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 closed planar path with $\dot{x}=u$, $\dot{y}=v$ and $\dot{z}=uy-vx$, the signed area satisfies $z(1)=-2$ times the usual oriented area. Periodic Wirtinger and Cauchy-Schwarz inequalities give $|z(1)|\le \frac{1}{2\pi }{\int }_0^1(u^2+v^2)dt$. Thus terminal $z(1)=1$ requires cost at least $\pi$. A clockwise circular path traversed once at constant speed attains equality, proving the global minimum rather than merely a stationary Pontryagin extremal.

For energy remaining $x$, $\dot{x}=-u$, $0\le u\le 1$, running cost $up(t)$ and terminal cost $-R(\tau )$, the normal maximizing Hamiltonian is $u(p_{\ast }-p(t))$, where the constant costate defines $p_{\ast }$. Hence full power is chosen below the threshold and zero power above it. Free completion time on an active terminal arc gives $p_{\ast }+R^{\prime }(\tau )=p(\tau )$. Global optimality still requires a comparison of feasible completion times; the maximum principle is only a necessary condition.

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.