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...

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"...