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.
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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.