# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2019035

## A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems

 Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

* Corresponding author: Mohammad Hadi Noori Skandari

Received  November 2017 Revised  February 2019 Published  August 2019

In this paper, we present a new method based on the Clenshaw-Curtis formula to solve a class of fractional optimal control problems. First, we convert the fractional optimal control problem to an equivalent problem in the fractional calculus of variations. Then, by utilizing the Clenshaw-Curtis formula and the Chebyshev-Gauss-Lobatto points, we transform the problem to a discrete form. By this approach, we get a nonlinear programming problem by solving of which we can approximate the optimal solution of the main problem. We analyze the convergence of the obtained approximate solution and solve some numerical examples to show the efficiency of the method.

Citation: Mohammad Hadi Noori Skandari, Marzieh Habibli, Alireza Nazemi. A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019035
##### References:
 [1] M. A. Aboelela, M. F. Ahmed and H. T. Dorrah, Design of aerospace control systems using fractional PID controller, Journal of Advanced Research, 3 (2012), 225-232. Google Scholar [2] A. Alizadeh and S. Effati, An iterative approach for solving fractional optimal control problems, Journal of Vibration and Control, 24 (2018), 18-36. doi: 10.1177/1077546316633391. Google Scholar [3] R. Almeida and D. F. M. Torres, A discrete method to solve fractional optimal control problems, Nonlinear Dynamics, 80 (2015), 1811-1816. doi: 10.1007/s11071-014-1378-1. Google Scholar [4] M. Beschi, F. Padula and A. Visioli, The generalised isodamping approach for robust fractional PID controllers design, International Journal of Control, 90 (2017), 1157-1164. doi: 10.1080/00207179.2015.1099076. Google Scholar [5] A. H. Bhrawy, S. S. Ezz-Eldien, E. H. Doha, M. A. Abdelkawy and D. Baleanu, Solving fractional optimal control problems within a Chebyshev Legendre operational technique, International Journal of Control, 90 (2017), 1230-1244. doi: 10.1080/00207179.2016.1278267. Google Scholar [6] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration Academic, New York-London, 1975. Google Scholar [7] Y. Ding, Z. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Transactions on Control Systems Technology, 20 (2012), 763-769. Google Scholar [8] D. Feliu-Talegon and V. Feliu-Batlle, Improving the position control of a two degrees of freedom robotic sensing antenna using fractional-order controllers, International Journal of Control, 90 (2017), 1256-1281. doi: 10.1080/00207179.2017.1281440. Google Scholar [9] E. Keshavarz, Y. Ordokhani and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, Journal of Vibration and Control, 22 (2016), 3889-3903. doi: 10.1177/1077546314567181. Google Scholar [10] A. Lotfi, M. Dehghan and S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Computers and Mathematics with Applications, 62 (2011), 1055-1067. doi: 10.1016/j.camwa.2011.03.044. Google Scholar [11] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993. Google Scholar [12] C. I. Muresan, A. Dutta, E. H. Dulf, Z. Pinar, A. Maxim and C. M. Ionescu, Tuning algorithms for fractional order internal model controllers for time delay processes, International Journal of Control, 89 (2016), 579-593. doi: 10.1080/00207179.2015.1086027. Google Scholar [13] A. Nemati, Numerical solution of 2D fractional optimal control problems by the spectral method combined with Bernstein operational matrix, International Journal of Control, 91 (2018), 2642-2645. doi: 10.1080/00207179.2017.1334267. Google Scholar [14] M. H. Noori Skandari, A. V. Kamyad and S. Effati, Smoothing approach for a class of nonsmooth optimal control problems, Applied Mathematical Modelling, 40 (2016), 886-903. doi: 10.1016/j.apm.2015.05.014. Google Scholar [15] K. B. Oldham and J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering, Academic Press, New York-London, 1974. Google Scholar [16] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. Google Scholar [17] S. Pooseh, R. Almeida and D. F. M. Torres, A numerical scheme to solve fractional optimal control problems, Conference Papers in Science, Hindawi Publishing Corporation, 2013 (2013), Article ID 165298, 10 pages. doi: 10.1155/2013/165298. Google Scholar [18] K. Rabiei, Y. Ordokhani and E. Babolian, The Boubaker polynomials and their application to solve fractional optimal control problems, Nonlinear Dynamics, 88 (2017), 1013-1026. doi: 10.1007/s11071-016-3291-2. Google Scholar [19] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. Google Scholar [20] N. H. Sweilam, T. M. Al-Ajami and R. H. W. Hoppe, Numerical solution of some types of fractional optimal control problems, The Scientific World Journal, 2013 (2013), 306237. doi: 10.1155/2013/306237. Google Scholar [21] L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598. Google Scholar [22] J. J. Trujillo and V. M. Ungureanu, Optimal control of discrete-time linear fractional-order systems with multiplicative noise, International Journal of Control, 91 (2018), 57-69. doi: 10.1080/00207179.2016.1266520. Google Scholar

