Articles - Mathematics and its Applications

Robust stability of discrete-time linear stochastic systems 153inequality in (32) one obtains the inequality from the statement with ρ =N∑νλ max Z(i) + ρ 21 . Thus the proof is complete.ε 2i=1If X τ (t), 0 ≤ t ≤ τ + 1 is the solution of the problem with given finalvalue (26)-(28) we define K(t) = (K(t, 1), ..., K(t, N)) byK(t, i) = X τ (τ + 1 − t, i). (33)We see that K(0, i) = X τ (τ + 1, i) = 0, 1 ≤ i ≤ N. Also, by direct calculationone obtains that K = {K(t)} t≥0 solves the following forward nonlinearequation on S N n :K(t + 1, i) = Π 1i K(t) + C T (i)C(i) − (Π 2i K(t) + C T (i)D(i))(Π 3i K(t) (34)+D T (i)D(i) − γ 2 I mv ) −1 (Π 2i K(t) + C T (i)D(i)) T .Let us denote K 0 (t) = (K 0 (t, 1), ..., K 0 (t, N)) the solution of (34) with giveninitial value K 0 (0, i) = 0, 1 ≤ i ≤ N.Several properties of the solution K 0 (t) are summarized in the next result:Proposition 3.3 Assume: a) the zero state equilibrium of (6) is ESMS.b) ‖T ‖ < γ.Then the solution K 0 (t) of the forward equation (34) with the given initialvalue K 0 (0, i) = 0 is defined for all t ≥ 0. It has the properties:(i)r∑Bk T (i)E i(K 0 (t))B k (i) + D T (i)D(i) − γ 2 I mv ≤ −ε 0 I mv (35)k=0where ε 0 ∈ (0, γ 2 − ‖T ‖ 2 ).(ii) 0 ≤ K 0 (τ, i) ≤ K 0 (τ + 1, i) ≤ cI n , (∀) t, i ∈ Z + × D, where c > 0 isa constant not depending upon t, i.Proof. Based on (10) we obtain that ‖T τ ‖ ≤ ‖T ‖ < γ for all τ ≥ 1.Therefore, we deduce, via Lemma 3.1, that for any integer τ ≥ 1 the solutionX τ (t) of the problem with given final value (26), (28) is well defined for0 ≤ t ≤ τ + 1 and it verifies (29). Thus we deduce via (33) that K 0 (t) is welldefined for all t ≥ 0. If 0 < ε 0 < γ 2 − ‖T ‖ 2 it follows that ε 0 < γ 2 − ‖T τ ‖ 2for all τ ≥ 1.Hence in (29) we may choose ε 0 independent of τ. Writing (29) for t = 0and taking into account that K 0 (τ, i) = X τ (1, i) we obtain that (i) is fulfilled.Further, from (35) and (34) we deduce that K 0 (t, i) ≥ 0 for all (t, i) ∈ Z + ×D.