An efficient hybrid pseudo-spectral method for solving optimal control of Volterra integral systems

Khosrow Maleknejad; Asyieh Ebrahimzadeh

Vol. 19 No. 2 (2014)

Regular Articles

An efficient hybrid pseudo-spectral method for solving optimal control of Volterra integral systems

Published 2014-10-15

Khosrow Maleknejad
Asyieh Ebrahimzadeh

Khosrow Maleknejad
School of Mathematics‎, ‎Iran University of Science and Technology‎

Asyieh Ebrahimzadeh
School of Mathematics‎, ‎Iran University of Science and Technology‎

Submission of an original manuscript to the Mathematical Communications will be taken to mean that it represents original work not previously published, that it is not being considered elsewhere for publication; and, if accepted for publication, it will be published in print and online form and it will not be published elsewhere in the same form, for commercial purposes, in any language, without the consent of the publisher. The journal takes the stance that the publication of scholarly research is meant to disseminate knowledge and, in a not-for-profit regime, benefits neither publisher nor author financially. It sees itself as having an obligation to its authors and to society to make content available online now that the technology allows for such a possibility.

Abstract

In this paper, a new pseudo-spectral (PS) method is developed for solving optimal controproblems governed by the non-linear Volterra integral equation(VIE). The novel method is based upon approximating the state and control variables by the hybrid of block pulse functions and Legendre polynomials. The properties of hybrid functions are presented. The numerical integration and collocation method is utilized to discretize the continuous optimal control problem and then the resulting large-scale finite-dimensional non-linear programming (NLP) is solved by the existing well-developed algorithm in Mathematica software. A set of sufficient conditions is presented under which optimal solutions of discrete optimal control problems converge to the optimal solution of the continuous problem. The error bound of approximation is also given. Numerical experiments confirm efficiency of the proposed method especially for problems with non-sufficiently smooth solutions belonging to class $C^1$ or $C^2$.

Keywords

Optimal control, hybrid, block-pulse function, Legendre polynomial, Volterra integral equation, collocation method

PDF

Supplementary File(s)

An efficient hybrid pseudo-spectral scheme for solving nonlinear optimal control problems governed by Volterra integral equation An efficient hybrid pseudo-spectral scheme for solving nonlinear optimal control problems governed by Volterra integral equation An efficient hybrid pseudo-spectral scheme for solving nonlinear optimal control problems governed by Volterra integral equation An efficient hybrid pseudo-spectral scheme for solving nonlinear optimal control problems governed by Volterra integral equation LaTeX file

Author Biography

Khosrow Maleknejad

Professor Khosrow

Maleknejad obtained his MSc degree in Applied

Mathematics from Tehran University, Iran, in

1972 and received his PhD degree in Applied

Mathematics in Numerical Analysis area from

the University of Wales, Aberystwyth, UK in

1980. In September 1972, he joined the School

of Mathematics at Iran University of Science and

Technology(IUST), Narmak, Tehran,Iran. Where

he was previously lecturer , assistant Professor

and associate professor at IUST. He has been a

professor since 2002 at IUST. He was a Visiting Professor at the University

of California at Los Angeles in 1990. His research interests include numerical

analysis in solving ill-posed problems and solving Fredholm and Volterra

integral equations. He has authored as a Editor-in-chief of the International

Journal of Mathematical Sciences, which publishers by Springer.He is a

member of the AMS. External link: http://webpages.iust.ac.ir/maleknejad/.

Asyieh Ebrahimzadeh

Asyieh Ebrahimzadeh is a

Ph.D student in Department of Applied Mathematics

at Iran university of Science and Technology, Tehran,

Iran. She obtained M.Sc in Mathematics in 2009

at Khaje Nasir Toosi University of Technology,

Tehran, Iran. She also obtained Bs.C in Mathematics

at ValiAsr University of Rafsanjan, Kerman, Iran.

Her scientific interests includes: Optimal control,

Wavelets, Integral Equations and Orthogonal functions.