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.
