Stability and estimate of solution to uncertain neutral delay systems

• Published in: Boundary value problems Vol. 2014; no. 1; pp. 1 - 14
• Main Authors: Domoshnitsky, Alexander, Gitman, Michael, Shklyar, Roman
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 12.03.2014; Hindawi Limited; BioMed Central Ltd
• Subjects: A Tribute to Professor Ivan Kiguradze; Analysis; Approximations and Expansions; Boundary value problems; Delay; Difference and Functional Equations; Estimates; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Operators; Ordinary Differential Equations; Partial Differential Equations; Stability; Uniqueness; estimates of solutions; neutral delay systems; Cauchy matrix; method; positivity of the Cauchy matrix
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/1687-2770-2014-55

The coefficients and delays in models describing various processes are usually obtained as a results of measurements and can be obtained only approximately. We deal with the question of how to estimate the influence of ‘mistakes’ in coefficients and delays on solutions’ behavior of the delay differential neutral system x i ′ ( t ) − q i ( t ) x i ′ ( t − θ i ( t ) ) + ∑ j = 1 n ( p i j ( t ) − Δ p i j ( t ) ) x j ( t − τ i j ( t ) − Δ τ i j ( t ) ) = f i ( t ) , i = 1 , … , n , t ∈ [ 0 , ∞ ) . This topic is known in the literature as uncertain systems or systems with interval defined coefficients. The goal of this paper is to obtain stability of uncertain systems and to estimate the difference between solutions of a ‘real’ system with uncertain coefficients and/or delays and corresponding ‘model’ system. We develop the so-called Azbelev W -transform, which is a sort of the right regularization allowing researchers to reduce analysis of boundary value problems to study of systems of functional equations in the space of measurable essentially bounded functions. In corresponding cases estimates of norms of auxiliary linear operators (obtained as a result of W -transform) lead researchers to conclusions about existence, uniqueness, positivity and stability of solutions of given boundary value problems. This method works efficiently in the case when a ‘model’ used in W -transform is ‘close’ to a given ‘real’ system. In this paper we choose, as the ‘models’, systems for which we know estimates of the resolvent Cauchy operators. We demonstrate that systems with positive Cauchy matrices present a class of convenient ‘models’. We use the W -transform and other methods of the general theory of functional differential equations. Positivity of the Cauchy operators is studied and then used in the analysis of stability and estimates of solutions. Results: We propose results about exponential stability of the given system and obtain estimates of difference between the solution of this uncertain system and the ‘model’ system x i ′ ( t ) − q i ( t ) x i ′ ( t − θ i ( t ) ) + ∑ j = 1 n p i j ( t ) x j ( t − τ i j ( t ) ) = f i ( t ) , i = 1 , … , n , t ∈ [ 0 , ∞ ) . New tests of stability and in the future of existence and uniqueness of boundary value problems for neutral delay systems can be obtained on the basis of this technique.