Summary:
Mathematical control theory is the area of application-oriented mathematics
that deals with the basic principles underlying the analysis and design of
control systems. To control an object means to influence its behavior so as to
achieve a desired goal. In order to implement this influence, engineers build
devices that incorporate various mathematical techniques. These devices range
from Watt’s steam engine governor, designed during the English Industrial
Revolution, to the sophisticated microprocessor controllers found in consumer
items —such as CD players and automobiles— or in industrial robots and airplane
autopilots. The study of these devices and their interaction with the object
being controlled is the subject of this work. More specifically, we will
address the basic reachability notions, time-invariant systems ,controllable
pairs of matrices, controllability under sampling, more on linear
controllability, bounded controls, first-order local controllability and
piecewise constant controls. Later on we shall prove the controllability of
some classical partial differential equations: the transport equation and the
heat equation. We first prove the well-posedness of the Cauchy problem of these
equations. Then we study the controllability by different methods, namely: an
explicit method, the extension method and the duality between controllability
and observability. We also present a classical general framework which includes
as special cases the study of the previous equations and their controllability
as well as of many other equations.
Keywords:
Reachability, Controllabity, Abstract systems, Transport equation, Heat equation
