Application to Optimal Control Problems

In an optimal control problem, we arrive at an integral objective functional having the general form 

the vector of certain undetermined multiplier functions or costates  

It is desired to determine the optimal control u that optimizes (i.e., either minimizes or maximizes) J. To that end, we need to obtain the variation of J to help determine the optimum. The variation of J is a straightforward generalization of Equation (2.16) and is given by 

A simpler version of the batch reactor problem (p. 2) for the reaction 

To proceed, we need to apply the Lagrange Multiplier Rule, the details of which will be provided later in Chapter 4. According to this rule, the above constrained problem is equivalent to the problem of finding the control T t) that maximizes the following augmented functional  

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