In the optimal control theory, the Hamiltonian formulation is a famous one convenient to find an optimal designed control force. However, when the performance index is a complicated function of control force, the Hamiltonian method is not easy to find the optimal solution, because one may encounter a twopoint boundary value problem of nonlinear differential algebraic equations (DAEs). In this paper we address this issue via a quite novel and effective approach, of which the optimally controlled vibration problem of Duffing oscillator is recast into a two-point nonlinear DAEs by identifying the unknown control force. We develop the corresponding SL(n, R) and GL(n, R) shooting methods, as well as a Lie-group differential algebraic equations (LGDAE) method to numerically solve the optimal control forces. Eight examples of a single Duffing oscillator and one coupled Duffing oscillators are used to test the performance of the present method.

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