Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 536–555

# Numerical Solution of Two-Dimensional Variable-Order Fractional Optimal Control Problem by Generalized Polynomial Basis

• Hossein Hassani
Article

## Abstract

This paper deals with an efficient numerical method for solving two-dimensional variable-order fractional optimal control problem. The dynamic constraint of two-dimensional variable-order fractional optimal control problem is given by the classical partial differential equations such as convection–diffusion, diffusion-wave and Burgers’ equations. The presented numerical approach is essentially based on a new class of basis functions with control parameters, called generalized polynomials, and the Lagrange multipliers method. First, generalized polynomials are introduced and an explicit formulation for their variable-order fractional operational matrix is obtained. Then, the state and control functions are expanded in terms of generalized polynomials with unknown coefficients and control parameters. By using the residual function and its 2-norm, the under consideration problem is transformed into an optimization one. Finally, the necessary conditions of optimality results in a system of algebraic equations with unknown coefficients and control parameters can be simply solved. Some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method.

## Keywords

Two-dimensional variable-order fractional optimal control problem Generalized polynomials Operational matrix Lagrange multipliers Optimization method

## Mathematics Subject Classification

34A08 49J20 41A58 49J21

## References

1. 1.
Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations. In: To Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York (1998)Google Scholar
2. 2.
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Langhorne (1993)
3. 3.
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2016)
4. 4.
Cattani, C., Guariglia, E., Wang, S., Han, L.: On the critical strip of the Riemann zeta fractional derivative. Fundam. Inf. 151(1–4), 459–472 (2017)
5. 5.
Srivastava, M.H., Kuma, D., Singh, H.: An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model. 45, 192–204 (2017)
6. 6.
Jajarmi, A., Baleanu, D.: Suboptimal control of fractional-order dynamic systems with delay argument. J. Vib. Control 24(12), 2430–2446 (2018)
7. 7.
Mohammadi, F., Mohyud-Din, S.T.: A fractional-order Legendre collocation method for solving the Bagley-Torvik equations. Adv. Differ. Equ. 2016(1), 269 (2016)
8. 8.
Dahaghina, MSh, Hassani, H.: A new optimization method for a class of time fractional convection–diffusion-wave equations with variable coefficients. Eur. Phys. J. Plus 132, 130 (2017)
9. 9.
Dehghan, M., Abbaszadeh, M., Deng, W.: Fourth-order numerical method for the space–time tempered fractional diffusion-wave equation. Appl. Math. Lett. 73, 120–127 (2017)
10. 10.
Ezz-Eldien, S.S., Hafez, R.M., Bhrawy, A.H., Baleanu, D., El-Kalaawy, A.A.: New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials. J. Optim. Theory Appl. 174(1), 295–320 (2017)
11. 11.
Bhrawy, A.H., Zaky, M.A., Machado, J.A.T.: Numerical solution of the two-sided space–time fractional telegraph equation via Chebyshev tau approximation. J. Optim. Theory Appl. 174(1), 321–341 (2017)
12. 12.
Fu, Z.J., Chen, W., Yang, H.T.: Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 235, 52–66 (2013)
13. 13.
Wei, S., Chen, W., Hon, Y.C.: Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations. Therm. Sci. 19, 59–67 (2015)
14. 14.
Sweilam, N.H., Khader, M.M., Almarwm, H.M.: Numerical studies for the variable-order nonlinear fractional wave equation. Fract. Calc. Appl. Anal. 15, 669–683 (2012)
15. 15.
Yang, X.J., Machado, J.A.T.: A new fractional operator of variable order: application in the description of anomalous diffusion. Phys. A 481, 276–283 (2017)
16. 16.
Fu, Z.J., Chen, W., Ling, L.: Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Eng. Anal. Bound. Elem. 57, 37–46 (2015)
17. 17.
Dahaghin, MSh, Hassani, H.: An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Nonlinear Dyn. 88(3), 1587–1598 (2017)
