Abstract
The method of radial basis functions (RBFs) is a method of scattered data interpolation that has many applications in different fields. Based on the RBFs approach, two simple and accurate methods are presented in this paper to favor the solution of fractional optimal control problems (FOCPs) which are nominated as indirect and direct methods. In the first one, the considered FOCP is converted into a system of fractional differential equations (FDEs) that is called two-point boundary value problem (TPBVP) of fractional order. Then, both the context of minimization the total error and a joint application of Volterra integral equation will be used to rewrite this problems as as an unconstrained optimization problem that can be solved by using RBFs approximation. In direct one, we rewrite the FOCP as a classical static optimization problem that can be easily solved using known formulas for computing fractional derivatives of RBFs. Numerical example are given to prove the accuracy, effectiveness and feasibility of these methods.
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References
Agrawal OP (2008) A quadratic numerical scheme for fractional optimal control problems. J Dyn Syst Meas Control 130(1):011010
Agrawal OP, Baleanu D (2007) A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J Vib Control 13(9–10):1269–1281
Alizadeh A, Effati S (2016) An iterative approach for solving fractional optimal control problems. J Vib Control 24:18. https://doi.org/10.1177/1077546316633391
Ben-Yu G (1998) Spectral methods and their applications. World Scientific, Singapore
Bhrawy AH, Assas LM, Tohidi E, Alghamdi MA (2013) A Legendre–Gauss collocation method for neutral functional–differential equations with proportional delay. Adv Differ Equ. https://doi.org/10.1186/1687-1847-2013-63
Biazar J, Ghazvini H (2009) He’s homotopy perturbation method for solving system of Volterra integral equations of the second kind. Chaos Solitons Fractals 39:770–777
Blaszczyk T, Ciesielski M (2014) Numerical solution of fractional Sturm–Liouville equation in integral form. Fract Calc Appl Anal 17(2):307–320
Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J Vib Control 14(9–10):1487–1498
Buhmann MD (2003) Radial basis functions: theory and implementations. Cambridge University Press, Cambridge
Caputo M (2008) Linear models of dissipation whose Q is almost frequency independent—II. Fract Calc Appl Anal 11(1):4–14
Ciarlet PG, Lions JL (1990) Handbook of numerical analysis, vol 1. North-Holland, New York, p 658
Dehghan M, Abbaszadeh M, Mohebbi A (2014) The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions. Comput Math Appl 68(3):212–237
Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J Math Anal Appl 265(2):229–248
Diethelm K, Ford NJ, Freed AD (2004) Detailed error analysis for a fractional Adams method. Numer Algorithms 36(1):31–52
Doha EH, Bhrawy AH (2012) An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method. Comput Math Appl 64:558–571
Doha EH, Bhrawy AH, Ezz-Eldien SS (2011) Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl Math Model 35:5662–5672
Doha EH, Bhrawy AH, Baleanu D, Ezz-Eldien SS, Hafez RM (2015) An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv Differ Equ 2015:15
Elansari M, Ouazar D, Cheng AD (2001) Boundary solution of Poisson’s equation using radial basis function collocated on Gaussian quadrature nodes. Commun Numer Methods Eng 17(7):455–464
Elnagar GN, Kazemi M (1996) Chebyshev spectral solution of nonlinear Volterra–Hammerstein integral equations. J Comput Appl Math 76(1–2):147–158
Fasshauer G, McCourt M (2015) Kernel-based approximation methods using Matlab, vol 19. World Scientific Publishing Company
Franke C, Schaback R (1998) Solving partial differential equations by collocation using radial basis functions. Appl Math Comput 93(1):73–82
Gelfand IM, Fomin SV (1963) Calculus of variation (R.A. Silverman, Trans.). Prentice Hall, Upper Saddle River
Kansa EJ (1990) Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput Math Appl 19(8–9):127–145
Kazemi BF, Ghoreishi F (2013) Error estimate in fractional differential equations using multiquadratic radial basis functions. J Comput Appl Math 245:133–147
Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier Science Limited, Amsterdam
Larsson E, Fornberg B (2003) A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput Math Appl 46(5):891–902
