Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales
© N. H. Du et al. 2011
Received: 29 September 2010
Accepted: 4 February 2011
Published: 27 February 2011
with being a given -matrix function, has been an intensively discussed field in both theory and practice. This problem can be seen in many real problems, such as in electric circuits, chemical reactions, and vehicle systems. März in  has dealt with the question whether the zero-solution of (1.1) is asymptotically stable in the Lyapunov sense with , with being constant and small perturbation .
This model appears in many practical areas, such as the Leontiev dynamic model of multisector economy, the Leslie population growth model, and singular discrete optimal control problems. On the other hand, SDEs occur in a natural way of using discretization techniques for solving DAEs and partial differential-algebraic equations, and so forth, which have already attracted much attention from researchers (cf. [2–4]). When , in , the authors considered the solvability of Cauchy problem for (1.2); the question of stability of the zero-solution of (1.2) has been considered in  where the nonlinear perturbation is small and does not depend on .
with in time scale and being the derivative operator on . When , (1.3) is (1.1); if , we have a similar equation to (1.2) if it is rewritten under the form .
The purpose of this paper is to answer the question whether results of stability for (1.1) and (1.2) can be extended and unified for the implicit dynamic equations of the form (1.3). The main tool to study the stability of this implicit dynamic equation is a generalized direct Lyapunov method, and the results of this paper can be considered as a generalization of (1.1) and (1.2).
with the assumption of differentiability for . The main results of this paper are established in Section 3 where we deal with the stability of (1.5). The technique we use in this section is somewhat similar to the one in [6–8]. However, we need some improvements because of the complicated structure of every time scale.
2. Nonlinear Implicit Dynamic Equations on Time Scales
2.1. Some Basic Notations of the Theory of the Analysis on Time Scales
A time scale is a nonempty closed subset of the real numbers , and we usually denote it by the symbol . We assume throughout that a time scale is endowed with the topology inherited from the real numbers with the standard topology. We define the forward jump operator and the backward jump operator by (supplemented by ) and (supplemented by ). The graininess is given by . A point is said to be right-dense if , right-scattered if , left-dense if , left-scattered if , and isolated if is right-scattered and left-scattered. For every , by , we mean the set . The set is defined to be if does not have a left-scattered maximum; otherwise, it is without this left-scattered maximum. Let be a function defined on , valued in . We say that is delta differentiable (or simply: differentiable) at provided there exists a vector , called the derivative of , such that for all there is a neighborhood around with for all . If is differentiable for every , then is said to be differentiable on . If , then delta derivative is from continuous calculus; if , the delta derivative is the forward difference, , from discrete calculus. A function defined on is rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point. The set of all rd-continuous functions from to a Banach space is denoted by . A matrix function from to is said to be regressive if for all , and denote the set of regressive functions from to . Moreover, denote the set of positively regressive functions from to , that is, the set .
The solution of the corresponding matrix-valued IVP , always exists for , even is not regressive. In this case, is defined only with (see [12, 13]) and is called the Cauchy operator of the dynamic equation (2.1). If we suppose further that is regressive, the Cauchy operator is defined for all .
2.2. Linear Equations with Small Nonlinear Perturbation
which has been well studied. If there is at least a such that is singular, we cannot solve explicitly the leading term . In fact, we are concerned with a so-called ill-posed problem where the solutions of Cauchy problem may exist only on a submanifold or even they do not exist. One of the ways to solve this equation is to impose some further assumptions stated under the form of indices of the equation.
where , are orthogonal matrices and is a diagonal matrix with singular values on its main diagonal. Since , on the above decomposition of , we can choose the matrix to be in (see ). Hence, by putting and , we obtain and as the requirement.
Under these notations, we have the following Lemma.
The following assertions are equivalent
Lemma 2.2 is proved.
Now, we add the following assumptions.
The homogeneous LIDE (2.4) is of index-1.
We now describe shortly the decomposition technique for (2.3) as follows.
Applying the global existence theorem (see ), we see that (2.31), with the initial condition has a unique solution .
Thus, we get the following theorem.
There hold the following statements:
Noting that if is a solution of (2.3) with the initial condition (2.33), then for all . Conversely, let and let , , be the solution of (2.3) satisfying the initial condition . We see that . This means that there exists a solution of (2.3) passing .
