4.2.5 Terminal cone

We now want to describe geometrically the combined effect of the temporal
and spatial control perturbations on the terminal state. The vector
describes
the infinitesimal (first-order in
) perturbation
of the terminal state caused by the needle perturbation with parameters
and
. (It corresponds to the vector labeled as
in
Figure 4.6.) From the definition (4.24) of
, it is clear that its *direction* depends only on
and
, but not on
. We let
denote the ray in this direction originating
at
. If we keep
,
, and
fixed,
consists of the points
for various values of
. The construction of
is analogous to that of
in Section 4.2.2, except that
is unidirectional (because both
and
are positive) whereas
was bidirectional. We also let
denote the union of the rays
for all possible values of
and
. Then
is a cone with vertex at
. Note that this cone is not convex; for example, if the control set
(in which
takes values) is finite, then
will in general be a union of isolated rays starting at
.

Let us now ask ourselves the following question: is there a
spatial control perturbation such that the corresponding
first-order perturbation of the terminal point is, say,
for some control values
and intervals
and
? We will now see that the right way
to ``add" two needle perturbations is to concatenate them, i.e.,
to perturb
*both* on
(by setting it equal to
there) and on
(by setting it equal to
). Here we
are assuming that
, so that for
small enough
and
do not overlap. Such a spatial perturbation
is shown in Figure 4.7.

The resulting first-order perturbation of the terminal point will then be the sum . This is true simply because the variational equation, which propagates first-order state perturbations up to the terminal time, is linear. Indeed, according to the formulas derived in the two previous subsections, we will have

at the end of the first perturbation interval, then

at the end of the second perturbation interval, and finally, since by the semigroup property of the transition matrix,

More generally, if we want to generate the infinitesimal perturbation for some , then it is easy to see that we need to adjust the lengths of the intervals on which the two needle perturbations are acting: we need to set on and on . It is also clear that this construction immediately extends to linear combinations (with positive coefficients) of more than two terms.

Recall that is the cone with vertex at formed by the rays corresponding to all simple (individual) needle perturbations. The preceding discussion demonstrates that by concatenating different needle perturbations, we generate a larger cone (with the same vertex) which consists exactly of convex combinations of points in . We denote this convex cone by co .

In Section 4.2.2 we also constructed the line of perturbation directions arising from the temporal perturbations of . We now add this line to the convex cone co , in the sense of adding vectors attached to the point . More precisely, we consider the set of points of the form

where , comes from (4.9) for some , comes from (4.24) for some and , and are arbitrary nonnegative numbers. We denote this set by and call it the

By the same reasoning as before, we can show that for every point given by (4.25) there exists a perturbation of such that the terminal point of the perturbed trajectory satisfies

To obtain the desired control perturbation, we need to apply a concatenated spatial perturbation as explained above, followed by the temporal perturbation that adjusts the terminal time by . Since the intervals are strictly inside , they do not interfere with the temporal perturbation (for small enough ). The fact that the resulting total first-order perturbation of the terminal point is indeed the correct one hinges on the linearity of the variational equation and on the linear dependence of on .