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## 4.2.3 Spatial control perturbation

We now construct control perturbations known as needle" perturbations, or Pontryagin-McShane perturbations. As the first name suggests, they will be represented by pulses of short duration; the reason for the second name is that perturbations of this kind were first used by McShane in calculus of variations (see Section 3.1.2) and later adopted by Pontryagin's school for the proof of the maximum principle.

Let be an arbitrary element of the control set . Consider the interval , where is a point of continuity4.2 of , is arbitrary, and is small. We define the perturbed control

Figure 4.5 illustrates this control perturbation and the resulting state trajectory perturbation. As the figure suggests, the perturbed trajectory corresponding to will deviate from on the interval and afterwards will run parallel" to . We now proceed to formally characterize the deviation over ; the behavior of over the interval will be studied in Section 4.2.4.

We will let denote equality up to terms of order . The first-order Taylor expansion of around gives

 (4.10)

Rearranging terms and using the fact that satisfies the differential equation (4.7) with , we have

 (4.11)

On the other hand, the first-order Taylor expansion of the perturbed solution around yields

where by we mean the right-sided derivative of at . Since by construction and satisfies (4.7) with , we obtain

 (4.12)

We now apply the Taylor expansion to the last term in (4.12):

 (4.13)

In view of (4.10), the second term on the right-hand side of (4.13) is of order ; hence we can omit it and the approximation will remain valid. Thus (4.12) simplifies to

Comparing this formula with (4.11), we arrive at

 (4.14)

where

 (4.15)

Intuitively, this result makes sense: up to terms of order , the difference between the two states and is the difference (4.15) between the state velocities at corresponding to and , multiplied by the length of the time interval on which the perturbation is acting.

Next: 4.2.4 Variational equation Up: 4.2 Proof of the Previous: 4.2.2 Temporal control perturbation   Contents   Index
Daniel 2010-12-20