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Numerical solution of the controlled Duffing oscillator by the pseudospectral method. (English) Zbl 0827.65074
The authors introduce a new direct numerical method for solving the controlled Duffing oscillator. The method is based on a pseudospectral method in which the authors construct the $$m$$th degree interpolation polynomials using the Legendre-Gauss-Lobatto collocation points to approximate the state and control functions.
With this method, the system dynamics, initial conditions and the integral expression are converted to a system of algebraic equations that can be solved for the unknown coefficients by the iterative Newton method. Finally, numerical results are presented and a comparison is made with an existing method of Chebyshev approximation to demonstrate the efficiency and the accuracy of the proposed numerical method.

##### MSC:
 65K10 Numerical optimization and variational techniques 93C15 Control/observation systems governed by ordinary differential equations
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##### References:
 [1] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer New York · Zbl 0658.76001 [2] Gille, J.C.; Decaulne, P.; Pelegrain, M., Systems asservis non linéaires, (1975), Bordas Paris [3] Gottlieb, D.; Hussaini, M.Y.; Orszag, S.A., Theory and applications of spectral methods, () [4] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), Wiley New York [5] Pontryagin, L.S.; Boltyanskii, V.; Gamkrelidze, R.; Mischenko, E., The mathematical theory of optimal processes, (1962), Interscience New York [6] Vlassenbroeck, J.; Van Dooren, R., Chebyshev series solution of the controlled Duffing oscillator, J. comput. phys., 47, 321-329, (1982) · Zbl 0493.65031
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