A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems: applications in electronics

The main object of this paper is to present a general mathematical theory applicable to the study of a large class of linear variational inequalities arising in electronics. Our approach uses recession tools so as to define a new class of problems that we call “semi-complementarity problems”. Then we show that the study of semi-complementarity problems can be used to prove new qualitative results applicable to the study of linear variational inequalities of the second kind.


Introduction
The first major contribution of this paper is to propose a new mathematical approach that can be used to derive conditions for the existence and uniqueness of a general class of variational inequalities of the form: Find u ∈ R n such that where M ∈ IR n×n is a real matrix, q ∈ R n a vector and Φ : R n → IR ∪{+∞} a proper convex and lower semicontinuous function.
In this paper, we first develop a topological approach using Brouwer degree theory that has been introcuded in [29] for the case Φ ≡ Ψ K where Ψ K denotes the indicator function of some nonempty closed convex set K. This approach reduces the study of problem (1) to the one of some semi-complementarity problems like: where D(Φ) ∞ is the recession cone of the domain of Φ and (D(Φ ∞ )) * is the dual cone of the domain of the recession function Φ ∞ of Φ.
Then we show that this approach can be used to state conditions for the existence and uniqueness of solutions of Problem (1) for various important classes of matrices M and functions Φ. In particular, we show that fundamental results known in complementarity theory and concerning several classes of matrices like positive definite matrices, P-matrices, weakly positive definite matrices, positive stable matrices, diagonally stable matrices, positive semidefinite matrices, P 0 -matrices, weakly positive semidefinite matrices, positive semi-stable matrices, copositive matrices and diagonally semi-stable matrices can be generalized to problem (1). Recession tools play a major role in this approach. Then we present how our theory for problem (1) can be used to derive conditions for the existence and uniqueness of solutions of the generalized equation Ax + Du ∈ B∂Ξ(Cx) where A ∈ IR n×n , B ∈ IR n×m , C ∈ IR m×n and D ∈ IR n×p are matrices, u ∈ IR p is a vector, Ξ : IR m → IR ∪{+∞} is a proper convex and lower semicontinuous function and ∂ denotes the subdifferential operator of convex analysis. The second major contribution of this paper consists to show that the mathematical model in (3) and consequently in (1) can be used to develop a suitable methodology for the formulation and mathematical analysis of circuits in electronics involving devices like diodes and operational amplifiers. As shown through several examples of classical static electrical circuits like clipping circuits, slicers, sampling gates, operational amplifiers, four-diode bridge full-wave rectifiers, such generalized equations are mandatory studying to characterize the well-posedness of the circuits (u may be a time-varying input signal t → u(t)) as well as to compute some defined output signal. We show also that the results which are presented in this paper can be useful for the determination of the equilibrium points of dynamical circuits, a topic of major importance for further dynamical analysis and control applications.

Mathematical Tools
The aim of this Section is to recall some notions and fundamental results in convex analysis which will be used throughout this paper.
For x, y ∈ R n , the notation x, y = n i=1 x i y i is used to denote the euclidean scalar product on R n and x = x, x to denote the corresponding norm. The identity mapping on R n will be denoted by id R n while the identity matrix of order n is denoted by I. We will also denote by {e 1 , ..., e n } the canonical basis of R n .
• [Closed convex set] Let K ⊂ R n be a nonempty closed convex set. We denote by Ψ K the indicator function of K, that is: Then The dual cone of K is the nonempty closed convex cone K * defined by • [Recession function] Let x 0 be any element in D(Φ). The recession function of Φ is defined by The function Φ ∞ : R n → R ∪ {+∞} is a proper convex and lower semicontinuous function which describes the asymptotic behavior of Φ.
• [Recession cone] Let x 0 be some arbitrary element of K. The recession cone of K is defined by The set K ∞ is a nonempty closed convex cone that is described in terms of the directions which recede from K.
Let us here recall some important properties of the recession function and recession cone (see Section 1.4 in [27]): Proposition 1 a) Let Φ : R n → R ∪ {+∞} be a proper convex and lower semicontinuous function. Then b) Let Φ 1 : R n → R ∪ {+∞} and Φ 2 : R n → R ∪ {+∞} be two proper, convex and lower semicontinuous functions. Then c) Let K ⊂ R n be a nonempty, closed and convex set. Then d) Let K ⊂ R n be a nonempty closed and convex cone. Then e) Let K ⊂ R n be a nonempty compact and convex set. Then • [Proximation operator] Let Φ ∈ Γ 0 (IR n ; IR ∪{+∞}) be given. It is known that for each y ∈ R n , there exists a unique x ∈ R n such that that is y ∈ x + ∂Φ(x).
The mapping P Φ : R n → R n ; y → P Φ (y), called the proximation operator (see e.g. [43]), and defined by P Φ (y) = (id R n + ∂Φ) −1 (y), (y ∈ R n ) is thus a well-defined singled-valued operator. Moreover, it is easy to check that: If K is a nonempty closed convex set, then P ΨK ≡ P K where P K denotes the projector from R n onto K, i.e.
• [Brouwer topological degree] Let D ⊂ R n be an open and bounded set. If f :D → R n is continuous and 0 / ∈ f (∂D) then the Brouwer topological degree of f with respect to D and 0 is well-defined (see e.g. [35]) and denoted by deg(f, D, 0). Let us here recall some properties of the topological degree we will use later in this paper. P1. Solution property: If 0 / ∈ f (∂D) and deg(f, D, 0) = 0 then there exists x ∈ D such that f (x) = 0.

