The first analysis is false, because no round square exists.
Definition 2.1 Given α > 0. The zero solution of a closed-loop system (2.7) is α-stabilizable in the mean square if there exists a positive number N > 0 such that every solution x ( t, ϕ ) satisfies the following condition: E { ∥ x ( t, ϕ ) ∥ } ≤ E { N e − α t ∥ ϕ ∥ }, ∀ t ≥ 0. Definition 2.2 Consider the control system (2.1).
The solution of system (2.2) with conditions (2.3) and (2.4) is said to be exponentially stable in mean square, if there exist two positive constants and such that (2.6).
The DSNN (1) is said to be robustly exponentially stable in the mean square if there exist constants α > 0 and μ ∈ ( 0, 1 ) such that every solution of the DSNN (1) satisfies that E { ∥ x ( k ) ∥ 2 } ≤ α μ k max − τ M ≤ i ≤ 0 E { ∥ x ( i ) ∥ 2 }, ∀ k ∈ N + (16).
Under Assumptions -, the discrete-time complex networks with randomly coupling and distributed time-varying delays are synchronized in mean square, if there exist positive definite matrices and diagonal matrices with appropriate dimensions such that the following matrix inequalities hold: (3..3).
Under assumptions -, the discrete-time complex networks with distributed time-varying delays is synchronized in mean square, if there exist positive definite matrices and diagonal matrices with appropriate dimensions such that the following matrix inequalities hold: (3..22).
Under Assumptions -, the discrete-time complex networks with randomly distributed time-varying delays are synchronized in mean square, if there exist positive definite matrices and diagonal matrices with appropriate dimensions such that the following matrix inequalities hold: (3..21).
The stochastic switched system with convex polytopic uncertainties (2.1) is robustly stable in the mean square if there exist symmetric matrices P j > 0, Q j > 0, R ≥ 0, j = 1, 2..., N and matrix S j, j = 1, 2..., N satisfying the following conditions.
Then under Assumptions 2.1 and 2.3, the recurrent neural network (2.9) with random delay and Markovian switching is exponentially stable in the mean square if there exist symmetric positive matrices and positive diagonal matrices such that the following LMIs holds (3.3).
Corollary 3.2 Assume p > 1. CGNNs (3.14) is global exponential robust stability in the mean square if there exist a positive scalar β > 0 and positive definite diagonal matrices P i ( i ∈ S ), L 1, L 2 and Q such that the following LMI conditions hold: ( Ω ˜ i 1 Ω ˜ i 2 ∗ Ω i 3 ) < 0, ∀ i ∈ S, where.
