軌跡最佳化是航太工業經常使用的技術，例如找出最佳軌跡使得酬載重量最大化或是縮短衛星到達目標軌道的時間等等，這種類型的問題可以用數學建模成連續時間的最佳化控制問題。本篇論文主要是研究full-space quasi Lagrange-Newton-Krylov method作為數值求解器，首先會引入拉格朗日乘子將有約束的最佳化問題轉為無約束的最佳化問題，並求解一階必要條件(KKT condition)，透過牛頓法結合回朔技術，在每一次牛頓迭代中，所有KKT系統的變數(狀態變數, 控制變數, 設計變數, 拉格朗日乘子) 會以 Krylov-subspace method 配合預處理技術同時被解出。本文會針對KKT系統中的 Hessian matrix 的建造找出提高計算效能的方法，並透過引入鬆弛變量來解決不等式限制條件問題。本文以三維座標的三節火箭問題和二維座標的金牛座火箭作為數值範例，相關的物理背景和數值結果將會在文章中呈現。

The trajectory optimization is a commonly-used technique of applications in aerospace engineering, such as find the optimal trajectory to maximize the weight of the payload or minimize the time for satellite insertion the target orbit, etc. This type of problem can be modeled mathematically as some form of continuous time optimal control problems. This thesis focuses on the study of the full-space quasi Lagrange-Newton-Krylov method as our numerical solver. First, we introduce the Lagrangian multiplier to convert the constrained optimization problem into unconstrained optimization problem, and solve the first-order necessary condition(KKT condition). By the Newton method combined with the backtrack technique, in each Newton iteration, all KKT system variables (state variables, control variables, design variables, Lagrangian multipliers) will be solved at the same time by Krylov-subspace method with the precondition technology. In this thesis, we improve the computational efficiency of Hessian matrix constructor in KKT system, and solve the problem of inequality constraint by introducing slack variable. The three-dimensional case of the three stage rocket problems and two-dimensional case of Taurus rocket as a numerical example, the relevant physical background and numerical results will be presented in the thesis.

[1] M. Benzi, G.H. Golub, and J. Liesen. Numerical solution of saddle point problems.
Acta Numer., 14:1–137, 2005.
[2] J.T. Betts. Survey of numerical methods for trajectory optimization. J. Guid. Contr.
Dynam., 21:193–207, 1998.
[3] J.T. Betts. Very low-thrust trajectory optimization using a direct SQP method. J.
Comput. Appl. Math., 120:27–40, 2000.
[4] J.T. Betts. Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming. SIAM, Philadelphia, 2nd edition, 2010.
[5] N. Biehn, S.L. Campbell, L. Jay, and T.Westbrook. Some comments on DAE theory
for IRK methods and trajectory optimization. J. Comput. Appl. Math., 120:109–131,
2000.
[6] G. Biros and O. Ghattas. Parallel Lagrange–Newton–Krylov–Schur methods for
PDE-constrained optimization. Part II: The Lagrange–Newton solver and its application
to optimal control of steady viscous flows. SIAM J. Sci. Comput., 27:714–739,
2005.
[7] G. Biros and O. Ghattas. Parallel Lagrange-Newton-Krylov-Schur methods for
PDE-constrained optimization. part I: The Krylov–Schur solver. SIAM J. Sci. Comput.,
27:687–713, 2005.
[8] A.E. Bryson and Y.-C. Ho. Applied Optimal Control: Optimization, Estimation, and
Control. Taylor & Francis Group, 1975.
[9] J.E. Dennis Jr. and R.B. Schnabel. Numerical methods for unconstrained optimization
and nonlinear equations. SIAM, Philadelphia, 1996.
[10] D.J. Estep, D.H. Hodges, and M. Warner. The solution of a launch vehicle trajectory
problem by an adaptive finite-element method. Computer methods in applied
mechanics and engineering, 190(35):4677–4690, 2001.