A Class of Variational Inequalities
Let Φ : R n → R ∪ {+∞} be a proper, convex and lower semicontinuous function with closed domain, i.e.
Let M ∈ R n×n be a given matrix and q ∈ R n a given vector. We consider the variational inequality problem: VI(M, q, Φ): Find u ∈ R n such that: The solution set of problem VI(M, q, Φ) will be denoted by SOL(M, q, Φ) and the resolvant set by R(M, Φ),  (16) is equivalent to the differential inclusion:

Remark 1 i) The variational inequality in
and R(M, Φ) is nothing else that the range of the set-valued mapping ii) If a solution of (16) exists, let us say u * , then: M u * + q, e + Φ(u * + e) − Φ(u * ) ≥ 0, ∀e ∈ R n and then using (8), we get: M u * + q, e + Φ ∞ (e) ≥ 0, ∀e ∈ R n . It results that necessarily: iii) Condition (15) is not really necessary to develop our approach but it is assumed in order to simplify its presentation.

Special classes of matrices M and functions Φ
In the study of Problem VI(M, q, Φ), the following definitions of various special matrices M and functions Φ will be used.
Let σ(M ) ⊂ C be the set of eigenvalues of M .
• We define by Q n the set of (M, Φ) ∈ R n×n × Γ(R n ; R ∪ {+∞}) such that i.e. for each q ∈ R n , problem VI(M, q, Φ) has at least one solution.
It is clear that PD n ⊂ PD0 n , P n ⊂ P0 n , PS n ⊂ PS0 n .
Subset-superset relationships cannot be expected as illustrated by the following simple examples.

Example 1 Let us consider the matrices
 and the functions: It is easy to see that (

Remark 4
The class of matrices satisfying condition (22) recovers the class of positive definite matrices, the class of matrices satisfying condition (25) recovers the class of P-matrices and the class of matrices satisfying condition (29) recovers the class of weakly positive definite matrices and consequently the class of positive stable matrices. The class of matrices satisfying condition (23) recovers the class of positive semidefinite matrices, the class of matrices satisfying condition (27) recovers the class of P 0 -matrices and the class of matrices satisfying condition (31) recovers the class of weakly positive semidefinite matrices and consequently the class of positive semi-stable matrices. The concepts introduced above can thus be used to recover and unify several situations involving matrices for which there is no subset-superset relationship.

The asymptotic continuation principle
• Recalling that D(Φ) is nonempty, convex and assumed to be closed, the set D(Φ) ∞ is then a well-defined nonempty closed and convex cone. Let us now consider the semi-complementarity problem: Remark 5 Let Φ : R n → R be a proper convex and lower semicontinuous function with closed domain.
i) The following inclusion holds: Indeed, let e ∈ D(Φ ∞ ) be given. Then, for some x 0 ∈ D(Φ), we have Then, remarking that is a nonempty closed convex set, then using (10), we see that In this case, problem SCP ∞ (M, Ψ K ) reduces to the complementarity problem: iii) Let Φ : IR → IR be the function defined by Let us also set: Proposition 3 Let Φ : R n → R be a proper convex and lower semicontinuous function and let M ∈ R n×n be a given matrix. If u 1 and u 2 denote two solutions of problem VI(M, Φ, q) then Proof: If u 1 and u 2 denote two solutions of VI(M, q, Φ) then M u 1 + q, The structure of the set B(M, Φ) can be specified in several situations that are described in the following proposition.

Proposition 4
Let Φ : R n → R be a proper convex and lower semicontinuous function with closed domain and let M ∈ R n×n be a given matrix.
Let (M, Φ) ∈ P0 n be given. Let w ∈ B(M, Φ) be given. It suffices to check that M w, w = 0. We know that: .., n} be given. We may set h = w, e j e j to get (M w) j w j ≥ 0. This last relation holds for all j ∈ {1, ..., n} and since 0 ≥ w, M w = n j=1 (M w) j w j we obtain finally that M w, w = 0. g) Part g) is a direct consequence of part c). In other words, we say that the couple (M, Φ) is of class AC n provided that, for all t ∈ [0, 1], 0 is the unique solution of problem SCP ∞ ((1 − t)I + tM, Φ).
This concept that may appear technical can in fact be used to recover various important situations. This is shown in the following proposition.
Let j ∈ {1, ..., n} be given. Using (25), we may set h = z, e j e j to get This last relation holds for all j ∈ {1, ..., n} and since we get the contradiction: and z ∈ B((1 − t)I + tM, Φ) be given. We claim that z = 0. Suppose on the contrary that z = 0. Here D(Φ ∞ ) * = {0} and the second relation in (32) yields If t = 0 then z = 0 and a contradiction. If 0 < t ≤ 1 then t ≤ 0 is a real eigenvalue of M and a contradiction to (29).
Let us now give some additional properties that will be used later.
The following Theorem is the basic result of this Section. It reduces the study of the general class of variational inequalities VI(M, q, Φ) to semi-complementarity problems SCP ∞ (tM + (1 − t)I, Φ) (t ∈ [0, 1]) involving the convex combinations of the matrix M and the identity matrix I. More precisely, we prove that if the couple (M, Φ) ∈ AC n then for each q ∈ R n , problem VI(M, q, Φ) has at least one solution. In other words we prove that AC n ⊂ Q n .

