在物理學和工程學中具有廣泛應用的軌跡優化問題可以在某種形式的連續時間最優控制問題中進行數學建模。在對最優控制問題進行離散化之後，我們使用拉格朗日 - 牛頓方法求解得到的參數約束優化問題。在這種方法中，我們將拉格朗日乘數引入目標函數，然後通過找到一階必要條件（KKT條件）的臨界解來求解約束優化問題。我們考慮兩類拉格朗日 - 牛頓方法：一類是全空間算法，另一類是簡化空間算法。全空間算法同時更新控制，狀態和拉格朗日乘數。另一方面，縮減空間算法按順序更新這些變量。在這項研究中，我們用數字表示，對於Hessian矩陣的構造，分析方法的計算時間小於有限差分法和BFGS方法的計算時間。值得注意的是，全空間拉格朗日 - 牛頓算法比簡化空間拉格朗日 - 牛頓算法更快，特別是對於精細網格情況。;The trajectory optimization problem with a wide range of applications in physics and engineering can be modeled mathematically in some form of continuous time optimal control problems. After discretizing the optimal control problem we solve the resulting parameter constrained optimization problem by using the Lagrange-Newton method. In this method, we introduce the Lagrange multiplier to the objective function and then solve the constrained optimization problem by finding the critical solution of the first-order necessary condition (KKT condition). We consider two classes of Lagrange-Newton method: one is the full space algorithm and the other is the reduced space algorithm. The full space algorithm updates the control, state, and Lagrange multipliers at the same time. On the other hand, the reduced space algorithm updates those variables sequentially. In this study, we show numerically that for the construction of the Hessian matrix, the computing time for the analytical method is less than that for the finite difference method and the BFGS method. Remarkably, the full space Lagrange-Newton algorithm is faster than the reduced space Lagrange-Newton algorithm, especially for refined mesh cases.
