Page 62 - Textos de Matemática Vol. 36

52 Chapter 2. Global aspects As a first result we obtain (see also [FJS97], [FJS00]):
Theorem 2.5.2. If for some a, γ > 0, a positive definite matrix C ∈ Rm×m and every t ≤ t0 there holds
−a(M (t)+MT(t)) MT(t)C−aM (t) η
(ηT,ηT) 11 11 21 12 1 (2.47)
1 2 CM (t)−aMT (t) CM (t)+MT (t)C η 21 12 22 22 2
≥ γ(|η1|2 + |η2|2)
for all η1 ∈ Rn,η2 ∈ Rm, then the set ||W||C < √a is negative invariant for
solutions W of RDE, where
||W||2C =max{σ(WTCW)}.
Theorem 2.5.2 allows us to include several parameters in order to obtain reasonable results. These parameters should be adapted somehow to the problem under consideration. It turns out to be an advantage of this approach that it applies also to non-linearizable differential equations as they will be attacked by another method in Theorem 2.5.10 of this section.
Proof. Let us consider the symmetric, positive semi-definite and piecewise con- tinuously differentiable (n × n)-matrix-function K (t) := W T (t)C W (t). Notice that the C-operator norm of W(t) is equal to the square root of the maximal eigenvalue of W T (t)C W (t), which is denoted by λ(t). Assume now λ(t) < a and x(t) ∈ Rn is a corresponding unit eigenvector, hence
K (t)x(t) = W T (t)C W (t)x(t) = λ(t)x(t), xT (t)x(t) = 1. λ(t) = xT (t)W T (t)C W (t)x(t)
Using
we infer from Theorem 2.5.1
λ˙ = xT (t)K˙ (t)x(t)
= 2xT (t)WT (t)CW˙ (t)x(t).
Inserting RDE (and suppressing time parameters) yields
1λ˙ = xT WT C(M21 + M22W − WM11 − WM12W)x 2
= xT WT CM21x + xT WT CM22Wx − λxT M11x − λxT M12Wx. Since this equality depends only on the symmetric parts of the matrices involved,
we may rewrite it in symmetric form,
λ˙ = xTWT(CM −λMT )x+xTWT(CM +MT C)Wx
21 12 22
22
− λxT(M +MT)x+xT(MTC−λM )Wx. 11 11 21 12