## ABSTRACT

In many applications, we are concerned with maximizing (or minimizing) an objective functional over a non-fixed time interval. If we return to our simple cancer example, Example 3.3, we could instead consider a slightly different problem. Before, we wanted to find a drug treatment over a given time frame [0, T ] which would minimize the final tumor cell concentration and total harmful effects of the drug. Suppose, instead, we want to find a time frame and a control that produce an objective functional value minimum among all time frames and all controls. Namely,

min u,T

x(T ) + ∫ T 0

u(t)2 dt

subject to x′(t) = αx(t)− u(t), x(0) = x0. Notice that the minimization is now considered over the variables u and T . This is the standard way of writing an optimal control problem when T is free. We now have more unknowns, with the optimal control and optimal termi-

nal time both to be determined. To handle this problem, and other problems where the terminal time is free, we must redevelop the necessary conditions. As you will see, having given up information, in some sense, by allowing T to be free, we will gain new information in the way of a necessary condition we did not have before. We note that we could just as easily allow the initial time, or both the

initial and terminal times, to be free. In most applications, though, it is the final time which is allowed to move, so we handle this case.