Research Article | Open Access
New Conditions for the Exponential Stability of Nonlinear Differential Equations
We develop a method for proving local exponential stability of nonlinear nonautonomous differential equations as well as pseudo-linear differential systems. The logarithmic norm technique combined with the “freezing” method is used to study stability of differential systems with slowly varying coefficients and nonlinear perturbations. Testable conditions for local exponential stability of pseudo-linear differential systems are given. Besides, we establish the robustness of the exponential stability in finite-dimensional spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. We illustrate the application of this test to linear approximations of the differential systems under consideration.
The stability and robustness of differential systems have been widely investigated over the past decades; see, for example, [1–7] and references therein. This is due to theoretical interests and to being a powerful tool for system analysis and control design. The stability and robustness are the basic requirements for controlled systems. In practice, to satisfy the performance specification and to have a good transient response of the system, the controlled system is often designed to possess a stability degree. If the controlled system has a stability degree , we say that the system is exponentially stable. The concept of -stability is related to the exponential stability with a convergence rate .
Unlike the situation for linear systems, where necessary and sufficient conditions for stability are provided, the nonlinear problem is not completely solved. In fact, in spite of recent efforts (see [8–13] and the references therein), the exponential stability problem of nonlinear nonautonomous systems can be considered largely open. The main technique to stability of differential systems is Lyapunov’s method and its variants (Razumikhin-type theorems, Lyapunov-Krasovskii functional techniques); see, for example, [10, 14–16]. In contrast, many alternative methods to Lyapunov’s functions have been successfully applied to the stability analysis of differential systems, for example, Ngoc , assuming that a nonlinear differential system with time-varying delay is bounded above by a positive linear time-invariant differential system and if this last system is exponentially stable then the nonlinear system under consideration is also exponentially stable. Anderson et al. , using the concept of Lyapunov exponents and Bohl exponents, discuss the problem of stabilization for linear time-varying systems with bounded matrices. Coppel , using the concept of ordinary and exponential dichotomy, establishes new results in stability theory, and also the “freezing” method became a fruitful tool among those alternative approaches; see, for example, Vinograd  and Gil and Medina . In particular, the latter has been applied to prove that exponential stability of linear time-invariant differential systems implies the exponential stability of the system under consideration, provided that the coefficients of the original differential system are slowly varying. Moreover, an important tool to obtain explicit stability criteria for linear differential systems is the logarithmic norm of matrices (measure of matrices), which were used effectively in the recent literature on investigations of equations with dissipative coefficient matrices and their perturbations; see, for example, Zevin and Pinsky . Besides, the logarithmic norm has been used to study the error bounds in the numerical integration of ordinary differential equations [20, 21], estimates or stability of differential equations , and the oscillatory behavior of retarded functional differential equations .
Banks et al.  and Martynyuk  derived new bounds for solutions of perturbed pseudo-linear differential equations, basically using Gronwall-type inequalities. Dvirnyi and Slyn’ko [23, 24], constructing a piecewise differential Lyapunov function, established the stability of solutions to impulsive differential equations with impulsive action in the pseudo-linear form. Banks et al. , using a Gronwall-type inequality and assuming that a matrix satisfies a jointly Lipchitz inequality in and , established the robust exponential stability of evolution differential equations of pseudo-linear form. In summary, in the existing literature there are many results concerning the stability or asymptotic behavior of pseudo-linear differential equations; however, in general, the assumptions are difficult to check or conservative.
The purpose of this paper is to establish explicit conditions for the exponential stability of nonlinear differential systems. This approach led to study special classes of control systems, for example, systems with linear compact operators. In fact, assuming appropriate conditions on the perturbation term, the exponential feedback stabilization of a class of time-varying nonlinear systems can be established, provided the rate of variation of the system coefficients operators is sufficiently small.
In this paper we consider differential systems defined in Euclidean spaces, with bounded operators on the right-hand side represented in the pseudo-linear form. New estimates for the norms of solutions are derived giving us explicit stability and boundedness conditions. The equations will be represented as a perturbation about a fixed value of the coefficient operator. Thus, applying norm estimates for the involved operator-valued functions, new stability results are established.
