Fractional Optimal Control of a Hollow Cylindrical Structure

This paper presents a general formulation and numerical scheme for Fractional Optimal Control Problem (FOCP) of a distributed system in cylindrical coordinate and uses a hollow cylinder with axial symmetry as the example to demonstrate the method. The fractional derivatives are expressed in the Caputo-Sense. The performance index of FOCP is considered as a function of both the state and the control variables and the dynamic constraints are expressed by a partial fractional differential equation. The method of separation of variables is employed to find the solution of the problem and the eigenfunction approach is used to decouple the equations. The Fractional Optimal Control (FOC) equations are reduced to the Volterra-type integral equations. Only a few eigenfunctions in both radial and axial directions are sufficient for the results to converge. The time domain is discretized into several subintervals and the result is more stable for smaller time steps. Various orders of fractional derivatives are analyzed and the numerical results converge toward the analytical solutions as the order of derivative goes toward the integer value of 1.