Chemistry Reference
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literature and only few results exist in quantum mechanics mainly for closed quan-
tum systems (see [14-19] to cite a few). The aim of this chapter is to present some
of the tools of geometric optimal control theory that can be used in quantum con-
trol. For that purpose, we consider some simple examples as the control by laser
fields of three-level conservative systems and of two-level dissipative systems. In
a first approach, we skip most of the mathematical details. The reader is referred
to standard mathematical textbooks for a complete and rigorous development of
geometric optimal control theory [1, 8-10, 20]. Finally, we point out that the con-
trollability analysis that is the first part of the study of an optimal control problem
will not be discussed in this chapter (see, e.g., [21-26]). The controllability prob-
lem consists in determining if there exists a path going from the initial to the target
state. This property will be assumed throughout this chapter.
This chapter is organized as follows: In Section II, we introduce the PMP
and we state the theorem in the different cases that can be encountered in prac-
tice. The techniques associated to the use of the PMP, that is, indirect and con-
tinuation methods or the second-order optimality conditions are also detailed.
Such tools allow us to compute the extremal solutions of the PMP and to de-
termine their local optimality. Sections III and IV are devoted to the application
of the PMP to the control of quantum systems. Section III deals with the opti-
mal control of a three-level quantum system by laser fields plus von Neumann
measurements. This three-level system corresponds to the Grushin model on the
sphere that is one of the basic models in geometric optimal control. In particu-
lar, global optimality results can be obtained for this model. In Section IV, we
solve the problem of the time-minimum control of two-level dissipative quantum
systems by one or two control fields. For this example, we use the continuation
techniques and the second-order optimality conditions. A realistic example in
nuclear magnetic resonance (NMR) of the control of a spin 1/2 particle in a dissi-
pative environment is also treated. Conclusion and prospective views are given in
Section V.
II. THE PONTRYAGIN MAXIMUM PRINCIPLE
A. The End-Point Mapping
We consider a controlled dynamical system governed by the differential equations
of the form
x
(
t
)
=
F
(
x
(
t
)
,u
(
t
))
n
is the state of the system,
F
is a smooth mapping, and
u
is the
control parameter that is a map: [0
,T
(
u
)]
where
x
(
t
)
∈ R
m
with
T>
0. The set
U
is
the set of possible values of the control. The time
T
is the control duration that can
→
U
⊂ R
