Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft

Abstract

In this article, the minimum time and fuel consumption of an aircraft in its climbing phase is studied. The controls are the thrust and the lift coefficient and state constraints are taken into account: air slope and speed limitations. The application of the Maximum Principle leads to parameterize the optimal control and the multipliers associated to the state constraints with the state and the costate and leads to describe a Multi- Point Boundary Value Problem which is solved by multiple shooting. This indirect method is the numerical implementation of the Maximum Principle with state-constraints and it is initialized by the direct method, both to determine the optimal structure and to obtain a satisfying initial guess. The solutions of the boundary value problems we define give extremals which satisfy necessary conditions of optimality with at most two boundary arcs. Note that the aircraft dynamics has a singular perturbation but no reduction is performed.