It has been shown in the previous section that the necessary conditions of optimality of the discretized problem reflect the necessary conditions of the original continuous problem. More precisely, it has been shown that Eq. (32) and Eq. (28), resp., are discretized versions of the adjoint differential equation (18) and the condition (19), respectively.
Therefore, we obtain an estimate of from the multipliers of the discretized problem by
where is a scaling factor depending on the discretization. In addition, an estimate of can be obtained from .
Another approach for estimating adjoint variables in combination with a direct collocation method has been used by Enright and Conway . They used the multipliers from Eq. (23) of the boundary conditions in the discretized problem in order to estimate . This estimate is then used as an initial value for the backward integration of the adjoint differential equations (18). It is a well-known matter of fact that this backward integration is crucial for highly nonlinear problems. Also, state constraints were not considered.
A further approach for estimating adjoint variables is based on an interpretation of the adjoint variables as sensitivities connected to the gradient of the cost function
where x satisfies the differential equations (2). This relation can be found, e. g., in Breakwell  or in Bryson, Ho . In a discretized version as, e. g.,
it can be used in combination with a direct shooting method as, e. g., , with a suitable steplength for the difference quotient. Here, the superscript denotes that the variable or value has been obtained numerically, e. g., by a direct shooting method. For more details, cf., e. g., Eq. (29) in .
The guess of adjoint variables by direct methods is usually affected by several sources of inaccuracies and troubles. First, the suboptimal control calculated by a direct method is often inaccurate and can differ significantly from the optimal control. Second, the accuracy of the calculated objective is often not better than one percent. In addition, the case of nearly active or inactive state variable inequality constraints has not yet been included in a reliable manner in previous attempts.
In contrast to the former approaches, the quality of the estimated adjoints does neither depend crucially on a highly accurate computation of the cost function or the calculated suboptimal control nor the appearance of active state constraints following our approach. As it is shown from the examples and the reported numerical results in this paper and in , , and , the new way of estimating adjoint variables herein proposed is very reliable and accurate even for complicated and highly nonlinear problems and problems including state constraints. Furthermore, convergence properties of the discretization scheme have been derived.