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## 4.2.2 Temporal control perturbation

Let us see what happens if we introduce a small change in the terminal time of the optimal trajectory, i.e., let the optimal control act on a little longer or a little shorter time interval. We formalize this as follows: for an arbitrary and a small , we consider the perturbed control

which is illustrated by the thick curves in Figure 4.3 (for the two cases depending on the sign of ).

We are interested in the value of the resulting perturbed trajectory at the new terminal time . For , the first-order Taylor expansion of around gives

 (4.9)

For , we have and the first-order Taylor expansion of around gives the same result. The vector describes the infinitesimal (first-order in ) perturbation of the terminal point. By definition, depends linearly on . As we vary over , keeping fixed, the points form a line through . We denote this line by ; see Figure 4.4. Every point on corresponds to a control for some . On the other hand, the approximation of by is valid only in the limit as . So, tells us the direction--but not the magnitude--of the terminal point deviation caused by an infinitesimal change in the terminal time. The arrow over is meant to indicate that points on the line correspond to perturbation directions. Note that we are describing deviations of the terminal point in the -space only, ignoring the differences in the terminal times; accordingly, the time axis is not included in the figures.

Next: 4.2.3 Spatial control perturbation Up: 4.2 Proof of the Previous: 4.2.1 From Lagrange to   Contents   Index
Daniel 2010-12-20