“…hj [(A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k)]) < 0, ∀h ∈ K(31) Since β hi ≥ 0, i, h ∈ K, and ∑ i∈K β hi > 0 according toLemma 10 and (31) one can conclude e(k) T P h e(k) > 1 ⇒ ∃i ∈ K such that ∑ j∈K λ hj [(A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k)] < 0, ∀h ∈ K(32)Again according to Lemma 10, since λ hj ≥ 0, j, h ∈ K and∑ j∈K λ hj > 0, Relation (32) implies e(k) T P h e(k) > 1 ⇒ ∃i, j ∈ K such that (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0, ∀h ∈ K (33)Now, according to the definition of the set V in (3), we havee(k) / ∈ V ⇒ ∃h ∈ K such that e(k) T P h e(k) > 1(34)From Relations(33) and(34), one can infere(k) / ∈ V ⇒ ∃i, j ∈ K such that (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0(35)since e(k) / ∈ V, according to Relation (35) and item 2) in the switching Algorithm 9, one can conclude there exist σ(e(k)), i, j ∈ K satisfying the following expression (A σ(e(k)) e(k) + l σ(e(k)) ) T P i (A σ(e(k)) e + l σ(e(k)) ) − e(k) T P σ(e(k)) e(k) ≤(A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k)(36)From Relations(35) and(36) one can conclude that there exist σ(e(k)), i ∈ K such thate(k) / ∈ V ⇒ ∃i, j, σ(e(k)) ∈ K such that (A σ(e(k)) e(k)+l σ(e(k)) ) T P i (A σ(e(k)) e(k)+l σ(e(k)) )−e(k) T P σ(e(k)) e(k) ≤ (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0 (37)or equivalentlye(k) / ∈ V ⇒ ∃i, j ∈ K such that v(e(k + 1)) − v(e(k)) = ∆v(e(k)) ≤ (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0 (38) since v(e(k)) ≥ min i∈K λ min (P i )∥e(k)∥ 2 > 0 when e(k) ̸ = 0 n×1 , then according item (d) of Lemma 6 the continuous, nondecreasing and positive definite scalar function w(∥e(k)∥) can be taken as w(∥e(k)∥) = min i∈K λ min (P i )∥e(k)∥ 2 . Moreover, according to item (c) of Lemma 6 we still need to find a nondecreasing and positive definite function ϕ(∥e(k)∥) such that ∆v(e(k)) ≤ −ϕ(∥e(k)∥) < 0 when e(k) ∈ D − V. In this regard, we define ϕ i,j (e(k)) = e(k) T P j e(k) − (A j e(k) + l j ) T P i (A j e(k) + l j ), (i, j) ∈ S (39) S = {(i, j) ∈ K × K|∆v(e(k)) ≤ −ϕ i,j (e(k)) < 0} (40) According to Relation (38), |S| ≥ 1 and ϕ i,j (e(k)) > 0 when e(k) / ∈ V. Now we define the function ϕ(s) as ϕ(s) = inf s ≤ ∥e(k)∥ The function ϕ(∥e(k∥) is nondecreasing and positive definite when e(k) / ∈ V. Moreover, according to Eq.…”