Delay-Dependent Dynamics of Switched Cohen-Grossberg Neural Networks with Mixed Delays

and Applied Analysis 3 As is well known, compared with linear matrix inequalities (LMIs) result, algebraic result is more conservative, and criteria in terms of LMI can be easily checked by using the powerful Matlab LMI toolbox. This motivates us to investigate the problems of the uniformly ultimate boundedness and the existence of an attractor for switchedCGNN in this paper. The illustrative examples are given to demonstrate the validity of the theoretical results. The paper is organized as follows. In Section 2, preliminaries and problem formulation are introduced. Section 3 gives the sufficient conditions of uniformly ultimate boundedness (UUB) and the existence of an attractor for switched CGNN. It is the main result of this paper. In Section 4, an example is given to illustrate the effectiveness of the proposed approach. The conclusions are summarized in Section 5. 2. Problem Formulation Throughout this paper, we use the following notations. The superscript “T” stands for matrix transposition; R denotes the n-dimensional Euclidean space; the notation P > 0 means that P is real symmetric and positive definite; I and O represent the identity matrix and a zero matrix; diag{⋅ ⋅ ⋅ } stands for a block-diagonal matrix; λmin(P) denotes the minimum eigenvalue of symmetric matrix P; in symmetric block matrices or long matrix expressions, “∗” is used to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. Consider the following Cohen-Grossberg neural network model with mixed delays (discrete delays and distributed delays): ?̇? (t) = − ?̂? (x (t)) × [ ̂ β (x (t)) − AF (x (t)) − BF (x (t − τ)) −C∫ t t−h F (x (s)) ds − J] ≜ −?̂? (x (t))H (t) , (6) where H(t) = ̂ β (x (t)) − AF (x (t)) − BF (x (t − τ)) − C∫ t t−h F (x (s)) − J. (7) The discrete delays and distributed delays are bounded as follows: 0 ≤ τ i , τ ∗ = max 1≤i≤n {τ i } ; 0 ≤ h i , h ∗ = max 1≤i≤n {h i } ; δ = max {τ∗, h∗} , (8) where τ, h, δ are scalars. As usual, the initial conditions associated with system (6) are given in the form x (t) = φ (t) , −δ ≤ t ≤ 0, (9) where φ(t) is a differentiable vector-valued function. Throughout this paper, we make the following assumptions. (H 1 ) For any j ∈ {1, 2, . . . , n}, there exist constants l j and