method for the thermal design of eutectic directional solidification processes in an open-boat configuration, -Numer.
i] may be calculated via the adjoint variable method (see, e.
where the adjoint variable vector [lambda] is computed from the linear system of equations 
The adjoint variable method is computationally efficient since it only requires the evaluation of one linear adjoint problem defined by Eq.
i]] are computed, the adjoint variable vector [lambda] is obtained with Eq.
This, on one hand, provides a closed-form expression for the optimal input as a function of the state and adjoint variables
, which will be used in solution methods based on shooting.
The solution to the optimization problem is made easier by introducing the Hamiltonian's function, H, which is a function of the adjoint variables, [lambda], which are functions of time.
f] of the adjoint variables are known and they must be zero.
The adjoint variables are the Lagrange multipliers, which are dependent upon time and independent of the state variables.