Distributed networks with small parasitic elements: Input-output stability

This paper considers a linear time-invariant network , made of lumped and distributed elements ( transformers, gyrators, controlled and independent sources, transmission lines). The positive number is a small number, proportional to the size of the stray lumped energy-storing elements: as the stray elements disappear: stray capacitors (inductors) become open (short, respectively) circuits. The problem is to find what additional condition is required to guarantee that if --the network , with set to zero-is input-output stable, then for any sufficiently small , is also input-output stable. The additional condition requires that some approximate "high-frequency" network be also input-output stable. It is also shown that if either of these conditions fail so does the input-output stability of for any sufficiently small.