| Summary: | 碩士 === 國立中央大學 === 數學系 === 107 === 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.
|
|---|
