- Open Access
Stability and estimate of solution to uncertain neutral delay systems
© Domoshnitsky et al.; licensee Springer. 2014
- Received: 16 December 2013
- Accepted: 27 February 2014
- Published: 12 March 2014
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 , , . 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 , , . 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.
- neutral delay systems
- Cauchy matrix
- positivity of the Cauchy matrix
- estimates of solutions
In systems of differential equations, modeling real processes, the coefficients and delays are usually known only approximately. For example, models in economics are obtained as a result of corresponding long time observation and regression procedures (see, for example, ). The coefficients and delays in model describing various processes in physics are usually obtained as a results of measurements and also can be obtained only approximately . The same situation can be noted in models of population dynamics . A natural question is to estimate the influence of ‘mistakes’ in coefficients and delays on solutions’ behavior of delay differential systems. This topic is known in the literature as uncertain systems or systems with interval defined coefficients.
Stability of systems with uncertain coefficients was studied in the papers [6–8, 11–14, 16–18, 23]. Stabilization of uncertain systems with unknown delay was studied in [4, 19, 22]. Stability of systems with uncertain delay is studied in [12, 15, 20, 21]. Stability of neutral uncertain systems was considered in [10, 16, 17]. The basic technique of these publications is the use of Lyapunov-Krasovskii functionals.
We assume here that , where is the space of essentially bounded functions on , and are measurable essentially bounded functions, . Concerning the operators we assume that are linear continuous Volterra operators and the spectral radius of the operator is less than 1 for . Our main purpose is to obtain conditions on the smallness of and such that the exponential stability of (2) is inherited by (1). Our second aim is to estimate the modulus of the difference of solutions to systems (1) and (2).
This equation is a particular case of (1).
where are essentially bounded measurable functions, are measurable functions for , and are summable with respect to s and measurable essentially bounded with respect to t for . All linear combinations of the operators (5) and (6) and their superpositions are also allowed.
is a sufficient condition that the spectral radii of the operators is less than 1. Below we assume that this inequality is fulfilled.
Various results on existence and uniqueness of solutions to boundary value problems for (4) and its stability were obtained in , where also the basic results about the representation of solutions were presented. Note also in this connection the papers [28–31], where results on nonoscillation and positivity of Green’s functions for neutral functional differential equations were obtained.
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 with fixed coefficients and delays. Instead of the traditional Lyapunov functionals technique, we propose an approach based on the idea of Azbelev’s W-transform presented in the book . This transform is a corresponding right regularization allowing researchers to reduce the analysis of boundary value problems to the 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’ uncertain system. In this paper we choose as a ‘model’, system for which we know solutions or estimates of the resolvent (Green’s) operators. In this paper we demonstrate that systems with positive Cauchy operators [32, 33] represent a class of very convenient ‘models’.
It should be noted that, although results are below formulated for the systems with neutral terms in the form (5), we can, using the approach of the papers [30, 31], obtain corresponding results with linear combinations and superpositions of the operators defined by (5) and (6).
Here , and is the space of essentially bounded functions on . are measurable essentially bounded functions for .
where is the Cauchy matrix of system (8) and is the fundamental matrix of system (8) satisfying the condition .
where and if .
Estimates of is the key problem in our approach. Now describe the cases in which the Cauchy matrix and the fundamental matrix of system (8) can be estimated.
, , , then all elements of the Cauchy matrix and all elements of the fundamental matrix of system (14) are nonnegative.
we can write (19) as and consequently every component of solution vector are nonnegative for nonnegative vectors and . As ω tends to infinity we complete the proof of Theorem 1. □
Definition 1 The Cauchy matrix of system (14) and its fundamental matrix satisfies the exponential estimates, if there exist real numbers such that and for , .
where be satisfied.
Proof Let us extend the coefficients and delays in system (14) on the interval , where as follows: , for , , for , , and consider system (14) on , calling it the extent system. The constant vector-function satisfies the inequality , , . All entries of the Cauchy matrix of the extent system are nonnegative according to Theorem 1. The Cauchy matrices of system (14) and the extent system coincide in . Using nonnegativity of all entries of the Cauchy matrix of system (14) we get the inequalities for , that implies inequalities (23). For every bounded right hand side, the solution of system (14) is bounded. From this the exponential estimates of entries of the Cauchy matrix and the fundamental matrix follow (see , Paragraph 5.3). □
Analogously we can obtain the following assertion.
All elements of the Cauchy matrix of system (27) are nonnegative. From the fact that satisfies the exponential estimate we obtain for . □
where the constants and are defined by inequalities (31) and (22) (where there are instead of ), respectively.
is bounded for every bounded right hand sides . Comparing the operators K and we see that = = = and consequently the solution vector is bounded for every bounded . These inequalities prove the estimates (32) and (33). According to Theorem 3.5 , the Cauchy and fundamental matrices satisfy the exponential estimates. □
for sufficiently large t.
In order to prove Theorem 6 we can note that in the case we get . A reference to (9) completes the proof.
where , is the Cauchy matrix of system (42) and is its fundamental matrix. Let us assume for simplicity that and for . If this is not fulfilled we can extend coefficients on the interval and consider the system on as in the Section 2.
where and the components of the vector satisfies the system of inequalities (31).
for sufficiently large t.
where , , are defined by inequalities (31).
for sufficiently large t.
This paper was prepared as an open auxiliary theoretical result in the frame of the project ‘Exploitation of a synergetic model for development of business of innovation type’ on the topic No. 2013/276-C ‘Development of a model of operation of innovation business as dynamic model with memory effect’, supported by Perm National Research Polytechnic University with financial support of Ministry of Science and Education of Russian Federation (agreement No. 02.G25.31.0068 of 23.05.2013).
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