- Open Access
Stability and estimate of solution to uncertain neutral delay systems
Boundary Value Problems volume 2014, Article number: 55 (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.
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.
Consider the system of the neutral functional differential equations
with uncertain coefficients and delays and the model system
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).
The studies of neutral functional differential equations have their own history. Equations in the form
were considered in the well-known books [3, 24, 25] (see also the bibliography therein), where existence and uniqueness of solutions and especially stability and oscillation results for these equations were obtained. There exist problems in applications whose models can be written in the form 
This equation is a particular case of (1).
Let us note here that the operator in (1) can be, for example, of the following forms:
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.
The study of the neutral functional differential equations is essentially based on the questions of the action and estimates of the spectral radii of the operators in the spaces of discontinuous functions, for example, in the spaces of summable or essentially bounded functions. The operator (5) is a key object in this topic. Properties of this operator were studied, for example, in [26, 27]. In order to achieve the action of the operator (5) in the space of essentially bounded functions , we have to assume that for every constant c. Let us suppose everywhere below that this condition is fulfilled. It is known that the spectral radius of the integral operator (6), considered on every finite interval , is equal to 0 (see, for example, ) and the inequality
implies that the spectral radii of the integral operators defined on the semiaxis are less than 1. Concerning the operator (5), we can note that the inequality
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).
2 Stability and estimation of solution of neutral systems with uncertain coefficients
Consider two systems
Here , and is the space of essentially bounded functions on . are measurable essentially bounded functions for .
It is known  that the general solution of (8) has the representation
where is the Cauchy matrix of system (8) and is the fundamental matrix of system (8) satisfying the condition .
Substituting representation (9) into (7), we get
where and if .
We can write (10) in the vector form
where the operator ( is the space of n vector-functions with measurable essentially bounded components) is defined as follows:
Estimating the norm of the operator , we see that
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.
Let us prove several assertions for the neutral system
Theorem 1 Let , ,
, , , then all elements of the Cauchy matrix and all elements of the fundamental matrix of system (14) are nonnegative.
Proof According to the result of , the conditions of Theorem 1 imply the positivity of the Cauchy functions and fundamental solutions of the diagonal equations
Let us consider the system of diagonal equations denoting by its Cauchy and fundamental matrices, respectively. These matrices are the diagonal ones, i.e. and for , , and the elements and coincide with the Cauchy functions and the fundamental solutions of the scalar diagonal equations, respectively. We can write the system
which is equivalent to system (14). On every finite interval the spectral radius of the integral operator
is 0. Denoting
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 , .
Theorem 2 Let the conditions of Theorem 1 be fulfilled for system (14) and there exist for , such that , and the system of inequalities
where be satisfied.
Then the Cauchy matrix of system (14) and its fundamental matrix satisfy the exponential estimates and the integral estimate
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.
Theorem 3 Let the conditions of Theorem 1 be fulfilled for system (14), let there exist a vector such that for , and
Then the Cauchy matrix of system (14) and its fundamental matrix satisfy the exponential estimates and the integral estimate
Theorem 4 Let all coefficients be constants and the conditions of Theorem 1 be fulfilled for system (14). Then the Cauchy matrix of system (14) and its fundamental matrix satisfy the exponential estimates if and only if all components of the solution vector of the algebraic system
Proof Sufficiency follows from Theorem 3. To prove necessity we extend the coefficients and delays in system (14) on the interval , where such that , , , for , . The constant vector-function is now the solution of the system
satisfying the condition , . Using the formula of the solution’s representation (see (9)) on , we obtain
All elements of the Cauchy matrix of system (27) are nonnegative. From the fact that satisfies the exponential estimate we obtain for . □
Remark 1 Under the conditions of Theorem 4, the equalities
Theorem 5 Let , the inequality (15) be fulfilled and let there exist a vector with all positive components such that
Then the Cauchy matrix of system (14) and its fundamental matrix satisfy the exponential estimates and the integral estimates
where the constants and are defined by inequalities (31) and (22) (where there are instead of ), respectively.
