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Semidefinite Optimization Estimating Bounds on Linear Functionals Defined on Solutions of Linear ODEs

Part of:Smooth dynamical systems: general theoryPartial differential equations

Published online by Cambridge University Press:  27 May 2016

Guangming Zhou,
Chao Deng and
Kun Wu
Guangming Zhou*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, ChinaHunan Key Laboratory for Computation and Simulation in Science and Engineering, Hunan 411105, China
Chao Deng*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
Kun Wu*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
*
*Corresponding author. Email:zhougmxtu@163.com (G. Zhou), dengchao1988xtu@163.com (C. Deng), wukun1988xtu@163.com (K. Wu)
*Corresponding author. Email:zhougmxtu@163.com (G. Zhou), dengchao1988xtu@163.com (C. Deng), wukun1988xtu@163.com (K. Wu)
*Corresponding author. Email:zhougmxtu@163.com (G. Zhou), dengchao1988xtu@163.com (C. Deng), wukun1988xtu@163.com (K. Wu)

Abstract

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In this paper, semidefinite optimization method is proposed to estimate bounds on linear functionals defined on solutions of linear ordinary differential equations (ODEs) with smooth coefficients. The method can get upper and lower bounds by solving two semidefinite programs, not solving ODEs directly. Its convergence theorem is proved. The theorem shows that the upper and lower bounds series of linear functionals discussed can approach their exact values infinitely. Numerical results show that the method is effective for the estimation problems discussed. In addition, in order to reduce calculation amount, Cheybeshev polynomials are applied to replace Taylor polynomials of smooth coefficients in computing process.

Keywords

Semidefinite optimizationboundlinear functionalordinary differential equation

MSC classification

Secondary:35A24: Methods of ordinary differential equations37C10: Vector fields, flows, ordinary differential equations90C22: Semidefinite programming

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 4 , August 2016, pp. 599 - 615
Copyright
Copyright © Global-Science Press 2016 

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