In this article, the minimum time and fuel consumption of an aircraft in its climbing phase is studied using direct and indirect schemes implemented in the Bocop and HamPath softwares. The controls are the thrust and the lift and state constraints are taken into account: air slope and speed limitations. The indirect method is the numerical implementation of the Maximum Principle with state-constraints and it is initialized by the direct method. This leads to describe, for prescribed boundary conditions, extremals which satisfy necessary conditions of optimality with at most two boundary arcs.