A general numerical scheme for the optimal control of fractional Birkhoffian systems

Abstract

This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.