Theorem 1 (Asymptotic continuation principle) If
(M, Φ) ∈ AC n then, for each q ∈ R n , problem VI(M, q, Φ) has at least one solution.
Proof: Let q ∈ R n be given. From (14) and (17), Problem VI(M, q, Φ) is equivalent to the fixed point problem: Let us now define by H : [0, 1] × R n → R n the continuous homotopy denoted as We claim that there exists R 0 > 0 such that for all R ≥ R 0 and for all t ∈ [0, 1], Indeed, if we suppose the contrary then we may find sequences Then It is clear from (41) that Moreover, for i large enough, ||u i || = 0 and we may set: There exists subsequences, again denoted by {t i } and {z i }, such that lim i→+∞ t i = t ∈ [0, 1] and lim i→+∞ z i = z with ||z|| = 1.
Let x 0 ∈ D(Φ) be any element in the domain of Φ. Let λ > 0 be given. For i large enough, λ ||ui|| < 1 and thus Recalling that the set D(Φ) is assumed to be closed and taking the limit as i → +∞, we get λz + x 0 ∈ D(Φ). This result holds for any λ > 0 and thus Let e ∈ D(Φ ∞ ) be given. Then from (34), e ∈ D(Φ) ∞ and from (11), u i +e ∈ D(Φ). We may thus set v = u i +e in (41) to get and thus using (8), we obtain Note that Φ ∞ (e) < +∞ since e ∈ D(Φ ∞ ) and we may therefore divide this last relation by ||u i || to get: Taking the limit as i → +∞, we get (1 − t)z + tM z, e ≥ 0. This holds for any e ∈ D(Φ ∞ ) and thus Setting now v = x 0 in (41), we obtain: The function Φ is proper, convex and lower semicontinuous, and thus (see e.g. Theorem 1.1.11 in [27]) there exists a ≥ 0 and b ∈ R such that: Dividing this last relation by ||u i || 2 , we get: Taking the limit as i → +∞, we get This last relation together with (42) and (43) imply that z ∈ B((1 − t)I + tM, Φ). Moreover z = 0 and we obtain a contradiction to our assumption requiring that (M, Φ) is AC well-posed.
Thus, for R ≥ R 0 , (40) holds and the Brouwer degree with respect to the set D R := {x ∈ R n : ||x|| < R} and 0 of the map u → u − H(t, u) is well-defined for all t ∈ [0, 1]. Set R 1 := P Φ (0) and let R > max{R 0 , R 1 } be given. Using the homotopy invariance property as well as the normalized property of Brouwer degree, we obtain: It results from solution property of Brouwer degree that SOL(M, q, Φ) = ∅ and the result follows.

Positivity and solvability conditions
Using Theorem 1 together with Proposition 5 we obtain B n ∪ PD n ∪ P n ∪ PS n ⊂ AC n ⊂ Q n and we get the following result ensuring that for each q ∈ R n , problem VI(M, q, Φ) has at least one solution: Each matrix discussed in Corollary 1 presents some "positivity property" and is nonsingular on D(Φ) ∞ in the sense that:

Nonnegativity and solvability conditions
Let Φ : R n → R ∪ {+∞} be a proper, convex and lower semicontinuous function with closed domain and let M ∈ R n×n be a matrix. In this Section, we will assume that (M, Φ) ∈ Q0 n .

Proof:
Here there exists λ 0 > 0 such that: It results that for all i ∈ N, i ≥ 1 λ0 , there exists u i ∈ R n such that We claim that the sequence {u i } ≡ {u i ; i ∈ N\{0}} is bounded. Suppose on the contrary that ||u i || → +∞ as i → +∞. Then, for i large enough, ||u i || = 0 and we may set z i := ui ||ui|| . There exists a subsequence, again denoted by {z i }, such that lim i→+∞ z i = z with ||z|| = 1.

It is clear from (46) that
Let x 0 ∈ D(Φ) be any element in the domain of Φ. Let λ > 0 be given. For i large enough, λ ||ui|| < 1 and thus since D(Φ) is convex and x 0 ∈ D(Φ). Taking the limit as i → +∞, we get λz + x 0 ∈ D(Φ) since D(Φ) is assumed to be closed. This result holds for any λ > 0 and thus Let e ∈ D(Φ ∞ ) be given. Then e ∈ D(Φ) ∞ and from (11), u i + e ∈ D(Φ). We may set v = u i + e in (46) to get and thus using (8), we obtain Remarking that Φ ∞ (e) < +∞ since e ∈ D(Φ ∞ ), we may divide this last relation by ||u i || to get: Taking the limit as i → +∞, we get M z, e ≥ 0. This holds for any e ∈ D(Φ ∞ ) and thus Setting now v = x 0 in (46), we obtain: The function Φ is proper, convex and lower semicontinuous, and thus (see e.g. Theorem 1.1.11 in [27]) there exists a ≥ 0 and b ∈ R such that: Dividing this last relation by ||u i || 2 , we get: Taking the limit as i → +∞, we get M z, z ≤ 0. Thus Using (47), (48) and (50) we obtain that z ∈ B(M, Φ).
Finally, we get a contradiction since we have proved that z = 0 and z ∈ B(M, Φ).
The sequence {u i } is thus bounded and there exists a subsequence, again denoted {u i } which converges. Let us set Let v ∈ R n be fixed. We have: Taking the limit inferior as i → +∞, and using the lower semicontinuity of Φ, we obtain: The vector v has been chosen arbitrarily in R n and thus the result in (51) holds for all v ∈ R n . The existence result follows.
From Theorem 1 and Proposition 6, we have This together with Proposition 4 give:

Remark 6
Various other classes of couples (M, Φ) can be studied in using the asymptotic continuation principle. This is however not the main deal of this paper and will be discussed in a future work.

