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This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

A comprehensive review on fractional-order optimal control problem and its solution