Articles - Mathematics and its Applications

Robust stability of discrete-time linear stochastic systems 159To prove the converse implication, (ii) → (i) we remark that if (ii) is fulfilledthen the (1;1) block of (48) is negative definite. Thus we obtained thatthere exists X = (X(1), ..., X(N)) ∈ Sn N with X(i) > 0, such that X(i) >r∑A T k (i)E i(X)A k (i), 1 ≤ i ≤ N. Applying Corollary 4.8 in [8] we deducek=0that the zero state equilibrium of the system (6) is ESMS. Further, applyingCorollary 3.3 in[9] for X(t, i) = X(i), 0 ≤ t ≤ τ, τ ≥ 1, 1 ≤ i ≤ N, andtaking the limit for τ → ∞ we have:˜J γ (∞; 0, v) =∞∑( x(t, 0, v)E[v(t)t=0) T ( x(t, 0, v)Q(X, η t )v(t))] (53)where Q(X, i) is the left hand side of (48). If X = (X(1), ..., X(N)) verifies(48) then for ε > 0 small enough we haveQ(X, i) ≤ −ε 2 I n+mv , 1 ≤ 1 ≤ N. (54)Combining (53) and (54) we deduceor equivalently∑˜J ∞ γ (∞; 0, v) ≤ −ε 2 E[|x(t, 0, v)| 2 ] + E[|v(t)| 2 ]t=0∑∞˜J˜γ (∞; 0, v) ≤ −ε 2 E[|x(t, 0, v)| 2 ] < 0t=0for all v ∈ l 2˜H{0, ∞; R mv } where ˜γ = (γ 2 − ε 2 ) 1 2 .The last inequality may be written:‖T v‖ 2 l 2˜H{0,∞;R nz } ≤ ˜γ2 ‖v‖ 2 l 2˜H{0,∞;R mv }for all v ∈ l 2˜H{0, ∞; R mv }. This leads to ‖T ‖ 2 ≤ γ 2 − ε 2 and thus theimplication (ii) → (i) is proved.To prove the equivalence (ii) ↔ (iii) let us consider the DTSGRE:whereX = Π 1 X + ˆM − (Π 2 X + ˆL)(Π 3 X + ˆR) −1 (Π 2 X + ˆL) T (55)ˆM(i) = −C T (i)C(i), ˆL(i) = −C T (i)D(i), ˆR(i) = γ 2 I mv − D T (i)D(i),