Consider the system
We know that the dynamics of the double integrator (4.47) are equivalently described by the state-space equations
Next, from the Hamiltonian maximization condition and the fact that we have
We conclude that the optimal control takes only the values and switches between these values at most once. Interpreted in terms of bringing a car to rest at the origin, the optimal control strategy consists in switching between maximal acceleration and maximal braking. The initial sign and the switching time of course depend on the initial condition. The property that only switches between the extreme values is intuitively natural and important; such controls are called bang-bang.
It turns out that for the present problem, the pattern identified above uniquely determines the optimal control law for every initial condition. To see this, let us plot the solutions of the system (4.47) in the -plane for . For , repeated integration gives and then for some constants and . The resulting relation (where ) defines a family of parabolas in the -plane parameterized by . Similarly, for we obtain the family of parabolas , . These curves are shown in Figure 4.15(a,b), with the arrows indicating the direction in which they are traversed. It is easy to see that only two of these trajectories hit the origin (which is the prescribed final point). Their union is the thick curve in Figure 4.15(c), which we call the switching curve and denote by ; it is defined by the relation . The optimal control strategy thus consists in applying or depending on whether the initial point is below or above , then switching the control value exactly on and subsequently following to the origin; no switching is needed if the initial point is already on . (Thinking slightly differently, we can generate all possible optimal trajectories--which cover the entire plane--by starting at the origin and flowing backward in time, first following and then switching at an arbitrary point on .) Recalling the interpretation of our problem as that of parking a car using bounded acceleration/braking, the reader can easily relate the optimal trajectories in the -plane with the corresponding motions of the car along the -axis. Note that if the car is initially moving away from the origin, then it begins braking until it stops, turns around, and starts accelerating (this is a ``false" switch because actually remains constant), and then switches sign and the car starts braking again.
The optimal control law that we just found has two important features. First, as we already said, it is bang-bang. Second, we see that it can be described in the form of a state feedback law. This is interesting because in general, the maximum principle only provides an open-loop description of an optimal control; indeed, depends, besides the state , on the costate , but we managed to eliminate this latter dependence here. It is natural to ask for what more general classes of systems time-optimal controls have these two properties, i.e., are bang-bang and take the state feedback form. The bang-bang property will be examined in detail in the next two subsections. The problem of representing optimal controls as state feedback laws is rather intricate and will not be treated in this book, except for the two exercises below.
The next exercise is along the same lines but the solution is less obvious.