Computation of viability kernels on grid computers for aircraft control in windshear

This paper is devoted to the analysis of aircraft dynamics in the cruise flight phase under windshear conditions. The study is conducted with reference to a point-mass aircraft model restricted to move in a vertical plane. We formulate the problem as a differential game against the wind disturbances: The first player, autopilot, manages, via additional smoothing filters, the aircraft's angle of attack and power setting. The second player, wind, produces disturbances that are transferred, also via smoothing filters, into most dangerous wind gusts. The state variables of the game are subject to state constraints representing aircraft safety conditions related, for example, to the altitude, path inclination and velocity. Viability theory is used to compute the so-called viability kernels, the maximal subsets of state constraints where an appropriate feedback strategy of the first player can keep aircraft trajectories arbitrary long for all admissible disturbances generated by the second player. A grid method is utilized, and challenging computations in seven dimensions are conducted on a supercomputer system.