4.2.2 Temporal control perturbation

Let us see what happens if we introduce a small change in the terminal time $t^*$ 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 $\tau\in\mathbb{R}$ and a small $\varepsilon >0$ , we consider the perturbed control

$$u_\tau(t):=u^*(\min\{t,t^*\}), \qquad t\in[t_0,t^*+\varepsilon \tau]$$

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

Figure: A temporal perturbation

We are interested in the value of the resulting perturbed trajectory $y$ at the new terminal time $t^*+\varepsilon \tau$ . For $\tau>0$ , the first-order Taylor expansion of $y$ around $t=t^*$ gives

$$\begin{split}y(t^*+\varepsilon \tau)&=y^*(t^*)+ \dot y(t^*)\v... ...:y^*(t^*)+\varepsilon \delta(\tau)+o(\varepsilon ). \end{split}$$ (4.9)

For $\tau<0$ , we have $y(t^*+\varepsilon \tau)=y^*(t^*+\varepsilon \tau)$ and the first-order Taylor expansion of $y^*$ around $t=t^*$ gives the same result. The vector $\varepsilon \delta(\tau)$ describes the infinitesimal (first-order in $\varepsilon$ ) perturbation of the terminal point. By definition, $\delta(\tau)$ depends linearly on $\tau$ . As we vary $\tau$ over $\mathbb{R}$ , keeping $\varepsilon$ fixed, the points $y^*(t^*)+\varepsilon \delta(\tau)$ form a line through $y^*(t^*)$ . We denote this line by $\vec\rho$ ; see Figure 4.4. Every point on $\vec\rho$ corresponds to a control $u_\tau$ for some $\tau$ . On the other hand, the approximation of $y(t^*+\varepsilon \tau)$ by $y^*(t^*)+\varepsilon \delta(\tau)$ is valid only in the limit as $\varepsilon \to 0$ . So, $\delta(\tau)$ 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 $\rho$ 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 $(x^0,x)$ -space only, ignoring the differences in the terminal times; accordingly, the time axis is not included in the figures.

Figure: The effect of a temporal control perturbation