Global Stability Analysis for Periodic Solution in Discontinuous Neural Networks with Nonlinear Growth Activations
© Y.Li and H.Wu. 2009
Received: 30 December 2008
Accepted: 18 March 2009
Published: 31 March 2009
This paper considers a new class of additive neural networks where the neuron activations are modelled by discontinuous functions with nonlinear growth. By Leray-Schauder alternative theorem in differential inclusion theory, matrix theory, and generalized Lyapunov approach, a general result is derived which ensures the existence and global asymptotical stability of a unique periodic solution for such neural networks. The obtained results can be applied to neural networks with a broad range of activation functions assuming neither boundedness nor monotonicity, and also show that Forti's conjecture for discontinuous neural networks with nonlinear growth activations is true.
The stability of neural networks, which includes the stability of periodic solution and the stability of equilibrium point, has been extensively studied by many authors so far; see, for example, [1–15]. In [1–4], the authors investigated the stability of periodic solutions of neural networks with or without time delays, where the assumptions on neuron activation functions include Lipschitz conditions, bounded and/or monotonic increasing property. Recently, in [13–15], the authors discussed global stability of the equilibrium points for the neural networks with discontinuous neuron activations. Particularly, in , Forti conjectures that all solutions of neural networks with discontinuous neuron activations converge to an asymptotically stable limit cycle whenever the neuron inputs are periodic functions. As far as we know, there are only works of Wu in [5, 7] and Papini and Taddei in  dealing with this conjecture. However, the activation functions are required to be monotonic in [5, 7, 9] and to be bounded in [5, 7].
In this paper, without assumptions of the boundedness and the monotonicity of the activation functions, by the Leray-Schauder alternative theorem in differential inclusion theory and some new analysis techniques, we study the existence of periodic solution for discontinuous neural networks with nonlinear growth activations. By constructing suitable Lyapunov functions we give a general condition on the global asymptotical stability of periodic solution. The results obtained in this paper show that Forti's conjecture in  for discontinuous neural networks with nonlinear growth activations is true.
For later discussion, we introduce the following notations.
Let , where the prime means the transpose. By (resp., ) we mean that (resp., ) for all . denotes the Euclidean norm of . denotes the inner product. denotes 2-norm of matrix , that is, , where denotes the spectral radius of .
The rest of this paper is organized as follows. Section 2 develops a discontinuous neural network model with nonlinear growth activations, and some preliminaries also are given. Section 3 presents the proof on the existence of periodic solution. Section 4 discusses global asymptotical stability of the neural network. Illustrative examples are provided to show the effectiveness of the obtained results in Section 5.
2. Model Description and Preliminaries
where is the vector of neuron states at time ; is an matrix representing the neuron inhibition; is an neuron interconnection matrix; , , represents the neuron input-output activation and is the continuous -periodic vector function denoting neuron inputs.
Throughout the paper, we assume that
Under the assumption , is undefined at the points where is discontinuous. Equation (2.1) is a differential equation with a discontinuous right-hand side. For (2.1), we adopt the following definition of the solution in the sense of Filippov  in this paper.
If is (i) regular in ; (ii) positive definite, that is, for , and ; (iii) radially unbounded, that is, as , then is said to be C-regular.
Lemma 2.4 (Chain Rule ).
3. Existence of Periodic Solution
By the assumption , it is easy to get that : is an upper semicontinuous set-valued map with nonempty, compact, and convex values. Hence, by Definition 2.1, the local existence of a solution for (2.1) on , , with , is obvious .
is a bounded subset of . Since is a bounded linear operator, is a bounded subset of . Noting that is compactly embedded in , is a relatively compact subset of . Hence by Proposition 3.3, is the upper semicontinuous set-valued map which maps bounded sets into relatively compact sets.
By the definition of , . Moreover, by Definition 2.1 and (3.29), is a solution of the boundary-value problem (2.5), that is, the neural network (2.1) has an -periodic solution. The proof is completed.
4. Global Asymptotical Stability of Periodic Solution
- (2)In , Forti et al. considered delayed neural networks modelled by the differential equation
has a unique periodic solution and all solutions converge to the asymptotically stable limit cycle when is a periodic function. When , the neural network (4.11) changes as the neural network (2.1) without delays. Thus, without assumptions of the boundedness and the monotonicity of the activation functions, Theorem 4.1 obtained in this paper shows that Forti's conjecture for discontinuous neural networks with nonlinear growth activations and without delays is true.
5. Illustrative Example
The authors are extremely grateful to anonymous reviewers for their valuable comments and suggestions, which help to enrich the content and improve the presentation of this paper. This work is supported by the National Science Foundation of China (60772079) and the National 863 Plans Projects (2006AA04z212).
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