A discrete orthogonal polynomials approach for fractional optimal control problems with time delay

Iranian Journal of Numerical Analysis and Optimization

مقاله 4، دوره 9، شماره 2 - شماره پیاپی 16، دی 2019، صفحه 49-75   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v9i2.73622

نویسنده

F. Mohammadi^*

University of Hormozgan

چکیده

An efficient direct and numerical method has been proposed to approximate a solution of time-delay fractional optimal control problems. First, a class of discrete orthogonal polynomials, called Hahn polynomials, has been introduced and their properties are investigated. These properties are employed to derive a general formulation of their operational matrix of fractional integration, in the Riemann–Liouville sense. Then, the fractional derivative of the state function in the dynamic constraint of time-delay fractional optimal control problems is approximated by the Hahn polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration time-delay fractional optimal control prob lems into an algebraic system. Some illustrative examples are given and the obtained numerical results are compared with those previously published in the literature.

کلیدواژه‌ها

Delay fractional optimal control problems؛ Riemann-Liouville integration؛ Hahn polynomials؛ Operational matrix

مراجع

1. Aiello, W.G., Freedman, H.I. and Wu, J. Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52(3), 855–869.

2. Asli, B.H.S. and Flusser, J. New discrete orthogonal moments for signal analysis, Signal Proc., 141 (2017), 57–73.

3. Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. Fractional calculus: Models and numerical methods, World Scientific, 2016.

4. Baleanu, D., Maaraba, T. and Jarad, F. Fractional variational principles with delay, J. Phys. A 41 (31) (2008), 315403, 8 pp.

5. Banks, H.T.J. and Burns, A. Hereditary control problems: Numerical methods based on averaging approximations, SIAM J. Control Optimization, 16(2) (1978), 169–208.

6. Basin, M. and Rodriguez-Gonzalez, J. Optimal control for linear systems with multiple time delays in control input, IEEE Trans. Autom. Control., 51(1)(2006), 91–97.

7. Bhrawy, A.H., and Ezz-Eldien, S.S. A new Legendre operational technique for delay fractional optimal control problems, Calcolo. 53(4) (2016), 521–543.

8. Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T. Spectral methods in fluid dynamics, Springer, 1988.

9. Cattani, C., Guariglia, E., and Wang, S. On the Critical Strip of the Riemann zeta Fractional Derivative, Fund. Inform. 151, (1-4) (2017), 459–472.

10. Chen, C.L., Sun, D.Y. and Chang, C.Y. Numerical solution of time delayed optimal control problems by iterative dynamic programming, Optimal Control Appl. Methods 21(3) (2000 ), 91–105.

11. Dadebo, S. and Luus, R. Optimal control of time-delay systems by dynamic programming, Optimal Control Appl. Methods 13(1) (1992), 29–41.

12. Dehghan, M., Abbaszadeh, M. and Deng, W. Fourth-order numerical method for the space-time tempered fractional diffusion-wave equation, Appl. Math. Lett. 73 (2017), 120–127.

13. Evans, D.J. and Raslan, K.R. The Adomian decomposition method for solving delay differential equation, Int. J. comput. Math. 82(2005), 49–54.

14. Ghomanjani, F., Farahi, M.H. and Gachpazan, M. Optimal control of time-varying linear delay systems based on the Bezier curves, Comput. Appl. Math. 33(3) (2014), 687–715.

15. Goertz, R. and Offner, P. ¨ On Hahn polynomial expansion of a continuous function of bounded variation, arXiv:1610.06748 (2016).

16. Goertz, R. and Offner, P. ¨ Spectral accuracy for the Hahn polynomials, arXiv:1609.07291 [math.NA].

17. Haddadi, N., Ordokhani, Y. and Razzaghi, M. Optimal control of delay systems by using a hybrid functions approximation, J. Optim. Theory Appl. 153(2) (2012), 338–356.

18. Hahn, W. Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, (German) Math. Nachr. 2, (1949). 4–34.

19. Hwang, C. and Chen, M.Y. Analysis of time-delay systems using the Galerkin method, Int. J. Control 44 (1986) 847–866.

