In this paper, a hybrid approximation \( F_{h\alpha} = a_1 + a_2 (fh\alpha) + a_3 (fh\alpha)^2 \) and prediction \(\min I (a_1, a_2, a_3) = \sum_{i=1}^n [a_1 + a_2 x_i + a_3 x_i^2 – y_i]^2\) model based on the generalized extension method is constructed for the optimization problem of nonlinear systems, and a Lyapunov stability analysis is carried out. The model is applied to the nonlinear gear micro-parameter optimization system and the trajectory tracking optimization system for optimal parameter solving and physical trajectory prediction reduction. During the gear microscopic parameter optimization process, most of the optimal parameter solving errors of the generalized extension method are between 0.00 and 0.02, which have high accuracy. The convex eigenfunction initial value construction and the spectral method discretization are utilized to solve the problem of efficiently solving the nonlinear algebraic equation system in the case of multiple eigenvalues, and the flight physical trajectory is effectively restored. The Lyapunov stability analysis shows that the generalized prolongation method satisfies the spectral stability condition under subharmonic perturbation.
