Method of Lyapunov functions for differential equations with piecewise constant delay

2011-06-15
Akhmet, Marat
ARUĞASLAN ÇİNÇİN, Duygu
Yılmaz, Elanur
We address differential equations with piecewise constant argument of generalized type [5-8] and investigate their stability with the second Lyapunov method. Despite the fact that these equations include delay, stability conditions are merely given in terms of Lyapunov functions; that is, no functionals are used. Several examples, one of which considers the logistic equation, are discussed to illustrate the development of the theory. Some of the results were announced at the 14th International Congress on Computational and Applied Mathematics (ICCAM2009), Antalya, Turkey, in 2009.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

Suggestions

LYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
Akhmet, Marat (2009-10-01)
At the first time, Razumikhin technique is applied for differential equations with piecewise constant argument of generalized type [1, 2]. Sufficient conditions are established for stability, uniform stability and uniform asymptotic stability of the trivial solution of such equations. We also provide appropriate examples to illustrate our results.
Stability analysis of recurrent neural networks with piecewise constant argument of generalized type
Akhmet, Marat; Yılmaz, Elanur (2010-09-01)
In this paper, we apply the method of Lyapunov functions for differential equations with piecewise constant argument of generalized type to a model of recurrent neural networks (RNNs). The model involves both advanced and delayed arguments. Sufficient conditions are obtained for global exponential stability of the equilibrium point. Examples with numerical simulations are presented to illustrate the results.
Inverse problems for a semilinear heat equation with memory
Kaya, Müjdat; Çelebi, Okay; Department of Mathematics (2005)
In this thesis, we study the existence and uniqueness of the solutions of the inverse problems to identify the memory kernel k and the source term h, derived from First, we obtain the structural stability for k, when p=1 and the coefficient p, when g( )= . To identify the memory kernel, we find an operator equation after employing the half Fourier transformation. For the source term identification, we make use of the direct application of the final overdetermination conditions.
Exponential stability of periodic solutions of recurrent neural networks with functional dependence on piecewise constant argument
Akhmet, Marat; Cengiz, Nur (null; 2015-08-25)
Akhmet [1] generalized differential equations with piecewise constant argument by taking any piecewise constant functions as arguments, and recently he introduced functional dependence on piecewise constant argument [2]. These equations play an important role in applications such as neural networks [3]. In this study, we develope a model of recurrent neural network with functional dependence on piecewise constant argument of generalized type given by x 0 (t) = −Ax (t) + Ex (γ (t)) + Bh (xt) + Cg xγ(t) + D...
Stability in cellular neural networks with a piecewise constant argument
Akhmet, Marat; Yılmaz, Elanur (2010-03-01)
In this paper, by using the concept of differential equations with piecewise constant arguments of generalized type [1-4], a model of cellular neural networks (CNNs) [5,6] is developed. The Lyapunov-Razumikhin technique is applied to find sufficient conditions for the uniform asymptotic stability of equilibria. Global exponential stability is investigated by means of Lyapunov functions. An example with numerical simulations is worked out to illustrate the results.
Citation Formats
M. Akhmet, D. ARUĞASLAN ÇİNÇİN, and E. Yılmaz, “Method of Lyapunov functions for differential equations with piecewise constant delay,” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, pp. 4554–4560, 2011, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30340.