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Abstract
Regular control problems in the sense of the Legendre condition are defined, and second-order necessary and sufficient optimality conditions in this class are reviewed. Adding a scalar homotopy parameter, differential pathfollowing is introduced. The previous sufficient conditions ensure the definiteness and regularity of the path. The role of AD for the implementation of this approach is discussed, and two examples excerpted from quantum and space mechanics are detailed.
Acknowledgements
J.-B. Caillau is supported by Conseil Régional de Bourgogne (contract no. 2009-160E-160-CE-160T) and ANR Geometric Control Methods (project no. NT09_504490). O. Cots is also supported by Conseil Régional de Bourgogne (contract no. 2009-160E-160-CE-160T).
Notes
Unless otherwise specified manifolds are supposed to be -smooth.
This does not exclude applications with bounded controls, though, as U may be a compact submanifold. See also Citation15 () and Citation16 (
).
That is -smooth.
A mapping between topological spaces is locally open at a point if it sends neighbourhood of the point onto neighbourhoods of its image.
That is
-smooth.
The application remains continuous though the topology is weakened because is a topological module over
.
In which case the manifolds X, U, and the data (f 0, f) have to be assumed real analytic as well.
A trajectory with the fixed endpoints generated by a bounded control valued in U.
Under our assumption, the absence of conjugate point even implies that we can parameterize by λ, at least locally.