A numerical approach for solving fractional optimal control problems using modified hat functions

Abstract

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems. More precisely, in this work we begin by considering the following fractional optimal control problem (FOCP):

Distributed-Order Non-Local Optimal Control

“…It follows from (12) that . 14Substituting (14) into 13 Example 2. The extremal (x,ũ, λ) given in Example 1 is a global minimizer for problem (7).…”

“…By applying such a result, it is possible to find and identify candidate solutions to the optimal control problem. For the state of the art on fractional optimal control, we refer the readers to [13][14][15] and references therein. Recently, distributed-order fractional problems of the calculus of variations were introduced and investigated in [16].…”

An efficient collocation method with convergence rates based on Müntz spaces for solving nonlinear fractional two-point boundary value problems

“…Example 5 (Salati et al 2019) Determine the state function x(t) and the control function u(t) that minimize the cost functional Table 4 Comparison of CPU time in seconds (CTs) and l 2 errors obtained by our method versus the proposed methods in Salati et al (2019) and Nemati et al (2019)…”

“…This problem is solved in Salati et al (2019), using the Grünwald−−Letnikov formula (GL method), trapezoidal formula (TR method), and the Simpson formula (SI method) to approximate the fractional integral. Also, in Nemati et al (2019), a numerical method based on modified hat functions is presented for solving this problem. In this paper, we derive the necessary optimality conditions and then in Table 4, we list the comparisons between the results obtained by our method and methods presented in Nemati et al (2019) and Salati et al (2019).…”

“…Also, in Nemati et al (2019), a numerical method based on modified hat functions is presented for solving this problem. In this paper, we derive the necessary optimality conditions and then in Table 4, we list the comparisons between the results obtained by our method and methods presented in Nemati et al (2019) and Salati et al (2019). To this aim, we define the l 2 norm of the error in control and state, i.e.,…”