ON THE OPTIMAL SPECTRAL CHEBYSHEV SOLUTION OF A CONTROLLED NONLINEAR DYNAMICAL SYSTEM

In a recent paper we considered the numerical solution of the controll ed Duffing oscillator: minimize J = 1/2 integral(-T)(0) U-2(tau) d tau , subject to X (tau) + w(2)X (tau) + epsilon X-2(tau) = U(tau) (-T les s than or equal to tau less than or equal to 0), where T is known, wit h X(-T) = x(0), X(0) = 0, by the pseudospectral Legendre method, which shows that in order to maintain spectral 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 ac curate computational method of solving the nonlinear controlled Duffin g oscillator. The method is based upon constructing the Mth-degree int erpolation polynomials, using Chebyshev nodes, to approximate the stat e and the control vectors. The differential and integral expressions w hich arise from the system dynamics and the performance index are conv erted 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.