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where
F
is a smooth mapping. We consider a smooth homotopy path
H
(
x, λ
), such
that
H
(
x,
0)
F
(
x
), where
G
is a map having known zero
points. In a smooth numerical continuation method (see [29]), the solutions are
computed iteratively along this path using, for example, a Newton-type algorithm,
the trial solutions at one step being given by the solutions of the previous step. This
general method can be applied in Optimal Control. Indeed, as mentioned above, the
optimal solution for a fixed set of boundary conditions can be computed by solving
a shooting equation and the smooth continuation method can be applied if one can
construct a smooth homotopy path between different optimal control problems.
Different continuation methods exist in the literature, such as the discrete one
which uses a Newton-type algorithm at each step, or the smooth one that uses the
derivatives of the Hamiltonian trajectories along the homotopy path. An example
will be given in Section IV for the optimal control of two-level dissipative quantum
systems, where we illustrate the practical implementation of this algorithm.
=
G
(
x
) and
H
(
x,
1)
=
E. Second-Order Optimality Conditions: The Concept
of Conjugate Points
The second-order optimality conditions are used to determine the local optimality
of extremal trajectories. By using the PMP for an open set
U
of values of the
control fields, an optimal control has to satisfy the conditions
∂H/∂u
=
0 and
∂
2
H/∂u
2
0, where
H
(
x, p, u
) is the pseudo-Hamiltonian. The condition on the
second derivative corresponds to the local optimality of the control, that is, when
one considers the extremals in a neighborhood of the reference trajectory (see Fig. 9
for an illustration). Now, we assume that
∂
2
H/∂u
2
<
0 along a given extremal
z
≤
(
x, p
). A conjugate point will be the first point of the trajectory for which this
relation does not hold. A conjugate point is therefore the point where the trajectory
=
x
Figure 9.
Plot of the extremals
in a neighborhood of a given extremal
(large solid black curve)
t
