Comparison on wavelets techniques for solving fractional optimal control problems. (English) Zbl 1400.93124
Summary: This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton’s method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods.
MSC:
| 93C23 | Control/observation systems governed by functional-differential equations |
| 65T60 | Numerical methods for wavelets |
Keywords:
fractional optimal control problem; fractional calculus; wavelet method; Legendre wavelets; Chebyshev wavelets; Laguerre wavelets; CAS waveletsReferences:
| [1] | Abd-Elhameed, WM, Doha, EH, Youssri, YH (2013) New wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth kinds. Abstract and Applied Analysis.Article ID: 542839, 1-9pp. . · Zbl 1291.65238 |
| [2] | Adibi, H, Assari, P (2010) Chebyshev wavelet method for numerical solution of Fredholm integral equations of the second kind. Mathematical Problems in Engineering. Article Id: 138408, 1-17pp. . · Zbl 1192.65154 |
| [3] | Agrawal, OP (1989) General formulation for the numerical solution of optimal control problems. International Journal of Control 50(2): 627-638. · Zbl 0679.49031 |
| [4] | Agrawal, OP (2004) A General formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics 38(1-2): 323-337. · Zbl 1121.70019 |
| [5] | Agrawal, OP (2006) A formulation and numerical scheme for fractional optimal control problems. Journal of Vibration and Control 14(9-10): 1291-1299. · Zbl 1229.49045 |
| [6] | Alizadeh, A, Effati, S (2016) An iterative approach for solving fractional optimal control problems. Journal of Vibration and Control 1: 1-19. · Zbl 1381.93054 |
| [7] | Babolian, E, Fattahzadeh, F (2007) Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration. Applied Mathematics and Computation 188: 1016-1022. · Zbl 1114.65366 |
| [8] | Biswas, RK, Sen, S (2010) Fractional optimal control problems: A pseudo-state-space approach. Journal of Vibration and Control 17: 1034-1041. · Zbl 1271.74330 |
| [9] | Bryson, AEJ, Ho, YC (1969) Applied Optimal Control: Optimization, Estimation, and Control, Waltham, MA: Blaisdell. |
| [10] | Caputo, M (1967) Linear models of dissipation whose Q is almost frequency independent, part II. Geophysical Journal of the Royal Astronomical Society 13: 529-539. . |
| [11] | Caputo, M (1969) Elasticita e Dissipazione, Bologna: Zanichelli. |
| [12] | Caputo, M, Mainardi, F (1971) A new dissipation model based on memory mechanism. Pure and Applied Geophysics 91: 134-147. · Zbl 1213.74005 |
| [13] | Chui, CK (1992) An Introduction to Wavelets, Wavelet Analysis and Its Applications (Vol 1), Massachusetts, MA: Academic Press. · Zbl 0925.42016 |
| [14] | Doha, EH, Abd-Elhameed, WM, Youssri, YH (2013) Second kind Chebyshev operational matrix for solving differential equations of Lane-Emden type. New Astronomy 23-24: 113-117. |
| [15] | Gelfand, IM, Fomin, SV (1963) Calculus of Variation. (RA Silverman Trans) Prentice-Hall Inc., Englewood Cliffs, NJ: Prentice Hall. |
| [16] | Hosseinpour, S, Nazemi, A (2016) Solving fractional optimal control problems with fixed or free final states by Haar wavelet collocation method. IMA Journal of Mathematical Control and Information 33: 543-561. · Zbl 1397.93104 |
| [17] | Iqbal, MA, Saeed, U, Mohyud-Din, ST (2015) Modified Laguerre wavelets method for delay differential equations of fractional order. Egyptian Journal of Basic and Applied Sciences 2: 50-54. |
| [18] | Jafari, H, Tajadodi, H (2014) Fractional order optimal control problems via the operational matrices of Bernstein polynomials. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics 76(3): 115-128. · Zbl 1313.49001 |
| [19] | Khader, MM, Hendy, AS (2012) An efficient numerical scheme for solving fractional optimal control problems. International Journal of Nonlinear Science 14(3): 287-296. · Zbl 1394.49032 |
| [20] | Kolwankar, KM, Gangal, AD (1996) Fractional differentiability of nowhere differentiable functions and dimensions. Chaos: An Interdisciplinary Journal of Nonlinear Science 6(4): 505-513. · Zbl 1055.26504 |
