Constrained Optimal Control of A Fractionally Damped Elastic Beam

Abstract

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 costate 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 Grünwald-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 Grünwald-Letnikov approach is exhibited by comparing it with an iterative method which is one-step Adams-Bashforth-Moulton method.