20. Jajarmi, A. and Baleanu, D. Suboptimal control of fractional-order dynamic systems with delay argument, J. Vib. Control 24(12) (2018), 2430–2446.

21. Jajarmi, A. and Hajipour, M. An efficient recursive shooting method for the optimal control of time-varying systems with state time-delay, Appl. Math. Model., 40(4) (2016), 2756–2769.

22. Jajarmi, A. and Hajipour, M. An efficient finite difference method for the time-delay optimal control problems with time-varying delay, Asian J. Control, 19(2) (2017), 1–10.

23. Jamshidi, M. and Wang, C.M. A computational algorithm for large scale nonlinear time-delay systems, IEEE Trans. Systems Man Cybernet, (1984) 14, 2–9.

24. Karlin, S. and McGregor, J.L. The Hahn polynomials, formulas, and an application, Scripta Math., 26 (1961), pp. 33–46.

25. Khellat, F. Optimal control of linear time-delayed systems by linear Legendre multiwavelets, J. Optim. Theory Appl., 143(1)(2009), 107–121.

26. Kuang, Y. Delay differential equations: with applications in population dynamics, Academic Press; 1993.

27. Malek-Zavarei, M. and Jamshidi, M. Time-delay systems: Analysis, optimization and applications, Elsevier Science Ltd, New York, 1987.

28. Marzban, H.R. and Razzaghi, M. Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials, J. Franklin Inst. 341(3) (2004), 279–293.

29. Marzban, H.R. and Razzaghi, M. Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series, J Sound. Vibration, 292 (2006), 954–963.

30. Mohammadi, F., Hosseini, M.M., and Mohyud-Din, S.T. Legendre wavelet galerkin method for solving ordinary differential equations with non-analytic solution, Internat. J. Systems Sci. 42(4) (2011), 579–585.

31. Mohammadi, F. and Mohyud-Din, S.T. A fractional-order Legendre collocation method for solving the Bagley-Torvik equations, Adv. Difference Equ. 2016(269) (2016), 14 pp.

32. Oldham, K.B. and Spanier, J. The fractional calculus, Academic Press, New York, 1974.

33. Palanisamy, K.R. and Rao, G.P. Optimal control of linear systems with delays in state and control via Wash functions, IEE Proc. 130(6) (1983), 300–312.

34. Rabiei, K., Ordokhani, Y. and Babolian, E. Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems, J. Vib. Control, 24(15) (2018), 3370–3383.

35. Rahimkhani, P., Ordokhani, Y. and Babolian, E. An efficient approximate method for solving delay fractional optimal control problems, Non linear Dynam. 86(3) (2016), 1649–1661.

36. Saeed, U. and Rehman, M.U. Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. (2014), 1–8.

37. Safaie, E., Farahi, M.H. and Farmani-Ardehaie, M. An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials, Comput. Appl. Math. 34(3) (2015), 831–846.

38. Samko, S.G., Kilbas, A.A. and Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Langhorne, 1993.

39. Sedaghat, S., Ordokhani, Y. and Dehghan, M. Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials, Commun. Nonl. Sci. Numer. Simul. 17 (2012), 4815–4830

40. Srivastava, H.M., Kumar, D. and Singh, J. An efficient analytical tech nique for fractional model of vibration equation, Appl. Math. Mod. 45 (2017), 192–204.

41. Wang, X.T. Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials, Appl. Math. Comput. 184(2)(2007), 849–856.

42. Wilson, M.W. On the Hahn Polynomials, SIAM J. Math. Anal., 1(1) (1970), 131–139.

43. Xiu, D. and Karniadakis, G.E. The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J Sci. comput. 24(2) (2002), 619–644.

44. Yang, Y. and Huang, Y. Spectral-collocation methods for fractional pan tograph delay-integrodifferential equations, Adv. Math. Phys. (2013), 1–14.

45. Yu, Z.H. Variational iteration method for solving the multi-pantograph delay equation, Phys. Lett. A 372 (2008), 6475–6479.