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Numerical solutions for distributed-order fractional optimal control problems by using Müntz–Legendre wavelets

In this paper, a numerical method using Münzt–Legendre wavelets for distributed-order fractional optimal control problems (DO-FOCPs) is presented. We first give an exact formula of the Riemann–Liouville fractional integral operator of the Münzt–Legendre wavelets by applying regularized beta function. By using the obtained exact formula, we transform the DO-FOCP to a new optimization problem which can be solved by existing techniques. The error analysis of the method is also given. Several illustrated examples are included to show the advantages of our method.