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This book proposes a semi-discrete version of the theory of Petitot
and Citti-Sarti, leading to a left-invariant structure over the
group SE(2,N), restricted to a finite number of rotations. This
apparently very simple group is in fact quite atypical: it is
maximally almost periodic, which leads to much simpler harmonic
analysis compared to SE(2). Based upon this semi-discrete model,
the authors improve on previous image-reconstruction algorithms and
develop a pattern-recognition theory that also leads to very
efficient algorithms in practice.
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.
This 2001 book presents a general theory as well as a constructive
methodology to solve 'observation problems', that is,
reconstructing the full information about a dynamical process on
the basis of partial observed data. A general methodology to
control processes on the basis of the observations is also
developed. Illustrative but also practical applications in the
chemical and petroleum industries are shown. This book is intended
for use by scientists in the areas of automatic control,
mathematics, chemical engineering and physics.
This work presents a general theory as well as constructive methodology in order to solve "observation problems," namely, those problems that pertain to reconstructing the full information about a dynamical process on the basis of partial observed data. A general methodology to control processes on the basis of the observations is also developed. Illustrative but practical applications in the chemical and petroleum industries are shown.
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