Existence and uniqueness results
In requiring some additional structural properties on Φ as specified in (19), (20) and (21), the uniqueness of the solution of problem VI(M, q, Φ) can be proved. The following result is a generalization of a well-known existence and uniqueness theorem in complementarity theory. Recall that Φ ∈ DΓ(R n ; IR ∪{+∞}) means that where, for all and and let M ∈ IR n×n be a P-matrix, i.e.
Then, for each q ∈ R n , problem VI(M, q, Φ) has a unique solution.
. This together with (55) ensure that (M, Φ) ∈ P n . The existence result is then a direct consequence of Corollary 1.
To prove the uniqueness, suppose by contradiction that problem VI(M, q, Φ) has two different solutions u and U . We set and We Thus, for all integers 1 ≤ j ≤ n: Using (57), we check in the same way that, for all integers 1 ≤ j ≤ n: Let and Setting v = 0 in (56), we get also and thus Using (57) we prove in the same way that Using (58) and (59) together with (62) and (63), we see that, for all integers 1 ≤ j ≤ n: Then, for all integers 1 ≤ j ≤ n: Note that Thus, for all integers 1 ≤ j ≤ n: Recalling that u − U = 0, a contradiction to (55) has been obtained.

Remark 7
The famous result on complementarity problems with P-matrices (see e.g. [18]) can be easily deduced from Theorem 3. Indeed, if Φ = Ψ (IR+) n then Φ can be written as in (52) and if M is a P-matrix then Theorem 3 can be applied to ensure that for each q ∈ R n , there exists a unique u ∈ (IR + ) n such that M u + q ∈ (IR + ) n and u, M u + q = 0.
The following variant of Theorem 3 can be used provided that the functions Φ i (1 ≤ i ≤ n) are strictly convex.
(a) Then, for each q ∈ R n , problem VI(M, q, Φ) has at most one solution.
has a unique solution.
Proof: As in the proof of Theorem 3, it is easy to check that (M, Φ) ∈ P0 n . The existence result in part (b) is then a direct consequence of Corollary 2.
To prove the uniqueness in parts (a) and (b), we suppose by contradiction that problem VI(M, q, Φ) has two different solutions u and U and we proceed as in the proof of Theorem 3 to see that, for all integers 1 ≤ j ≤ n: Using the strict convexity of the functions Φ j (1 ≤ j ≤ n), we obtain finally that for all integers α ∈ {1, ..., n} such that u α = U α : and a contradiction to (65) has been obtained.

Semicoercivity and solvability conditions
The If there exists x 0 ∈ D(Φ) such that: then problem VI(M, q, Φ) has at least one solution.
Proof: Let q ∈ R n be given. Here (M, Φ) ∈ PD0 n and using Proposition 6 and Theorem 1, we see that for all We claim that the sequence {u i } ≡ {u i ; i ∈ N\{0}} is bounded. Suppose on the contrary that ||u i || → +∞ as i → +∞. Then, for i large enough, ||u i || = 0 and we may set: z i := ui ||ui|| . There exists a subsequence, again denoted by {z i }, such that lim i→+∞ z i = z with ||z|| = 1.
As in the proof of Theorem 2 we check that z ∈ B(M, Φ). Using now (67) with v = x 0 , we get also: Here Dividing this last relation by ||u i ||, we get: Taking the limit inferior as i → +∞, we get: and thus, using (7), we obtain: This is a contradiction to condition (66) since we have proved above that z ∈ B(M, Φ) and z = 0.
The sequence {u i } is thus bounded and we may conclude as in the proof of Theorem 2. (66) is trivially satisfied on the empty set.

Remark 9
If 0 ∈ D(Φ) (which is the case for most practical problems) then we may choose x 0 = 0 to see (66) in the more legible form Theorem 5 may obviously be applied to the class of positive semidefinite matrices, i.e.
This last class of (not necessarily symmetric) matrices is of particular interest for various problems in engineering and it is then worthwhile to specify our results in this framework.
then problem VI(M, q, Φ) has at least one solution.
c) If u 1 and u 2 denote two solutions of problem VI(M, q, Φ) then Proof: Setting X 1 = ker{M + M T }, we may write R n = X 1 ⊕ X ⊥ 1 . We denote by P X1 (resp. P X ⊥ 1 ) the orthogonal projector from R n onto X 1 (resp. X ⊥ 1 ). The matrix M is positive semidefinite and thus (see e.g. Proposition 3.4.3 in [27]) there exists c > 0 such that: Thus N − (M ) = X 1 . Part a) is then a direct consequence of Corollary 2, part b) follows from Theorem 5 and part c) is a consequence of Proposition 3.