Lotfi A, Dehghan M, Yousefi SA (2011) A numerical technique for solving fractional optimal control problems. Comput Math Appl 62(3):1055–1067
Lubich C (1983) On the stability of linear multistep methods for Volterra convolution equations. IMA J Numer Anal 3(4):439–465
Lubich C (1986) Discretized fractional calculus. SIAM J Math Anal 17(3):704–719
Magin RL (2006) Fractional calculus in bioengineering. Begell House Publisher, Inc., Danbury
Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77
Mokhtari R, Mohseni M (2012) A meshless method for solving mKdV equation. Comput Phys Commun 183(6):1259–1268
Mukherjee YX, Mukherjee S (1997) On boundary conditions in the element-free Galerkin method. Comput Mech 19(4):264–270
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic Press, New York
Podlubny I (1999) Fractional differential equations. Academic Press, New York
Pooseh S, Almeida R, Torres DF (2013a) Fractional order optimal control problems with free terminal time. arXiv preprint arXiv:1302.1717
Pooseh S, Almeida R, Torres DFM (2013b) A numerical scheme to solve fractional optimal control problems. In: Conference papers in mathematics, Article ID 165298, 10 Pages
Sahn N, Yzba S, Gulsu M (2011) A collocation approach for solving systems of linear Volterra integral equations with variable coefficients. Comput Math Appl 62(2):755–769
Sahu PK, Ray SS (2016) Comparison on wavelets techniques for solving fractional optimal control problems. J Vib Control 24:1185. https://doi.org/10.1177/1077546316659611
Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Yverdon
Sanz-Serna JM (1988) A numerical method for a partial integro-differential equation. SIAM J Numer Anal 25(2):319–327
Schaback R (1995) Error estimates and condition numbers for radial basis function interpolation. Adv Comput Math 3:251–64
Schaback R (1999) Native Hilbert spaces for radial basis functions. I. New developments in approximation theory (Dortmund, 1998). Int Ser Numer Math 132:255–282
Sweilam NH, Al-Ajami TM, Hoppe RH (2013) Numerical solution of some types of fractional optimal control problems. The Scientific World Journal, 2013
Tohidi E, Samadi ORN (2012) Optimal control of nonlinear Volterra integral equations via Legendre polynomials. IMA J Math Control Inf 30:67–83
Toutounian F, Tohidi E, Kilicman A (2013) Fourier operational matrices of differentiation and transmission: introduction and applications. Abstr Appl Anal 2013:1–11
Wendland H (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4:389–96
Wendland H (1999) Meshless Galerkin method using radial basis functions. Math Comput 68:1521–31
Wendland H (2005) Scattered data approximation. Cambridge University Press, New York
Wu ZM, Schaback R (1993) Local error estimates for radial basis function interpolation of scattered data. IMA J Numer Anal 13:13–27
Yoon J (2003) \( L_p \)-error estimates for shifted surface spline interpolation on Sobolev space. Math Comput 72(243):1349–1367
Zamani AA, Tavakoli S, Etedali S (2017) Fractional order PID control design for semi-active control of smart base-isolated structures: a multi-objective cuckoo search approach. ISA Trans 67:222–232
Zeid SS, Yousefi M (2016) Approximated solutions of linear quadratic fractional optimal control problems. J Appl Math 12:83–94
Zeid SS, Kamyad AV, Effati S, Rakhshan SA, Hosseinpour S (2017) Numerical solutions for solving a class of fractional optimal control problems via fixed-point approach. SeMA J 74(4):585–603
Zeid SS, Effati S, Kamyad AV (2018) Approximation methods for solving fractional optimal control problems. Comput Appl Math 37(1):158–182
Zhang X, Liu X, Lu MW, Chen Y (2001) Imposition of essential boundary conditions by displacement constraint equations in meshless methods. Commun Numer Methods Eng 17(3):165–178
Zongmin W (1992) Hermite–Birkhoff interpolation of scattered data by radial basis functions. Approx Theory Appl 8(2):1–10
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Communicated by Vasily E. Tarasov.
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Soradi-Zeid, S. Efficient radial basis functions approaches for solving a class of fractional optimal control problems. Comp. Appl. Math. 39, 20 (2020). https://doi.org/10.1007/s40314-019-1003-5
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DOI: https://doi.org/10.1007/s40314-019-1003-5
Keywords
- Optimal control problems
- Radial basis functions
- Direct and indirect methods
- Fractional two-point boundary value problem