Abstract
This paper presents three direct methods based on Grünwald–Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald–Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCPs are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.
Highlights
Fractional calculus may be considered an old and yet an interesting topic [1]
fractional optimal control problems (FOCPs) are a generalization of the integer-order optimal control problems, which are obtained by replacing integer-order derivatives with fractional ones
FOCPs are solved by transcribing them into Nonlinear Programming problems (NLP)
Summary
Fractional calculus may be considered an old and yet an interesting topic [1]. It deals with the investigation of integrals and derivatives of an arbitrary order. The solution is obtained by solving a fractional Hamiltonian boundary-value problem, which is derived from the optimality conditions. Direct methods do not rely on optimality conditions In these approaches, FOCPs are solved by transcribing them into Nonlinear Programming problems (NLP). We extend these methods to fractional order optimal control problems For this purpose, the matrix form of Grunwald-Letnikov, trapezoidal and Simpson approximation formulas for fractional integrals, which are called fractional integral matrices, are derived. The reliability, efficiency and accuracy of the proposed direct method are demonstrated with four test problems, including a nonlinear and complex FOCP, a FOCP with path and terminal constraints, a bang-bang FOCP and an applied optimal control problem of a fractional order HIV-immune system with memory.
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