In this talk we will first review optimal control and approaches to solve optimal control problems based on the Hamilton-Jacobi-Bellman equation, its symmetries, and also based on Pontryagin's Maximum Principle. Then we will formulate the problem of Flight Management Systems (FMS) in cruise economy mode as an optimal control problem and find approximate analytical solutions that depend on a scalar parameter called the coefficient index (used by pilots in airlines). We will show that when the coefficient index converges to zero the analytical solution is exact and corresponds to the maximum range solution. We then proceed to show that the solution of the climb and descent FMS problem in economy mode using Bellman's Principle of Optimality verifies the expected vertical flight profile as described in FMS documentation by Airbus and Boeing. No analytical solutions are found for climb and descent but it will be shown that the optimal control solution corresponds to the roots of a fifth order polynomial that can be found numerically using Newton's Method for a particular model of an aircraft. Flight simulation data from an FMS of an Airbus A320 (courtesy of TRU Simulation & Training) will be shown to agree very closely with the theoretical predictions.
Groupe d’études et de recherche en analyse des décisions