  We study a coupled system that describes the interacting dynamics between a
bulk field, confined to a finite region with timelike boundary, and a boundary
observable. In our system the dynamics of the boundary observable prescribes
dynamical boundary conditions for the bulk field. We cast our classical system
in the form of an abstract linear Klein-Gordon equation, in an enlarged Hilbert
space for the bulk field and the boundary observable. This makes it possible to
apply to our coupled system the general methods of quantization. In particular,
we implement the Fock quantization in full detail. Using this quantization we
study the Casimir effect in our coupled system. Specifically, we compute the
renormalized local state polarization and the local Casimir energy, which we
can define for both the bulk field and the boundary observable of our system.
Numerical examples in which the integrated Casimir energy is positive or
negative are presented.