18. 18.
Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control. 14(9–10), 1487–1498 (2008)
19. 19.
Zamani, M., Karimi-Ghartemani, M., Sadati, N.: FOPID controller design for robust performance using particle swarm optimization. Fract. Calcul. Appl. Anal. 10(2), 169–187 (2007)
20. 20.
Tripathy, M.C., Mondal, D., Biswas, K., Sen, S.: Design and performance study of phase-locked loop using fractional-order loop filter. Int. J. Circuit Theory Appl. 43(6), 776–792 (2015)
21. 21.
Khader, M.M., Hendy, A.S.: An efficient numerical scheme for solving fractional optimal control problems. Int. J. Nonlinear Sci. 14(3), 287–296 (2012)
22. 22.
Biswas, R.K., Sen, S.: Fractional optimal control problems: a pseudo-state-space approach. J. Vib. Control. 17(7), 1034–1041 (2011)
23. 23.
Jafari, H., Ghasempour, S., Baleanu, D.: On comparison between iterative methods for solving nonlinear optimal control problems. J. Vib. Control. 22(9), 2281–2287 (2016)
24. 24.
Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62(3), 1055–1067 (2011)
25. 25.
Nemati, A., Yousefi, S.A.: A numerical method for solving fractional optimal control problems using Ritz method. J. Comput. Nonlinear Dyn. 11(5), 051015 (2016)
26. 26.
Nemati, A., Yousefi, S., Soltanian, F., Ardabili, J.S.: An efficient numerical solution of fractional optimal control problems by using the Ritz method and Bernstein operational matrix. Asian J. Control 18(6), 2272–2282 (2016)
27. 27.
Rabiei, K., Ordokhani, Y., Babolian, E.: The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dyn. 88(2), 1013–1026 (2017)
28. 28.
Dehghan, M., Hamedi, E.A., Khosravian-Arab, H.: A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vib. Control 22(6), 1547–1559 (2016)
29. 29.
Ejlali, N., Hosseini, S.M.: A pseudospectral method for fractional optimal control problems. J. Optim. Theory Appl. 174(1), 1–25 (2016)
30. 30.
Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M., Cattani, C.: Wavelets method for solving fractional optimal control problems. Appl. Math. Comput. 286, 139–154 (2016)
31. 31.
Doha, E.H., Bhrawy, A.H., Baleanu, D., Ezz-Eldien, S.S., Hafez, R.M.: An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv. Differ. Equ. 1, 1–17 (2015)
32. 32.
Almeida, R., Torres, D.F.: A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80(4), 1811–1816 (2015)
33. 33.
Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S., Abdelkawy, M.A.: An accurate numerical technique for solving fractional optimal control problems. Proc. Roman. Acad. A 16, 47–54 (2015)
34. 34.
Gugat, M., Hante, F.M.: Lipschitz continuity of the value function in mixed-integer optimal control problems. Math. Control Signals Syst. 29, 3 (2017)
35. 35.
Alipour, M., Rostamy, D., Baleanu, D.: Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J. Vib. Control 19(16), 2523–2540 (2013)
36. 36.
Tsai, J.S.H., Li, J., Shieh, L.S.: Discretized quadratic optimal control for continuous-time two-dimensional system. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(1), 116–125 (2002)
37. 37.
Hasan, M.M., Tangpong, X.W., Agrawal, O.P.: Fractional optimal control of distributed systems in spherical and cylindrical coordinates. J. Vib. Control 18, 1506–1525 (2009)
38. 38.
Özdemir, N., Agrawal, O.P., Iskender, B.B., Karadeniz, D.: Fractional optimal control of a 2-dimensional distributed system using eigenfunctions. Nonlinear Dyn. 55, 251–260 (2009)
39. 39.
Nemati, A., Yousefi, S.A.: A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix. IMA J. Math. Control Inf. 34(4), 1079–1097 (2017)
40. 40.
Nemati, A.: Numerical solution of 2D fractional optimal control problems by the spectral method along with Bernstein operational matrix. Int. J. Control. (2017).
41. 41.
Mamehrashi, K., Yousefi, S.A.: A numerical method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation. Int. J. Control 90(2), 298–306 (2017)
42. 42.
Rahimkhani, P., Ordokhani, Y.: Generalized fractional-order Bernoulli-Legendre functions: an effective tool for solving two-dimensional fractional optimal control problems. IMA J. Math. Control Inf. (2017).