The new initiatives of air traffic management (ATM) imposed by EUROCONTROL and EU Commission aim to eliminate the drawbacks of the current ATM system in Europe. Triggered by the imperative need for increasing the air sectors' capacity to face the future increase in the flight demand, innovative methodologies are developed aligned with the modernisation of ATM procedures. This study proposes a two-level hierarchical framework based on mixed integer nonlinear programming (MINLP) in the context of free flight. The first level aims to minimise the costs derived from ground - holding and cancellation policies, and from fuel consumption due to airborne delays and speed deviations. The second level focuses on minimising the costs derived from the fuel consumption for travelling in specified direction, speed and flight levels. The framework generates the optimal 4D trajectories of each airplane for a given time frame. It is designed as a two-level hierarchical architecture model to decrease the computing demands of NP hard formulations of current ATM mathematical models.

The growth in demand for air transport has generated new challenges for capacity and safety. In response, manufacturers develop new types of aircraft while airlines open new routes and adapt their fleet. This excessive demand for air transport also leads to the need for further investments in airport expansion and ATM modernization. The current work was focused on the ATM problem with respect to new procedures, such as free flight, for addressing the air capacity issues in an environmental approach. The study was triggered by and aligned with the following performance objectives set by EUROCONTROL and the European Commission: (1) to improve ATM safety whilst accommodating air traffic growth; (2) to increase the ATM network efficiency; (3) to strengthen ATM’s contribution to aviation security and to environmental objectives; (4) to match capacity and air transport growth. The proposed mathematical model covers the aforementioned objectives by focusing on energy losses and costs of flights under the scenario of a controlled free flight and a unified airspace. The factors enhanced in the model were chosen based on their impact on the ATM energy efficiency, such as the airborne delays and flight duration, the delays due to ground holding, the flight cancellation, the flight speed deviations and the flight level alterations. Therefore, the presented mathematical model minimizes the energy costs due to the above terms under certain assumptions and constraints. Finally, simulation case studies, used as proof tests, have been conducted under different ATM scenarios to examine the complexity and the efficiency of the developed model.

We consider an optimal control problem for systems defined by nonlinear Volterra integral equations, with state constraints. No convexity assumptions are made on the data, and the problem is transformed into its relaxed form. We prove the existence of an optimal relaxed control and derive necessary conditions for optimality in the form of a relaxed minimum principle of Pontryagin type. We then apply a mixed Frank-Wolfe penalty method which constructs sequences of relaxed controls converging to extremal controls for this problem. A numerical example is given.

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 a second order elliptic partial differential equation, jointly nonlinear in the state and control with high monotone nonlinearity in the state, with control and state constraints, where the state constraints and the cost functional involve also the state gradient. Since this problem may have no classical solutions, it is also formulated in the relaxed form. The existence of an optimal relaxed control is proved in the relaxed case, without convexity assumptions, and various necessary conditions for optimality are established for the classical and the relaxed problem. For the numerical solution of these problems, we propose a penalized gradient projection method generating classical controls, and a penalized conditional descent method generating relaxed controls. Using also relaxation theory, the behavior in the limit of sequences generated by these methods is examined. Finally, numerical examples are given.

An optimal control problem is considered, for systems defined by nonlinear ordinary differential equations, with control and pointwise state constraints. Since the problem may have no classical solutions, it is also formulated in the relaxed form. Various necessary/sufficient conditions for optimality are first given for both formulations. In order to solve these problems numerically, we then propose a discrete penalized gradient projection method generating classical controls, and a discrete penalised conditional descent method generating relaxed controls. In both methods, the discretization procedure is progressively refining in order to achieve efficiency with reduced computational cost. Results are given concerning the behaviour in the limit of these methods. Finally, numerical examples are provided.

We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. The state equation is discretized by the explicit fourth order Runge-Kutta scheme and the controls are approximated by discontinuous piecewise affine ones. We then propose an approximate gradient projection method that generates sequences of discrete controls and progressively refines the discretization during the iterations. Instead of using the exact discrete directional derivative, which is difficult to calculate, we use an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation by the same Runge-Kutta scheme and the integral involved by Simpson's integration rule, both involving intermediate approximations. The main result is that accumulation points, if they exist, of sequences constructed by this method satisfy the weak necessary conditions for optimality for the continuous problem. Finally, numerical examples are given.

We consider an optimal control problem for systems defined by nonlinear hyperbolic partial differential equations with state constraints. Since no convexity assumptions are made on the data, we also consider the control problem in relaxed form. We discretize both the classical and the relaxed problems by using a finite element method in space and a finite difference scheme in time, the controls being approximated by piecewise constant ones. We develop the existence theory and the necessary conditions for optimality, for the continuous and the discrete problems. Finally, we study the behaviour in the limit of discrete optimality, admissibility and extremality properties.

We consider a general optimization problem which is an abstract formulation of a broad class of state-constrained optimal control problems in relaxed form. We describe a generalized mixed Frank–Wolfe penalty method for solving the problem and prove that, under appropriate assumptions, accumulation points of sequences constructed by this method satisfy the necessary conditions for optimality. The method is then applied to relaxed optimal control problems involving lumped as well as distributed parameter systems. Numerical examples are given.