Optimality of Periodic Controls

The above problem is equivalent to minimizing the augmented functional 

Note that the above equation has one additional term, A (0) (fy(0), when compared to Equation (6.9) on p. 156. The reason is that dy(0) in the present problem is not zero since y(0) is not fixed but equal to y(tf). This periodicity condition for the general case of unspecified tf means that y(0) is equal to the final state yf. As a result 

Considering this fact, the necessary conditions for the minimum of J, and 

Consider the periodic operation of the batch reactor in Example 2.10 (p. 45) with the objective to minimize 

Note that if t is not fixed, then we have the additional necessary condition 

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