Solution of the minimum time-to-climb problem by matched asymptotic expansions

TLDR

Singular perturbation techniques are applied to trajectory optimization in flight mechanics. The method of matched asymptotic expansions is employed to derive approximate minimum‑time‑to‑climb solutions and a stability criterion for the zeroth‑order boundary‑layer solutions. The resulting zeroth‑ and first‑order approximations show that the first‑order solution accurately predicts trajectory and minimum time, satisfies the stability criterion, and achieves lower computational cost than steepest‑descent methods, highlighting the promise of singular perturbation approaches for optimal control problems.

Abstract

Application of singular perturbation techniques to trajectory optimization problems of flight mechanics is discussed. The method of matched asymptotic expansions is used to obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer, boundary-layer, and composite solutions are obtained to zeroth and first orders. A stability criterion is derived for the zeroth-order boundary-layer solutions (the theory requires a form of boundary-layer stability). A numerical example is considered for which it is shown that the stability criterion is satisfied and a useful numerical solution is obtained. The zeroth-order solution proves to be a poor approximation, but the first-order solution gives a good approximation for both the trajectory and the minimum time-to-climb. The computational cost of the singular perturbation solution is considerably less than that of a steepest descent solution. Thus singular perturbation methods appear to be promising for the solution of optimal control problems.