  A representation of a fundamental solution group for a class of wave
equations is constructed by exploiting connections between stationary action
and optimal control. By using a Yosida approximation of the associated
generator, an approximation of the group of interest is represented for
sufficiently short time horizons via an idempotent convolution kernel that
describes all possible solutions of a corresponding short time horizon optimal
control problem. It is shown that this representation of the approximate group
can be extended to arbitrary longer horizons via a concatenation of such short
horizon optimal control problems, provided that the associated initial and
terminal conditions employed in concatenating trajectories are determined via a
stationarity rather than an optimality based condition. The long horizon
approximate group obtained is shown to converge strongly to the exact group of
interest, under reasonable conditions. The construction is illustrated by its
application to the solution of a two point boundary value problem.
