order only possibility model minimax problem method aviation safety velocity components solution aircraft theorem approximation optimal control serious situation constraints force occurrence connection only control variable order three boundary subarcs control variables minimax optimal control problem paper control means additional constraints complexity conditions abort landing point inequality Part 1 presence multipliers distribution account standard form regular Hamiltonian multiple shooting method multiple shooting boundary arcs present paper kind articles maneuvers flight maneuvers existence theorem Hamiltonian control problem boundary points 1991-07-01 variable constraints Abort landing in the presence of windshear as a minimax optimal control problem, part 1: Necessary conditions homotopy strategy shooting optimal control problem https://scigraph.springernature.com/explorer/license/ situation sort data presence of windshear strategies signs passenger aircraft windshear landing wind velocity components variables subarcs threat first-order state constraint user guide guide The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting. arc three certain multipliers safety features singular subarcs single problem state constraints necessary condition problem sharp necessary conditions components shooting method 2022-08-04T16:51 bang-bang control variable constraints 1-23 1991-07 https://doi.org/10.1007/bf00940502 false aerodynamic forces possibility article form Department of Mathematics, University of Technology, Munich, Germany Department of Mathematics, University of Technology, Munich, Germany Applied Mathematics H. J. Pesch Springer Nature - SN SciGraph project pub.1047532819 dimensions_id R. Bulirsch 1 70 10.1007/bf00940502 doi 0022-3239 1573-2878 Springer Nature Journal of Optimization Theory and Applications Mathematical Sciences Montrone F.