Free-terminal-time transversality condition (${\min }_{u}H(T,x(T),u,p(T))+{\Phi }_T(T,x(T))=0$)

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.