4.3.1.3 Terminal cost

Since , there is no need to consider the -coordinate. Temporal and spatial control perturbations can be used to construct the terminal cone as before, except that it now lives in the original -space: . By optimality, and since the terminal state is free, no perturbed state trajectory can have a terminal cost lower than , the terminal cost of the candidate optimal trajectory. Thus we expect that should not decrease along any direction in , a property that we can write as

Geometrically, this means that lies on one side of the hyperplane passing through with normal , which we henceforth assume to be a nonzero vector. A comparison of (4.42) with (4.29) unmistakably suggests that we should define

Indeed, we know that in a free-endpoint problem the final value of the costate should be completely constrained (to give the correct total number of boundary conditions for the system of canonical equations). With the above definition of the inequality (4.42) matches (4.29), following which the adjoint equation and the analysis of the Hamiltonian

are developed in the same fashion as before. The companion inequality (4.30) in our earlier proof of the maximum principle was only needed to show that ; here we do not have such an inequality, and we do not need it because there is no .

The above argument is of course not rigorous, so let us again validate our conjecture by formally reducing the present scenario to the Basic Variable-Endpoint Control Problem. A transformation that accomplishes this goal is provided by the formula

Ignoring which is a known constant, we arrive at an equivalent problem in the Lagrange form with . For this modified problem, the Hamiltonian is where is the costate. Applying the maximum principle, we obtain the differential equation with the boundary condition . The latter tells us, by the way, that , hence from now on we assume and to have been normalized so that . Now, matching the expression for with (4.44), let us define the costate for our original problem to be

Its final value is consistent with (4.43). We can check that defined in this way satisfies the correct canonical equation.

By construction, is maximized and equals 0 along the optimal trajectory because these properties hold for . We thus conclude that the statement of the maximum principle for the present case, with respect to the Hamiltonian (4.44), is obtained from that for the Basic Variable-Endpoint Control Problem by eliminating and changing the transversality condition to . Recall that we already saw such a boundary condition for the costate in (3.37).