Remark 10
Recession tools like recession functions and recession cones can also be used to develop powerful methods, called recession methods, that can be used to the study the solvability of general noncoercive variational inequalities in Hilbert spaces. The idea of this approach goes back to G. Fichera [19]. Various solvability results using recession tools have then been developed in the field of semicoercive linear and semicoercive nonlinear variational and hemivariational inequalities in Hilbert spaces with applications in unilateral mechanics (see e.g. [9], [23]- [28] and the references cited therein).
If in addition the matrix M is symmetric, then u is a solution of problem VI(M, q, Φ) if and only if u is a solution of the optimization problem: This last case is also of particular interest.
then problem VI(M, q, Φ) has at least one solution.
c) If u 1 and u 2 denote two solutions of problem VI(M, q, Φ) then and Proof: Let us first remark that ker{M + M T } = ker{M } and parts a) and the first relation in part c) are direct consequences of part a) and c) in Corollary 3. Using the equivalence of problem VI(M, q, Φ) with problem (72), we get also from which we deduce the second relation in part c). It is easy to check that part d) is a direct consequence of part c).
Moreover, let x 0 ∈ D(Φ) be given. Then, for all v ∈ ker{M }, we have It results that condition (69) is thus here equivalent to condition (73). Part b) is then a direct consequences of part b) in Corollary 3.
Finally, if Φ is strictly convex then the function x → 1 2 M x, x + q, x + Φ(x) is strictly convex too and thus problem (72) has at most one solution. The result in part e) follows since problem (72) is equivalent to problem VI(M, q, Φ).

Copositivity and solvability conditions
Our aim in this section is to show that our results established in Corollary 2 and Theorem 5 recover some results established in the framework of complementarity systems on K = (R + ) n involving copositive plus matrices.
Let K ⊂ R n be a nonempty closed convex cone. We set: Here B(M, Ψ K ) ≡ B(M, K) and problem VI(M, q, Ψ K ) is equivalent to the complementarity problem CP(M, q, K): Our results in Corollary 2 and Theorem 5 read here: then problem CP(M, q, K) has at least one solution.
c) Moreover, if u 1 and u 2 denote two solutions of problem CP(M, q, K) then Remark 11 i) Recall that one says that a matrix M ∈ R n×n is copositive plus on K provided that In this case B(M, K) = {x ∈ K : M x ∈ K * and x ∈ ker{M + M T }}.
ii) Note that the approach developed in [20] and [21] for complementarity systems originated the approach developed in [23] for variational inequalities. We note that Theorem 5 recovers both solvability results and unifies both approaches (anyway in the framework of finite dimensional problems).

Variational inequalities method in electronics
Several researchers have recently shown that mathematical tools from complementarity systems theory and variational inequalities theory may be used to develop rigorous mathematical study of circuits in electronics involving devices like diodes, Zener diodes and varistors that are characterized by set-valued ampere-volt characteristics. Mathematical approaches using tools from mathematical programming and complementarity systems theory have been particularly developed in [15], [16], [31], [34], [37] and the references cited therein while mathematical approaches using tools from set-valued analysis and variational inequalities theory have been studied in [7], [8], [10], [13], [17] and [22].

Set-valued Ampere-Volt Characteristics in Electronics
Electrical devices like diodes are described in terms of Ampere-Volt characteristics (i, V ) that is a graph expressing the difference of potential V across the device as a function of current i through the device. The schematic symbol of a circuit element is given in Figure 1. The conventional current flow i will be depicted on the conductor in the direction of the arrow and the potential V := V A − V B (V A (resp. V B )) potential of point A (resp. B) across the device will be denoted alongside the device. Experimental measures as well as empirical and physical models lead to a variety of monotone graphs that may present vertical branches. The reader can find general descriptions of devices and Ampere-Volt characteristics either in the appropriate electronics literature or in electronics society catalogs (see e.g. [1], [2], [3], [4], [5], [6], [36]).
Let us so suppose here that we may write: for some set-valued function F : R ⇒ R. The domain D(F) of F is defined by: We assume that F is maximal monotone, i.e.
and the graph G(F) of F, i.e.
is not properly included in any other monotone subset of R × R.

Remark 12 (Terminology) We will say that an electrical device is VAM-admissible provided that its ampere-volt characteristic graph (i, V ) is maximal monotone. Then there exists a proper convex and lower semicontinuous function
The function ϕ will be called the electrical superpotential (determined up to an additive constant) of the device. Roughly speaking, the electrical superpotential ϕ appears as a "primitive" of F in the sense that the "derivative" (in the generalized sense) of ϕ recovers the set-valued function F.

Diode models
The diode is a device that constitutes a rectifier which permits the easy flow of charges in one direction but restrains the flow in the opposite direction. Diodes are used in power electronics applications like rectifier circuits, switching inverter and converter circuits. Figure 2 illustrates the ampere-volt characteristic of an ideal diode. This is a model in which the diode is a simple switch. If V < 0 then i = 0 and the diode is blocking. If i > 0 then V = 0 and the diode is conducting. We first see that the ideal diode is described by the complementarity relation

Example 2 (IDEAL MODEL)
The electrical superpotential of the ideal diode is Then ϕ * D (z) = Ψ R− (z), (z ∈ R) and the recession function of the electrical superpotential is: We have also The complementarity relation can be written as Figure 3 illustrates the ampere-volt characteristic of a practical diode model. There is a voltage point, called the knee voltage V 1 , at which the diode begins to conduct and a maximum reverse voltage, called the peak reverse voltage V 2 , that will not force the diode to conduct. When this voltage is exceeded, the depletion may breakdown and allow the diode to conduct in the reverse direction. Note that usually | V 2 |>>| V 1 | and the model is locally ideal.

