About Robust Stability of Dynamic Systems with Time Delays through Fixed Point Theory
© M. De la Sen. 2008
Received: 22 September 2008
Accepted: 19 November 2008
Published: 4 December 2008
This paper investigates the global asymptotic stability independent of the sizes of the delays of linear time-varying systems with internal point delays which possess a limiting equation via fixed point theory. The error equation between the solutions of the limiting equation and that of the current one is considered as a perturbation equation in the fixed- point and stability analyses. The existence of a unique fixed point which is later proved to be an asymptotically stable equilibrium point is investigated. The stability conditions are basically concerned with the matrix measure of the delay-free matrix of dynamics to be negative and to have a modulus larger than the contribution of the error dynamics with respect to the limiting one. Alternative conditions are obtained concerned with the matrix dynamics for zero delay to be negative and to have a modulus larger than an appropriate contributions of the error dynamics of the current dynamics with respect to the limiting one. Since global stability is guaranteed under some deviation of the current solution related to the limiting one, which is considered as nominal, the stability is robust against such errors for certain tolerance margins.
Time-delay dynamic systems are an interesting field of research in dynamic systems and functional differential equations. Their intrinsic related theoretical interest is due to the fact that the necessary formalism lies in that of functional differential equations, being infinite dimensional. Another reason for their interest relies on the wide range of their applicability in modelling a number of physical systems like, for instance, transportation systems, queuing systems, teleoperated systems, war/peace models, biological systems, finite impulse response filtering, and so on [1–4]. Important particular interest has been devoted to stability, stabilization, and model-matching of control systems where the object to be controlled possesses delayed dynamics and the controller is synthesized incorporating delayed dynamics or its structure may be delay-free (see, e.g., [1, 4–14]). The properties are formulated as either being independent of or dependent on the sizes of the delays. An intrinsic problem which generated analysis complexity is the presence of infinitely many characteristic zeros because of the functional nature of the dynamics. This fact generates difficulties in the closed-loop pole-placement problem compared to the delay-free case , as well as in the stabilization problem [2, 4–6, 8–11, 13, 15–20], including the case of singular time-delay systems where the solution is sometimes nonunique and impulsive because of the dynamics associated to a nilpotent matrix . The properties of the associated evolution operators have been investigated in [2, 6, 11]. This paper is devoted to obtain results relying on a comparison and an asymptotic comparison of the solutions between a nominal (unperturbed) functional differential equation involving wide classes of delays and a perturbed version (describing the current dynamics) with some smallness in the limit assumptions on the perturbed functional differential equation. The nominal equation is defined as the limiting equation of the perturbed one since the parameters of the last one converge asymptotically to those of its limiting counterpart. The problem of interest arises since very often the perturbations related to a nominal model in dynamic systems occur during the transients while they are asymptotically vanishing in the steady state or, in the most general worst case, they grow at a smaller rate than the solution of the nominal differential equation. In this context, the nominal differential equation may be viewed as the limiting equation of the perturbed one. The comparison between the solutions of the limiting differential equation and those of the perturbed one based on Perron-type results has been studied classically for ordinary differential equations and more recently for the case of functional equations [10, 21, 22]. Particular functional equations of interest are those involving both point and distributed delays potentially including the last ones Volterra-type terms [2, 5–7, 23]. On the other hand, fixed point theory [2, 21, 24] is a very powerful mathematical tool to be used in many applications where stability is required. At a theoretical level, fixed point theory is being of an increasing interest along the last years. For instance, the concept of weak contractiveness is discussed in  where the contraction constant is allowed to be unity but a negative vanishing term associated with some continuous nondecreasing function is also allowed. Weak contractiveness still ensures the existence of a unique fixed point. The existence of a unique fixed point has also been proved for asymptotic contractions . Also, the existence of a nonempty fixed point set in a self-map of where is a complete metric space allows guaranteeing the -stability of iteration procedures . In this paper, linear time-varying functional differential equations with point constant delays are investigated. Based on the contraction mapping principle, it is first proved the existence of a unique fixed point. The related proofs are based on the convergence of the parameters of the current equation to their counterparts of the limiting equation. The existence of such a fixed point requires that a relevant matrix of the limiting equation (either that of the delay-free dynamics or that of the zero-delay dynamics) be a stability matrix. Furthermore, an inequality concerning the parameters of the absolute value of such a matrix with a measure of all the remaining dynamics (formulated in terms of norms) has to be fulfilled. Once the existence of a unique fixed point has been proved, simple extra conditions ensure that such a point is a globally stable zero equilibrium point of the state-trajectory solution. This leads immediately to prove the global asymptotic stability independent of the sizes of the delays of the dynamic system. The analysis is then extended to the case of closed-loop systems obtained via state or output linear feedback from the original uncontrolled dynamic system. A method to synthesize both the time-invariant parts and the incremental ones of the controller matrices is given so that the existence of a fixed point of the closed-loop system is guaranteed. The obtained results are of robust stability type since the global asymptotic stability is guaranteed under a certain deviation from the current solution with respect to the limiting one, which is considered the nominal dynamics.
