Fractional optimal control of distributed systems in spherical and cylindrical coordinates

This paper presents a general formulation and numerical scheme for the fractional optimal control problem (FOCP) of distributed systems in spherical and cylindrical coordinates. 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. A method of separation of variables is employed to separate the time and space terms, and the eigenfunction approach is used to eliminate the terms containing space parameter and define the formulation in terms of countable number of infinite state and control variables. The fractional optimal control equations are reduced to the Volterra-type integral equations. For the problems considered, only a few eigenfunctions in each direction are sufficient for the calculations to converge. The time domain is discretized into several subintervals and the result is more stable for a larger number of time segments. Various orders of fractional derivatives are analyzed and the results converge toward those of integer optimal control problems as the order approaches the integer value of 1.