A New Approach for Solving Optimal Control Problem by Using Orthogonal Function

Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)

Abstract

In the present paper we introduce a numerical technique for solving fractional optimal control problems (FOCP) based on an orthonormal wavelet. First we approximate the involved functions by Sine-Cosine wavelet basis; then, an operational matrix is used to transfer the given problem in to a linear system of algebraic equations. In fact operational matrix of the Riemann-Liouville fractional integration and derivative of Sine-Cosine wavelet are employed to achieve a linear algebraic equation, in place of the dynamical system in terms of the unknown coefficients. The solution of this system, gives us the solution of original problem. A numerical example is also given.

Keywords

Fractional optimal control problem Sine-Cosine wavelet Operational matrix Caputo derivative Riemann-Liouville fractional integration

References

1. 1.
Agrawal OP (2008) A formulation and numerical scheme for fractional optimal control problems. J Vibr Control 14(9–10):1291–1299
2. 2.
Bagley RL, Torvik P (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27(3):201–210
3. 3.
Baillie RT (1996) Long memory processes and fractional integration in econometrics. J Econometrics 73(1):5–59
4. 4.
Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J Vibr Control 14(9–10):1487–1498
5. 5.
Canuto C, Hussaini MY, Quarteroni A, Zang TA Jr (1988) Spectral methods in fluid dynamics. Annu Rev Fluid Mech 57(196):339–367Google Scholar
6. 6.
Dehghan M, Manafian J, Saadatmandi A (2010) The solution of the linear fractional partial differential equations using the homotopy analysis method. Z Naturforsch A 65(Z. Naturforsch):935–949Google Scholar
7. 7.
Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ 26(2):448–479
8. 8.
He J (1998) Nonlinear oscillation with fractional derivative and its applications. In: International conference on vibrating engineering, Dalian, China, vol 98, pp 288–291Google Scholar
9. 9.
He J (1999) Some applications of nonlinear fractional differential equations and their approximations. Bull Sci Technol 15(2):86–90
10. 10.
Jafari H, Tajadodi H (2014) Fractional order optimal control problems via the operational matrices of bernstein polynomials. Upb Sci Bull 76(3):115–128
11. 11.
Kajani MT, Ghasemi M, Babolian E (2006) Numerical solution of linear integro-differential equation by using sine-cosine wavelets. Appl Math Comput 180(2):569–574
12. 12.
Li Y, Sun N (2011) Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Comput Math Appl 62(3):1046–1054
13. 13.
Liu F, Agrawal OP et al (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic PressGoogle Scholar
14. 14.
Lotfi A, Dehghan M, Yousefi SA (2011) A numerical technique for solving fractional optimal control problems. Comput Math Appl 62(3):1055–1067
15. 15.
Momani S, Odibat Z (2007) Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31(5):1248–1255
16. 16.
Odibat ZM, Shawagfeh NT (2007) Generalized taylor’s formula. Appl Math Comput 186(1):286–293
17. 17.
Panda R, Dash M (2006) Fractional generalized splines and signal processing. Sig Process 86(9):2340–2350
18. 18.
Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50:15–67
19. 19.
Saadatmandi A, Dehghan M (2010) A new operational matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336
20. 20.
Saez D (2009) Analytical solution of a fractional diffusion equation by variational iteration method. Comput Math Appl 57(3):483–487
21. 21.
Sen S (2011) Fractional optimal control problems: A pseudo-state space approach. J Vibr Control 17(17):1034–1041
22. 22.
Shawagfeh NT (2002) Analytical approximate solutions for nonlinear fractional differential equations. Appl Math Comput 131(2–3):517–529
23. 23.
Sohrabi S (2011) Comparison chebyshev wavelets method with bpfs method for solving abel’s integral equation. Ain Shams Eng J 2(3–4):249–254
24. 24.
Sweilam NH, Alajami TM, Hoppe RHW (2013) Numerical solution of some types of fractional optimal control problems. Sci World J 2013(2):306237Google Scholar
25. 25.
Tangpong XW, Agrawal OP (2009) Fractional optimal control of continuum systems. J Vibr Acoust 131(2):557–557

© Springer International Publishing AG 2018

Authors and Affiliations

• 1
Email author 