| [21] | Larsson, S, Racheva, M, Saedpanah, F (2015) Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity. Computer Methods in Applied Mechanics and Engineering 283: 196-209. · Zbl 1423.74900 |
| [22] | Lotfi, A, Dehghan, M, Yousefi, SA (2011) A numerical technique for solving fractional optimal control problems. Computers and Mathematics with Applications 62: 1055-1067. · Zbl 1228.65109 |
| [23] | Magin, RL (2004) Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering 32: 1-104. |
| [24] | Mainardi, F (1997) Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and Fractional Calculus in Continuum Mechanics 378: 291-348. · Zbl 0917.73004 |
| [25] | Miller, KS, Ross, B (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley. · Zbl 0789.26002 |
| [26] | Podlubny I (1994) The Laplace transform method for linear differential equations of the fractional order. In: UEF-02-94, arXiv:funct-an/9710005. |
| [27] | Podlubny, I (1999) Fractional Differential Equations, San Diego, CA: Academic. · Zbl 0924.34008 |
| [28] | Pooseh, S, Almeida, R, Torres, DFM (2014) Fractional order optimal control problems with free terminal time. Journal of Industrial and Management Optimization 10(2): 363-381. · Zbl 1278.26013 |
| [29] | Razzaghi, M, Yousefi, S (2001) Legendre wavelets method for the solution of nonlinear problems in the calculus of variations. Mathematical and Computer Modelling 34: 45-54. · Zbl 0991.65053 |
| [30] | Saeedi, H, Moghadam, MM (2011) Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets. Communications in Nonlinear Science and Numerical Simulation 16: 1216-1226. · Zbl 1221.65140 |
| [31] | Sage, AP, White, CC (1977) Optimum Systems Control, Englewood Cliffs, NJ: Prentice-Hall. · Zbl 0388.49002 |
| [32] | Saha Ray, S (2015) A novel method for travelling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water. Mathematical Methods in the Applied Sciences 38(7): 1352-1368. · Zbl 1325.35266 |
| [33] | Sahu, PK, Saha Ray, S (2015a) Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Applied Mathematics and Computation 256: 715-723. · Zbl 1338.65208 |
| [34] | Sahu, PK, Saha Ray, S (2015b) Two dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations. Journal of Intelligent & Fuzzy Systems 28: 1271-1279. · Zbl 1351.45009 |
| [35] | Samko, SG, Kilbas, AA, Marichev, OI (1993) Fractional Integrals and Derivatives: Theory and Applications, London: Gordon and Breach. · Zbl 0818.26003 |
| [36] | Sierociuk D and Vinagre BM (2010) Infinite horizon state feedback LQR controller for fractional systems. In: 49th IEEE conference on decision and control (CDC), Atlanta, Georgia, USA, 15-17 December 2010, volume 1, pp.6674-6679. |
| [37] | Sierociuk, D, Dzielinski, A, Sarwas, MG (2013) Modelling heat transfer in heterogeneous media using fractional calculus. Philosophical Transactions of the Royal Society of London 371: 1-10. . · Zbl 1382.80004 |
| [38] | Sweilam, NH, Al-Ajami, TM, Hoppe, RHW (2013) Numerical solution of some types of fractional optimal control problems. The Scientific World Journal. Article Id: 306237, 1-9pp. . |
| [39] | Smith, W, De Vries, H (1970) Rheological models containing fractional derivatives. Rheologica Acta 9: 525-534. |
| [40] | Tangpong, XW, Agrawal, OP (2009) Fractional optimal control of a continuum system. ASME Journal of Vibration and Acoustics 131(2): 021012. |
| [41] | Tricaud, C, Chen, YQ (2010) An approximate method for numerically solving fractional order optimal control problems of general form. Computers & Mathematics with Applications 59: 1644-1655. · Zbl 1189.49045 |
| [42] | Wang, Y, Fan, Q (2012) The second kind Chebyshev wavelet method for solving fractional differential equations. Applied Mathematics and Computation 218(17): 8592-8601. · Zbl 1245.65090 |
| [43] | Yi, M, Huang, J (2015) CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel. Learn. International Journal of Computers for Mathematical Learning 92(8): 1715-1728. · Zbl 1317.65261 |
| [44] | Yousefi, SA, Lotfi, A, Dehghan, M (2011) The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. Journal of Vibration and Control 13: 1-7. · Zbl 1271.65105 |
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