2.3. Quasilinear Implicit Dynamic Equations
Suppose that (2.43) is of index-1. Then, by Lemma 2.2, this condition is equivalent to one of the following conditions:
See [17, Lemma 1].
is bounded for all and . Then, the right-hand side of (2.56) also satisfies the Lipschitz condition. Thus, from the global existence theorem (see ), (2.56) with the initial condition (2.57) has a unique solution defined on .
Therefore, we have the following theorem.
3. Stability Theorems of Implicit Dynamic Equations
has a unique solution defined on . The condition ensuring the existence of a unique solution can be refered to Section 2. We denote the solution with the initial condition (3.2) by . Remember that we look for the solution of (3.1) in the space . Let for all , which follows that (3.1) has the trivial solution .
(3) -asymptotically (resp., -asymptotically) stable if it is stable and for each , there exist positive such that the inequality (resp., ) implies . If is independent of , then the corresponding stability is -uniformly asymptotically ( -uniformly asymptotically) stable,
(5)P-exponentially stable if there is positive constant with such that for every there exists an , the solution of (3.1) with the initial condition satisfies . If the constant can be chosen independent of , then this solution is called -uniformly exponentially stable.
From and , the notions of -stable and -stable as well as -asymptotically stable and -asymptotically stable are equivalent. Therefore, in the following theorems we will omit the prefixes and when talking about stability and asymptotical stability. However, the concept of -uniform stability implies -uniform stability if the matrices are uniformly bounded and -uniform stability implies -uniform stability if the matrices are uniformly bounded.
For , we denote . If , then by (remember that does not imply that ). Consider the case where . If , then by the relations (3.6) and (3.8) we have . In particular, which is a contradiction. Thus , this implies , provided .
The proposition is proved.
Similarly, we have the following proposition.
Note that when the function is independent of and even if the vector field associated with the implicit dynamic equation (3.1) is autonomous, the derivative may depend on .
Assume further that (3.1) is locally solvable. Then the trivial solution of (3.1) is stable.
By virtue of Theorem 3.6 and the conditions (2) and (3), it follows that (3.1) is globally solvable. Suppose on the contrary that the trivial solution of (3.1) is not stable. Then, there exists an such that for all there exists a solution of (3.1) satisfying and for some . Put .
satisfying the conditions
Further, (3.1) is locally solvable. Then the trivial solution of (3.1) is asymptotically stable.
Also from Theorem 3.6 and the conditions (2) and (3), it implies that (3.1) is globally solvable.
The proof is complete.
We present a theorem of uniform global asymptotical stability.
By the same argument as Theorem 3.6, we have the following theorem.
Assume further that (3.1) is locally solvable. Then, (3.1) is globally solvable.
Assume that there is a function satisfying the assertions (1), (2), and (3) but the trivial solution of (3.1) is not stable. Then, there exist a positive and a such that ; there exists a solution of (3.1) satisfying and , for some . Let . Since , it is possible to find a satisfying when . Consider the solution satisfying and for a .
The proof of the theorem is complete.
The proof is similar to the one of Theorem 3.9.
Similarly to the proof of Theorem 3.11.
It is difficult to establish the inverse theorem for Theorems from 3.7 to 3.15, that is, if the trivial solution of (3.1) is stable, there exists a function satisfying the assertions in the above theorems. However, if the structure of the time scale is rather simple we have the following theorem.
Suppose that contains no right-dense points and the trivial solution of (3.1) is -uniformly stable. Then, there exists a function being rd-continuous satisfying the conditions (1), (2), and (3) of Theorem 3.13, where is an open neighborhood of 0 in .
The proof is complete.
Now we give an example on using Lyapunov functions to test the stability of equations. The following result finds out that the stability of a linear equation will be ensured if nonlinear perturbations are sufficiently small Lipschitz.
Denote . It is easy to show that the trivial solution of (3.35) is -uniformly exponentially stable if and only if , where is the domain of uniform exponential stability of . On the exponential stable domain of a time scale, we can refer to [10, 18, 19]. By the definition of exponential stability, it implies that the graininess function of the time scale is upper bounded. Let .