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##### References:
 [1] M. A. Aboelela, M. F. Ahmed and H. T. Dorrah, Design of aerospace control systems using fractional PID controller, Journal of Advanced Research, 3 (2012), 225-232. Google Scholar [2] A. Alizadeh and S. Effati, An iterative approach for solving fractional optimal control problems, Journal of Vibration and Control, 24 (2018), 18-36. doi: 10.1177/1077546316633391. Google Scholar [3] R. Almeida and D. F. M. Torres, A discrete method to solve fractional optimal control problems, Nonlinear Dynamics, 80 (2015), 1811-1816. doi: 10.1007/s11071-014-1378-1. Google Scholar [4] M. Beschi, F. Padula and A. Visioli, The generalised isodamping approach for robust fractional PID controllers design, International Journal of Control, 90 (2017), 1157-1164. doi: 10.1080/00207179.2015.1099076. Google Scholar [5] A. H. Bhrawy, S. S. Ezz-Eldien, E. H. Doha, M. A. Abdelkawy and D. Baleanu, Solving fractional optimal control problems within a Chebyshev Legendre operational technique, International Journal of Control, 90 (2017), 1230-1244. doi: 10.1080/00207179.2016.1278267. Google Scholar [6] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration Academic, New York-London, 1975. Google Scholar [7] Y. Ding, Z. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Transactions on Control Systems Technology, 20 (2012), 763-769. Google Scholar [8] D. Feliu-Talegon and V. Feliu-Batlle, Improving the position control of a two degrees of freedom robotic sensing antenna using fractional-order controllers, International Journal of Control, 90 (2017), 1256-1281. doi: 10.1080/00207179.2017.1281440. Google Scholar [9] E. Keshavarz, Y. Ordokhani and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, Journal of Vibration and Control, 22 (2016), 3889-3903. doi: 10.1177/1077546314567181. Google Scholar [10] A. Lotfi, M. Dehghan and S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Computers and Mathematics with Applications, 62 (2011), 1055-1067. doi: 10.1016/j.camwa.2011.03.044. Google Scholar [11] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993. Google Scholar [12] C. I. Muresan, A. Dutta, E. H. Dulf, Z. Pinar, A. Maxim and C. M. Ionescu, Tuning algorithms for fractional order internal model controllers for time delay processes, International Journal of Control, 89 (2016), 579-593. doi: 10.1080/00207179.2015.1086027. Google Scholar [13] A. Nemati, Numerical solution of 2D fractional optimal control problems by the spectral method combined with Bernstein operational matrix, International Journal of Control, 91 (2018), 2642-2645. doi: 10.1080/00207179.2017.1334267. Google Scholar [14] M. H. Noori Skandari, A. V. Kamyad and S. Effati, Smoothing approach for a class of nonsmooth optimal control problems, Applied Mathematical Modelling, 40 (2016), 886-903. doi: 10.1016/j.apm.2015.05.014. Google Scholar [15] K. B. Oldham and J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering, Academic Press, New York-London, 1974. Google Scholar [16] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. Google Scholar [17] S. Pooseh, R. Almeida and D. F. M. Torres, A numerical scheme to solve fractional optimal control problems, Conference Papers in Science, Hindawi Publishing Corporation, 2013 (2013), Article ID 165298, 10 pages. doi: 10.1155/2013/165298. Google Scholar [18] K. Rabiei, Y. Ordokhani and E. Babolian, The Boubaker polynomials and their application to solve fractional optimal control problems, Nonlinear Dynamics, 88 (2017), 1013-1026. doi: 10.1007/s11071-016-3291-2. Google Scholar [19] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. Google Scholar [20] N. H. Sweilam, T. M. Al-Ajami and R. H. W. Hoppe, Numerical solution of some types of fractional optimal control problems, The Scientific World Journal, 2013 (2013), 306237. doi: 10.1155/2013/306237. Google Scholar [21] L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598. Google Scholar [22] J. J. Trujillo and V. M. Ungureanu, Optimal control of discrete-time linear fractional-order systems with multiplicative noise, International Journal of Control, 91 (2018), 57-69. doi: 10.1080/00207179.2016.1266520. Google Scholar