The structure of this paper is as follows: in Section 2, we introduce some notations, the concept of stability with respect to a ball, and the definition and its properties of the logarithmic norm functions. In Section 3, the main exponential stability results and its consequences are established for nonlinear differential equations. In Section 4, we extend the main results to pseudo-linear differential systems. In Section 5, we applied the results of Section 4 to a linear approximation of the considered nonlinear system. Finally, Section 6 is devoted to the discussions of our results: in fact, the results are interpreted appropriately and robust conclusions are drawn.
Let be the set of all -valued continuous functions on .
Let us consider a system described by the following equation in the Euclidean space :where is a matrix-valued function, continuous and uniformly bounded on , and is a nonlinear and continuous vector function. The existence of solutions is assumed.
For a number , putWe will use to denote norms in and , respectively.
Definition 1. The zero solution of system (1) is exponentially stable with respect to a ball if there are constants , such that for any solution of (1), with initial condition , the following inequality holds:
Remarks 1. (1) The stability analysis with respect to a ball has been considered by many researchers (see, e.g., Furuta and Kim  and Hsiao et al. ). However, this kind of stability is defined in terms of the set of roots of characteristic polynomials corresponding to linear autonomous delay equations. In contrast, the concept of stability with respect to a ball used in this paper allows us to characterize the region of attraction of exponential stability of nonautonomous differential equations. Furthermore, in our case, the radius of the ball can be explicitly calculated in terms of known quantities.
(2) We want to point out that, considering solutions with initial functions into the region , we will ensure reasonable dynamics, for example, exponential decay rates.
Definition 2. System (1) is said to have a stability degree (or to be exponentially stable), with , if is a bounded function, with a solution of (1). In this case, the parameter is called the convergence rate.
It is assumed that, for a positive number , there is a constant , such that the growth condition
Remark 3. If, for some , there is a constant such thatthen for any fixed , condition (4) holds with .
The logarithmic norm of a square matrix is defined byThis logarithmic norm is often used as measure of stability and asymptotic decay in analytic and numerical studies concerning to ordinary differential equations (see [14, 20]). Its dependence upon the vector norm and matrix norms under consideration is clear.
Remark 4. Different norms in give rise to different logarithmic norms. However, independently of the considered norm, a logarithmic norm always has the following properties (see ).
Lemma 5. Let be a square matrix. Then
(i) , , and if and only if ;
(ii) for any norm, we have(iii) is a stable matrix if and only if there exists a logarithmic norm such that .
For the -norm , the induced matrix measure is given byFor the -norm , the induced logarithmic norm is given by
Remark 6. Although logarithmic norm is only defined for constant fixed matrices, it can be applied to any matrix, either time-invariant or time-varying. Thus, logarithmic norm technique can be used to study the stability of linear time-varying systems (Coppel [7, 15]).
3. Main Results
The results described here are based upon the following Coppel’s inequality (): where is a logarithmic norm of a square matrix .
To establish our main results we make two basic assumptions on the coefficients of system (1):()There is a positive real number such that()For any logarithmic norm , the matrix satisfies
Remark 7. Condition (12) works on matrices , for every fixed .
Proof. Let us take an initial value , satisfying inequality (14).
Rewrite system (1) in the formregarding an arbitrary as fixed. If is a solution of system (15), thenThere are two cases to consider: and , respectively. First, assume that (4) is valid with ; then taking norms of both sides and using conditions (), (4), and (12), we obtain the relationTaking , we getHence, for any finite , with , we haveThe last inequalities yieldBut HenceProceeding in a similar way, we haveBy (17), (22), and (23), we getButConsequently,The right-hand side of (26) does not depend on . Hence, condition , with , ensures this bound for all . Thus, it follows thatBound (27) proves the Lyapunov stability with respect to the ball , with .
