Page 186 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 177

1
n
n
Corollary 16. Let F ∈ C (R ;R ) be a Lipschitz function with F(0) = 0. Let
K be a nonempty closed convex cone. Let us denote by J F (0) the Jacobian
matrix of F at 0, that is,
⎛ ⎞
∂F 1 ∂F 1 ∂F 1
(0) ... (0) ... (0)
∂x 1 ∂x j ∂x n
⎜ ⎟
⎜ . . . . ⎟
. . . .
⎜ . . . . ⎟
⎜ ⎟
⎜ ⎟
J F (0) = ⎜ ∂F i (0) ... ∂F i (0) ... ∂F i (0) ⎟ .
⎜ ∂x 1 ∂x j ∂x n ⎟
⎜ ⎟
. . . .
⎜ . . . . ⎟
⎜ . . . . ⎟
⎝ ⎠
(0) ... (0) ... (0)
∂F n ∂F n ∂F n
∂x 1 ∂x j ∂x n
Suppose that there exists a positive definite matrix G ∈ R n×n such that
T
(∀x ∈ K,x = 0) :
J F (0)x,[G + G ]x > 0
and
T
(∀x ∈ K) :[I −[G + G ]]x ∈ K.
Then the trivial stationary solution of (5.37)–(5.40) with ϕ = K is (a) isolated
in S(F,ψ K ) and (b) asymptotically stable.
Proof. By Taylor’s formula we may write
F(x) = Ax + F 1 (x),
where A = J F (0), and F 1 satisfies

||F 1 (x)||
lim = 0. (5.62)
||x||→0 ||x||
Moreover, F 1 is Lipschitz continuous since F 1 (·) = F(·)−A, and F is assumed
to be Lipschitz continuous.
Our aim is to verify that all conditions of Corollary 14 are satisfied with
1
n
V ∈ C (R ;R) defined by
1 T
V(x) =
[G + G ]x,x .
2
Denoting by λ 1 > 0 the first eigenvalue of the symmetric positive definite matrix
T
G + G , we obtain
n 2
(∀x ∈ R ) : V(x) =
Gx,x ≥ λ 1 ||x|| .
It is also clear that V(0) = 0. We have
T
n
(∀x ∈ R ) :∇V(x) =[G + G ]x