A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations

Zahra Pirouzeh; Mohammad Hadi Noori Skandari; Kamele Nassiri Pirbazari; Stanford Shateyi; Zahra Pirouzeh; Mohammad Hadi Noori Skandari; Kamele Nassiri Pirbazari; Stanford Shateyi

doi:10.3934/math.20241151

AIMS Mathematics

2024, Volume 9, Issue 9: 23692-23710. doi: 10.3934/math.20241151

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A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations

1.
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2.
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
3.
Department of Applied Mathematics, University of Venda, P. Bag X5050, Thohoyandu, Limpopo 950, South Africa

Received: 16 February 2024 Revised: 12 June 2024 Accepted: 14 June 2024 Published: 08 August 2024
MSC : 34K37, 49M25, 49M37

Nonlinear optimal control problems governed by variable-order fractional integro-differential equations constitute an important subgroup of optimal control problems. This group of problems is often difficult or impossible to solve analytically because of the variable-order fractional derivatives and fractional integrals. In this article, we utilized the expansion of Lagrange polynomials in terms of Chebyshev polynomials and the power series of Chebyshev polynomials to find an approximate solution with high accuracy. Subsequently, by employing collocation points, the problem was transformed into a nonlinear programming problem. In addition, variable-order fractional derivatives in the Caputo sense were represented by a new operational matrix, and an operational matrix represented fractional integrals. As a result, the mentioned integro-differential optimal control problem becomes a nonlinear programming problem that can be easily solved with the repetitive optimization method. In the end, the proposed method is illustrated by numerical examples that demonstrate its efficiency and accuracy.
Citation: Zahra Pirouzeh, Mohammad Hadi Noori Skandari, Kamele Nassiri Pirbazari, Stanford Shateyi. A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations[J]. AIMS Mathematics, 2024, 9(9): 23692-23710. doi: 10.3934/math.20241151

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Abstract

Nonlinear optimal control problems governed by variable-order fractional integro-differential equations constitute an important subgroup of optimal control problems. This group of problems is often difficult or impossible to solve analytically because of the variable-order fractional derivatives and fractional integrals. In this article, we utilized the expansion of Lagrange polynomials in terms of Chebyshev polynomials and the power series of Chebyshev polynomials to find an approximate solution with high accuracy. Subsequently, by employing collocation points, the problem was transformed into a nonlinear programming problem. In addition, variable-order fractional derivatives in the Caputo sense were represented by a new operational matrix, and an operational matrix represented fractional integrals. As a result, the mentioned integro-differential optimal control problem becomes a nonlinear programming problem that can be easily solved with the repetitive optimization method. In the end, the proposed method is illustrated by numerical examples that demonstrate its efficiency and accuracy.

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