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## 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

 (4.25)

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 terminal cone. It is easy to check that is again a convex cone, with vertex at . This construction is illustrated in Figure 4.8, where is the infinite wedge" between the two half-planes.

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 .

Next: 4.2.6 Key topological lemma Up: 4.2 Proof of the Previous: 4.2.4 Variational equation   Contents   Index
Daniel 2010-12-20