Proof Consider the system
We can write the equivalent integral system
is positive. Condition (31) implies according to Theorem 5.6  that its spectral radius . According to Theorem 5.3  we obtain the inequality for the spectral radius of the operator K defined by equality (20) for . The exponential estimates of and imply that
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. □
Theorem 6 Let the conditions of Theorem 5 be fulfilled and the norm of the operator Ω, defined by equality (12), satisfy the inequality , then the Cauchy matrix of system (14) and its fundamental matrix satisfy the exponential estimates and the difference Δx of solution x of system (7) and of system (14) satisfies the inequality
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.
3 Stability and estimation of solution of neutral systems with uncertain delays
Consider the system
Rewrite (39) in the form
It is clear that the solution of (42) has the representation
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.
It is clear that and
From (42) we get . Substituting this into (43), we obtain
where the operator is defined by (5), and after substituting x from (43) we get
Here the operator , is defined as , where every component is defined as follows:
We can write (46) in the vector form
It is clear that
where and the components of the vector satisfies the system of inequalities (31).
Theorem 7 Let the conditions of Theorem 5 be fulfilled and , then the Cauchy matrix of system (39) and its fundamental matrix satisfy the exponential estimates and the difference Δx of solution x of system (39) and of system (14) satisfies the estimate
for sufficiently large t.
4 Stability and estimation of solution of neutral systems with uncertain coefficients and delays
Consider the system
and the auxiliary system
The solution of (52) has the representation
Rewrite (51) in the form
Using the technique of two previous sections, we obtain the system of the following form:
where when , and estimating the norm of the corresponding operator Ω, we get
where , , are defined by inequalities (31).
Theorem 8 Let the conditions of Theorem 5 be fulfilled and , then the Cauchy matrix of system (51) and its fundamental matrix satisfy the exponential estimates and the difference Δx of solution x of system (51) and of system (14) satisfies the inequality
for sufficiently large t.
Azbelev NV, Maksimov VP, Rakhmatullina LF Advanced Series in Mathematical Science and Engineering 3. In Introduction to the Theory of Linear Functional-Differential Equations. World Federation Publishers Company, Atlanta; 1995.
Kolmanovskii V, Myshkis AD: Introduction to Theory and Applications of Functional Differential Equations. Kluwer Academic, Dordrecht; 1999.
Gopalsamy K: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht; 1992.
Cao Y, Sun Y, Cheng C: Delay-dependent robust stabilization of uncertain systems with multiple state delays. IEEE Trans. Autom. Control 1998, 43: 1608-1612. 10.1109/9.728880
Chen XW, Liu B: Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim. Decis. Mak. 2010, 9: 69-81. 10.1007/s10700-010-9073-2
Dong Y, Liu J: Exponential stabilization of uncertain nonlinear time-delay systems. Adv. Differ. Equ. 2012., 2012: Article ID 180 10.1186/1687-1847-2012-180
Dong Y, Mei S, Wang X: Novel stability criteria of nonlinear uncertain systems with time-varying delay. Abstr. Appl. Anal. 2011., 2011: Article ID 969674
Dzhalladova IA, Bastinec J, Diblik J, Khusainov DY: Estimates of exponential stability for solutions of stochastic control systems with delay. Abstr. Appl. Anal. 2011., 2011: Article ID 920412
Yao K, Gao JW, Gao Y: Some stability theorems of uncertain differential equation. Fuzzy Optim. Decis. Mak. 2013, 12: 3-13. 10.1007/s10700-012-9139-4
Khusainov D, Diblik J, Bavstinec J, Ryvolova J: Interval stability of linear systems neutral type. Vyčisl. Prikl. Mat. (Kiev) 2011, 3(106):148-160. (in Russian)
Kwon OM, Park JH: Exponential stability of uncertain dynamic systems including state delay. Appl. Math. Lett. 2006, 19: 901-907. 10.1016/j.aml.2005.10.017
Kwon OM, Park JH: Exponential stability analysis for uncertain neural networks with interval time-varying delays. Appl. Math. Comput. 2009, 212: 530-541. 10.1016/j.amc.2009.02.043
Phat VN, Nam PT: Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function. Int. J. Control 2007, 80: 1333-1341. 10.1080/00207170701338867
Phat VN, Niamsup P: Stability analysis for a class of functional differential equations and applications. Nonlinear Anal., Theory Methods Appl. 2009, 71: 6265-6275. 10.1016/j.na.2009.06.028
Shao H-Y: New delay-dependent stability criteria for systems with interval delay. Automatica 2009, 45: 744-749. 10.1016/j.automatica.2008.09.010
Shatyrko AV, Khusainov DY, Diblik J, Bastinec J, Rivolova A: Estimates of perturbation of nonlinear indirect interval control system of neutral type. J. Autom. Inf. Sci. 2011, 43: 13-28. 10.1615/JAutomatInfScien.v43.i1.20
Shatyrko AV, Khusainov DY: Absolute interval stability of indirect regulating systems of neutral type. J. Autom. Inf. Sci. 2010, 42: 43-54.