Example 3 (PRACTICAL DIODE MODEL)
For general purpose diodes used in low frequency/speed applications, | V 1 | 0.7 − 2.5 V and | V 2 | 5 kV; for high voltage rectifier diodes, | V 1 | 10 V and | V 2 | 30 kV; for fast diodes used in switched mode power supply and inverter circuits, | V 1 | 0.7 − 1.5 V and | V 2 | 3 kV and for Schottky diodes used in high frequency applications, | V 1 | 0.2 − 0.9 V and | V 2 | 100 V. The electrical superpotential of the practical diode is Then ϕ * P D (z) = Ψ [V2,V1] (z), (z ∈ R) and the recession function of the electrical superpotential is given by: We see that , (z ∈ R). Figure 4 illustrates a complete diode model which includes the effect of the natural resistance of the diode, called the bulk resistance, the reverse current I R1 , the diode capacitance and the diffusion current. This last model is more accurate and represents the true operating characteristics of the diode.

Example 4 (COMPLETE DIODE MODEL)
Note that | V 4 |<<| V 1 |. For example, the 10ETS.. rectifier (SAFEIR series [3]) has been designed with | V 1 |= 1.1 V, | V 4 |= 800 − 1600 V, I R1 = 0.05 mA and with a bulk resistance equal to 20 mΩ. Let us use the notation of Figure 4. It is implicitly assumed that Let us also set: The electrical superpotential of the complete diode is and simple calculations yield: On the other hand, we may compute the conjugate function: The recession function of the electrical superpotential is here: The ampere-volt characteristic of the complete diode can then be written as

Example 5 (EMPIRICAL DIODE MODEL) An empirical model used in electronics to describe the amperevolt characteristic of a diode is:
where I S is the saturating reverse current (10 −15 ≤ . ≤ 10 −12 A), V T is the thermodynamic voltage (25 mV) and η is the emission coefficient (1 ≤ . ≤ 2). This model is usually considered in the engineering literature when a rigorous mathematical analysis taking care of the domain of V is not required. It is however possible to proceed to a suitable mathematical treatment as above in defining the set-valued function V : R ⇒ R by Then setting

Zener diode models
The Zener diodes are made to permit current to flow in the reverse direction if the voltage is larger than the rated breakdown or "Zener voltage" V 2 . For example, for a common Zener diode, V 1 0.7 volts and V 2 −7 volts.
The Zener diode (see Figure 5) is a good voltage regulator to maintain a constant voltage regardless of minor variations in load current or input voltage. There is a current point I Z , called the Zener knee current, which is the minimum value of the Zener current required to maintain voltage regulation and a maximum allowable value of Zener current I M . Currents above this value will damage or destroy the system. The graph corresponding to the ampere-volt characteristic (i, V ) is maximal monotone and there exists a proper convex and continuous electrical superpotential ϕ : IR → IR such that V ∈ ∂ϕ(i), (i ∈ IR).

Example 6 (IDEAL ZENER DIODE MODEL)
The ideal Zener diode model (see Figure 6) is given by the complete diode model (see Figure 3) with the appropriate values for V 1 and V 2 . This means that the voltage across the diode is constant over a wide range of device current values. Figure 7) is a piecewise linear model that includes the effects of the Zener impedance. Let us use the notation of Figure 7. It is here implicitly assumed that

Example 7 (PRACTICAL ZENER DIODE MODEL) This model (see
The electrical superpotential of the Zener diode is The recession function of the electrical superpotential is given by: The ampere-volt characteristic of the complete diode can thus be written as

Example 8 (VARISTOR) A varistor is a nonlinear device that has an electrical behavior similar to the Zener
More precisely, the varistor (see Figure 8) is a voltage-dependent resistor with a symmetrical monotone ampere-volt characteristic. It is used connected in parallel with the electronic device or circuit that is to be guarded in order to form a lowresistance shunt when voltage increases and thus prevent any further rise in the overvoltage.

Mathematical Formulation of a Class of Non-regular Models in Electronics
A circuit in electronics is formed by the interconnection of electrical devices like generators, resistors, capacitors, inductors, transistors, diodes and various others. The behaviour 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 to formulate the ampere-volt characteristic of each electrical device, to write the Kirchoff's voltage law expressing that the algebraic sum of the voltages between successive nodes in all mesches in the circuit are zero and to write the Kirchhoff's current law stating that the algebraic sum of the currents in all branches which converge to a common node equal zero.
The practice (see [8] and [12]) shows that a large class of circuits can be studied via the following general mathematical formalism.
Let A ∈ R n×n , B ∈ R n×m , C ∈ R m×n and D ∈ R n×p be given matrices. Let Ξ : R m → R ∪ {+∞} be a given proper, convex and lower semicontinuous function. Let u ∈ R p be given, we consider the problem: and y L ∈ ∂Ξ(y).
The matrices A, B, C and D in (81) are structural matrices used to state Kirchoff's voltage laws and Kirchoff's current laws in matrix form. The matrix A depends of electrical parameters like resistances, capacitances and inductances. Usually u is a control vector that drives the system, x denotes a current vector and y L is a voltage vector corresponding to electrical devices like diodes whose (possibly set-valued) ampere-volt characteristics can be described as in (83).
It is noteworthy that (81)-(83) may represent the equations of a static circuit, but also the generalized equation that is to be satisfied by the equilibrium points of a dynamical circuit, or more generally of a class of differential inclusions (see [12] for applications in the absolute stability problem).
Let us now make the following two assumptions: Assumption (H2): There existsx 0 ∈ R n such that Ξ is finite and continuous atȳ 0 = Cx 0 .
Assumption (H3): There exists an invertible matrix P ∈ R n×n such that Then The existence of a vectorȳ 0 = Cx 0 at which Ξ is finite and continuous ensures that (see e.g. Proposition 1.3.11 in [27]): Thus Suppose now that x is solution of Problem (86). We see as above that: It results that there exists y L ∈ ∂Φ(Cx) such that: Then we obtain the relations in (81)-(83) by setting y = Cx.
Proposition 7 means that if assumptions (H1)−(H3) hold then problem (81)-(83) can be studied via the variational inequality VI(M, q, Φ) with The results discussed in Section 3 can thus be used here too. The following result is then of particular interest to calculate the recession tools involved in B(M, q, Φ), that are here D(Φ) ∞ , Φ ∞ and D(Φ ∞ ) with Φ ≡ Ξ • C. (H1) and (H3) are satisfied and let Φ be defined as in (84). Then

