March 1, 2009
The aim of this tutorial is to present basic results (e.g., on
controllability, observability, feedback laws...) in control theory of
systems modeled by ordinary differential equations. First, we give the
classical results for linear control systems. Then, we give the direct
applications of this linear theory to local results for nonlinear
control systems. Finally, we present some more advanced tools to deal
with global properties and with the case where the linearized control
system is not controllable. We illustrate the methods presented on
simple physical systems.
Since the development of the laser some 40 years ago, a long standing dream has been to utilize this special source of radiation to manipulate dynamical events at the atomic and molecular scales. Hints that this goal may become a reality began to emerge in the 1990's, due to a confluence of concepts and technologies involving (a) control theory, (b) ultrafast laser sources, (c) laser pulse shaping techniques, and (d) fast pattern recognition algorithms. These concepts and tools have resulted in a high speed instrument configuration capable of adaptively changing the driving laser pulse shapes, approaching the performance of thousands of independent experiments in a matter of minutes. Each particular shaped laser pulse acts as a “Photonic Reagent” much as an ordinary reagent would at the molecular scale. Although a Photonic Reagent has a fleeting existence, it can leave a permanent impact. Current demonstrations have ranged from manipulating simple systems (atoms) out to the highly complex (biomolecules), and applications to quantum information sciences are being pursued. In all cases, the fundamental concept is one of adaptively manipulating quantum systems. The principles involved will be discussed, along with the presentation of the state of the field.
We address in this talk some practical issues that occur in the design of (optimal) control field that manipulate quantum phenomena.
After discussing modelization issues (which functional to optimize, to what goal it corresponds etc) several algorithms will be discussed: genetic/evolutionary algorithm, adjoint state (optimal control) approaches and stabilization (Lyapounov) algorithms.