Shatyrko AV, Khusainov D: Interval stability investigations of different regulator systems with time delay by Lyapunov-Krasovskii functional. Bull. Taras Shevchenko Natl. Univ. Kyiv. Ser., Phys. Math. 2009, 3: 212-221. (in Ukrainian)
Sun YJ: Global stabilizability of uncertain systems with time-varying delays via dynamic observer-based output feedback. Linear Algebra Appl. 2002, 353: 91-105. 10.1016/S0024-3795(02)00292-6
Tian J, Zhou X: Improved asymptotic stability criteria for neural networks with interval time-varying delay. Expert Syst. Appl. 2010, 37: 7521-7525. 10.1016/j.eswa.2010.04.093
Verriest EI, Florchinger P: Stability of stochastic systems with uncertain time delays. Syst. Control Lett. 1995, 24: 41-47. 10.1016/0167-6911(94)00030-Y
Yue D: Robust stabilization of uncertain systems with unknown input delay. Automatica 2004, 40: 331-336. 10.1016/j.automatica.2003.10.005
Wang TY, Xie LH, De Souz CE: Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 1992, 19: 139-149. 10.1016/0167-6911(92)90097-C
Erbe LN, Kong Q, Zhang BG: Oscillation Theory for Functional Differential Equations. Dekker, New York; 1995.
Gyori I, Ladas G: Oscillation Theory of Delay Differential Equations. Clarendon, Oxford; 1991.
Drakhlin ME: Operator of the internal superposition in the space of summable functions. Izv. Vysš. Učebn. Zaved., Mat. 1986, 5: 18-23.
Drakhlin ME, Plyshevskaya TK: To the theory of functional differential equations. Differ. Uravn. 1978, 14: 1347-1361.
Berezansky L, Braverman E: Non-oscillation properties of a linear neutral differential equations. Funct. Differ. Equ. 2002, 9(3-4):275-288.
Berezansky L, Braverman E: Oscillation criteria for linear neutral differential equations. J. Math. Anal. Appl. 2003, 286: 601-617. 10.1016/S0022-247X(03)00502-X
Domoshnitsky A: Extension of Sturm’s theorem to equations with time-lag. Differ. Uravn. 1983, 19: 1475-1482.
Gusarenko SA, Domoshnitskii AI: Asymptotic and oscillation properties of the first order scalar functional-differential equations. Differ. Uravn. 1989, 25: 1480-1491.
Agarwal RP, Berezansky L, Braverman E, Domoshnitsky A: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York; 2012.
Domoshnitsky A, Sheina MV: Nonnegativity of Cauchy matrix and stability of systems with delay. Differ. Uravn. 1989, 25: 201-208.
Krasnosel’skii MA, Vainikko GM, Zabreiko PP, Rutitskii JB, Stezenko VJ: Approximate Methods for Solving Operator Equations. Nauka, Moscow; 1969. (in Russian)
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).
The authors declare that they have no competing interests.
All authors worked on this paper together and have the same rights on the content of the paper. All authors read and approved the final manuscript.