本論文主要在於分析民航機飛行之最省燃料軌跡，最省燃料軌跡之分析在於以Euler-Lagrange所導出之最佳化必要條件包含奇異弧(singular arc)}。
因以數值分析方法分析奇異弧問題，尚有困難，故於研究中，係以奇攝動法的概念，設速度與飛行路徑角為快變數，將原來運動方程式降階，其目的在於消除奇異弧之問題。
研究中發現，若將速度與飛行路徑角之兩個最佳控制條件聯立，以解兩個控制變數，則在反覆計算中無法收歛。因此，於研究中，
乃設飛行路徑角為時間之多項式函數，以避開數值計算之敏感度問題。又因零升力阻力係數在馬赫數接近穿音速時急遽增加，馬赫數很難以最佳控制條件獲得，故在研究中係假設巡航馬赫數為常數參數，算出最佳軌跡後再變化其值，
以求最佳馬赫數。本論文分析方法較傳統之工程方法更進一步的是，前者允許飛行途中高度變化，且高度符合端點條件；而後者若允許飛行途中高度變化，則高度無法符合端點條件，
若欲高度符合端點條件，則飛行途中高度無法變化。

The principal objective of this thesis is to analyze the minimum-fuel flight
trajectory of civil aircraft. The difficulty in the analysis of minimum-fuel flight
trajectory is that a singular arc condition is involved in
the necessary conditions derived with the Euler-Lagrange method.
Since it is always very difficult to analyze the singular-arc problem numerically,
a singular perturbation concept is introduced in the analysis. By using this concept,
the velocity and the flight path angle are assumed to be fast variables.
As long as the zeroth-order outer solution is studied only,
the order of original equations of motion can be reduced and the singular arc condition
disappears. However, after a further study, it is found that if the velocity
and the flight path angle are solved simultaneously from the two optimal control
conditions, the two control variables are very difficult to converge.
In order to circumvent this difficulty, the flight path angle are modeled as
a polynomial function of the time. Also, since the zero-lift drag coefficient
increases rapidly as the Mach number approaches in the transonic range, it is
very difficult to compute the optimal Mach number from the optimal velocity condition.
In this study, the Mach number is assumed to be constant and an optimal trajectory
is determined correspondingly. The Mach number is then changed to another constant value
and the
optimal trajectory is determined again correspondingly. After several trajectories
are determined, the optimal Mach number can be readily deduced. The analysis method
in this thesis is better than that in the traditional engineering. The
former allows the altitude to vary in between the boundary
and to satisfy the two boundary
conditions while the latter does not. By using the latter method,
if the altitude is to satisfy the boundary conditions, then it can not vary with
the time in between the boundary. However, if the altitude is to vary in between the boundary,
then it can not satisfy the two boundary conditions.

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