as The constraints on the system dynamics can be adjoined to the Lagrangian by introducing time-varying Lagrange multiplier vector , whose elements are called the costates of the system. This motivates the construction of the Hamiltonian defined for all by: where is the transpose of . Pontryagin's minimum principle states that the optimal state trajectory , optimal control , and corresponding Lagrange multiplier vector must minimize the Hamiltonian so that for all time and for all permissible control inputs . Additionally, the costate equation and its terminal conditions must be satisfied. If the final state is not fixed (i.e., its differential variation is not zero), it must also be that These four conditions in (1)-(4) are the necessary conditions for an optimal control. Note that (4) only applies when is free. If it is fixed, then this condition is not necessary for an optimum. See also Lagrange multipliers on Banach spaces, Lagrangian method in calculus of variations Notes References Further reading External links Optimal control Principles fr:Commande optimale#Principe du maximum ru:Оптимальное управление#Принцип максимума Понтрягина