In this paper, an efficient method based on rationalized Haar (RH) wavelets is proposed for the numerical solution of optimal control problem for systems governed by Volterra integral equations with a quadratic performance index. Many problems in economics, biology, epidemiology and memory effects can be modeled as Volterra control problems. The main advantage of the RH wavelet is based on its efficiency and simple applicability. The properties of RH wavelets are represented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the solution of the optimization problem to a nonlinear programming one to which well-developed algorithms may be applied. The convergence analysis of the method and illustrative examples are included to demonstrate the validity and applicability of the technique.
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.
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