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Extra info for Continuous-time Markov jump linear systems

Example text

Suppose that (iii) holds. 38), for any H ∈ HnC , we can √ 1 2 3 4 find Hi ∈ Hn+ C , i = 1, 2, 3, 4, such that H = (H − H ) + −1(H − H ). From the linearity of the semigroup eLt we have from (iii) that, as t → ∞, eLt (H) 4 ≤ 1 e Lt Hi 1 → 0. i=1 Thus, we have that (iiia) holds, and since (iiia) implies (ii), we have that (iii) implies (ii). Therefore, we have that (i), (ii), and (iii) are all equivalent. Using similar reasoning, we can show that (ii) is equivalent to (iv). Suppose now that the operator eLt is positive for every t ∈ R+ .

B) For some Gj > 0 in B(Cn ), j ∈ S, we have Li (G) < 0, i ∈ S. 4 Mean-Square Stability for the Homogeneous Case 49 (c) For any Si > 0 in B(Cn ), i ∈ S, there is a unique G = (G1 , . . , GN ), Gi > 0 in B(Cn ), i ∈ S, such that L(G) + S = 0. 45) Moreover, Gi = ϕˆ −1 −A−1 ϕ(S) ˆ , i i ∈ S. Furthermore, the above results also hold if we replace L by T and A by A∗ , or C by R. Proof Clearly (c) implies (b). Suppose now that (b) holds. 46) where 2 ϕˆ Hn+ = y ∈ CN n ; y = ϕ(Q), ˆ Q ∈ Hn+ . 8 we have that −1 ϕˆj−1 y(t) ˙ = Tj ϕˆ 1−1 y(t) , .

Throughout the book we will consider a homogeneous Markov chain θ = {(θ (t), Ft ); t ∈ R+ } taking values in S as presented in Sect. 5. Let A := (A1 , . . , AN ) ∈ Hn and J := (J1 , . . , JN ) ∈ Hr,n . We deal in this chapter with three types of linear systems with Markov jump parameters. First, we consider the homogeneous system x(t) ˙ = Aθ(t) x(t), t ∈ R+ x(0) = x0 , θ (0) = θ0 , P (θ0 = i) = νi . 2) where {w(t); t ∈ R+ } is any Lr2 (Ω, F, P )-function, which is the usual scenario for the H∞ approach, to be considered in Chap.

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