denotes any vector or induced matrix norm. Also, and are the -norms of the vector or (induced) real or complex matrix and denote the measure of the square matrix . The matrix measure is defined as the existing limit which has the property for any square -matrix of spectrum An important property for the investigation of this paper is that if is a stability matrix, that is, if
denotes the supremum norm on or its induced supremum metric, for functions or vector and matrix functions without specification of any pointwise particular vector or matrix norm for each If pointwise vector or matrix norms are specified, the corresponding particular supremum norms are defined by using an extra subscript. Thus, and are, respectively, the supremum norms on for vector and matrix functions of domains in respectively, in defined from their pointwise respective norms for each
2. Linear Systems with Point Constant Delays and the Contraction Mapping Theorem
According to Lyapunov's stability theory, global stability means that the state-trajectory solution is uniformly bounded for any bounded function of initial conditions. Global asymptotic stability implies also that there is a unique asymptotically stable equilibrium point which is then a global attractor. See, for instance, [2, 4–11, 13, 16, 21, 24, 28]. Generic relations of stability with fixed point theory have been reported in [2, 21, 24, 27, 29, 30]. It turns out that a system whose state-trajectory solutions are all bounded and converge to a unique point is globally asymptotically stable to its equilibrium in Lyapunov's sense, provided that such equilibrium is unique. The following simple result is well known. Assume the system (2.1) with then the resulting linear time-invariant delay-free system (2.1) is globally asymptotically stable if is a stability matrix so that if Nonasymptotic stability is guaranteed if The subsequent result is concerned with global stability independent of the sizes of the delays and it is obtained from the contraction mapping theorem for the case when (2.1) has a limiting equation with a unique asymptotically stable equilibrium point. It is assumed that the matrices defining the delayed dynamics have sufficiently small norms and that the norm of the error matrix of the delay-free dynamics with respect to its limiting value is also sufficiently small.
Assume that is a stability matrix of -matrix measure and that for some real constants and and any real constant such that the -semigroup of the infinitesimal generator satisfies Assume also that and that is nonsingular. Then, the system (3.1) is globally asymptotically stable independent of the sizes of the delays.
for some where is the spectrum of of cardinal and any given vector norm and corresponding induced matrix norm. The limiting equation of (3.1) is Since is nonsingular, Thus, the unique fixed point in of the limiting equation, and then that of (3.1) whose uniqueness follows from the contraction mapping theorem, is As a result, the unique fixed point is a global attractor so that (2.1) is globally asymptotically stable. Property (i) has been proven.
Assume that is a stability matrix of -matrix measure and that Assume also that and that is nonsingular. Then, the system (2.1) is globally asymptotically stable independent of the sizes of the delays.
so that is a contraction if since may be chosen either fulfilling (the stability abscissa of is associated with a multiple eigenvalue), but arbitrarily close to or (the dominant eigenvalue of is single). As a result, has a unique fixed point in Since any positive and arbitrarily close to zero real constant may be used, is a contraction if In addition, since (3.1) converges to a limiting equation and since the unique fixed point is zero which is a global asymptotic attractor independent of the sizes of the delays. Therefore, no state-trajectory solution may converge to a distinct point or to be oscillatory since the attractor is global and asymptotic, and no state-trajectory solution may be unbounded (since is bounded). Therefore, the constraint leads to a contraction and then to a fixed point for the mapping and the result has been proven.
