Before revision:  (2) the Principal Orthogonal Decomposition (POD) is performed to reduce these models to ones with merely two variables. The minimum-time entrainment of the circadian phase is performed in a feedback form; the minimum-time entrainment of the second-order model is treated as a two-point boundary valued problem(TPBVP), which is solved by a direct shooting algorithm in terms of the Hamiltonian and Pontryagin Minimum Principle. Eventually, the results in low-order models are used for initializing a gradient descent algorithm , which is introduced in solving the entrainment problem of the full models .
After revision:  (2) the Principal Orthogonal Decomposition (POD) . The variational calculus and a gradient descent algorithm are applied for solving the optimal light input in the high-order models. As the results of the gradient descent algorithm , which is introduced in solving the entrainment problem of the full models .