Global Convergence for Cohen-Grossberg Neural Networks with Discontinuous Activation Functions

and Applied Analysis 3 not imply convergence of the outputs. In addition, in the practical applications, the result of the neural computation is usually the steady-state neuron output, rather than the asymptotic value of the state. Hence, in this paper, we will study global convergence of CGNNs with discontinuous activation functions, where the interconnection matrix is assumed to be anMmatrix orH-matrix. Firstly, using the property ofM-matrix and a generalized Lyapunov-like approach, we prove the uniqueness of state solutions and corresponding output solutions, and equilibrium point and corresponding output equilibrium point for the considered neural networks. Then, global exponential stability of unique equilibrium point is discussed and exponential convergence rate is estimated. Also, by contraction mapping principle, the globally exponential stability of limit cycle is given. Finally, we use a numerical example to illustrate the effectiveness of the theoretical results. The rest of the paper is organized as follows. In Section 2, model description and preliminaries are presented. The main results are stated in Section 3. In Section 4, an example is given to show the validity of the obtained results. Finally, in Section 5, the conclusions are drawn. Notations. Throughout the paper, the transpose of and inverse of any square matrix A are expressed as A and A−1, respectively. For α α1, . . . , αn T ∈ R, α > 0 denotes αi > 0 for i 1, 2, . . . , n. For x, y ∈ R, 〈x, y〉 ni 1 xiyi denotes the scalar product of x, y. 2. Model Description and Preliminaries In this paper, we consider the CGNNs 1.1 with discontinuous right-hand side. The compact form of model 1.1 is expressed as follows: du t dt −A u t [Bu t −Wf u t − I], 2.1 where u t u1 t , u2 t , . . . , un t T ∈ R,A u t diag a1 u1 t , a2 u2 t , . . . , an un t , B diag b1, b2, . . . , bn ,W wij n×n, I I1, I2, . . . , In T ∈ R, and f u t f1 u1 t , . . . , fn un t T . Throughout this paper, we make the following assumptions. A1 The function ai r is continuous, 0 < ǎi ≤ ai r ≤ âi for all r ∈ R, where ǎi and âi are positive constants, i 1, 2, . . . , n. A2 The matrix W wij n×n is nonsingular, that is, detW / 0. Moreover, f f1, . . . , fn is supposed to belong to the following class of discontinuous functions. Definition 2.1 see 18 Function Class FD . f x ∈ FD if and only if for i 1, 2, . . . , n, the following conditions hold: i fi is bounded on R; ii fi is piecewise continuous onR; namely, fi is continuous onR except a countable set of points of discontinuity pki, where there exist finite right and left limits fi p ki and fi p− ki , respectively; moreover, fi has finite discontinuous points in any compact interval of R; iii fi is nondecreasing on R. 4 Abstract and Applied Analysis Denote the set of discontinuous points of fi, i 1, 2, . . . , n, by Si { pki ∈ R : fi ( p ki ) > fi ( p− ki )} . 2.2 Sometimes, f f1, . . . , fn is supposed to belong to the next class of discontinuous functions, which is included in FD. Definition 2.2 see 18 Function Class FDL . f x ∈ FDL if and only if f x ∈ FD and for i 1, 2, . . . , n, fi is locally Lipschitz with Lipschitz constant li xi ≥ 0 for all xi ∈ R \ Si. Furthermore, we have li xi ≥ Li < ∞ for all xi ∈ R \ Si. For model 1.1 or model 2.1 with discontinuous right-hand side, a solution of Cauchy problem need to be explained. In this paper, solutions in the sense of Filippov 21 are considered whose definition will be given next. Let K f u K f1 u1 , K f2 u2 , . . . , K fn un T , where K fi ui fi ui , fi u i . Definition 2.3. A function u t , t ∈ t1, t2 , where t1 < t2 ≤ ∞ is a solution in the sense of Filippov of 2.1 in the interval t1, t2 , with initial condition u t1 u0 ∈ R, if u t is absolutely continuous on t1, t2 and u t1 u0, and for almost all a.a. t ∈ t1, t2 we have du t dt ∈ −A u t [Bu t −WK[f u t ] − I]. 