Thus, Theorem 2.1 and Corollary 2.2 are modified as follows.
Assume that is a stability matrix of -matrix measure and that for some real constants and and any real constant such that the -semigroup of the infinitesimal generator satisfies Assume also that Then, the system (2.1) is globally asymptotically stable independent of the sizes of the delays.
Note that the requirement that is nonsingular is imposed in Theorem 2.3 and Corollary 2.4, since is directly nonsingular as it is a stability matrix. Note also that being a stability matrix is also a direct consequence of Theorem 2.1 and Corollary 2.2, which give a result of asymptotic stability independent of the delays thus being valid for zero delays. However, such condition of nonsingularity of (and even the strongest one of being a stability matrix) is neither required to apply of the contraction mapping principle [21, 24], nor a direct consequence of it in Theorem 2.1 and Corollary 2.2. As a result, it cannot be invoked prior to stability but only being a consequence after stability has been proven.
As a result, a stronger result than Theorem 2.1(i) holds by replacing and also a stronger result than Theorem 2.1(ii) holds by replacing In the same way, a stronger result than Theorem 2.3(i) holds by replacing and a stronger result than Theorem 2.3(ii) holds by replacing Then, Corollaries 2.2 and 2.4 follow directly as stronger results than Theorem 2.1, respectively, Theorem 2.3 via a very short modified proof by using simple Lyapunov's theory. In other words, global asymptotic stability of the current system holds under asymptotic stability of the respective auxiliary limiting delay-free systems by taking and in the relevant inequalities of norms of Theorems 2.1 and 2.3 just as proven in Corollaries 2.2 and 2.4.
3. Feedback Linear Systems with Point Constant Delays and the Contraction Mapping Theorem
the dynamic system is controllable (or, simply, the pair is controllable) if and only if An equivalent test is that is controllable if and only if The meaning of this property is that for any bounded there exists a piecewise continuous control such that for some finite An equivalent property is the existence of a controller of gain such that a linear state-feedback control defined by makes the matrix feedback obtained system to possess a prescribed spectrum
The dynamic system is stabilizable (or, simply, the pair is stabilizable) if and only if Its meaning is that there exists such that the matrix of dynamics of the closed-loop feedback system is a stability matrix; that is, and any state-trajectory solution with bounded initial conditions is uniformly bounded and converges asymptotically to the zero equilibrium, as a result. By comparing the controllability and stabilizability tests, it turns out that controllability implies stabilizability but the converse is not true in general.
The dynamic system is observable (or, simply, the pair is observable) if and only if the pair is controllable. If and are admitted to be complex matrices, then transposes are replaced with conjugate transposes. Observability is related to the ability of calculating the past-state vector from output measurements (usually Similarly, the dynamic system is detectable (or, simply, the pair is detectable) if and only if the pair is stabilizable.
The above concepts are extendable with more involved tests to time-varying and nonlinear dynamic systems. Related results have also been investigated related to fixed point theory (see, e.g., [32, 33]). Recent stability results based on fixed point theory are provided in [34, 35]. The following result follows directly from the controllability property of linear systems. It will be then used for obtaining small left-hand side terms in the norms inequalities of Theorems 2.1 and 2.3 via feedback under assumptions of controllability of relevant matrix pairs.
Assume that the pair is controllable for any given Then, for any prescribed set of nonnecessarily distinct complex numbers there exists a controller matrix such that the spectrum of the obtained via (3.4) is As a result, is also predefined according to the prescribed set
Corollary 2.2, through Lemma 3.1(i), leads directly to the subsequent result.
Since is controllable for there exists such that has any prescribed spectrum Then, fix Since the eigenvalues of are distinct, it always exists a nonsingular real -matrix such that where As a result, one gets from (3.6) that for some real -matrices provided that under the incremental controller gains choice The proof follows from Theorem 2.1(i), since is independent of and thus on so that is independent of so that it can be fixed fulfilling by appropriately selecting for any given
Theorem 3.2 is useful to guarantee closed-loop stabilization of the system (2.1) under controllability conditions of the time-invariant dynamics by first stabilizing the delay-free dynamics of the limiting equation via linear feedback with sufficiently large stability abscissa. The result is achievable irrespective of the norms of the incremental matrices of dynamics of (2.1) with respect to its limiting equation. Theorem 3.2 is now extended by replacing the time-invariant controllers by time-varying ones.
