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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 fixedtime problem to a
freetime 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 VariableEndpoint
Control Problem can now be applied. The Hamiltonian for the new
problem is
,
where
is the Hamiltonian for the original fixedtime
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 Timedependent system and
Up: 4.3.1 Changes of variables
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Daniel
20101220