A formulation and numerical scheme for fractional optimal control of cylindrical structures subjected to general initial conditions

Abstract

A general formulation and numerical scheme for Fractional Optimal Control (FOC) of cylindrical structures are presented in this chapter. Examples of solid cylinder and hollow cylinder with axial symmetry are discussed to demonstrate the method. The fractional derivatives (FDs) are expressed in the form of Caputo derivatives. The performance index of the FOC problem is considered as a function of both the state and the control variables, and the dynamic constraint is 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. Convergence studies are conducted to determine the number of eigenfunctions in the radial and axial directions. The results also converge as the time step decreases. Various orders of FDs are analyzed and the numerical results converge toward the analytical solutions as the order of derivative goes toward the integer value of 1, and therefore verifies the numerical scheme. Parameter studies of problems with different initial conditions indicate that the method applies to systems that are subjected to general initial conditions.