Suppose that the assumptions (1)–(3) of Theorem 3.2 hold. Then, there exist nonzero (nonunique) controller gain matrix functions such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays.
Furthermore, if then the controller gains defined with any members of sets of constant controller gains chosen according to Theorem 3.2 and incremental controller gains guarantee the global asymptotic stability independent of the sizes of the delays of the closed-loop system (3.1)-(3.2).
Note by construction, (3.6) and Theorem 3.2, that Thus, it follows from Theorem 3.2 that there is an open ball of centered at cero fulfilling so that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the delays for sets of nonzero incremental controller gains since the same property is fulfilled for sets of constant controller gains.
guarantee the global asymptotic stability independent of the sizes of the delays of the closed-loop system (3.1)–(3.3).
Note that if and then and guarantee the closed-loop stability from Corollary 3.3. If and then from Kronecker-Capelli's theorem, see, for instance, [11, 15], there exist infinitely many solutions of the incremental controller gains which make so that the closed-loop stability is guaranteed under with On the other hand, Corollary 3.3 allows obtaining a subsequent direct result under a weaker Condition 2 of Theorem 3.2. In particular, only the controllability of and not that of the remaining pairs is requested for selecting an appropriate negative value of and the static controller gains are chosen for least-squares minimization of the associated term in
In the same way as Theorem 3.2 is obtained from Corollary 2.2 (a refinement of Theorem 2.1), Theorem 2.3 leads to the subsequent result which is obtained based on a comparison of the delayed dynamics with the delay-free limiting dynamics.
Then, there exist sets of nonunique controller gain matrix functions defined by such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays and has an arbitrary spectrum of distinct eigenvalues of modulus not larger than a prescribed upper bound
the set of static controller gain matrices satisfies that consists of distinct complex numbers of modulus not larger than any prescribed and the nonsingular matrix defines the similarity transformation
Corollaries to Theorem 3.5 might be obtained directly based on the ideals of Corollaries 3.3-3.4 for Theorem 3.2.
4. Further Extensions
Stabilizability of is a weaker property than the controllability of such a pair what means that such that for each prescribed set of numbers (possibly repeated) An equivalent characterization of the controllability of is
If an open-loop system is stabilizable but not controllable, all its uncontrollable open-loop modes are invariant and stable under any state-feedback law.
This section gives extensions of some results of Section 3 for the case when some controllability conditions are lost but stabilizability still holds and some further extensions for the case of output feedback controllers. The following result is weaker but more general than Theorem 3.2.
Then, there exist (in general, nonunique) constant controller gains such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that where is defined in (3.6) and is the stability abscissa of the uncontrollable dominant eigenvalue of The stability property also holds if with being defined in the proof of Theorem 3.2.
Theorem 3.2 may also be extended straightforwardly by replacing controllable by stabilizable, and Corollaries 3.3-3.4 are also directly extendable based on Corollary 4.1. On the other hand, Theorem 3.5 extends directly to the subsequent result which is weaker in the sense that the matrix measure cannot be prefixed since controllability of is replaced by its stabilizability.
Then, there exist sets of nonunique controller gain matrix functions defined by such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that with defined in (3.12) and is the stability abscissa of the uncontrollable (stable and invariant under state feedback) dominant eigenvalue of
The triple is said to be controllable and observable if is observable and is controllable. If the is controllable and observable, then there exists such that has values arbitrarily close to any prescribed subset of of cardinal with possibly repeated members provided that and are full rank. The remaining members of cannot be allocated arbitrarily close to prefixed values.
The above properties lead to the fact that the control output feedback law (3.1) is unable to reallocate all the eigenvalues of respectively, those of to exact prescribed positions if the triples respectively, are controllable and observable even if and and are both full rank. However, under this constraint, all the eigenvalues of the matrix respectively, of the matrix can be allocated arbitrarily close to any prefixed set of complex numbers, by some choice of the static controller gain Also, if and are full rank and respectively, are controllable and observable triples, then of the eigenvalues of respectively, of may be allocated arbitrarily close to prescribed complex sets by some
Corollary 4.1 is reformulated as follows for the case of linear output feedback (4.1) by taking into account the above properties of linear time-invariant output feedback.
