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Honoring Andrei Agrachev's 60th birthday, this volume presents
recent advances in the interaction between Geometric Control Theory
and sub-Riemannian geometry. On the one hand, Geometric Control
Theory used the differential geometric and Lie algebraic language
for studying controllability, motion planning, stabilizability and
optimality for control systems. The geometric approach turned out
to be fruitful in applications to robotics, vision modeling,
mathematical physics etc. On the other hand, Riemannian geometry
and its generalizations, such as sub-Riemannian, Finslerian
geometry etc., have been actively adopting methods developed in the
scope of geometric control. Application of these methods has led to
important results regarding geometry of sub-Riemannian spaces,
regularity of sub-Riemannian distances, properties of the group of
diffeomorphisms of sub-Riemannian manifolds, local geometry and
equivalence of distributions and sub-Riemannian structures,
regularity of the Hausdorff volume, etc.
Honoring Andrei Agrachev's 60th birthday, this volume presents
recent advances in the interaction between Geometric Control Theory
and sub-Riemannian geometry. On the one hand, Geometric Control
Theory used the differential geometric and Lie algebraic language
for studying controllability, motion planning, stabilizability and
optimality for control systems. The geometric approach turned out
to be fruitful in applications to robotics, vision modeling,
mathematical physics etc. On the other hand, Riemannian geometry
and its generalizations, such as sub-Riemannian, Finslerian
geometry etc., have been actively adopting methods developed in the
scope of geometric control. Application of these methods has led to
important results regarding geometry of sub-Riemannian spaces,
regularity of sub-Riemannian distances, properties of the group of
diffeomorphisms of sub-Riemannian manifolds, local geometry and
equivalence of distributions and sub-Riemannian structures,
regularity of the Hausdorff volume, etc.
Mathematical Control Theory is a branch of Mathematics having as
one of its main aims the establishment of a sound mathematical
foundation for the c- trol techniques employed in several di?erent
?elds of applications, including engineering, economy,
biologyandsoforth. Thesystemsarisingfromthese- plied Sciences are
modeled using di?erent types of mathematical formalism, primarily
involving Ordinary Di?erential Equations, or Partial Di?erential
Equations or Functional Di?erential Equations. These equations
depend on oneormoreparameters thatcanbevaried, andthusconstitute
thecontrol - pect of the problem. The parameters are to be chosen
soas to obtain a desired behavior for the system. From the many
di?erent problems arising in Control Theory, the C. I. M. E. school
focused on some aspects of the control and op- mization
ofnonlinear, notnecessarilysmooth, dynamical systems. Two points of
view were presented: Geometric Control Theory and Nonlinear Control
Theory. The C. I. M. E. session was arranged in ?ve six-hours
courses delivered by Professors A. A. Agrachev (SISSA-ISAS, Trieste
and Steklov Mathematical Institute, Moscow), A. S. Morse (Yale
University, USA), E. D. Sontag (Rutgers University, NJ, USA), H. J.
Sussmann (Rutgers University, NJ, USA) and V. I. Utkin (Ohio State
University Columbus, OH, USA). We now brie?y describe the
presentations. Agrachev's contribution began with the investigation
of second order - formation in smooth optimal control problems as a
means of explaining the variational and dynamical nature of
powerful concepts and results such as Jacobi ?elds, Morse's index
formula, Levi-Civita connection, Riemannian c- vature.
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