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 continuity^{4.2} 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
where by we mean the right-sided derivative of at . Since by construction and satisfies (4.7) with , we obtain
Comparing this formula with (4.11), we arrive at