We now turn to the examination of optimal control problems that are linear in the control variables. A prominent feature of this class of problems is that the optimal control often turns out to be a piecewise continuous function of time. Recall that in Chapter 4, we defined an admissible pair of curves (x(t), u(t)) by allowing the control vector to be a piecewise continuous function of time and the state vector to be a piecewise smooth function of time. Though we allowed for this possibility in the theorems of Chapter 4, we did not solve or confront an optimal control problem whose solution exhibited these properties. You may recall, however, that in Example 4.6, where we solved the ubiquitous inventory accumulation problem subject to a nonnegativity constraint on the production rate, the optimal production rate was a continuous but not a differentiable function of time, that is, it was a piecewise smooth function of time. There are two reasons why the optimal production rate turned out to be a piecewise smooth function of time, namely, (i) the nonnegativity constraint on the production rate and (ii) the assumption of a “long” production period. One important lesson from this example, therefore, is that once an inequality constraint is imposed on the control variable, the differentiability of an optimal control function with respect to time may not hold.