Solution of Fractional Optimal Control Problems with Noise Function Using the Bernstein Functions

Document Type : بنیادی - نظری

Authors

1 Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran.

2 Department of Mathematics, University of Payame Noor, Tehran, Iran

10.30473/coam.2020.33909.1078

Abstract

This paper presents a numerical solution of a class of fractional optimal control problems (FOCPs) in a bounded domain having a noise function by the spectral Ritz method‎. ‎The Bernstein polynomials with the fractional operational matrix are applied to approximate the unknown functions‎. ‎By substituting these estimated functions into the cost functional‎, ‎an unconstrained nonlinear optimization problem is achieved‎. In order to solve this optimization problem‎, ‎the Matlab software and its optimization toolbox are used‎. ‎In the considered FOCP‎, ‎the performance index is expressed as a function of both state and control functions‎. ‎The method is robust enough because of its computational consistency in the presence of the noise function‎. ‎Moreover‎, ‎the proposed scheme has a good pliability satisfying the given initial and boundary conditions‎. ‎At last‎, ‎some test problems are investigated to confirm the efficiency and applicability of the new method.

Keywords


‎bibitem{1} Magin R. L‎. ‎(2006)‎. ‎``Fractional calculus in bioengineering''‎, ‎Chicago‎: ‎Begell House‎.
 
‎bibitem{2} Jumarie G‎. ‎(2008)‎. ‎``Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions''‎, ‎Applied Mathematical Modelling‎, ‎32(5)‎, ‎836--859‎.
 
‎bibitem{3} Tricaud C.‎, ‎Chen Y. Q‎. ‎(2012)‎. ‎``Optimal Mobile Sensing and Actuation Policies in Cyber-physical Systems''‎, ‎Londo‎: ‎Springer-Verlag‎.
 
‎bibitem{4} Oldham K. B.‎, ‎Spanier J‎. ‎(1974)‎. ‎``The fractional calculus''‎, ‎Academic Press‎, ‎New York‎.
 
‎bibitem{5} Monje C. A.‎, ‎Chen Y.‎, ‎Vinagre B. M.‎, ‎Xue D.‎, ‎Feliu-Batlle V‎. ‎(2010)‎. ‎``Fractional-order Systems and Controls‎: ‎Fundamentals and Applications''‎, ‎Springer-Verlag‎, ‎London‎.
 
‎bibitem{6} Jajarmi A.‎, ‎Baleanu D‎. ‎(2017)‎. ‎``Suboptimal control of fractional-order dynamic systems with delay argument''‎, ‎Journal of Vibration and Control‎, ‎24(12)‎, ‎2430--2446‎. %doi: ‎10.1177/1077546316687936‎.
 
‎bibitem{7} Kilbas A. A.‎, ‎Srivastava‎, ‎H. M.‎, ‎Trujillo J. J‎. ‎(2006)‎. ‎``Theory and Applications of Fractional Differential Equations''‎, ‎204‎, ‎Elsevier Science‎.
 
‎bibitem{8} Kowalski E‎. ‎(2006)‎. ‎``Bernstein Polynomials and Brownian Motion''‎, ‎The American Mathematical Monthly‎, ‎113(10)‎, ‎865--886‎. %doi: ‎10.2307/27642086‎.
 
 
‎bibitem{9} Hasan M. M.‎, ‎Tangpong X. W.‎, ‎Agrawal O. P‎. ‎(2011)‎. ‎``Fractional optimal control of distributed systems in spherical and cylindrical coordinates"‎, ‎Journal of Vibration and Control‎, ‎18‎, ‎1506--1525‎.
 
 
 
‎bibitem{10} "{O}zdemir N.‎, ‎Agrawal O. P.‎, ‎Karadeniz D.‎, ‎İskender B. B‎. ‎(2009)‎. ‎``Fractional optimal control problem of an axis-symmetric diffusion-wave propagation"‎, ‎Physica Scripta‎, ‎T136 014024‎.
 
 
‎bibitem{10b} Nemati A.‎, ‎Yousefi S. A.‎, ‎Soltanian F.‎, ‎Saffar-Ardabili J‎. ‎(2016)‎. ‎``An efficient numerical solution of fractional optimal control problems by using the Ritz method and Bernstein operational matrix''‎, ‎Asian Journal of Control‎, ‎18(6)‎, ‎2272--2282‎.
 