Then, there exist (in general, nonunique) constant controller gains such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that If the above Condition (2), replaced with is a controllable and observable triple instead of stabilizable and detectable and, furthermore, and then constant controller gains can be found so that the closed-loop stability is guaranteed if for a prefixed
Corollary 4.2 is reformulated as follows for the case of linear output feedback (4.1).
Then, there exist sets of nonunique controller gain matrix functions defined by such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that The stability property of the closed-loop system also holds if If the above Condition (2), replaced with is a controllable and observable triple instead of stabilizable and detectable and, furthermore, and then constant controller gains can be found so that the closed-loop stability is guaranteed if for a prefixed matrix measure.
The author is very grateful to the Spanish Ministry of Education for its partial support of this work through Project DPI2006-00714. He is also grateful to the Basque Government for its support through GIC07143-IT-269-07, SAIOTEK SPED06UN10, and SPE07UN04. The author is grateful to the reviewers who helped very much in clarifying the manuscript content in the revised version.
- De la Sen M, Luo N: Discretization and FIR filtering of continuous linear systems with internal and external point delays. International Journal of Control 1994, 60(6):1223-1246. 10.1080/00207179408921518MATHMathSciNetView ArticleGoogle Scholar
- Burton TA: Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, Mathematics in Science and Engineering. Volume 178. Academic Press, Orlando, Fla, USA; 1985:x+337.Google Scholar
- Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.View ArticleGoogle Scholar
- Niculescu S-I: Delay Effects on Stability. A Robust Control Approach, Lecture Notes in Control and Information Sciences. Volume 269. Springer, London, UK; 2001:xvi+383.Google Scholar
- De la Sen M, Luo N: On the uniform exponential stability of a wide class of linear time-delay systems. Journal of Mathematical Analysis and Applications 2004, 289(2):456-476. 10.1016/j.jmaa.2003.08.048MATHMathSciNetView ArticleGoogle Scholar
- De la Sen M: Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces. Journal of Mathematical Analysis and Applications 2006, 321(2):621-650. 10.1016/j.jmaa.2005.08.038MATHMathSciNetView ArticleGoogle Scholar
- Datko RF: Time-delayed perturbations and robust stability. In Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Applied Mathematics. Volume 152. Marcel Dekker, New York, NY, USA; 1994:457-468.Google Scholar
- Alastruey CF, De la Sen M, González de Mendívil JR: The stabilizability of integro-differential systems with two distributed delays. Mathematical and Computer Modelling 1995, 21(8):85-94. 10.1016/0895-7177(95)00041-YMATHMathSciNetView ArticleGoogle Scholar
- Ahmed NU: Optimal control of infinite-dimensional systems governed by integrodifferential equations. In Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Applied Mathematics. Volume 152. Marcel Dekker, New York, NY, USA; 1994:383-402.Google Scholar
- Kulenović MRS, Ladas G, Meimaridou A: Stability of solutions of linear delay differential equations. Proceedings of the American Mathematical Society 1987, 100(3):433-441.MATHMathSciNetView ArticleGoogle Scholar
- De la Sen M: On impulsive time-varying systems with unbounded time-varying point delays: stability and compactness of the relevant operators mapping the input space into the state and output spaces. The Rocky Mountain Journal of Mathematics 2007, 37(1):79-129. 10.1216/rmjm/1181069321MATHMathSciNetView ArticleGoogle Scholar
- Jiang Y, Jurang Y: Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. Journal of Mathematical Analysis and Applications 1997, 207(2):388-396. 10.1006/jmaa.1997.5276MATHMathSciNetView ArticleGoogle Scholar
- Sengadir T: Asymptotic stability of nonlinear functional-differential equations. Nonlinear Analysis: Theory, Methods & Applications 1997, 28(12):1997-2003. 10.1016/S0362-546X(96)00029-6MATHMathSciNetView ArticleGoogle Scholar