Proposition 8 Suppose that assumptions
and It is easy to see that Thus e ∈ D ∞ (C, Ξ).
ii) We have: iii) The relation in (90) is a direct consequence of (89).

Diode Circuits
We are now in position to study diode circuits like amplitude selectors that are used to transmit the part of a given waveform which lies above or below some given reference level, double-diode clippers that are used to limit the input amplitude at two independent levels, sampling gates which are transmission circuits in which the output is a reproduction of an input waveform during a selected time interval and is zero otherwise and other circuits involving both diodes and operational amplifiers. We will present some electrical systems involving ideal diodes that can be studied by means of classical tools from complementarity theory and then discuss some extensions to systems involving practical components. Let us first start with two simple clipping circuits to illustrate the essence of our methodology. where U R = Ri denotes the difference of potential across the resistor and V ∈ ∂Ψ IR + (i) is the difference of potential across diode. Thus which is equivalent to VI(R, E − u, Ψ IR+ ), i.e.
Here R > 0 and for each E, u ∈ R, we may apply Corollary 1 to assert that (92) has a unique solution. Moreover: If u ≤ E then the diode is blocking while if u > E then the diode is conducting. Let us now consider a driven time depending input t → u(t) and define the output The time depending current t → i(t) is given by and thus This shows that the circuit in figure 9 can be used to transmit the part of a given input-signal u which lies below some given reference level E.

Example 10 (Clipping circuit 1 / General diode)
In the case of a diode with electrical superpotential ϕ, we may follow the same steps as above to get: and  Here we have Here R > 0 and for each E, u ∈ R, we may apply Corollary 1 to assert that (97) has a unique solution. Moreover: If u < E then the diode is conducting while if u ≥ E then the diode is blocking. For a driven time depending input t → u(t) the time depending current t → i(t) is given by and the output- and thus the circuit in figure 12 can be used to transmit the part of a given input-signal u which lies above some given reference level E.

Example 12 (Clipping circuit 2 / General diode)
In the case of a general diode with superpotential ϕ, we set: Suppose there exists a point y 0 = −x 0 at which ϕ is finite and continuous. Note that this is the case for each model discussed in Section 4. Then: .
For a driven time depending input t → u(t) the time depending current t → i(t) is given by and the output-signal t → V o (t) = V (t) + E can be computed by the formula Let us now discuss two circuits, a double-diode clipper and a sampling gate, to illustrate the results established in Proposition 7 and Corollary 4. Figure 15 involving a load resistance R > 0, two ideal diodes, an input-signal source and two supply voltages E 1 and E 2 . It is assumed that E 1 < E 2 . We denote by i the current through the resistor R and we set i = i 1 + i 2 where −i 1 denotes the current through diode D 1 and i 2 is the current through diode Using Kirchoff's voltage laws, we get the system:

