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### 4.3.1.1 Fixed terminal time

Suppose that the terminal time is fixed, so that the terminal time of the optimal trajectory considered in the proof must equal a given value . Temporal control perturbations are then no longer admissible. Accordingly, the line formed by the perturbation directions defined in (4.9) must not be used when generating the terminal cone . We now invite the reader to check exactly where these perturbation directions were used in the proof of the maximum principle. As a matter of fact, they were used in one place only: to show that . Thus we conclude that the Hamiltonian will now be constant but not necessarily 0 along the optimal trajectory, while all the other conditions remain unchanged.

Let us confirm this fact by reducing the fixed-time problem to a free-time one via the familiar trick of introducing the extra state variable . The system becomes

 (4.39)

If the original target set was , then for the new system we can write the target set as , with the terminal time no longer explicitly constrained. The maximum principle for the Basic Variable-Endpoint Control Problem can now be applied. The Hamiltonian for the new problem is , where is the Hamiltonian for the original fixed-time problem. Clearly, the differential equation for is the same as before and the Hamiltonian maximization condition for implies the one for . We know that . Moreover, we have (since does not depend on ) hence is constant. Thus is indeed equal to a constant, namely, . It is no longer guaranteed to be 0; in fact, since the final value of is fixed, the transversality condition gives us no information about . This property of the Hamiltonian is also consistent with what we saw in Section 3.4.4.

Next: 4.3.1.2 Time-dependent system and Up: 4.3.1 Changes of variables Previous: 4.3.1 Changes of variables   Contents   Index
Daniel 2010-12-20