- De la Sen M: An algebraic method for pole placement in multivariable systems with internal and external point delays by using single rate or multirate sampling. Dynamics and Control 2000, 10(1):5-31. 10.1023/A:1008351026971MATHMathSciNetView ArticleGoogle Scholar
- De la Sen M: On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays. Applied Mathematics and Computation 2007, 190(1):382-401. 10.1016/j.amc.2007.01.053MATHMathSciNetView ArticleGoogle Scholar
- Chen F, Shi C: Global attractivity in an almost periodic multi-species nonlinear ecological model. Applied Mathematics and Computation 2006, 180(1):376-392. 10.1016/j.amc.2005.12.024MATHMathSciNetView ArticleGoogle Scholar
- Meng X, Xu W, Chen L: Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion. Applied Mathematics and Computation 2007, 188(1):365-378. 10.1016/j.amc.2006.09.133MATHMathSciNetView ArticleGoogle Scholar
- White L, White F, Luo Y, Xu T: Estimation of parameters in carbon sequestration models from net ecosystem exchange data. Applied Mathematics and Computation 2006, 181(2):864-879. 10.1016/j.amc.2006.02.014MATHMathSciNetView ArticleGoogle Scholar
- Sarkar RR, Mukhopadhyay B, Bhattacharyya R, Banerjee S: Time lags can control algal bloom in two harmful phytoplankton-zooplankton system. Applied Mathematics and Computation 2007, 186(1):445-459. 10.1016/j.amc.2006.07.113MATHMathSciNetView ArticleGoogle Scholar
- Bakule L, Rossell JM: Overlapping controllers for uncertain delay continuous-time systems. Kybernetika 2008, 44(1):17-34.MATHMathSciNetGoogle Scholar
- Burton TA: Stability by Fixed Point Theory for Functional Differential Equations. Dover, Mineola, NY, USA; 2006:xiv+348.MATHGoogle Scholar
- Pituk M: A Perron type theorem for functional differential equations. Journal of Mathematical Analysis and Applications 2006, 316(1):24-41. 10.1016/j.jmaa.2005.04.027MATHMathSciNetView ArticleGoogle Scholar
- Da Prato G, Iannelli M: Linear integro-differential equations in Banach spaces. Rendiconti del Seminario Matematico della Università di Padova 1980, 62: 207-219.MATHMathSciNetGoogle Scholar
- Burton TA: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Dover, Mineola, NY, USA; 2005.MATHGoogle Scholar
- Dutta PN, Choudhury BS: A generalisation of contraction principle in metric spaces. Fixed Point Theory and Applications 2008, 2008(1):-8.
- Arav M, Santos FEC, Reich S, Zaslavski AJ: A note on asymptotic contractions. Fixed Point Theory and Applications 2007, 2007:-6.Google Scholar
- Qing Y, Rhoades BE: -stability of Picard iteration in metric spaces. Fixed Point Theory and Applications 2008, 2008:-4.Google Scholar
- Mackenroth U: Robust Control Systems. Theory and Case Studies. Springer, Berlin, Germany; 2004:xvi+519.MATHView ArticleGoogle Scholar
- Jung S-M, Rassias JM: A fixed point approach to the stability of a functional equation of the spiral of Theodorus. Fixed Point Theory and Applications 2008, 2008:-7.Google Scholar
- Wu M, Huang N-J, Zhao C-W: Fixed points and stability in neutral stochastic differential equations with variable delays. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
- Ionescu V, Oară C, Weiss M: Generalized Riccati Theory and Robust Control. A Popov Function Approach. John Wiley & Sons, Chichester, UK; 1999:xxii+380.MATHGoogle Scholar
- Chalishajar DN: Controllability of nonlinear integro-differential third order dispersion system. Journal of Mathematical Analysis and Applications 2008, 348(1):480-486. 10.1016/j.jmaa.2008.07.047MATHMathSciNetView ArticleGoogle Scholar
- Sakthivel K, Balachandran K, Sritharan SS: Exact controllability of nonlinear diffusion equations arising in reactor dynamics. Nonlinear Analysis: Real World Applications 2008, 9(5):2029-2054. 10.1016/j.nonrwa.2007.06.013MATHMathSciNetView ArticleGoogle Scholar
- Cho YJ, Kang SM, Qin X: Convergence theorems of fixed points for a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2008, 2008:-7.Google Scholar
- Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:-15.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.