Discretization-Optimization Methods for Nonlinear Parabolic Relaxed Optimal Control Problems with State Constraints

We consider an optimal control problem described by a semilinear parabolic partial differential equation, with control and state constraints, where the state constraints and cost involve also the state gradient. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized by using a finite element method in space involving numerical integration and an implicit theta-scheme in time for space approximation, while the controls are approximated by blockwise constant relaxed controls. Under appropriate assumptions, we prove that relaxed accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem. We then apply a penalized conditional descent method to each discrete problem, and also a progressively refining version of this method to the continuous relaxed problem. We prove that accumulation points of sequences generated by the first method are extremal for the discrete problem, and that relaxed accumulation points of sequences of discrete controls generated by the second method are admissible and extremal for the continuous relaxed problem. Finally, numerical examples are given

Discretization-Optimization Methods for Nonlinear Elliptic Relaxed Optimal Control Problems with State Constraints.

We consider an optimal control problem described by a second order elliptic boundary value problem, jointly nonlinear in the state and control with high monotone nonlinearity in the state, with control and state constraints, where the state constraints and cost functional involve also the state gradient. Since no convexity assumptions are made, the problem may have no classical solutions, and so it is reformulated in the relaxed form using Young measures. Existence of an optimal control and necessary conditions for optimality are established for the relaxed problem. The relaxed problem is then discretized by using a finite element method, while the controls are approximated by elementwise constant Young measures. We show that relaxed accumulation points of sequences of optimal (resp. admissible and extremal) discrete controls are optimal (resp. admissible and extremal) for the continuous relaxed problem. We then apply a penalized conditional descent method to each discrete problem, and also a progressively refining version of this method to the continuous relaxed problem. We prove that accumulation points of sequences generated by the first method are admissible and extremal for the discrete relaxed problem, and that accumulation points of sequences of discrete controls generated by the second method are admissible and extremal for the continuous relaxed problem. Finally, numerical examples are given

Discrete relaxed method for semilinear parabolic optimal control problem

We consider an optimal control problem for systems governed by semilinear parabolic partial differential equations with control and state constraints, without any convexity assumptions. A discrete optimization method is proposed to solve this problem in its relaxed form which combines a penalized Armijo type method with a finite element discretization and constructs sequences of discrete Gamkrelidze relaxed controls. Under appropriate assumptions, we prove that accumulation points of these sequences satisfy the relaxed Pontryagin necessary conditions for optimality. Moreover, we show that the Gamkrelidze controls thus generated can be replaced by simulating piecewise constant classical controls

Approximate relaxed descent method for optimal control problems

We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. Since no convexity assumptions are made on the data, the problem is reformulated in relaxed form. The relaxed state equation is discretized by the implicit trapezoidal scheme and the relaxed controls are approximated by piecewise constant relaxed controls. We then propose a combined descent and discretization method that generates sequences of discrete relaxed controls and progressively refines the discretization. Since here the adjoint of the discrete state equation is not defined, we use, at each iteration, an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation and the integral involved by appropriate trapezoidal schemes. It is proved that accumulation points of sequences constructed by this method satisfy the strong relaxed necessary conditions for optimality for the continuous problem. Finally, the computed relaxed controls can be easily approximated by piecewise constant classical controls

A dynamic integer programming approach for free flight air traffic management (ATM) scenario with 4D-trajectories and energy efficiency aspects

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. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature