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This paper is concerned with real-time generation of optimal flighttrajectories for Minimum-Effort Control Problems (MECPs), which isfundamentally important for autonomous flight of aerospace vehicles. Althoughexisting optimal control methods, such as indirect methods and direct methods,can be amended to solve MECPs, it is very challenging to obtain, in real time,the solution trajectories since those methods suffer the issue of convergence.As the artificial neural network can generate its output within a constanttime, it has been alternative for real-time generation of optimal trajectoriesin the literature. The usual way is to train neural networks by solutions fromindirect or direct methods, which, however, cannot ensure sufficient conditionsfor optimality to be met. As a result, the trained neural networks cannot beguaranteed to generate at least locally optimal trajectories. To address thisissue, a parametrization approach is developed in the paper so that not onlynecessary but also sufficient conditions are embedded into a parameterized setof differential equations. This allows generating the data set of at leastlocally optimal trajectories through solving some initial value problems. Oncea neural network is trained by the data set, it can not only generate optimaltrajectories within milliseconds but also ensure the generated trajectories tobe at least locally optimal, as finally demonstrated by two conventional MECPsin aerospace engineering.

Parametrization Approach for Real-Time Generation of Minimum-Effort Trajectories via Neural Network