Pontryagin maximum principle ($H=L+p\cdot f$)

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.

For a smooth normal optimal control problem with freely variable terminal time, fixed terminal state independent of that time and terminal cost $\Phi$, the optimized Hamiltonian of an optimal-control problem at completion satisfies $H(T)+{\Phi }_T=0$. It follows by varying the terminal time in the augmented objective; restrictions on terminal time or a moving endpoint introduce additional terms or inequalities.