Summary: QUASI-EXTREMALITY FOR CONTROL SYSTEMS
A. A. Agrachev and R. V. Gamkrelidze UDC 517.971:514.7
A full exposition of the authors' previously announced results about the
extremality index of controls in smooth control systems and a generalization
of these results to systems with constraints on the controls.
This paper will present proofs of some results announced in [I], as well as generaliza-
tions of these results to systems in which constraints are imposed on the control parameters,
as promised in [i].
Throughout the paper (and the following paper by S. A. Vakhrameev) the functional nota-
tion introduced in the first paper of this volume will be employed without special mention.
The main object of study will be the Hessian of the "input-output" map of a control
system at a certain critical point (extremal of the system). Let us recall, therefore,
the definition of the Hessian of a smooth map. Let r ~ § M be a smooth map of some smooth
Banach manifold into a finite-dimensional manifold and let ~0 ~ ~. The differential of r
at D0 is the linear map D$0~:TB0 ~ + T~(~0)M of the tangent spaces. If we fix local coordi-
nates in the neighborhoods of ~0 and r we can also define the second differential (a
symmetric bilinear map of a Banach space into a finite-dimensional space). However, this
procedure does not yield a well-defined bilinear map of T~o~XT~o~ into Tr since the
quadratic part of a smooth map depends essentially on the choice of local coordinates (for