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Solving optimal control problems using the Picard’s iteration method

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ssue RAIRO-Oper. Res.

Volume 54, Number 5, September-October 2020 Page(s) 1419 - 1435

DOI https://doi.org/10.1051/ro/2019057

Published online 23 July 2020

RAIRO-Oper. Res. 54 (2020) 1419–1435

Solving optimal control problems using the Picard’s

iteration method

Abderrahmane Akkouche1,2* and Mohamed Aidene2

1 Département de Mathématiques, Faculté des Sciences et des Sciences Appliquées, Université Akli Mohand Oulhadj de Bouira, 10 000 Bouira, Algeria

2 Laboratoire de Conception et Conduite des Systèmes de Production Université Mouloud Mammeri de Tizi-Ouzou, 15 000 Tizi-Ouzou, Algeria

* Corresponding author: akkouche.abdo@yahoo.fr Received: 13 June 2018 Accepted: 27 May 2019 Abstract

In this paper, the Picard’s iteration method is proposed to obtain an approximate analytical solution for linear and nonlinear optimal control problems with quadratic objective functional. It consists in deriving the necessary optimality conditions using the minimum principle of Pontryagin, which result in a two-point-boundary-value-problem (TPBVP). By applying the Picard’s iteration method to the resulting TPBVP, the optimal control law and the optimal trajectory are obtained in the form of a truncated series. The efficiency of the proposed technique for handling optimal control problems is illustrated by four numerical examples, and comparison with other methods is made.

Mathematics Subject Classification: 49J15 / 93C15

Key words: Optimal control / Pontryagin’s minimum principle / Hamilton–Pontryagin equations / Picard’s iteration method / ordinary differential equations

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