Andrei A. Agrachev
SISSA and Steklov Math. Inst.
On the Curvature of Control Systems
The curvature tensor on a Riemannian manifold reflects intrinsic
properties of the geodesic flow, i.e. the properties which do
not depend on the choice of local coordinates. In particular,
nonzero curvature is an obstruction for the simultaneous
rectification of the geodesics by a change of coordinates.
The knowledge of the curvature gives a fundamental information
about the behavior of geodesics and about the structure of
optimal synthesis. On the other hand, in order to evaluate the curvature
we don't need to solve the geodesics differential equation,
it is enough to compute certain polynomials of the partial
derivatives of its right-hand side.
Riemannian geodesics are just extremals of a special optimization
problem, the problem to find a shortest path between to points
on the manifold.
Why not to try to find analogues of the curvature for general optimal
control problems? This talk is an attempt of a light presentation of the
large program going in that direction.
Abstract in Postscript
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