Solution of delay fractional optimal control problems using a hybrid of block pulse functions and orthonormal Taylor polynomials Mohammad Mehdi Soleymani Far m soleymanifar@math iut ac ir January 11 2021 Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Hamid Reza Marzban hmarzban@iut ac ir2000 MSC 93C23 34K37 34A08 26A33 90C30Keywords Delay fractional optimal control operational matrix of fractional integration Riemann Liouville integraloperator Caputo fractional derivative block pulse functions orthonormal Taylor polynomials Abstract Time delays occur so often in almost every situation that to ignore them is to ignore reality Time delays have a greatinfluence on the stability and controllability of the system under consideration As a consequence the presence and thesignificant effects of time delays cannot be ignored Indeed the existence of delay makes the method of solution muchmore complicated The purpose of this thesis is to introduce an efficient numerical scheme for solving delay fractionaloptimal control problems The foundation of the proposed approach is based on a hybrid of block pulse functionsand orthonormal Taylor polynomials Combining the two mentioned bases provides a flexible framework for solvingthe problem under consideration The developed framework is flexible as the order of block pulse functions andthe degree of the orthonormal Taylor polynomials can be chosen arbitrarily The structure of the proposed hybridfunctions is much simpler than the structure of those which have been introduced in the literature It is worth notingthat the analytic solution of a delay optimal control problem is a piecewise smooth function Based on this fact theexact response of an arbitrary delay fractional optimal control problem cannot be obtained solely either by piecewiseconstant basis functions such as block pulse functions or by smooth basis functions Indeed smooth functions likeLegendre polynomials and Bernstein polynomials are appropriate for the approximation of problems whose exactsolutions are infinitely smooth functions The main contributions of the current work are summarized as follows Theoperational matrix of fractional integration corresponding to the proposed hybrid functions is constructed To obtainthis matrix we employ the Laplace transform method This matrix plays an important role for solving the optimalcontrol problem under discussion This matrix is an extension of the operational matrix of integration associatedwith the hybrid of block pulse functions and orthonormal Taylor polynomials in the case of 1 In other words if 1 then the operational matrix of fractional integration denoted by P coincides with the conventional one It is demonstrated that the combination of block pulse functions and orthonormal Taylor polynomials constitutes acomplete set in the Hilbert space L2 0 1 This property enables one to expand any function in the Hilbert spaceL2 0 1 in terms of the hybrid functions An upper error bound for the proposed hybrid functions with respect to theL2 norm is obtained This result verifies that the convergence rate of the hybrid functions is faster than that of theconventional orthonormal Taylor polynomials The convergence of the proposed hybrid functions with respect to theL2 norm is proved Two excellent features of the proposed approximation scheme are explained as follows Firstly prior to applying our method based on the values of time delays we can determine the exact locations of the pointat which behavioral changes arise in delay systems Secondly the proposed approach is capable of providing exactsolution for a delay fractional optimal control problem whose solution is a piecewise polynomial function Severalexamples are tested to eva‎luate the performance of the proposed method

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