[11] F. Fahroo and I.M. Ross. Costate estimation by a Legendre pseudospectral method.
J. Guid., Contr. Dynam, 24:270–277, 2001.
[12] M. Fink. Automatic Differentiation for Matlab. MATLAB Central File Exchange.
Retrieved May 18, 2015, 2007.
[13] P.E. Gill, L.O. Jay, M.W. Leonard, L.R. Petzold, and V. Sharma. An SQP method
for the optimal control of large-scale dynamical systems. J. Comput. Appl. Math.,
120:197–213, 2000.
[14] D.G. Hull. Optimal Control Theory for Applications. Springer-Verlag, New York,
2003.
[15] C.T. Kelley. Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia,
1995.
[16] Wilfried Ley, KlausWittmann, andWilli Hallmann. Handbook of space technology,
volume 22. John Wiley & Sons, 2009.
[17] P. Lu and M.A. Khan. Nonsmooth trajectory optimization-an approach using continuous
simulated annealing. J. Guid. Contr, Dynam., 17:685–691, 1994.
[18] R Timothy Marler and Jasbir S Arora. Survey of multi-objective optimization methods
for engineering. Struct. Multidisc. Optim., 26(6):369–395, 2004.
[19] J. Nocedal and S.J. Wright. Numerical Optimization. Springer-Verlag, New York,
2006.
[20] M.A. Patterson and A.V. Rao. GPOPS-II: A MATLAB software for solving
multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation
methods and sparse nonlinear programming. ACM Trans. Math. Software,
41:1, 2014.
[21] M. Pontani. Particle swarm optimization of ascent trajectories of multistage launch
vehicles. Acta Astronaut., 94:852–864, 2014.
[22] E.E. Prudencio, R. Byrd, and X.C. Cai. Parallel full space SQP Lagrange-Newton-
Krylov-Schwarz algorithms for PDE-constrained optimization problems. SIAM J.
Sci. Comput., 27:1305–1328, 2006.
[23] A.V. Rao. Trajectory optimization: A survey. In Optimization and Optimal Control
in Automotive Systems, pages 3–21. Springer, 2014.
[24] W. Roh and Y. Kim. Trajectory optimization for a multi-stage launch vehicle using
time finite element and direct collocation methods. Eng. Optim., 34:15–32, 2002.
[25] Woongrae Roh and Youdan Kim. Trajectory optimization for a multi-stage launch
vehicle using time finite element and direct collocation methods. Engineering optimization,
34(1):15–32, 2002.
[26] Y. Saad. Iterative methods for sparse linear systems. SIAM, Philadelphia, 2nd
edition, 2003.
[27] M.R. Sentinella and L. Casalino. Cooperative evolutionary algorithm for space trajectory
optimization. Celestial Mech. Dyn. Astron., 105:211–227, 2009.
[28] K. Subbarao and B.M. Shippey. Hybrid genetic algorithm collocation method for
trajectory optimization. J. Guid. Contr, Dynam., 32:1396–1403, 2009.
[29] S. Subchan and R. ˙ Zbikowski. Computational Optimal Control: Tools and Practice.
John Wiley & Sons, 2009.
[30] BN Suresh and K Sivan. Integrated Design for Space Transportation System.
Springer, 2015.
[31] O. von Stryk and R. Bulirsch. Direct and indirect methods for trajectory optimization.
Annals Oper. Res., 37:357–373, 1992.
[32] P. Williams. Jacobi pseudospectral method for solving optimal control problems. J.
Guid. Contr. Dynam., 27:293–297, 2004.
[33] A. Wuerl, T. Crain, and E. Braden. Genetic algorithm and calculus of variationsbased
trajectory optimization technique. J. Spacecraft Rockets, 40:882–888, 2003.
[34] H. Yang, F.-N. Hwang, and X-.C. Cai. Nonlinear preconditioning techniques for
full-space Lagrange-Newton solution of PDE-constrained optimization problems.
SIAM J. Sci. Comput., to appear, 2016.