Second Chebyshev wavelets (SCWs) method for solving finite-time fractional linear quadratic optimal control problems

Abstract In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.

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

[2]  A. Nazemi,et al.  A New Approach for Solving a Class of Delay Fractional Partial Differential Equations , 2018, Mediterranean Journal of Mathematics.

[3]  Existence, uniqueness, and relaxation results in initial value type problems for nonlinear fractional differential equations , 2021 .

[4]  Carlo Cattani,et al.  Wavelets method for solving fractional optimal control problems , 2016, Appl. Math. Comput..

[5]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[6]  Li Zhu,et al.  SCW method for solving the fractional integro-differential equations with a weakly singular kernel , 2016, Appl. Math. Comput..

[7]  Delfim F. M. Torres,et al.  Fractional order optimal control problems with free terminal time , 2013, 1302.1717.

[8]  Fengying Zhou,et al.  Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets , 2014, Appl. Math. Comput..

[9]  Yadollah Ordokhani,et al.  An efficient approximate method for solving delay fractional optimal control problems , 2016, Nonlinear Dynamics.

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

[11]  Sohrab Effati,et al.  A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems , 2013 .

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

[13]  Qibin Fan,et al.  Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet , 2012 .

[14]  Rubén Morales-Menéndez,et al.  Glucose optimal control system in diabetes treatment , 2009, Appl. Math. Comput..

[15]  Yongsheng Ding,et al.  Optimal Control of a Fractional-Order HIV-Immune System With Memory , 2012, IEEE Transactions on Control Systems Technology.

[16]  Hamid Reza Marzban,et al.  Optimal control of linear multi-delay systems with piecewise constant delays , 2016, IMA J. Math. Control. Inf..

[17]  Hamid Reza Marzban,et al.  A hybrid approximation scheme for discretizing constrained quadratic optimal control problems , 2014, J. Frankl. Inst..

[18]  K. Atkinson,et al.  Theoretical Numerical Analysis: A Functional Analysis Framework , 2001 .

[19]  M. Dehghan,et al.  Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations , 2011 .

[20]  Delfim F. M. Torres,et al.  Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives , 2010, 1007.2937.

[21]  Yanxin Wang,et al.  The second kind Chebyshev wavelet method for solving fractional differential equations , 2012, Appl. Math. Comput..

[22]  Dumitru Baleanu,et al.  A new approach for the nonlinear fractional optimal control problems with external persistent disturbances , 2018, J. Frankl. Inst..

[23]  Sohrab Effati,et al.  Solving a class of nonlinear optimal control problems via he’s variational iteration method , 2012 .

[24]  Li Zhu,et al.  Numerical solutions of Volterra integral equation with weakly singular kernel using SCW method , 2015, Appl. Math. Comput..

[25]  H. Baghani,et al.  A new contraction condition and its application to weakly singular Volterra integral equations of the second kind , 2017 .

[26]  H. R. Marzban,et al.  A composite Chebyshev finite difference method for nonlinear optimal control problems , 2013, Commun. Nonlinear Sci. Numer. Simul..

[27]  Delfim F. M. Torres,et al.  Calculus of variations with fractional derivatives and fractional integrals , 2009, Appl. Math. Lett..

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

[29]  A. Kamyad,et al.  Approximate Solutions for a Class of Fractional-Order Model of HIV Infection via Linear Programming Problem , 2016 .

[30]  Om P. Agrawal,et al.  Formulation of Euler–Lagrange equations for fractional variational problems , 2002 .

[31]  Omid Baghani,et al.  Solving state feedback control of fractional linear quadratic regulator systems using triangular functions , 2019, Commun. Nonlinear Sci. Numer. Simul..