Page 45 - Complementarity and Variational Inequalities in Electronics

The Variational Inequality Problem Chapter | 3 35

satisfies the variational inequality

n
F(x),v − x + (v) − (x) ≥ 0, ∀v ∈ R . (3.6)
Indeed, if
n
∗
∗
(∀v ∈ R ) : G(x ) + (x ) ≤ G(v) + (v),
n
then for all λ ∈[0,1] and h ∈ R ,
∗
∗
∗
G(x ) + (x ) ≤ G(x + λ(h − x )) + (x + λ(h − x ))
∗
∗
∗
∗
∗
∗
≤ G(x + λ(h − x )) + λ (h) + (1 − λ) (x ).
Thus
∗
∗
∗
G(x + λ(h − x )) − G(x )
+ (h) − (x ) ≥ 0.
∗
λ
Taking the limit as λ → 0+, we obtain
∗
∗
∗
∇G(x ),h − x + (h) − (x ) ≥ 0.
It results in that x is a solution of (3.6). The converse is also true, provided that
∗
G is convex. Indeed, if x is a solution of (3.6), then (see Remarks 2 and 4)
∗
∗
∗
∗
0 ∈∇G(x ) + ∂ (x ) = ∂(G + )(x ),
and thus
n ∗ ∗
(∀v ∈ R ) : G(x ) + (x ) ≤ G(v) + (v).
3.2 THE VARIATIONAL INEQUALITY MODEL IN
ELECTRONICS
A circuit in electronics is formed by the interconnection of electrical devices
like generators, resistors, capacitors, inductors, transistors, diodes, and vari-
ous others. The behavior of a circuit is usually described in terms of currents
and voltages that can be specified through each involved electrical device. The
approach to state a mathematical model that can be used to determine these
currents and voltages consists in formulating the ampere–volt characteristic of
each electrical device, to write the Kirchhofs voltage law expressing that the
algebraic sum of the voltages between successive nodes in all meshes in the cir-
cuit is zero, and to write the Kirchhoff current law stating that the algebraic sum
of the currents in all branches that converge to a common node equals zero. We
will see in this section that general electrical circuits with diodes and transistors
can be studied by using the variational inequality modeling approach.