This book presents some facts and methods of the Mathematical Control
Theory treated from the geometric point of view. The book is mainly
based on graduate courses given by the first coauthor in the years
2000-2001 at the International School for Advanced Studies, Trieste,
Italy. Mathematical prerequisites are reduced to standard courses of
Analysis and Linear Algebra plus some basic Real and Functional
Analysis. No preliminary knowledge of Control Theory or Differential
Geometry is required. What this book is about? The classical
deterministic physical world is described by smooth dynamical systems:
the future in such a system is com- pletely determined by the initial
conditions. Moreover, the near future changes smoothly with the initial
data. If we leave room for "free will" in this fatalistic world, then we
come to control systems. We do so by allowing certain param- eters of
the dynamical system to change freely at every instant of time. That is
what we routinely do in real life with our body, car, cooker, as well as
with aircraft, technological processes etc. We try to control all these
dynamical systems! Smooth dynamical systems are governed by differential
equations. In this book we deal only with finite dimensional systems:
they are governed by ordi- nary differential equations on finite
dimensional smooth manifolds. A control system for us is thus a family
of ordinary differential equations. The family is parametrized by
control parameters.
