Chemistry Reference
In-Depth Information
The solutions (
x, p
)of
H
r
are called extremals of order zero and the surface
is called the switching surface. In order to be optimal, they have to satisfy
H
r
≥
0
and those where
H
r
=
0 are called abnormal. In this case, the singular extremals
belong to the surface
.
C. Geometric Aspects of the Optimal Control Theory
This chapter gives a heuristic introduction to the geometric tools of optimal control
theory.
The goal of geometric optimal control theory is to analyze the trajectories of
the Hamiltonian system given by the PMP from a geometric point of view in order
to deduce the geometric properties of the extremals: smooth or broken, number of
switchings
(a switching time is a time where the control field is discontinuous).
An important point of this analysis is the fact that the PMP is only a necessary
condition of optimality and one difficulty of the problem is to determine the cut
point, that is, the first point along a given trajectory where the extremal ceases
to be optimal. The computation of the cut point is a very difficult problem since
it is a global problem, where all the Hamiltonian trajectories starting from the
same initial point can play a role. In particular cases, one can determine geomet-
rically the cut points that correspond to the points where two extremals with the
same cost intersect. Figure 6 displays this situation (see below for a more con-
crete example). The optimal control problem consists in minimizing the length of
the trajectory going from
A
to the final point. In Fig. 6, we consider two different
optimal trajectories starting from point A. We can show that the red extremal is
not optimal to go from A to D since in the BCD triangle, the BD length is smaller
than the sum of the lengths BC and CD.
The computation of the cut points is global, but it is related to the computation
of conjugate points. A conjugate point is a point where the extremal loses its local
optimality when considering only extremals in a neighborhood of the first extremal
(see Fig. 9 for an illustration). From a physical point of view, a conjugate point
corresponds to a point where the extremals starting from the same initial point
···
D
A
B
C
Figure 6.
Broken solution to go from A to
D with the shortest length. The extremals in red
and blue starting from the point A intersect in
C with the same length. The red extremal is not
optimal from the point C since one can construct
a broken solution ABD with a shortest length.
(See insert for color representation of the figure.)
