A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

Abstract

This paper aims to demonstrate the superiority of the discrete Chebyshev polynomials over the classical Chebyshev polynomials for solving time-delay fractional optimal control problems (TDFOCPs). The discrete Chebyshev polynomials have been introduced and their properties are investigated thoroughly. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by these polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algabric system. A comparison has been made between the required CPU time and accuracy of the discrete and continuous Chebyshev polynomials methods. The obtained numerical results reveal that utilizing discrete Chebyshev polynomials is more efficient and less time-consuming in comparison to the continuous Chebyshev polynomials.

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method

“…As can be seen in Figure 1 , the initial condition is achieved with the proposed method. By contrast, that condition was not reached in previous works [ 26 , 28 , 38 , 39 ], thus increasing their error.…”

“…This problem was introduced by Moradi and Mohammadi [ 38 ], who proposed a solution based on discrete Chebyshev polynomials. More precisely, the authors solved this problem for different choices of [ 26 , 28 ].…”

“…Consider the following time-varying DFOCP, subject to: where . This example have been solved by Rahimkhani et al [ 28 ], Haddadi et al [ 41 ], Ordukhani et al [ 42 ], Moradi et al [ 27 , 38 ], and Rabiei et al [ 29 ], but any of them reached the initial condition . A comparison of the values of J obtained by these methods and that reported by the proposed scheme is presented in Table 5 .…”

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

“…Note that depending on the type of the inner product defined over the solution space, the orthogonal polynomials are classified into continuous and discrete polynomials. 16 In dealing with the contin-uous polynomials, we have to compute an integral in the inner product, whereas a summation should be evaluated for the discrete polynomials. It is worth noting that although the continuous polynomials have been extensively utilized for solving different kinds of functional equations, but the discrete polynomials have many advantages that cause them to be successfully used for solving different problems.…”

“…The significance is even more obvious in dealing with more complicated operators such as variable order fractional derivatives and integrals. Note that depending on the type of the inner product defined over the solution space, the orthogonal polynomials are classified into continuous and discrete polynomials 16 . In dealing with the continuous polynomials, we have to compute an integral in the inner product, whereas a summation should be evaluated for the discrete polynomials.…”