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

Abstract

In this manuscript, we investigate two categories of optimal control problems (OCPs), OPCs via fractional Volterra integro-differential equations and Volterra integral equations. Touchard wavelets as an appropriate class of bases are defined to develop a new hybrid scheme for the considered problems. To this approach, Riemann–Liouville fractional integral operator (RLFIO) of Touchard wavelets is achieved exactly using the Hypergeometric functions. Next, by approximating the fractional derivative of the state variables and control variables using the mentioned wavelet functions, applying RLFIO, collocation method, and Gauss–Legendre quadrature formula, the considered problems are inserted into systems of algebraic equations, which can be solved using “FindRoot” package in Mathematica software. Numerical results are presented that validate the theory and show the effectiveness of the established technique.

Solving distributed-order fractional optimal control problems via the Fibonacci wavelet method

“…Lemma 1. If RL 0 I ν t is the Riemann-Liouville integral operator, the following relation is established (Rabiei and Razzaghi, 2021;Sabermahani et al, 2022…”

“…If 0RLfrakturItν is the Riemann–Liouville integral operator, the following relation is established (Rabiei and Razzaghi, 2021; Sabermahani et al, 2022)where χ c is the unit step function, 2frakturF1 is the hypergeometric function, andwhere |η|<1 and…”