The exact and approximate optimal state for $N = 4$ in Example 6.1
The exact and approximate optimal control for $N = 4$ in Example 6.1
The absolute error of the approximate optimal state in Example 6.1
The absolute error of the approximate optimal control in Example 6.1
The exact and approximate optimal state for $N = 6$ in Example 6.2
The exact and approximate optimal control for $N = 6$ in Example 6.2
The absolute error of approximate optimal state in Example 6.2
The absolute error of the approximate control in Example 6.2
The exact and approximate optimal state for $N = 6$ in Example 6.3
The exact and approximate optimal control for $N = 6$ in Example 6.3
The absolute error of the approximate optimal state in Example 6.3
The absolute error of the approximate optimal control in Example 6.3
The maximum absolute error for $N = 4$ in Example 6.1
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $6.077585\times 10^{-5}$ $2.236463 \times 10^{-5}$ $1.662597\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $2.031847\times 10^{-5}$ $6.593832 \times 10^{-5}$ $4.616561\times 10^{-4}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $6.077585\times 10^{-5}$ $2.236463 \times 10^{-5}$ $1.662597\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $2.031847\times 10^{-5}$ $6.593832 \times 10^{-5}$ $4.616561\times 10^{-4}$
The optimal value of the objective functional for $N = 4$ in Example 6.name-style="western"
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $7.986571 \times 10^{-10}$ $1.200881\times 10^{-8}$ $6.080480\times 10^{-7}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $7.986571 \times 10^{-10}$ $1.200881\times 10^{-8}$ $6.080480\times 10^{-7}$
The maximum absolute error for $N = 6$ in Example 6.2
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $1.128957\times 10^{-3}$ $2.967422 \times 10^{-4}$ $1.393032\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $1.508986\times 10^{-2}$ $5.269326 \times 10^{-3}$ $2.011737\times 10^{-3}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $1.128957\times 10^{-3}$ $2.967422 \times 10^{-4}$ $1.393032\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $1.508986\times 10^{-2}$ $5.269326 \times 10^{-3}$ $2.011737\times 10^{-3}$
The optimal value of the objective function for $N = 6$ in Example 6.2
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $4.699837 \times 10^{-6}$ $5.516687\times 10^{-7}$ $8.165173\times 10^{-8}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $4.699837 \times 10^{-6}$ $5.516687\times 10^{-7}$ $8.165173\times 10^{-8}$
The maximum absolute error for $N = 6$ in Example 6.3
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $2.475186\times 10^{-3}$ $3.593088 \times 10^{-4}$ $1.005558\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $5.873404\times 10^{-2}$ $5.118654 \times 10^{-3}$ $5.399343\times 10^{-3}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $2.475186\times 10^{-3}$ $3.593088 \times 10^{-4}$ $1.005558\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $5.873404\times 10^{-2}$ $5.118654 \times 10^{-3}$ $5.399343\times 10^{-3}$
The optimal value of the objective functional for $N = 6$ in Example 6.3
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $1.080409 \times 10^{-7}$ $2.074337\times 10^{-9}$ $4.013053\times 10^{-9}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $1.080409 \times 10^{-7}$ $2.074337\times 10^{-9}$ $4.013053\times 10^{-9}$
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