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The book is devoted to systems with discontinuous control. The
study of discontinuous dynamic systems is a multifacet problem
which embraces mathematical, control theoretic and application
aspects. Times and again, this problem has been approached by
mathematicians, physicists and engineers, each profession treating
it from its own positions. Interestingly, the results obtained by
specialists in different disciplines have almost always had a
significant effect upon the development of the control theory. It
suffices to mention works on the theory of oscillations of
discontinuous nonlinear systems, mathematical studies in ordinary
differential equations with discontinuous righthand parts or
variational problems in nonclassic statements. The unremitting
interest to discontinuous control systems enhanced by their
effective application to solution of problems most diverse in their
physical nature and functional purpose is, in the author's opinion,
a cogent argument in favour of the importance of this area of
studies. It seems a useful effort to consider, from a control
theoretic viewpoint, the mathematical and application aspects of
the theory of discontinuous dynamic systems and determine their
place within the scope of the present-day control theory. The first
attempt was made by the author in 1975-1976 in his course on "The
Theory of Discontinuous Dynamic Systems" and "The Theory of
Variable Structure Systems" read to post-graduates at the
University of Illinois, USA, and then presented in 1978-1979 at the
seminars held in the Laboratory of Systems with Discontinous
Control at the Institute of Control Sciences in Moscow.
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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