An efficient approximate method for solving delay fractional optimal control problems

In this paper, a new numerical method for solving the delay fractional optimal control problems (DFOCPs) with quadratic performance index is presented. In the discussed DFOCP, the fractional derivative is considered in the Caputo sense. The method is based upon the Bernoulli wavelets basis. To solve the problem, first the DFOCP is transformed into an equivalent problem with dynamical system without delay. Then, an operational matrix of Riemann–Liouville fractional integration based on Bernoulli wavelets is introduced and is utilized to reduce the problem to the solution of a system of algebraic equations. With the aid of Gauss–Legendre integration formula and Newton’s iterative method for solving a system of algebraic equations, the problem is solved approximately. Also, the convergence of the Bernoulli wavelet basis is obtained. Finally, some examples are given to demonstrate the validity and applicability of the new technique and a comparison is made with the existing results.

[1]  Shaher Momani,et al.  Generalized differential transform method: Application to differential equations of fractional order , 2008, Appl. Math. Comput..

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

[3]  Mo M. Jamshidi,et al.  A computational algorithm for large-scale nonlinear time-delay systems , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[4]  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..

[5]  Wichai Witayakiattilerd.  Optimal Regulation of Impulsive Fractional Differential Equation with Delay and Application to Nonlinear Fractional Heat Equation , 2013 .

[6]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[7]  Mohsen Razzaghi,et al.  Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials , 2004, J. Frankl. Inst..

[8]  Gennady N. Chuev,et al.  A CAS Wavelet Method for Solving Nonlinear Fredholm Integro- Differential Equations of Fractional Order , 2011 .

[9]  Yadollah Ordokhani,et al.  A numerical solution for fractional optimal control problems via Bernoulli polynomials , 2016 .

[10]  Dumitru Baleanu,et al.  Fractional variational optimal control problems with delayed arguments , 2010 .

[11]  Yadollah Ordokhani,et al.  Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet , 2017, J. Comput. Appl. Math..

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

[13]  Necati Özdemir,et al.  Fractional optimal control of a 2-dimensional distributed system using eigenfunctions , 2008 .

[14]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

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

[16]  Yadollah Ordokhani,et al.  Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations , 2014 .

[17]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

[18]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[19]  Om P. Agrawal,et al.  A Formulation and Numerical Scheme for Fractional Optimal Control Problems , 2008 .

[20]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[21]  Zhiqiang Zhou,et al.  Convergence analysis of moving finite element methods for space fractional differential equations , 2014, J. Comput. Appl. Math..

[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]  Yuanlu Li,et al.  Solving a nonlinear fractional differential equation using Chebyshev wavelets , 2010 .

[24]  O. Agrawal.  Fractional Optimal Control of a Distributed System Using Eigenfunctions , 2007 .

[25]  Yadollah Ordokhani,et al.  Optimal Control of Delay Systems by Using a Hybrid Functions Approximation , 2012, J. Optim. Theory Appl..

[26]  Esmail Babolian,et al.  Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration , 2007, Appl. Math. Comput..

[27]  Xing Tao Wang,et al.  Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials , 2007, Appl. Math. Comput..

[28]  Qibin Fan,et al.  Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW , 2013, Commun. Nonlinear Sci. Numer. Simul..

[29]  Yadollah Ordokhani,et al.  Fractional-order Bernoulli wavelets and their applications , 2016 .

[30]  S. A. Yousefi,et al.  A numerical technique for solving a class of fractional variational problems , 2013, J. Comput. Appl. Math..

[31]  Yuriy Povstenko,et al.  Time-fractional radial diffusion in a sphere , 2008 .

[32]  Rahmat Ali Khan,et al.  The Legendre wavelet method for solving fractional differential equations , 2011 .

[33]  F. Khellat,et al.  Optimal Control of Linear Time-Delayed Systems by Linear Legendre Multiwavelets , 2009 .

[34]  Suayip Yüzbasi,et al.  Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials , 2013, Appl. Math. Comput..

[35]  Feng Chen,et al.  Maximum principle for optimal control problem of stochastic delay differential equations driven by fractional Brownian motions , 2016 .

[36]  T. S. Chow,et al.  Fractional dynamics of interfaces between soft-nanoparticles and rough substrates , 2005 .

[37]  Haitao Qi,et al.  Time-fractional radial diffusion in hollow geometries , 2010 .

[38]  S. S. Ezz-Eldien,et al.  A new Legendre operational technique for delay fractional optimal control problems , 2016 .

[39]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[40]  E. Safaie,et al.  An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials , 2015 .

[41]  Omar Abu Arqub,et al.  New Results on Fractional Power Series: Theory and Applications , 2013, Entropy.

[42]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

[43]  Dumitru Baleanu,et al.  Fractional variational principles with delay , 2008 .

[44]  Mohammed Al-Smadi,et al.  A general form of the generalized Taylor's formula with some applications , 2015, Appl. Math. Comput..

[45]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

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

[47]  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 .