Constrained optimal control of a fractionally damped elastic beam

Abstract

This work presents the constrained optimal control of a fractionally damped elastic beam in which the damping characteristic is described with the Caputo fractional derivative of order 1/2. To achieve the optimal control that involves energy optimal control index with fixed endpoints, the fractionally damped elastic beam problem is first converted to a state space form of order 1/2 by using a change of coordinates. Then, the state and the cost-ate equations are set in terms of Hamiltonian formalism and the constrained control law is acquired from Pontryagin Principle. The numerical solution of the problem is obtained with Grunwald-Letnikov approach by utilizing the link between the Riemann-Liouville and the Caputo fractional derivatives. Application of the formulations is demonstrated with an example and the illustrations are figured by MATLAB. Also, the effectiveness of the Griinwald-Letnikov approach is exhibited by comparing it with an iterative method which is one-step AdamsBashforth-Moulton method.