Lagrange Multiplier and Objective Functional

In most problems, a Lagrange multiplier can be shown to be related to the rate of change of the optimal objective functional with respect to the constraint value. This is an important result, which will be utilized in developing the necessary conditions for optimal control problems having inequality constraints. 

For simplicity, consider an objective functional J dependent on a control function u t) and subject to the constraint 

We assume that both J and K are Gateaux differentiable. This is a modest assumption, which is valid in most optimal control problems we encounter. The reason for this assumption is the need for the linearity of the differentials in the following three-step derivation of the relation between a Lagrange multiplier p and the objective functional J  

Step 1 Let J be optimal at u, which depends on ko, the value of the constraint. Then at any t in the t-interval, Taylor s first order expansion gives 

We already know that if J is Gateaux differentiable, then its variation 6J exists and is equal to d J. Therefore, 

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