Numerical Solution of Fractional Optimal Control Problems with Inequality Constraint Using the Fractional-Order Bernoull
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RESEARCH PAPER
Numerical Solution of Fractional Optimal Control Problems with Inequality Constraint Using the Fractional-Order Bernoulli Wavelet Functions Forugh Valian1 • Yadollah Ordokhani2 • Mohammad Ali Vali1 Received: 19 October 2019 / Accepted: 24 February 2020 Ó Shiraz University 2020
Abstract This paper studies the fractional optimal control problems (FOCPs) with inequality constraints. Using the Caputo definition, an optimization method based on a set of basis functions, namely the fractional-order Bernoulli wavelet functions (F-BWFs), is proposed. The solution is expanded in terms of the F-BWFs with unknown coefficients. In the first step, we convert the inequality conditions to equality conditions. In the second step, we use the operational matrix (OM) of fractional integration and the product OM of F-BWFs, with the help of the Lagrange multipliers technique for converting the FOCPs into an easier one, described by a system of nonlinear algebraic equations. Finally, for illustrating the efficiency and accuracy of the proposed technique, several numerical examples are analysed and the results compared with the analytical or the approximate solutions obtained by other techniques. Keywords Fractional optimal control problems Fractional-order Bernoulli wavelet functions Operational matrix of fractional integration Product operational matrix Lagrange multipliers
1 Introduction Fractional differential equations (FDEs) occur in the modelling of many phenomena in various fields of science and engineering. Several studies by some researchers (Bagley and Torvik 1985; Kulish and Lage 2002; Oldham 2010; Dahaghin and Hassani 2017; Bhrawy and Zaky 2017; Parsa Moghaddam and Tenreiro Machado 2017; Karamali et al. 2018; Hassani and Naraghirad 2019; Hassani et al. 2019a; Heydari et al. 2019) have shown that many complex physical and engineering problems can be described with great success via FDEs. We refer the interested reader to refer (Tenreiro Machado et al. 2011) for a historical perspective on fractional calculus. Most of FDEs do not have analytic solutions, so approximate and numerical techniques must be used. Several & Yadollah Ordokhani [email protected] 1
Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
2
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
analytical and numerical methods to solve FDEs have been given such as extrapolation method (Diethelm and Walz 1997), Predictor-Corrector method (Diethelm et al. 2002), Adomian decomposition method (Elsayed and Gaber 2006), multistep method (Galeone and Garrappa 2006), Homotopy perturbation method (Odibat et al. 2010), linear B-spline method (Lakestani et al. 2012), product integration method (Garrappa and Popolizio 2012) and wavelets method (Rehman and Khan 2011; Heydari et al. 2013). Optimal control theory is a branch of optimization theory concerned with minimizing a cost or maximizing a pay-off. Optimal control theory has
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