An efficient computational method based on the hat functions for solving fractional optimal control problems

Abstract

In this paper, an efficient and accurate computational method based on the hat functions (HFs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the fractional optimal control problem under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional state and control variables are expanded by the HFs with unknown coefficients. Then, the operational matrix of fractional integration of the HFs with some properties of these basis functions are employed to achieve a nonlinear algebraic equation, replacing the performance index and a nonlinear system of algebraic equations, replacing the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

A new Wavelet Method for Variable‐Order Fractional Optimal Control Problems

“…Finally, the method of constrained extremum is applied. By comparing the numerical results obtained in Table with the ones in , one can simply see that our new results are more accurate than the results obtained in . However, it can be expressed that the proposed method requires less computational efforts and CPU time to compute the solution.…”

“…Also, for this case Heydari et al. in have proposed a computational method based on the HFs to obtain an approximate solution for the problem. In their proposed method, the problem is reduced to a system of nonlinear algebraic equations.…”

“…From this table, it can be stated that the numerical results are in a good agreement with the exact solutions. For the case α ( t )=1.9, the problem is solved by numerical methods in . One can simply see that the computational work of the proposed method is less than the method in and also the new results obtained are more accurate than the ones in and requires less time to compute the solution.…”

“…For the case α ( t )=1.9, the problem is solved by numerical methods in . One can simply see that the computational work of the proposed method is less than the method in and also the new results obtained are more accurate than the ones in and requires less time to compute the solution. However, it can be seen that the proposed method requires less computational efforts and CPU time to compute the solution.…”

An efficient computational approach for linear and nonlinear fractional differential equations

“…The theory of fractional calculus is defined manly two types Riemann-Liouville and Caputo in differential and integral operators. The fractional derivatives and integrals are the generalization of the standard derivatives and integrals having genetic and nonlocal effects in material properties, studied and described by number of researchers such as Caputo [4], Heydari et al [12,13], Oldham and Spanier [23], Podlubny [24], Miller and Ross [22], Samko et al [30], Kilbas et al [15], Yang et al [34] and many others. In order to solve nonlinear differential equations various techniques have been used such as variational iteration scheme is employed to examine three-dimensional Navier-Stokes equations of flow between two stretchable disks [27], differential transform method for solving Burgers' and nonlinear heat transfer equations [28], homotopy perturbation technique for nonlinear vibration of von karman rectangular plates [29], etc.…”