Solution of a specific class of nonlinear fractional optimal control problems including multiple delays

Abstract

SummaryThis research provides a new framework based on a hybrid of block‐pulse functions and Legendre polynomials for the numerical examination of a special class of scalar nonlinear fractional optimal control problems involving delay. The concepts of the fractional derivative and the fractional integral are employed in the Caputo sense and the Riemann‐Liouville sense, respectively. In accordance with the notion of the Riemann‐Liouville integral, we derive a new integral operator related to the proposed basis called the operational matrix of fractional integration. By employing two significant operators, namely, the delay operator and the integral operator connected to the hybrid basis, the system dynamics of the primal optimal control problem converts to another system involving algebraic equations. Consequently, the optimal control problem under study is reduced to a static optimization one that is solved by existing well‐established optimization procedures. Some new theoretical results regarding the new basis are obtained. Various kinds of fractional optimal control problems containing delay are examined to measure the accuracy of the new method. The simulation results justify the merits and superiority of the devised procedure over the existing optimization methods in the literature.