Page 157 - Complementarity and Variational Inequalities in Electronics

148 Complementarity and Variational Inequalities in Electronics

Proof. Indeed, we have already seen that if (x,y L ) is solution of problem
P(x 0 ,A,B,C,D,u, ), then z = Rx is solution of problem Q(Rx 0 ,RAR −1 ,
RDu, ). Suppose now that z is solution of problem Q(Rx 0 ,RAR −1 ,RDu, ).
Then, setting x = R −1 z, we see as before that

dx
∈ Ax − B∂ (Cx) + Du.
dt

Therefore, there exists a function y L ∈ ∂ (Cx) such that

dx
= Ax − By L + Du.
dt
Note that
dx 1 n
By L =− + Ax + Du ∈ L (0,+∞;R ).
loc
dt
Then we obtain relations (5.4)–(5.10) by setting

y = Cx.

So, using assumptions (G 1 ) and (G 2 ), we may reduce the study of problem
P(x 0 ,A,B,C,D,u, ) to that of problem Q(Rx 0 ,RAR −1 ,RDu, ), which
can be investigated by means of mathematical tools from set-valued analysis,
theory of maximal monotone operators, and variational inequality theory (see
e.g. [13], [17], [18], [27], [34], [36], [44], [46], [57], [68], [72]). The equivalence
between complementarity systems, projected systems, and unilateral differential
inclusions is recapitulated in [22]. General results allowing a stability analysis of
the stationary solutions of nonregular dynamical systems can be found in [24]
and [42]. A generalization of Krakovskii–LaSalle invariance theory for non-
regular systems can be found in [25], and related results in [26]. The stability
analysis applicable to the study of a DC–DC Buck converter is detailed in [9].
Piecewise affine dynamical systems and linear complementarity systems with
applications in electronics are given in [16], [21], [28], [29], [30], [31], [32].
Numerical methods have been proposed in [2] and [1] so as to study switched
circuits. The nonsmooth approach applied to simulating integrated circuits and
power electronics is detailed in [35]. We also refer the readers to [3] for a book
on numerical methods for nonsmooth dynamical systems with applications in
electronics. Let us also mention a study including mathematical formulation
and numerical simulations of higher-order Moreau’s sweeping process in elec-
tronics [4], [23].