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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.
Control theory provides a large set of theoretical and
computational tools with applications in a wide range of ?elds,
running from "pure" branches of mathematics, like geometry, to more
applied areas where the objective is to ?nd solutions to "real
life" problems, as is the case in robotics, control of industrial
processes or ?nance. The "high tech" character of modern business
has increased the need for advanced methods. These rely heavily on
mathematical techniques and seem indispensable for competitiveness
of modern enterprises. It became essential for the ?nancial analyst
to possess a high level of mathematical skills. C- versely, the
complex challenges posed by the problems and models relevant to
?nance have, for a long time, been an important source of new
research topics for mathematicians. The use of techniques from
stochastic optimal control constitutes a well established and
important branch of mathematical ?nance. Up to now, other branches
of control theory have found comparatively less application in ?n-
cial problems. To some extent, deterministic and stochastic control
theories developed as di?erent branches of mathematics. However,
there are many points of contact between them and in recent years
the exchange of ideas between these ?elds has intensi?ed. Some
concepts from stochastic calculus (e.g., rough paths)
havedrawntheattentionofthedeterministiccontroltheorycommunity.Also,
some ideas and tools usual in deterministic control (e.g.,
geometric, algebraic or functional-analytic methods) can be
successfully applied to stochastic c- trol.
Control theory provides a large set of theoretical and
computational tools with applications in a wide range of ?elds,
running from "pure" branches of mathematics, like geometry, to more
applied areas where the objective is to ?nd solutions to "real
life" problems, as is the case in robotics, control of industrial
processes or ?nance. The "high tech" character of modern business
has increased the need for advanced methods. These rely heavily on
mathematical techniques and seem indispensable for competitiveness
of modern enterprises. It became essential for the ?nancial analyst
to possess a high level of mathematical skills. C- versely, the
complex challenges posed by the problems and models relevant to
?nance have, for a long time, been an important source of new
research topics for mathematicians. The use of techniques from
stochastic optimal control constitutes a well established and
important branch of mathematical ?nance. Up to now, other branches
of control theory have found comparatively less application in ?n-
cial problems. To some extent, deterministic and stochastic control
theories developed as di?erent branches of mathematics. However,
there are many points of contact between them and in recent years
the exchange of ideas between these ?elds has intensi?ed. Some
concepts from stochastic calculus (e.g., rough paths)
havedrawntheattentionofthedeterministiccontroltheorycommunity.Also,
some ideas and tools usual in deterministic control (e.g.,
geometric, algebraic or functional-analytic methods) can be
successfully applied to stochastic c- trol.
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