Global Synchronization of Neutral-Type Stochastic Delayed Complex Networks

and Applied Analysis 3 It is still very difficult to calculate expectations of these stochastic cross terms up to now. The results in 28–31 resorted to bounding techniques, which obviously can bring the conservatism. Some papers such as 32–34 considered that expectations of these stochastic cross terms are all equal to zero. However, these results are not given by strict mathematical proofs, and we can find examples to illustrate that expectations of some stochastic cross terms are not equal to zero in Remark 3.3. Therefore, in order to obtain the delay-dependent synchronization criterion with less conservatism for neutral-type stochastic delayed complex networks, there is a strong need to investigate the expectations of stochastic cross terms containing the Itô integral firstly. Motivated by the discussion mentioned above, this paper investigates the delaydependent synchronization problem for neutral-type stochastic delayed complex networks. The main contributions of this paper are summarized as follows. 1 Expectations of stochastic cross terms containing the Itô integral are investigated by stochastic analysis techniques in Lemma 3.1 and Corollary 3.2. We prove that the expectation of x t − h K ∫ t t−h μ s, xs dw s is equal to zero and expectations of other stochastic cross terms are not. 2 Based on this conclusion, this paper establishes a delay-dependent synchronization criterion that guarantees the globally asymptotic synchronization of neural-type stochastic delayed complex networks. In the derivation process, the mathematical development avoids bounding stochastic cross terms. Thus, thismethod leads to a criterionwith less conservatism. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach. Notation. Throughout the paper, unless otherwise specified, we will employ the following notation. Let Ω,F, {Ft}t≥0,P be a complete probability space with a natural filtration {Ft}t≥0, and let E · be the expectation operator with respect to the probability measure. If A is a vector or matrix, its transpose is denoted by A . If P is a square matrix, then P > 0 P < 0 means that it is a symmetric positive negative definite matrix of appropriate dimensions while P ≥ 0 P ≤ 0 is a symmetric positive negative semidefinite matrix. I stands for the identity matrix of appropriate dimensions. Denote by λmin · the minimum eigenvalue of a given matrix. Let | · | denote the Euclidean norm of a vector and its induced norm of a matrix. Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions. L2 Ω denotes the space of all random variables X with E|X|2 < ∞, it is a Banach space with norm ‖X‖2 E|X|2 . Let h > 0 and C −h, 0 ;Rn denote the family of all continuous Rn-valued functions φ on −h, 0 with the norm ‖φ‖ sup{|φ θ | : −h ≤ θ ≤ 0}. Let LF0 −h, 0 ;Rn be the family of all F0-measurable C −h, 0 ;Rn -valued random variables φ such that E ‖φ‖2 < ∞, and let L2 a, b ;Rn be the family of all Rn-valued Ftadapted processes {f t }a≤t≤b such that ∫b a |f t |2dt < ∞ a.s. Let M2 a, b ;Rn be the family of processes {f t }a≤t≤b in L2 a, b ;Rn such that E ∫b a |f t |2dt < ∞, and M2 a, b is the 1-dimensional case of M2 a, b ;Rn . 2. Problem Formulation and Preliminaries In this paper, we consider the following neutral-type stochastic delayed complex networks consisting of N identical nodes: 4 Abstract and Applied Analysis d xi t −Dxi t − h ⎡ ⎣Axi t Bf xi t Cf xi t − h N ∑