Page 64 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 55

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Proposition 12. Let 1 , 2 ∈ 0 (R ;R ∪{+∞}). Suppose that 1 + 2 is
proper. Then

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(∀x ∈ R ) : ( 1 + 2 ) ∞ (x) = ( 1 ) ∞ (x) + ( 2 ) ∞ (x).
Proof. Since the function 1 + 2 is proper, there exists x 0 ∈ D( 1 )∩D( 2 ).
We have

1
( 1 + 2 ) ∞ (x) = lim ( 1 + 2 )(x 0 + λx)
λ→+∞ λ
1 1
= lim 1 (x 0 + λx) + lim 2 (x 0 + λx)
λ→+∞ λ λ→+∞ λ
= ( 1 ) ∞ (x) + ( 2 ) ∞ (x).

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Remark 15. Let ∈ 0 (R ;R ∪{+∞}) and suppose that K ⊂ R is a closed
convex set such that D( ) ∩ K
= ∅.Using Propositions 11 and 12, we get
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(∀x ∈ R ) : ( + K ) ∞ (x) = ∞ (x) + K ∞ (x).
Example 20 (Ideal diode). The recession function of the electrical superpoten-
tial of the ideal diode (see Section 2.3.1 in Chapter 2)is

(x).
(∀x ∈ R) : (ϕ D ) ∞ (x) = ϕ D (x) = R +
Example 21 (Practical diode and ideal Zener diode). The recession function of
the electrical superpotential of the practical diode (see Section 2.3.2 in Chap-
ter 2)is

V 1 x if x ≥ 0,
(∀x ∈ R) : (ϕ D ) ∞ (x) = ϕ D (x) =
V 2 x if x< 0.
The same result is valid for the ideal Zener diode (see Section 2.3.4 in Chap-
ter 2). The recession function of the electrical superpotential of the ideal Zener
diode is

V 1 x if x ≥ 0,
(∀x ∈ R) : (ϕ Z ) ∞ (x) = ϕ Z (x) =
V 2 x if x< 0.
Example 22 (Complete diode model). The recession function of the electrical
superpotential of the complete diode (see Section 2.3.3 in Chapter 2)is

V 4 x if x ≤ I R2 ,
(∀x ∈ R) : (ϕ CD ) ∞ (x) =
+∞ if x> I R2 .