Halanay-type Inequality Research Articles

Global exponential stability of hybrid bidirectional associative memory neural networks with discrete delays.

In this paper, the dynamical characteristics of hybrid bidirectional associative memory neural networks with constant transmission delays are investigated. Without assuming symmetry of synaptic connection weights and monotonicity and differentiability of activation functions, Halanay-type inequalities (which are different from the approach of constructing Lyapunov functionals) are employed to derive the delay-independent sufficient conditions under which the networks converge exponentially to the equilibria associated with temporally uniform external inputs. Our results are less conservative and restrictive than previously known results.

Extreme stability and almost periodicity in continuous and discrete neuronal models with finite delays

We consider the dynamical characteristics of a continuous-time isolated Hopfield-type neuron subjected to an almost periodic external stimulus. The model neuron is assumed to be dissipative having finite time delays in the process of encoding the external input stimulus and recalling the encoded pattern associated with the external stimulus. By using non-autonomous Halanay-type inequalities we obtain sufficient conditions for the hetero-associative stable encoding of temporally non-uniform stimuli. A brief study of a discrete-time model derived from the continuous-time system is given. It is shown that the discrete-time model preserves the stability conditions of the continuous-time system.

Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks

We formulate discrete-time analogues of integrodifferential equations modelling bidirectional neural networks studied by Gopalsamy and He. The discrete-time analogues are considered to be numerical discretizations of the continuous-time networks and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the equilibria of the continuous-time networks. By constructing a Lyapunov-type sequence, we obtain easily verifiable sufficient conditions under which every solution of the discrete-time analogue converges exponentially to the unique equilibrium. The sufficient conditions are identical to those obtained by Gopalsamy and He for the uniqueness and global asymptotic stability of the equilibrium of the continuous-time network. By constructing discrete-time versions of Halanay-type inequalities, we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete-time analogue. The latter sufficient conditions have not been obtained in the literature of continuous-time bidirectional neural networks. Several computer simulations are provided to illustrate the advantages of our discrete-time analogue in numerically simulating the continuous-time network with distributed delays over finite intervals.

Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks

Convergence dynamics of continuous-time bidirectional neural networks with constant transmission delays are studied. Without assuming the symmetry of synaptic connection weights and the monotonicity and differentiability of activation functions, Lyapunov functionals and Halanay-type inequalities are constructed and employed to derive delay independent sufficient conditions under which the continuous-time networks converge exponentially to the equilibria associated with temporally uniform external inputs to the networks. Discrete-time analogues of the continuous-time networks are formulated and we study their dynamical characteristics. It is shown that the convergence dynamics of the continuous-time networks are preserved by the discrete-time analogues without any restriction on the discretization step-size. Several examples are given to illustrate the advantages of the discrete-time analogues in numerically simulating the continuous-time networks.

Exponential stability of continuous-time and discrete-time cellular neural networks with delays

Convergence characteristics of continuous-time cellular neural networks with discrete delays are studied. By using Lyapunov functionals, we obtain delay independent sufficient conditions for the networks to converge exponentially toward the equilibria associated with the constant input sources. Halanay-type inequalities are employed to obtain sufficient conditions for the networks to be globally exponentially stable. It is shown that the estimates obtained from the Halanay-type inequalities improve the estimates obtained from the Lyapunov methods. Discrete-time analogues of the continuous-time cellular neural networks are formulated and studied. It is shown that the convergence characteristics of the continuous-time systems are preserved by the discrete-time analogues without any restriction imposed on the uniform discretization step size.

Continuous and discrete Halanay-type inequalities

We consider continuous time and discrete time Halanay-type inequalities for nonautonomous scalar systems with discrete and distributed delays. The results obtained generalise the existing results of Halanay and improve certain results of Baker and Tang. Furthermore, it is shown that the discrete time inequalities which are analogues of continuous time inequalities preserve the stability conditions corresponding to the continuous time Halanay-type inequalities.