  In this article, we are interested in optimal aircraft trajectories in
climbing phase. We consider the cost index criterion which is a convex
combination of the time-to-climb and the fuel consumption. We assume that the
thrust is constant and we control the flight path angle of the aircraft. This
optimization problem is modeled as a Mayer optimal control problem with a
single-input affine dynamics in the control and with two pure state
constraints, limiting the Calibrated AirSpeed (CAS) and the Mach speed. The
candidates as minimizers are selected among a set of extremals given by the
maximum principle. We first analyze the minimum time-to-climb problem with
respect to the bounds of the state constraints, combining small time analysis,
indirect multiple shooting and homotopy methods with monitoring. This
investigation emphasizes two strategies: the common CAS/Mach procedure in
aeronautics and the classical Bang-Singular-Bang policy in control theory. We
then compare these two procedures for the cost index criterion.
