A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems

The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes an integer derivative in addition to the fractional derivative. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Based on the shifted Legendre orthonormal polynomials, we employ the operational matrix of fractional derivatives, the Legendre–Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations. The convergence of the proposed method is analyzed. For confirming the validity and accuracy of the proposed numerical method, a numerical example is presented along with a comparison between our numerical results and those obtained using the Legendre spectral-collocation method.

[1]  W. Varnhorn,et al.  An approximation method using approximate approximations , 2006 .

[2]  O. Agrawal,et al.  A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems , 2007 .

[3]  Gisèle M. Mophou,et al.  Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation , 2011, Comput. Math. Appl..

[4]  E. H. Doha,et al.  A NEW JACOBI OPERATIONAL MATRIX: AN APPLICATION FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS , 2012 .

[5]  J. A. Tenreiro Machado,et al.  Fractional calculus analysis of the cosmic microwave background , 2013 .

[6]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[7]  J. Tenreiro Machado,et al.  Efficient Legendre spectral tau algorithm for solving the two-sided space–time Caputo fractional advection–dispersion equation , 2016 .

[8]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[9]  S. Das,et al.  Functional Fractional Calculus for System Identification and Controls , 2007 .

[10]  Ali H. Bhrawy,et al.  An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Subdiffusion and Reaction Subdiffusion Equations , 2015 .

[11]  Dumitru Baleanu,et al.  On shifted Jacobi spectral approximations for solving fractional differential equations , 2013, Appl. Math. Comput..

[12]  Richard T. Baillie,et al.  Long memory processes and fractional integration in econometrics , 1996 .

[13]  Mehdi Dehghan,et al.  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 , 2013, J. Comput. Appl. Math..

[14]  Gisèle M. Mophou,et al.  Optimal control of fractional diffusion equation , 2011, Comput. Math. Appl..

[15]  Ronald H. W. Hoppe,et al.  Numerical Solution of Some Types of Fractional Optimal Control Problems , 2013, TheScientificWorldJournal.

[16]  Dumitru Baleanu,et al.  A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations , 2015, J. Comput. Phys..

[17]  O. Agrawal.  A Quadratic Numerical Scheme for Fractional Optimal Control Problems , 2008 .

[18]  Mataz Alcoutlabi,et al.  Application of fractional calculus to viscoelastic behaviour modelling and to the physical ageing phenomenon in glassy amorphous polymers , 1998 .

[19]  Yangquan Chen,et al.  Computers and Mathematics with Applications an Approximate Method for Numerically Solving Fractional Order Optimal Control Problems of General Form Optimal Control Time-optimal Control Fractional Calculus Fractional Order Optimal Control Fractional Dynamic Systems Riots_95 Optimal Control Toolbox , 2022 .

[20]  R. Feynman,et al.  RECENT APPLICATIONS OF FRACTIONAL CALCULUS TO SCIENCE AND ENGINEERING , 2003 .

[21]  Mehdi Dehghan,et al.  Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets , 2014 .

[22]  Ali H. Bhrawy,et al.  A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations , 2015, J. Comput. Phys..

[23]  Saudi Arabia,et al.  NEW NUMERICAL APPROXIMATIONS FOR SPACE-TIME FRACTIONAL BURGERS' EQUATIONS VIA A LEGENDRE SPECTRAL-COLLOCATION METHOD , 2014 .

[24]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[25]  Nasser H. Sweilam,et al.  Legendre spectral-collocation method for solving some types of fractional optimal control problems , 2014, Journal of advanced research.

[26]  Benoit B. Mandelbrot,et al.  Some noises with I/f spectrum, a bridge between direct current and white noise , 1967, IEEE Trans. Inf. Theory.

[27]  M. Zaky,et al.  Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation , 2014, Nonlinear Dynamics.

[28]  Yao-Lin Jiang,et al.  Waveform relaxation methods for fractional differential equations with the Caputo derivatives , 2013, J. Comput. Appl. Math..

[29]  Mehdi Dalir,et al.  Applications of Fractional Calculus , 2010 .

[30]  D. Benson,et al.  Fractional calculus in hydrologic modeling: A numerical perspective. , 2013, Advances in water resources.

[31]  G. Bohannan.  Analog Fractional Order Controller in Temperature and Motor Control Applications , 2008 .

[32]  Dumitru Baleanu,et al.  A NEW GENERALIZED LAGUERRE-GAUSS COLLOCATION SCHEME FOR NUMERICAL SOLUTION OF GENERALIZED FRACTIONAL PANTOGRAPH EQUATIONS , 2014 .

[33]  Mehdi Dehghan,et al.  A numerical technique for solving fractional optimal control problems , 2011, Comput. Math. Appl..

[34]  Dominik Sierociuk,et al.  Some applications of fractional order calculus , 2010 .

[35]  Dumitru Baleanu,et al.  A Spectral Legendre–Gauss–Lobatto Collocation Method for a Space-Fractional Advection Diffusion Equations with Variable Coefficients , 2013 .