In the analysis above, the spatial domain is assumed unbounded (i.e., we took ). It is useful to examine the energetic behavior of (3.1) if this is not the case. Integrating (3.3) over , we get

where , and where we have defined

If the boundary of is sufficently smooth, then upon applying the Divergence Theorem, we get

where is the boundary of , is defined as the unit outward normal (assumed unique everywhere on except over a set of measure zero), and is a surface element of . If we define the total energy by

then we have

If is positive semi-definite, then a simple condition for passivity is

and the system is lossless if is antisymmetric and (3.8) holds with equality.

This analysis is grossly incomplete, however, because we have not said anything about which boundary conditions ensure the existence and uniqueness of a solution; this analysis is rather involved, and we refer the reader to [82] for an introduction. The basic issue is the over- or under-specification of on the boundary. We will consider only lossless, memoryless boundary conditions in this thesis.