A new class of orthonormal basis functions: application for fractional optimal control problems

This study aims to generate a novel set of basis functions called the orthonormal piecewise Chelyshkov functions to solve a certain category of optimal control problems whose dynamical system is governed by a nonlinear fractional differential equation. A new fractional integral matrix associated with these basis functions is derived. This matrix significantly reduces the computations in solving such problems. The proposed approach transforms the original problem into a nonlinear programming one by expanding the control and state variables in terms of the orthonormal piecewise Chelyshkov functions and employing the derived fractional integral matrix. Some numerical problems are examined for verification of the proposed method.

Automated summary

This study introduces a novel approach to solving optimal control problems governed by nonlinear fractional differential equations, reducing computations significantly by introducing orthonormal piecewise Chelyshkov functions and a fractional integral matrix. The original problem is transformed into a nonlinear programming one through expanding control and state variables in terms of these functions and using the derived matrix. Numerical problems are examined to verify the proposed method.