= Pontryagin maximum principle
{c}
{title2=$H=L+p\cdot f$}
For a regular finite-horizon <optimal control> problem, an optimal trajectory admits nontrivial multipliers and a <costate> obeying adjoint equations, together with pointwise optimization of the <Hamiltonian of an optimal-control problem>. In the normal minimization convention $H=L+p\cdot f$, the conditions are $\dot x=H_p$, $\dot p=-H_x$ and minimization of $H$ over admissible controls; endpoint and state constraints change the terminal conditions. Abnormal extremals use a zero cost multiplier and must not be silently excluded.
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