4.4.2 Bang-bang principle for linear systems

Consider now a system with general linear time-invariant dynamics

where and . We want to investigate under what conditions time-optimal controls for this system have a bang-bang property along the lines of what we saw in the previous example. Of course, we need to specify what the control set is and exactly what we mean by a bang-bang property for controls taking values in this . As a natural generalization of the interval to higher dimensions, we take to be an -dimensional hypercube:

This is a reasonable control set representing independent control actuators. We take the magnitude constraints on the different components of to be the same just for simplicity; the extension to different constraints is immediate.

Suppose that the control objective is to steer from a given initial state to a given final state in minimal time. To be sure that this problem is well posed, we assume that there exists some control that achieves the transfer from to (in some time). As we will see in Section 4.5 (Theorem 4.3), this guarantees that a time-optimal control exists. We now use the maximum principle to characterize it. The Hamiltonian is . The Hamiltonian maximization condition implies that

for all . We can rewrite this formula in terms of the input components as

where are the columns of . Since the components of the optimal control can be chosen independently, it is clear that each term in the summation must be maximized:

It has now become obvious that we must have

for each . Observe that (4.54) resembles (4.50). Similarly to how we proceeded from that formula, we now need to investigate the possibility that on some interval of time for some , as this would prevent us from determining on that interval.

The adjoint equation is , which gives . From this we obtain . This is a real analytic function of ; hence, if it vanishes on some time interval, then it vanishes for all , together with all its derivatives. Calculating these derivatives at , we arrive at the equalities

In other words, is orthogonal to the vectors , , , . As we know, itself cannot be 0 in a time-optimal control problem (see page ). Therefore, to rule out (4.55) it is enough to assume that is a controllable pair. We need this to be true for each ; i.e., the system should be controllable with respect to each individual input channel. Such linear control systems are called

Let us now collect the properties of the optimal control
that we are able to derive under the above normality assumption.
None of the functions
equal 0 on any
time interval; being real analytic functions, they only have
finitely many zeros on the interval
. Using the
formula (4.54), we see that each function
only takes the values
and switches between these values
finitely many times. Away from these switching times,
is
uniquely determined by (4.54). We conclude that the
overall optimal control
takes values only in the set of
*vertices* of the hypercube
, has finitely many
discontinuities (switches), and is unique everywhere else.
Generalizing the earlier notion, we say that controls taking
values in the set of vertices of
are *bang-bang* (or
have the bang-bang property); the result that we have just
obtained is a version of the **bang-bang principle** for
linear systems.

Before closing this discussion, it is instructive to see how the above bang-bang property can be established in a self-contained way, without relying on the maximum principle. More precisely, the argument outlined next essentially rederives the maximum principle from scratch for the particular problem at hand (in the spirit of Section 4.3.1). Solutions of the system (4.51) take the form

For , let us introduce the set of points reachable from at time :

We know that

In fact, the optimal time is the smallest time such that . It follows that must belong to the

which, in view of the formula , is equivalent to

From this it is not difficult to recover the fact that (4.53) must hold for (almost) all , and we can proceed from there as before.

The assumption of normality, which was needed to prove the
bang-bang property of time-optimal controls for
a hypercube,
is quite strong.
A different, weaker
version of the bang-bang principle could be formulated as follows.
Rather than wishing for *every* time-optimal control to be
bang-bang, we could ask whether every state
reachable from
by some control is also reachable from
*in the
same time* by a bang-bang control; in other words, whether
reachable sets for bang-bang controls coincide with reachable sets
for all controls. This would imply that, even though not all
time-optimal controls are necessarily bang-bang, we can always
select one that is bang-bang. It turns out that this modified
bang-bang principle holds for every linear control system (no
controllability assumption is necessary) and every control set
that is a convex polyhedron. The proof requires a refinement
of the above argument and some additional
steps; see [Sus83, Section 8.1]
for details.