conjugate gradients
cites work: The Conjugate Gradient Method for Linear and Nonlinear Operator Equations
cites work: Q5606154
cites work: A newton-type computing technique for optimal control problems
cites work: Q4751263
cites work: On the Ritz penalty method for solving the control of a diffusion equation
cites work: Software for a method of finite approximations for the numerical solution of differential equations
cites work: Numerical solution of high order boundary value problems for ordinary differential equations with an estimation of the error
description: scientific article; zbMATH DE number 3881369