For optimal control problems with ordinary
differential equations where the
-norm of the control is
minimized, often bang-bang principles hold. For systems that are
governed by a hyperbolic partial differential equation, the
situation is different:
even if a weak form of the bang-bang principle still holds for the wave equation,
it implies no restriction on the form of the optimal control.
To illustrate that
for the Dirichlet boundary control of the wave equation
in general not even feasible controls
of bang-bang type exist,
we examine the states that can be reached by bang-bang-off
controls, that is controls that are allowed to attain only three
values: Their maximum and minimum values and the value zero.
We show that for certain control times,
the difference between the initial and the terminal state can only
attain a finite number of values.
For the problems of optimal exact and approximate
boundary control of the wave equation
-norm of the control is minimized,
we introduce dual problems and present the weak form of a bang-bang
principle, that states that the values of
controls are constrained by the sign of the dual solutions. Since
these dual solutions are in general given as measures, this
is no restriction on the form of the control function:
the dual solution may have a finite support, and when the dual solution vanishes,
the control is allowed to attain all values from the interval
between the two extremal control values.