Page 10 - Complementarity and Variational Inequalities in Electronics

Introduction

The objective of this book is to explain to engineers and mathematicians how
advanced tools from convex analysis can be used to build rigorous mathematical
models for the qualitative study and numerical simulation of electrical networks
involving devices like diodes and transistors. Our objective is also to show that
mathematical models like complementarity problems, variational inequalities,
and differential inclusions can be used to analyze diverse problems in electron-
ics. These last models are indeed well known for their applications in mechanics
and economics (see e.g. [36], [37], [44], [55], [57] [61], [62], [70], [72], [73],
[74], [82]), and we show here that electronics is also an important source of
applications.
In this work, we review and discuss some methodology that has been re-
cently developed by several authors for the rigorous formulation and mathemat-
ical analysis of circuits in electronics like slicers, amplitude selectors, sampling
gates, operational amplifiers, four-diode bridges, full-wave rectifiers, and so on.
All these circuits use semiconductors like diodes and transistors leading to some
highly nonlinear phenomena like switching and clipping. The peculiarity of de-
vices like diodes is that their ampere–volt characteristics are described by graphs
including vertical branches. Such graphs are thus set-valued, and their mathe-
matical treatment requires the use of appropriate tools.
Mathematical models like complementarity problems, variational inequali-
ties, and nonregular dynamical systems are indeed particulary useful to charac-
terize the qualitative properties of the circuits (see [5], [6], [7], [8], [24], [28],
[29], [31], [32], [39], [49], [51], [52], [66]) and to compute some defined output
signals (see [1], [2], [3], [4], [30], [35], [38]). Such mathematical models are also
useful for the determination of the stationary points of dynamical circuits and to
determine the corresponding Lyapunov stability and attractivity properties (see
[9], [25], [26], [42]), a topic of major importance for further dynamical analysis
and control applications (see [10], [11], [16], [20], [21], [27], [50], [81], [82]).
In this book, we review the main mathematical models applicable to the
study of electrical networks involving devices like diodes and transistors. We
also discuss theoretical mathematical results ensuring the existence and unique-
ness of a solution, stability of a stationary solution, and invariance properties.

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