An Efficient Numerical Scheme for Solving Multi‐Dimensional Fractional Optimal Control Problems With a Quadratic Performance Index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

[1]  M. Shitikova,et al.  Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results , 2010 .

[2]  M. Dehghan,et al.  The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems , 2011 .

[3]  Micael S. Couceiro,et al.  Application of fractional algorithms in the control of a robotic bird , 2010 .

[4]  Abbas Saadatmandi,et al.  Bernstein operational matrix of fractional derivatives and its applications , 2014 .

[5]  Mehdi Dehghan,et al.  A tau approach for solution of the space fractional diffusion equation , 2011, Comput. Math. Appl..

[6]  Delfim F. M. Torres,et al.  The calculus of variations and optimal control , 2015 .

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

[8]  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..

[9]  Dumitru Baleanu,et al.  Efficient generalized Laguerre-spectral methods for solving multi-term fractional differential equations on the half line , 2014 .

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

[11]  Mehdi Dehghan,et al.  A new operational matrix for solving fractional-order differential equations , 2010, Comput. Math. Appl..

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

[13]  The operational matrix of fractional integration for shifted Legendre polynomials , 2013 .

[14]  Ali H. Bhrawy,et al.  Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method , 2013 .

[15]  Nasser Sadati,et al.  Fopid Controller Design for Robust Performance Using Particle Swarm Optimization , 2007 .

[16]  Hossein Jafari,et al.  FRACTIONAL ORDER OPTIMAL CONTROL PROBLEMS VIA THE OPERATIONAL MATRICES OF BERNSTEIN POLYNOMIALS , 2014 .

[17]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

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

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

[20]  J. Gregory,et al.  Constrained optimization in the calculus of variations and optimal control theory , 1992 .

[21]  B A.H.,et al.  A New Operational Matrix of Fractional Integration for Shifted Jacobi Polynomials , 2014 .

[22]  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..

[23]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[24]  Ali H. Bhrawy,et al.  The operational matrix of fractional integration for shifted Chebyshev polynomials , 2013, Appl. Math. Lett..

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

[26]  R. Haydock,et al.  Vector continued fractions using a generalized inverse , 2003, math-ph/0310041.

[27]  P. Berck,et al.  Calculus of variations and optimal control theory , 1993 .

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

[29]  José António Tenreiro Machado,et al.  The effect of fractional order in variable structure control , 2012, Comput. Math. Appl..

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

[31]  YangQuan Chen,et al.  A Fractional Adaptation Scheme for Lateral Control of an AGV , 2008 .

[32]  Eid H. Doha,et al.  A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order , 2011, Comput. Math. Appl..

[33]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[34]  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 .

[35]  Isabel S. Jesus,et al.  Fractional control of heat diffusion systems , 2008 .

[36]  Ozlem Defterli,et al.  A numerical scheme for two-dimensional optimal control problems with memory effect , 2010, Comput. Math. Appl..