Abort Landing of an Airplane under Diierent Windshear Conditions: an Optimal Control Problem with a Third-order State Constraint and a Multifarious Switching Structure

B Abstract. Optimal abort landing trajectories of an aircraft under diier-ent windshear-downburst situations are computed and discussed. In order to avoid an airplane crash due to severe winds inadvertently encountered by the aircraft during the landing approach, the minimum altitude obtained during the abort landing maneuver is to be maximized. This maneuver is mathematically described by a Chebyshev optimal control problem. By a transformation to an optimal control problem of Mayer type, an additional state variable inequality constraint for the altitude has to be taken into account which is of order three here. Due to this altitude constraint, the optimal trajector-ies exhibit, depending on the windshear parameters, up to four touch points and also up to one boundary arc of the level of minimum altitude. The control variable is the angle of attack rate which enters the equations of motion linearly. Therefore, the Hamiltonian of the problem is nonregular. The complicated switching structures which also include up to three singular subarcs and up to two boundary subarcs of an angle of attack constraint being of rst order, can only be obtained by carefully applying some advanced necessary conditions known in optimal control theory and by means of the multiple shooting method known to be particularly well suited for precise computations of optimal trajectories. In summary, the optimal solutions show an oscillat-ory ight which reaches the level of minimum altitude several times. Those oscillatory ights have been reported also from real aircraft crashes in strong microbursts. By the optimization, the maximum survival capability, too, can be determined, i.e., the maximum diierence between maximum headwind and maximum tailwind of a windshear from which recovery is just possible. Since the computed optimal trajectories may serve as benchmark trajectories, both for guidance laws that is desirable to approach in actual ight and for pilot training in ight simulators, a more realistic windshear model is considered. This model also takes into account regions with upwinds as occur for aviation-hazardous low altitude windshears accompanied by vortices.