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4.3.1.1 Fixed terminal time

Suppose that the terminal time $ t_f$ is fixed, so that the terminal time $ t^*$ of the optimal trajectory $ x^*$ considered in the proof must equal a given value $ t_1$ . Temporal control perturbations are then no longer admissible. Accordingly, the line $ \vec\rho$ formed by the perturbation directions $ \delta(\tau)$ defined in (4.9) must not be used when generating the terminal cone $ C_{t^*}$ . We now invite the reader to check exactly where these perturbation directions $ \delta(\tau)$ were used in the proof of the maximum principle. As a matter of fact, they were used in one place only: to show that $ \left.H\right\vert _{*}(t^*)=0$ . 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 $ x_{n+1}:=t$ . The system becomes

\begin{displaymath}\begin{split}&\dot x=f(x,u),\qquad x(t_0)=x_0\\ &\dot x_{n+1}=1, \qquad x_{n+1}(t_0)=t_0 \end{split}\end{displaymath} (4.39)

If the original target set was $ \{t_1\}\times S_1$ , then for the new system we can write the target set as $ [t_0,\infty)\times
S_1\times \{t_1\}$ , 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 $ \overline H=\langle p,f\rangle +p_{n+1}+p_0L=H+p_{n+1} $ , where $ H=\langle p,f\rangle +p_0L$ is the Hamiltonian for the original fixed-time problem. Clearly, the differential equation for $ p^*$ is the same as before and the Hamiltonian maximization condition for $ \overline H$ implies the one for $ H$ . We know that $ \left.\overline H\right\vert _{*}=\left.H\right\vert _{*}+p_{n+1}^*\equiv 0$ . Moreover, we have $ \dot
p^*_{n+1}=\left.-{\overline H}_{x_{n+1}}\right\vert _{*}=\left.-{H}_{t}\right\vert _{*}=0$ (since $ H$ does not depend on $ t$ ) hence $ p^*_{n+1}$ is constant. Thus $ \left.H\right\vert _{*}$ is indeed equal to a constant, namely, $ -p^*_{n+1}$ . It is no longer guaranteed to be 0; in fact, since the final value of $ x_{n+1}$ is fixed, the transversality condition gives us no information about $ p^*_{n+1}$ . This property of the Hamiltonian is also consistent with what we saw in Section 3.4.4.


next up previous contents index
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