The insufficient air routes combined with the adverse weather and congestion to air sectors lead to economic, environmental and safety problems to political aviation in Europe. This situation creates negative aspects to airlines and airports, as well. Furthermore, according to recent studies over 40,000 daily flights are predicted for 2020, and therefore the current ATM system will not be able to handle this volume of traffic in an efficient manner. A new promising approach of solving these problems in the future consists of transforming the ATM system from an 'airport-centered' to an 'airplane-centered' system so it can: (i) increase safety and energy efficiency, (ii) support the free flight concept, (iii) distribute fairly ground-holding and air delays among the flights, (iv) minimize the volume of work of ATCs as an observer, (v) relax the existing distance limits between airplane since the human factor has been annihilated, and therefore, (vi) increase the air sectors' capacity avoiding congestions and (vii) prioritize the airline preferences. Our attempt will be to develop a mathematical model for a support system for the free flight concept. We divide the problem into two sub-problems (upper and lower level) in order to decrease the computational efforts and the complexity of the air traffic flow management problem and to allow flexibility, supporting in the same time the free flight scenario.
We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints, including pointwise state constraints. Because no convexity assumptions are made, the problem may have no classical solutions, and it is reformulated in relaxed form. The relaxed control problem is then discretized by using the implicit midpoint scheme, while the controls are approximated by piecewise constant relaxed controls. We first study the behavior in the limit of properties of discrete relaxed optimality, and of discrete relaxed admissibility and extremality. We then apply a penalized conditional descent method to each discrete relaxed problem, and also a corresponding discrete method to the continuous relaxed problem that progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that accumulation points of sequences generated by these methods are admissible and extremal for the discrete or the continuous problem. Finally, numerical examples are given.
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