Optimal control of fractional integro-differential systems based on a spectral method and grey wolf optimizer

Abstract

This paper elaborated an effective and robust metaheuristic algorithm with acceptable performance based on solution accuracy. The algorithm applied in solution of the optimal control of fractional Volterra integro-differential (FVID) equation which be substituted by nonlinear programming (NLP). Subsequently the FIVD convert the problem to a NLP by using spectral collocation techniques and thereafter we execute the grey wolf optimizer (GWO) to improve the speed and accuracy and find the solutions of the optimal control and state as well as the optimal value of the cost function. It is mentioned that the utilization of the GWO is simple, due to the fact that the GWO is global search algorithm, the method can be applied to find optimal solution of the NLP. The efficiency of the proposed scheme is shown by the results obtained in comparison with the local methods. Further, some illustrative examples introduced with their approximate solutions and the results of the present approach compared with those achieved using other methods.

Optimal control of system governed by nonlinear volterra integral and fractional derivative equations

“…At the end, a comparison between the obtained optimal performance indicator J with the suggested method and the other ones reported in Khanduzi et al (2020) for N = 7, where α = 1 and M = N + 2 is reported in Table 4 (for Examples 1, 2 and 3). As can be seen, the superiority of the method for solving NVIFs is clear that the implementation of Chelyshkov polynomials is efficient and accurate.…”

Solution of optimal control problems governed by volterra integral and fractional integro-differential equations

“…Moreover, some numerical schemes are proposed for solving OCPs by integer and non-integer integro-differential equations such as the methods based on fractional-order Legendre functions (Rabiei et al (2018)), discrete Hahn polynomials (Mohammadi et al (2022)), Orthonormal piecewise Bernoulli functions (Heydari et al (2022)), hybrid of block-pulse functions and Legendre polynomials (Marzban (2020)), spectral method and grey wolf optimizer (Khanduzi et al (2020)), Chebyshev approximations (El-Kady and Moussa (2013)), etc.…”

Existence of optimal control for an integro‐differential Bolza problem

“…Most of the existing results concern the existence of a solution to (), its uniqueness and continuous dependence on parameters, as well as numerical methods of finding it. Numerical solutions to optimal control problems governed by integro‐differential equations of Volterra type were studied, for example, in Reference 2 (case of nonlinear control system and nonlinear integral cost functional), and in References 3 and 4 (case of linear system and quadratic cost functional). Optimal control of feedback type involving also the past history of the solution has been obtained in Reference 5 for a scalar linear system (with f 2 = u ) and quadratic cost functional.…”