Article Transactions of the Institute of Measurement and Control 1–13 Ó The Author(s) 2020 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/0142331219898343 journals.sagepub.com/home/tim A new operational matrix based on Boubaker wavelet for solving optimal control problems of arbitrary order Kobra Rabiei and Yadollah Ordokhani Abstract This paper presents numerical solution for solving the nonlinear one and two-dimensional optimal control problems of arbitrary order. First, we have constructed Boubaker wavelet for the first time and defined a general formulation for its fractional derivative operational matrix. To solve the one- dimensional problem, we have transformed the problems into an optimization one. The similar process together with the Ritz method are applied to find a solution for two-dimensional problems as well. Then, the necessary conditions of optimality result in a system of algebraic equations with unknown coefficients and then control parameters can be simply solved. The error vector is considered to show the convergence of the used approxi- mation in this method. Finally, some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method. Keywords Boubaker wavelet, nonlinear optimal control problems, two-dimensional optimal control problems, Caputo derivative, operational matrix, convergence analysis Introduction Nowadays, it has been realized that fractional calculus is more appropriate tool to model complex phenomena in science and engineering and has many applications in different fields such as signal processing, electrochemistry, statistical mechanics and viscoelasticity (Baleanu et al., 2016; Cattani et al., 2017; Srivastava et al., 2017). So, careful attention is paid to approx- imate solutions of fractional problems. The application of fractional optimal control problems can be seen in engineering and physics and the aim of solving an optimal control prob- lem is extremizing a cost function over an admissible set of control and state functions. Several numerical methods are applied to find an approximate solution to one-dimensional fractional optimal control problems, such as eigen functions method (Agrawal, 2008), rational approximation method (Tricaud and Chen, 2010), Legendre orthonormal basis method (Lotfi et al., 2013), Legendre operational technique (Bhrawy and Ezz-Eldien, 2016), Bernoulli polynomials method (Rabiei et al., 2018b), Hybrid of block-pulse functions and Bernoulli polynomials (Mashayekhi and Razzaghi, 2018), hybrid Chelyshkov functions (Mohammadi et al., 2018), frac- tional order Lagrange polynomials (Sabermahani et al., 2019), Adomian decomposition method (Alizadeh and Effati, 2018), Chebysheve collocation method (Rabiei and Parand, 2019), GrunwaldLetnikov, trapezoidal and Simpson fractional integral formulas (Salati et al., 2019), low dimensional approx- imations (Peng et al., 2019) and Spectral Galerkin approxima- tion (Zhang and Zhou, 2019). But there are few researches devoted to two-dimensional problem especially in fractional area; for example, the authors in Nemati and Yousefi (2017) used the Ritz method to solve a class of these problems. In (Nemati, 2017), the spectral method and Bernstein operational matrix are applied to solve these problems. Generalized fractional-order Bernoulli-Legendre functions method is pro- posed by Rahimkhani and Ordokhani (2019) and the Jacobi polynomials are used to solve them in Zaky and Tenreiro Machado (2017). Here, we consider both one-dimensional and two-dimensional optimal control problems and try to solve them via Boubaker wavelet. Boubaker polynomials are intro- duced in Boubaker (2007). The Boltzmann diffusion equation was solved using these polynomials in Boubaker (2011) and Kumar (2010) solved the Love’s equation via the Boubaker polynomials expansion scheme. The application of these poly- nomials to solve the mathematical problems in fractional area was presented in Rabiei et al. (2017a). The authors in Rabiei and Ordokhani (2019) and Rabiei et al. (2017b) introduced the fractional Boubaker polynomials and used them to solve the delay fractional optimal control problems and fractional delay differential equations. Boubaker hybrid functions are constructed in Rabiei et al. (2018a) and used to solve the inequality constrained optimal control problems. The application of different wavelets such as Legendre wavelet, Chebyshev wavelet and Bernoulli wavelet shows that Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Iran Corresponding author: Yadollah Ordokhani, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Vanak Street, Tehran, 1993893973, Iran. Email: ordokhani@alzahra.ac.ir