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A chosen input to a dynamical system, commonly written $u(t)$ in the state equation $\dot{x}=f(t,x,u)$. An optimal control problem specifies both the admissible input set and any information or regularity constraints on the choice.

An adjoint variable enforcing the state dynamics in optimal control. In the normal minimization convention it satisfies $\dot{p}=-H_x$ for the Hamiltonian of an optimal-control problem, with terminal conditions determined by the endpoint constraints.

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.

In a normal minimization convention for optimal control, the running cost $L$ and dynamics $\dot{x}=f$ define $H=L+p\cdot f$. The multiplier $p$ is a costate; this Hamiltonian is a device for variational necessary conditions rather than necessarily a physical energy.

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.