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The mathematical theory of "open" dynamical systems is a creation
of the twentieth century. Its humble beginnings focused on ideas of
Laplace transforms applied to linear problems of automatic control
and to the analysis and synthesis of electrical circuits. However
during the second half of the century, it flowered into a field
based on an array of sophisticated mathematical concepts and
techniques from algebra, nonlinear analysis and differential
geometry. The central notion is that of a dynamical system that
exchanges matter, energy, or information with its surroundings,
i.e. an "open" dynamical system. The mathema tization of this
notion evolved considerably over the years. The early development
centered around the input/output point of view and led to important
results, particularly in controller design. Thinking about open
systems as a "black box" that accepts stimuli and produces
responses has had a wide influence also in areas outside
engineering, for example in biology, psychology, and economics. In
the early 1960's, especially through the work of Kalman,
input/state/output models came in vogue. This model class
accommodates very nicely the internal initial conditions that are
essentially always present in a dynamical system. The introduction
of input/state/output models led to a tempestuous development that
made systems and control into a mature discipline with a wide range
of concepts, results, algorithms, and applications.
Ordinary differential control thPory (the classical theory) studies
input/output re lations defined by systems of ordinary differential
equations (ODE). The various con cepts that can be introduced
(controllability, observability, invertibility, etc. ) must be
tested on formal objects (matrices, vector fields, etc. ) by means
of formal operations (multiplication, bracket, rank, etc. ), but
without appealing to the explicit integration (search for
trajectories, etc. ) of the given ODE. Many partial results have
been re cently unified by means of new formal methods coming from
differential geometry and differential algebra. However, certain
problems (invariance, equivalence, linearization, etc. ) naturally
lead to systems of partial differential equations (PDE). More
generally, partial differential control theory studies input/output
relations defined by systems of PDE (mechanics, thermodynamics,
hydrodynamics, plasma physics, robotics, etc. ). One of the aims of
this book is to extend the preceding con cepts to this new
situation, where, of course, functional analysis and/or a dynamical
system approach cannot be used. A link will be exhibited between
this domain of applied mathematics and the famous 'Backlund
problem', existing in the study of solitary waves or solitons. In
particular, we shall show how the methods of differ ential
elimination presented here will allow us to determine compatibility
conditions on input and/or output as a better understanding of the
foundations of control the ory. At the same time we shall unify
differential geometry and differential algebra in a new framework,
called differential algebraic geometry."
Ordinary differential control thPory (the classical theory) studies
input/output re lations defined by systems of ordinary differential
equations (ODE). The various con cepts that can be introduced
(controllability, observability, invertibility, etc. ) must be
tested on formal objects (matrices, vector fields, etc. ) by means
of formal operations (multiplication, bracket, rank, etc. ), but
without appealing to the explicit integration (search for
trajectories, etc. ) of the given ODE. Many partial results have
been re cently unified by means of new formal methods coming from
differential geometry and differential algebra. However, certain
problems (invariance, equivalence, linearization, etc. ) naturally
lead to systems of partial differential equations (PDE). More
generally, partial differential control theory studies input/output
relations defined by systems of PDE (mechanics, thermodynamics,
hydrodynamics, plasma physics, robotics, etc. ). One of the aims of
this book is to extend the preceding con cepts to this new
situation, where, of course, functional analysis and/or a dynamical
system approach cannot be used. A link will be exhibited between
this domain of applied mathematics and the famous 'Backlund
problem', existing in the study of solitary waves or solitons. In
particular, we shall show how the methods of differ ential
elimination presented here will allow us to determine compatibility
conditions on input and/or output as a better understanding of the
foundations of control the ory. At the same time we shall unify
differential geometry and differential algebra in a new framework,
called differential algebraic geometry."
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