Now, let . Then by a simple application of the Urysohn’s lemma [28, p.15]. and taking into account the relationwe get the Lyapunov stability in this case.
To establish the exponential stability with respect to the ball of the zero solution of system (1), let us define a new variablewith a small enough . Here is a solution of (1). Substituting (29) in (1), we obtainwhere , andThe growth condition (4) yieldsApplying our reasoning above to (30), we get according to inequality (27) that is a bounded function. Hence (29) yields the exponential stability.
Remarks 2. (a) Notice that this theorem is valid for an arbitrary logarithmic norm. For the specific case of , Theorem 8 extends some results given by [3, 10, 25].
(b) Theorem 8 asserts that any initial condition , satisfying the conditionbelongs to the region of attraction.
(c) If , then any solution of (1) with initial valueis bounded.
Example 9. Let us illustrate the obtained results by the following example:
Consider the nonlinear system in given bywhere and and are constants. are bounded nonnegative functions.
Rewrite system (35) in the vector form of (1):where , , , andDefine
Theorem 10. Assume that there exist constants such thatIn addition, letThen the zero solution of system (36) is exponentially stable with respect to a ball , with , provided that , where
Proof. Define the matrix norm by ; thenHence, for small , we have Thus,To prove the exponential stability of the zero solution of (36), it suffices that condition (4) of Theorem 8 holds for any vector , and the coefficients , are slowly varying; that is, there exists such thatBy (39), simple calculations show thatOn the other hand, by (37), it follows that Hence, by Theorem 8 the zero solution of system (35) is exponentially stable.
4. Pseudo-Linear Systems
Pseudo-linear systems are an important class of nonlinear systems. Theorem 8 will be a fundamental theorem to establish the stability and robustness of this kind of differential systems.
Consider in the equationwhere continuously depends on and Everywhere below, it is assumed that are bounded linear operators in for each , . The existence of solutions of (48) is assumed.
It is assumed that there are constants and such thatfor ; .and .
Assume that, for any logarithmic norm , the matrix satisfies for every fixed . Namely,
Remark 11. Here we will consider system (48) as a perturbation of a fixed operator . Thus, our results can be applied to robust stability; in fact, we can regardas the nominal system andas a system with state-dependent parametric perturbations.
Proof. Let us introduce the linear equationwhere is a differentiable function. If is a solution of (48), then (48) and (56) coincide. Further, the continuous dependence of solution on initial data implies that, under condition , there is , such thatfor a solution of (48). If we put with some function , then (48) takes the formDue to (49), we have Hence, But according to (48),Thus,Applying Theorem 8, with to (48), we have the boundBut condition (55) allows us to extend this bound to all . Hence, estimate (63) is proved for all . It yields the Lyapunov stability. Following the lines of Theorem 8, the exponential stability is proven.
Consider the following approximation to (1):Denote by the fundamental matrix of this system. We assume thatMake the change of variables in (1). Hence, (1) reduces to the formAssume that the right-hand side of (67) reduces to the formWe additionally assume that satisfies conditions (49)-(50) applied to (67); that is,
Proof. The proof follows directly from Theorem 12.
New conditions for the exponential stability for nonlinear finite-dimensional differential systems as well as a class of finite-dimensional pseudo-linear systems are derived. We establish the robustness of the exponential stability, in the sense that the exponential stability for a given pseudo-linear equation persists under sufficiently small perturbations. It is shown for finite-dimensional systems that the local frozen time analysis is justifiable for the systems with Hölder-like continuity which is broader than the class of slow-varying systems. The proofs are carried out using the semigroup theory combined with the freezing method and the logarithmic technique. That is, the equation is represented as a perturbation about a fixed value of the operator and then applying norm estimates for operator-valued functions the results follow. We have presented an example which shows how this approach bring out different aspects of the stability problem of pseudo-linear equations. Finally, an application of the exponential stability results for pseudo-linear differential systems is applied to an approximation to (1).
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
This research was supported by Direccion de Investigacion under Grant NU 06/16 (Chile).
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