On the optimal spectral Chebyshev solution of a controlled nonlinear dynamical system

In a recent paper we considered the numerical solution of the controlled Duffing oscillator minimize J= 1 2 ∫ −T 0 U 2 (τ)dτ, subject to X(τ)+ w 2 X(τ)+ε X 2 (τ)=U(τ) (−T≤τ≤0), where T is known, with X(-T)= x 0 , X(0)=0 by the pseudospectral Legendre method, which shows that in order to maintain spectra accuracy the grids on which a physical problem is to be solved must also be obtained by spectrally accurate techniques. This paper presents an alternative spectrally accurate computational method of solving the nonlinear controlled Duffing oscillator. The method is based upon constructing the Mth-degree interpolation polynomials, using Chebyshev nodes, to approximate the state and the control vectors. The differential and integral expressions which arise from the system dynamics and the performance index are converted into an algebraic nonlinear programming problem. The results of computer-simulation studies compare favourably with optimal solutions obtained by closed-form analysis and/or by other numerical schemes.