2.3.5 Variable-endpoint problems

So far we have been considering the Basic Calculus of Variations Problem, in which the curves have both their endpoints fixed by the boundary conditions (2.8). Accordingly, the class of admissible perturbations is restricted to those vanishing at the endpoints. This fact, reflected in (2.11), was explicitly used in the derivation of the Euler-Lagrange equation (2.18). Indeed, the first-order necessary condition (1.37)--which serves as the basis for the Euler-Lagrange equation--need only hold for admissible perturbations.

If we change the boundary conditions for the curves of interest, then the class of admissible perturbations will also change, and in general the necessary condition for optimality will be different. To give an example of such a situation, we now consider a simple variable-endpoint problem. Suppose that the cost functional takes the same form (2.9) as before, the initial point of the curve is still fixed by the boundary condition $y(a)=y_0$ , but the terminal point $y(b)$ is free. The resulting family of curves is depicted in Figure 2.9.

Figure: Variable terminal point

The perturbations $\eta$ must still satisfy $\eta(a)=0$ but $\eta(b)$ can be arbitrary. In view of (2.15), the first variation is then given by

$$\left.\delta J\right\vert _{y}(\eta)=\int_a^b\Big({L}_{y}(x,y(x),... ...))-\frac d{dx}{L}_{ z}(x,y(x),y'(x))\Big)\eta(x)dx+{L}_{z}(b,y(b),y'(b))\eta(b)$$ (2.25)

and this must be 0 if $y$ is to be an extremum. Perturbations such that $\eta(b)=0$ are still allowed; let us consider them first. They make the last term in (2.26) disappear, leaving us with (2.16). Exactly as before, we deduce from this that the Euler-Lagrange equation must hold, i.e., it is still a necessary condition for optimality. The Euler-Lagrange equation says that the expression in the large parentheses inside the integral in (2.26) is 0. But this means that the entire integral is 0, for all admissible $\eta$ (not just those vanishing at $x=b$ ). The last term in (2.26) must then also vanish, which gives us an additional necessary condition for optimality:

$${L}_{ z}(b,y(b),y'(b))\eta(b)=0$$

or, since $\eta(b)$ is arbitrary,

$${L}_{ z}(b,y(b),y'(b))=0.$$ (2.26)

We can think of (2.27) as replacing the boundary condition $y(b)=y_1$ . Recall that we want to have two boundary conditions to uniquely specify an extremal. Comparing with the Basic Calculus of Variations Problem, here we have only one endpoint fixed a priori, but on the other hand we have a richer perturbation family which allows us to obtain one extra condition (2.27).

Working on this exercise, the reader will realize that obtaining a transversality condition in the specified form requires a somewhat more advanced analysis than what we have done so far. We will employ similar techniques again soon when deriving conditions for strong minima in Section 3.1.1. The transversality condition itself is essentially a preview of what we will see later in the context of the maximum principle. More general variable-endpoint problems in which the initial point is allowed to vary as well, and the resulting transversality conditions, will also be mentioned in the optimal control setting (at the end of Section 4.3).