Optimal control problems fixed final time

We explain the shooting Newton Raphson method with the help of an optimal control problem having one state, one control, and fixed final time. The objective of the problem is to find the control function u t) that minimizes the functional... 

We first consider optimal control problems in which the final time is free, i.e., not specified or fixed. Thus, in addition to finding optimal control functions, we need to determine the optimal final time as well in these problems. 

Consider again the optimal control problem of Section 6.1.1 (p. 153). If the final time is fixed, then its variation Stf must be zero in Equation (6.9) on p. 156. Moreover, yf = Sy tf) from Equation (6.8). Consequently, Equation (6.9) simplifies to... 

In this case, we have the optimal control problem of Section 6.1.3 (p. 158), but with fixed final time. This problem is equivalent to that of minimizing... 

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