Non-linear optimal control of a Duffing system

The Duffing oscillator is a useful current model for the behavior of structural systems including columns, gyroscopes, plates, pendulums and certain types of bridges. This paper studies a class of non-linear optimal controls for these oscillators, and determines the effects of higher-order feedback corrections based upon series expansions of the optimal cost function and the optimal control function in a Hamilton-Jacobi context. A novel representation of the solution is presented, in terms of the indicial formulation of tensor algebra. For the case in point, the indicial approach offers a conceptually attractive alternative to the general tensor solutions discussed by Buric in 1978 and by O'Sullivan and Sain in 1985. Numerical studies are provided, and an analysis of the effect of higher-order feedbacks upon the stability region of the controlled system is presented.