 
Summary: QUASIEXTREMALITY 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.
INTRODUCTION
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 "inputoutput" 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 finitedimensional 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 finitedimensional space). However, this
procedure does not yield a welldefined 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
