Lossless Nonlinear Networks

Over the past decade a number of results concerning the relationship between causality and various energy concepts such as losslessness, passivity, and stability have been obtained [l][4] for linear networks. For nonlinear networks, however, no analogous results have been established except for the case of weakly additive networks [S], [6], wherein the linear theory holds without modification.’ In the present correspondence it is shown that an arbitrary solvable network is lossless if and only if it is isometric and causal in an appropriate sense. The key to the characterization of lossless and passive networks is the concept ot causality. In fact, the primary difficulty encountered in extending the linear results to the nonlinear case is due to the fact that the “same-input same-output” and ‘(zero-input zero-output’ definitions for causality are not equivalent in the nonlinear case. As such, we must deal with several distinct modes of causality, our theorem being predicated on the properties of a new causality concept which falls between the two classical concepts (and is therefore equivalent to them in the linear case). Following the formulation of [4] we adopt a scattering formalism with the space of admissible signals (incident and reflected waves) for the network taken to be locally integrable (n-vector valued) functions which have support bounded on the left (i.e., for each admissible input or output f there exists a real number r, such that f(t) = 0 for almost all t< r,). A solvable network may then be represented by an operator S mapping this space of admissible signals into itself in such a way that a (globally) square integrable incident wave a is mapped into a (globally) square integrable reflected wave b =Sa. For any admissible signalf we denote byft its truncate defined by