2.3 Let u t , t ∈ t1, t2 , be a solution of model 2.1 . For a.a. t ∈ t1, t2 , one can obtain du t dt −A u t [Bu t −Wγ t − I], 2.4 where γ t W−1 ( A−1 u u̇ t Bu t − I ) ∈ K[f u t ] 2.5 is the output solution of model 2.1 corresponding to u t . And γ t is a boundedmeasurable function 11 , which is uniquely defined by the state solution u t for a.a. t ∈ t1, t2 . Definition 2.4 equilibrium point . u∗ ∈ R is an equilibrium point of model 2.1 if and only if the following algebraic inclusion is satisfied: 0 ∈ A u∗ (−Bu∗ WK[f u∗ ] I). 2.6 Definition 2.5 output equilibrium point . Let u∗ be an equilibrium point of model 1.1 ; γ∗ W−1 Bu∗ − I ∈ K[f u∗ ] 2.7 is the output equilibrium point of model 2.1 corresponding to u∗. In this paper, we also need the following definitions and lemma. Abstract and Applied Analysis 5 Definition 2.6 see 18 . Let Q ∈ Rn×n be a square matrix. Matrix Q is said to be anM-matrix if and only if Qij ≤ 0 for each i / j, and all successive principal minors of Q are positive. Definition 2.7 see 18 . Let Q ∈ Rn×n be a square matrix. Matrix Q is said to be an H-matrix if and only if the comparison matrix of Q, which is defined byand Applied Analysis 5 Definition 2.6 see 18 . Let Q ∈ Rn×n be a square matrix. Matrix Q is said to be anM-matrix if and only if Qij ≤ 0 for each i / j, and all successive principal minors of Q are positive. Definition 2.7 see 18 . Let Q ∈ Rn×n be a square matrix. Matrix Q is said to be an H-matrix if and only if the comparison matrix of Q, which is defined by M Q ij { |Qii|, i j, −∣Qij ∣∣, i / j, 2.8 is an M-matrix. Lemma 2.8 see 18 . Suppose that Q is an M-matrix. Then, there exists a vector ξ > 0 such that Qξ > 0. All results of this paper are under one of the following assumptions: a −W is an M-matrix; b −W is an H-matrix such that Wii < 0. a and b can be applied to cooperative neural networks 22 and cooperativecompetitive neural networks, respectively. From 23 , the result that −W is LDS under a or b can be obtained; hence, all results in 20 hold. So, for any u0 ∈ R, model 2.1 has a bounded absolutely continuous solution u t for t ≥ 0 which satisfies u 0 u0. Meanwhile, there exists an equilibrium point u∗ ∈ R of model 2.1 . If −W is an M-matrix, then, there exists ξ ξ1, . . . , ξn T > 0 such that −W T ξ β > 0. 2.9 If −W is an H-matrix, then, there exists ξ ξ1, . . . , ξn T > 0 such that M −W T ξ β > 0. 2.10 Using the positive vector ξ, we define the distance in R as follows: for any x, y ∈ R, define ∥∥x − y∥ξ n ∑ i 1 ξi ∣xi − yi ∣∣. 2.11 Definition 2.9. The equilibrium point u∗ of 2.1 is said to be globally exponentially stable, if there exist constants α > 0 and M > 0 such that for any solution u t of model 2.1 , we have ‖u t − u‖ξ ≤ M‖u0 − u‖ξ exp{−αt}. 2.12 Also, we can consider the CGNNs with periodic input: du t dt −A u t [Bu t −Wf u t − I t ], 2.13 where I t I1 t , I2 t , . . . , In t T is periodic input vectors with period ω. 6 Abstract and Applied Analysis Definition 2.10. A periodic orbit u∗ t of Cohen-Grossberg networks is said to be globally exponentially stable, if there exist constants α > 0 and M > 0 such that such that for any solution u t of model 2.13 , we have ‖u t − u∗ t ‖ ≤ M∥u0 − u0 ∥∥ ξ exp{−αt}. 2.14 3. Main Results In this section, we shall establish some sufficient conditions to ensure the uniqueness of solutions, equilibrium point, output equilibrium point, and limit cycle as well as the global exponential stability of the state solutions. Because Filippov solution includes set-valued function, in the general case, for a given initial condition, a discontinuous differential equation has multiple solutions starting at it 16 . Next, it will be shown that the uniqueness of solutions of model 2.1 can be obtained under the assumptions A1 and A2 . Theorem 3.1. Under the assumptions (A1) and (A2), if f ∈ FD and −W is an M-matrix or −W is an H-matrix such that Wii < 0, then, for any u0 there is a unique solution u t of model 2.1 with initial condition u 0 u0, which is defined and bounded for all t ≥ 0. Meanwhile, the corresponding output solution γ t of model 2.1 is uniquely defined and bounded for a.a. t ≥ 0. Proof. We only need to prove the uniqueness. Let u t and ũ t , t ≥ 0 are two solutions of model 2.1 with the initial condition u 0 ũ 0 u0. Define