4.2.4 Variational equation

We are now interested in how the difference between the trajectory arising from a spatial (needle) perturbation and the optimal trajectory propagates after the perturbation stops acting, i.e., for . To study this question, let us begin by writing

for , where is a quantity that we want to characterize and is the same as in Section 4.2.3. We know from (4.14) that exists and is given by

We now derive a differential equation that must satisfy on the time interval , where both and have the same corresponding control . It is clear from (4.16) that

(the existence of this partial derivative, and therefore of , for will become evident in a moment). Let us rewrite the system (4.7) as an integral equation:

Differentiating both sides of this equation with respect to at and using (4.16) with and (4.17), we obtain

which, in view of (4.16) and (4.18), amounts to

Taking the derivative with respect to , we conclude that satisfies the differential equation

We will use this equation to describe how spatial perturbations propagate with time. Pictorially, the role of is illustrated in Figure 4.6, with the understanding that the labels involving are accurate only up to terms of order .

The equation (4.19) can be written in the form

where . This system is simply the linearization of the original system (4.7) around the optimal trajectory . Recall that a more detailed description of the system (4.7) is given by (4.5). Letting be the corresponding components of and writing out the variational equation (4.19) in terms of these components, we easily arrive at

(because and do not depend on ). Here is a row vector (the transpose of the gradient of with respect to ) and is an matrix (the Jacobian matrix of with respect to ). The resulting more explicit formula for is

The equation (4.19) is
in fact the *variational equation* corresponding to the system (4.7), according to the terminology
introduced in Section 2.6.2. The equation (3.27) in Section 3.4 is also similar and was similarly derived, although it served a different purpose (it described the effect of a small
control perturbation on a given trajectory, while here we are studying
the difference between two nearby trajectories with the same control).
Also, we see that
in the definition of a
needle perturbation
essentially corresponds to
in the variational approach; we chose to use different symbols for these two parameters because of the different specific ways in which they are introduced.

The value of at the terminal time gives us an approximation of the terminal point of the perturbed trajectory. Namely, from (4.16) evaluated at we have

Let us denote by the transition matrix for the linear time-varying system (4.20), so that

The initial value is given by (4.17), hence

where was defined in (4.15). Plugging this expression into (4.22), we obtain

Let us introduce the notation

to arrive at the more compact formula

(here the interval used for constructing the needle perturbation encodes information about the values of and ).