 
 
‎bibitem{10c} Mamehrashi K.‎, ‎Nemati A‎. ‎(2019)‎. ‎``A new approach for solving infinite horizon optimal control problems using Laguerre functions and Ritz spectral method''‎, ‎International Journal of Computer Mathematics‎. %DOI: ‎10.1080/00207160.2019.162‎,
 
 
 
‎bibitem{11} Bhrawy A. H.‎, ‎Doha‎, ‎E. H.‎, ‎Tenreiro Machado‎, ‎J. A.‎, ‎Ezz-Eldien‎, ‎S. S‎. ‎(2015)‎. ‎``An Efficient Numerical Scheme for Solving Multi-Dimensional Fractional Optimal Control Problems With a Quadratic Performance Index‎. ‎Asian Journal of Contro''l‎, ‎17(6)‎, ‎2389--2402‎. %doi: ‎10.1002/asjc.1109‎.
 
 
 
‎bibitem{12} Nemati A.‎, ‎Mamehrashi‎, ‎K‎. ‎(2019)‎. ‎``The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional Order Optimal Control Problems Described by the Roesser Model''‎, ‎Asian Journal of Control‎, ‎21(4)‎, ‎1--13‎.
 
 
 
‎bibitem{12b} Taherpour V.‎, ‎Nazari M.‎, ‎Nemati A‎. ‎(2020)‎. ‎``A new numerical Bernoulli polynomial method for solving fractional optimal control problems with vector components"‎, ‎Computational Methods for Differential Equations‎, ‎Accepted paper.%DOI:10.22034/cmde.2020.34992.1598‎‎
 
‎bibitem{13} Agrawal O. P.‎, ‎Baleanu D‎. ‎(2007)‎. ‎``A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems''‎, ‎Journal of Vibration and Control‎, ‎13(9-10)‎, ‎1269--1281‎. %doi: ‎10.1177/1077546307077467‎
 
‎bibitem{14} Maleki M.‎, ‎Hashim I.‎, ‎Abbasbandy S.‎, ‎Alsaedi A‎. ‎(2015)‎. ‎``Direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method''‎, ‎Journal of Computational and Applied Mathematics‎, ‎283‎, ‎41--57‎. %doi: ‎https://doi.org/10.1016/j.cam.2015.01.019‎
‎bibitem{14b} Nemati A.‎, ‎Yousefi S. A‎. ‎(2016)‎. ‎``A numerical method for solving fractional optimal control problems using Ritz method''‎, ‎Journal of Computational and Nonlinear Dynamics‎, ‎11‎, ‎1--7‎.
 
