Metrically Regular Generalized Equations: A Case Study in Electronic Circuits

TL;DR

This thesis investigates how perturbations in input signals affect the output of electronic circuits modeled by set-valued maps, using variational analysis to study stability and regularity properties in static and dynamic cases.

Contribution

It introduces a framework applying variational analysis to generalized equations modeling semiconductor circuits, analyzing stability and regularity under input perturbations in static and dynamic scenarios.

Findings

Conditions for local stability properties like Aubin property and calmness are established.

Necessary and sufficient conditions for metric regularity are derived.

Descriptions of solution trajectory regularity in dynamic cases are provided.

Abstract

In this thesis, set-valued maps are considered to model the $i-v$ characteristics of semiconductors like diode, and transistor. Using the circuit theory laws, a generalized equation is obtained. The main concern of the thesis is to investigate how perturbing the input signal will affect the output variables. The problem is studied in two cases: the static case, where the input signal is a DC source; and the dynamic case, where there exists an AC source in the circuit. In the static case, the problem can be reduced to the existence or absence of local stability properties of the solution map, like the Aubin property, isolated calmness, and calmness, or metric regularity for the inverse map. Some tools from variational analysis are used to provide necessary and/or sufficient conditions that guarantee these properties. In the dynamic case, those pointwise results are used to obtain descriptions for regularity properties of the solution trajectories in function spaces.