Example 13 (Double-diode clipper / Ideal diode) Let us consider the circuit in
we see that the system in (102) is equivalent to the variational inequality VI(M, q, Φ), i.e.
Here the matrix M is positive semidefinite and symmetric, We may apply Corollary 4 which ensures that the system in (104) has at least one solution.
It results that if Υ * = (i * 1 i * 2 ) T denotes a solution of the system in (104) then the current through the resistor R, i.e. i * = i * 1 + i * 2 is uniquely determined. Using now the second relation of part (c) of Corollary 4, we obtain also that: Here E 2 = E 1 and thus the system in (106)-(107) yields i 1 =ī 1 and i 2 =ī 2 . The solution of (104) is thus unique.
Using the relations in (102) we see that : from which we deduce, after elementary calculations, that: So, for a driven time depending input t → u(t) the time depending current t → i * (t) through the resistor R is given by is then given by the expression: This shows that the circuit can be used to transmit the part of a given input-signal u that lies above some level E 1 and below some level E 2 .  Figure 15 and suppose that the electrical superpotential of each diodes D 1 and D 2 is given by (practical diode model): where ν 2 < 0 < ν 1 . We suppose also that We setφ Kirchoff's laws yield the system which is equivalent to the variational inequality VI(M, q, Φ), i.e.
with M and q as in (103) and Φ as in (111).
Let v ∈ ker{M }, v = 0, be given. Then: It results that if v 2 > 0 then We may then apply Corollary 4 which ensures that the system in (113) has at least one solution.
If Υ * = (i * 1 i * 2 ) T denotes a solution of the system in (113) then from the first relation in part c) of Corollary 4, we deduce that the current i * = i * 1 + i * 2 through the resistor R is uniquely determined. Moreover, the function Φ is strictly convex and part e) of Corollary 4 ensures that the solution Υ * is unique and given by: So, for a driven time depending input t → u(t) the time depending current t → i * (t) through the resistor R is given by and the output-signal V o can then be determined by the formula:  Vc Vc x 7 x 5 x 6 Figure 18: four-diode-bridge sampling gate We denote by V j the voltage of the diode D j and by x i the current across the diode D j (1 ≤ j ≤ 4). Moreover, x 5 denotes the current through the left resistor R c , x 6 is the current through the right resistor R c and x 7 denotes the current trough resistor R L . Kirchoff's laws yield Setting x 7 x 6 It results from Corollary 4 that the system in NRM(A, B, C, D, u, Ξ) has a at least one solution x * . The matrix A is symmetric and thus: for some α ∈ R. It results that x * 7 =x 7 and x * 6 =x 6 , i.e. the current through the resistor R and the current through the resistor R c are uniquely determined. So, for a driven time depending input t → V s (t) and control gate signals t → V c (t) and t → −V c (t), the output time depending voltage t → V o (t) through the resistor R L is uniquely determined by: where the current function t → x 7 (t) is uniquely determined in solving the variational inequality VI(−A, −Du(t), Φ).  The previous examples show that the place filled by the class of symmetric and positive semidefinite matrices is as important in Electronics as it is in Mechanics. However, the mathematical formulation of circuits involving devices like operational amplifiers involves matrices that are neither symmetric nor positive semidefinite. The following example constitutes a nice illustration of Theorem 3. Figure 20 involving an ideal Zener diode D and a non-inverting amplifier circuit with resistors R 1 > 0 and R 2 > 0. We denote by R i > 0 the input resistor, R o the output resistor and γ > 0 the differential gain of the operational amplifier. Note that R i is very large (a few MΩ) while R o is very small (a few Ω). The differential gain γ is very large ( 100000).

Example 16 (Operational amplifier) An operational amplifier is a great versatile circuit element based on transistors which is used to design audio equipments, oscillators and waveform generators, filters and many other applications. Let us consider the system of
with currents I i , I 1 , I 2 and voltages V D , U i as defined in Figure 21. Therefore: where ϕ ZD denotes the electrical superpotential of the Zener diode, i.e. Here we have: The matrix −P A is a P-matrix and Φ ∈ DΓ(IR 3 ; IR ∪{+∞}) (with Φ 1 ≡ Φ, Φ 2 ≡ Φ 3 ≡ 0). We may thus apply Theorem 3 to ensure that the system in NRM(A, B, C, D, U i , ϕ ZD ) has a unique solution. Thus, for a driven time depending input t → U i (t) the output time depending voltage t → U o (t) defined by (see Figure 20): is uniquely defined with the current functions t → I i (t) and t → I 2 (t) that are uniquely determined in solving VI(−PA, −PDU i (t), Φ 1 ).
In the next example we analyse the stationary solutions of a bridge rectifier. This application constitutes an example that does not fit within this class of generalized equations in (81)-(83). Here the unknown in (16) is not x but y L (see (81)-(83)).
It results that the dynamical behavior of the circuit in Figure 22 are described by the system: and y ∈ −Φ(y L ).
Assuming that u remains constant, i.e. u(.) ≡ u, the stationary solutions (or fixed points) of (120)-(122) satisfy the problem: with a := 1 RC > 0. From the first equation of (123) one deduces that x = 1 a By L , so that y = (N + 1 a CB)y L + F u and our problem reduces to problem VI((N + 1 a CB), Φ, Fu), i.e: The matrix N ∈ R 4×4 is skew-symmetric and the matrix Indeed, x ∈ ker{M + M T } yields x 1 = −x 3 and thus x 1 = x 3 = 0 since x ∈ (IR + ) 4 . Using then x ∈ K(M, Φ), we get also −x 2 ≥ 0 and −x 4 ≥ 0 and thus x 2 = x 4 = 0 since x ∈ (IR + ) 4 . It results from Corollary 2 that for each u ∈ IR, problem VI((N + 1 a CB), Φ, Fu) has at least one solution. Case 2. Suppose that diodes D 1 and D 4 are ideal and consider for diodes D 2 and D 3 the practical model with electrical superpotential Indeed, here x 2 = 0 and x ∈ ker{M +M T } yields x 1 = −x 3 and thus x 1 = x 3 = 0 since x 1 ≥ 0. Using then x ∈ K(M, Φ), we get also x 4 = 0. It results from Corollary 2 that for each u ∈ IR, problem VI((N + 1 a CB), Φ, Fu) has at least one solution.
Various cases can be so studied and this shows that studying the fixed points of an interesting class of electrical circuits can be cast into a variational inequality problem such as (16), written as: Find y L ∈ R n such that

Conclusions
In the first part of this paper, it is shown that some classical existence and uniqueness results for complementarity problems can be recovered and can be generalized to variational inequalities using recession tools from convex analysis. Various classes of matrices M are introduced which allow one to state well-posedness results for variational inequalities of the form M u + q, v − u + Φ(v) − Φ(u) ≥ 0 for all v ∈ IR n , thus recovering the well-known results for P-, P 0 , positive definite and semi positive definite matrices in linear complementarity problems. In the second part of the paper, it is shown that the study of electrical circuits with diodes, a topic of major importance in Electrical Engineering, may benefit a lot from the results of the first part. This work also paves the way to numerical tools that may be used to determine the equilibrium points of electrical circuits with non-smooth elements and external excitation.