Suppose that and the homogeneous equation (3.35) is of index-1 and the constant is sufficiently small. Then, the trivial solution of (3.33) is -uniformly globally asymptotically stable.
Therefore, having the above result is obvious.
We have studied some criteria ensuring the stability for a class of quasilinear dynamic equations on time scales. So far, the inverse theorem of the theorems of the stability in Section 3 of this paper is still an open problem for an arbitrary time scale meanwhile it is true for discrete and continuous time scales.
This work was done under the support of NAFOSTED no 101.02.63.09.
- März R: Criteria for the trivial solution of differential algebraic equations with small nonlinearities to be asymptotically stable. Journal of Mathematical Analysis and Applications 1998,225(2):587–607. 10.1006/jmaa.1998.6055MATHMathSciNetView ArticleGoogle Scholar
- Campbell SL: Singular Systems of Differential Equations I, II, Research Notes in Mathematics. Volume 40. Pitman, London, UK; 1980:vii+176.Google Scholar
- Griepentrog E, März R: Differential-Algebraic Equations and Their Numerical Treatment. Volume 88. Teubner, Leipzig, Germany; 1986:220.MATHGoogle Scholar
- Kunkel P, Mehrmann V: Differential-Algebraic Equations, EMS Textbooks in Mathematics. European Mathematical Society House, Zürich, Switzerland; 2006:viii+377.View ArticleGoogle Scholar
- Loi LC, Du NH, Anh PK: On linear implicit non-autonomous systems of difference equations. Journal of Difference Equations and Applications 2002,8(12):1085–1105. 10.1080/1023619021000053962MATHMathSciNetView ArticleGoogle Scholar
- Anh PK, Hoang DS: Stability of a class of singular difference equations. International Journal of Difference Equations 2006,1(2):181–193.MATHMathSciNetGoogle Scholar
- Kloeden PE, Zmorzynska A: Lyapunov functions for linear nonautonomous dynamical equations on time scales. Advances in Difference Equations 2006, 2006: 1–10.MathSciNetView ArticleGoogle Scholar
- Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and its Applications. Volume 370. Kluwer Academic, Dordrecht, The Netherlands; 1996:x+285.View ArticleGoogle Scholar
- Advances in Dynamic Equations on Time Scales. Birkhäuse, Boston, Mass, USA; 2003:xii+348.Google Scholar
- Bohner M, Stević S: Linear perturbations of a nonoscillatory second-order dynamic equation. Journal of Difference Equations and Applications 2009,15(11–12):1211–1221. 10.1080/10236190903022782MATHMathSciNetView ArticleGoogle Scholar
- Bohner M, Stević S: Trench's perturbation theorem for dynamic equations. Discrete Dynamics in Nature and Society 2007, 2007:-11.Google Scholar
- Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.MATHMathSciNetView ArticleGoogle Scholar
- Pötzsche C: Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients. Journal of Mathematical Analysis and Applications 2004,289(1):317–335. 10.1016/j.jmaa.2003.09.063MATHMathSciNetView ArticleGoogle Scholar
- Pötzsche C: Chain rule and invariance principle on measure chains. Journal of Computational and Applied Mathematics 2002,141(1–2):249–254. 10.1016/S0377-0427(01)00450-2MATHMathSciNetView ArticleGoogle Scholar
- Pötzsche C: Analysis Auf Maßketten. Universität Augsburg; 2002.Google Scholar
- Dai L: Singular Control Systems, Lecture Notes in Control and Information Sciences. Volume 118. Springer, Berlin, Germany; 1989:x+332.Google Scholar
- Du NH, Linh VH: Stability radii for linear time-varying differential-algebraic equations with respect to dynamic perturbations. Journal of Differential Equations 2006,230(2):579–599. 10.1016/j.jde.2006.07.004MATHMathSciNetView ArticleGoogle Scholar
- Doan TS, Kalauch A, Siegmund S, Wirth FR: Stability radii for positive linear time-invariant systems on time scales. Systems & Control Letters 2010,59(3–4):173–179. 10.1016/j.sysconle.2010.01.002MATHMathSciNetView ArticleGoogle Scholar
- Pötzsche C, Siegmund S, Wirth F: A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete and Continuous Dynamical Systems 2003,9(5):1223–1241.MATHMathSciNetView ArticleGoogle 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.