‎bibitem{15} Lotfi A.‎, ‎Yousefi S. A.‎, ‎Dehghan M‎. ‎(2013)‎. ‎``Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule''‎, ‎Journal of Computational and Applied Mathematics‎, ‎250‎, ‎143--160‎. %doi: ‎https://doi.org/10.1016/j.cam.2013.03.003‎
‎bibitem{16} Doha E. H.‎, ‎Bhrawy A. H.‎, ‎Baleanu D.‎, ‎Ezz-Eldien S. S.‎, ‎Hafez R. M‎. ‎(2015)‎. ‎``An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems''‎, ‎Advances in Difference Equations‎, ‎2015(1)‎, ‎15‎. %doi: ‎10.1186/s13662-014-0344-z‎
‎bibitem{17} Lotfi A.‎, ‎Dehghan M.‎, ‎Yousefi S. A‎. ‎(2011)‎. ‎``A numerical technique for solving fractional optimal control problems''‎, ‎Computers & Mathematics with Applications‎, ‎62(3)‎, ‎1055--1067‎. %doi: ‎https://doi.org/10.1016/j.camwa.2011.03.044‎
‎bibitem{18} Trujillo J. J.‎, ‎Ungureanu V. M‎. ‎(2018)‎. ‎``Optimal control of discrete-time linear fractional-order systems with multiplicative noise''‎, ‎International Journal of Control‎, ‎91(1)‎, ‎57--69‎. %doi: ‎10.1080/00207179.2016.1266520‎
‎bibitem{18a} Ahadpour S.‎, ‎Nemati A.‎, ‎Mirmasoudi F.‎, ‎Hematpour N‎. ‎(2018)‎. ‎``Projective synchronization of piecewise nonlinear Chaotic maps''‎, ‎Theoretical and Mathematical Physics‎, ‎197(3)‎, ‎1856--1864‎.
‎bibitem{19} FujishlGe S.‎, ‎Sawaragl Y‎. ‎(1975)‎. ‎``Optimal control for linear continuous-time systems with general noises based upon sampled data‎. ‎International Journal of Systems Science''‎, ‎6(12)‎, ‎1111--1118‎. %doi: ‎10.1080/00207727508941890‎
‎bibitem{20} Saadatmandi A‎. ‎(2014)‎. ‎``Bernstein operational matrix of fractional derivatives and its applications''‎, ‎Applied Mathematical Modelling‎, ‎38(4)‎, ‎1365--1372‎. %doi: ‎https://doi.org/10.1016/j.apm.2013.08.007‎
‎bibitem{20b} Nemati A‎. ‎(2017)‎. ‎``Numerical solution of 2D fractional optimal control problems by the spectral method along with Bernstein operational matrix''‎, ‎International Journal of Control‎, %https://doi.org/10.1080/00207173429.2017.1367.
‎bibitem{21} Farin G‎. ‎(1993)‎. ‎``Curves and Surfaces for CAGD''‎, ‎Academic Press‎, ‎Boston‎.
 
 
 
‎bibitem{22} R. T‎. ‎F.‎, ‎T. N. T‎. ‎G‎. ‎(2003)‎. ‎``Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains''‎.
 
‎bibitem{23} G‎. ‎M‎. ‎P‎. ‎(2003)‎. ‎``Interpolation and Approximation by Polynomials''‎.
 
‎bibitem{24} Samko S. G.‎, ‎Kilbas A. A.‎, ‎Marichev O. I‎. ‎(1993)‎. ‎``Fractional Integrals and Derivatives‎. ‎Theory and Applications''‎, ‎Gordon and Breach Science Publishers‎, ‎Yverdon‎.
 
‎bibitem{25} Wu Z.‎, ‎Sun X.‎, ‎Ma L‎. ‎(2013)‎. ‎``Sampling scattered data with Bernstein polynomials‎: ‎stochastic and deterministic error estimates‎. ‎Advances in Computational Mathematics''‎, ‎38(1)‎, ‎187--205‎. %doi: ‎10.1007/s10444-011-9233-0‎
‎bibitem{26} Rivlin T. J‎. ‎(1981)‎. ‎``An Introduction to the Approximation of Functions''‎, ‎Courier Corporation‎, ‎New York‎.
 
‎bibitem{27} J"{u}ttler B‎. ‎(1998)‎. ‎``The dual basis functions for the Bernstein polynomials‎. ‎Advances in Computational Mathematics''‎, ‎8(4)‎, ‎345--352‎. %doi: ‎10.1023/A:1018912801267‎
‎bibitem{28} Rudas I. J.‎, ‎Tar J. K.‎, ‎Patkai B‎. ‎(2006)‎. ‎``Compensation of dynamic friction by a fractional order robust controller''‎, ‎Paper presented at the IEEE International Cconference on Computational Cybernetics‎, ‎Budapest‎.
 
 
 
‎bibitem{29} Petrovacki N.‎, ‎Jelicic Z. D‎. ‎(2006)‎. ‎``Optimal transient response of erbium-doped fiber amplifiers Paper presented at the IEEE international conference on numerical simulation of semiconductor optoelectronic devices''‎, ‎Singapore‎.
 
 
 
‎bibitem{30} Kreyszig E‎. ‎(1978)‎. ‎``Introductory Functional Analysis with Applications''‎, ‎John Wiley and Sons‎, ‎Inc.‎, ‎New York‎.
 
‎bibitem{31} Shakoor P.‎, ‎Ricardo A.‎, ‎Delfim F. M. T‎. ‎(2013)‎. ‎``Fractional order optimal control problems with free terminal time''‎, ‎Journal of Industrial & Management Optimization‎, ‎10(2)‎, ‎363--381‎.