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.