New Conditions for the Exponential Stability of Nonlinear Differential Equations

and Applied Analysis 3 Remark 4. Different norms in Rn×n give rise to different logarithmic norms. However, independently of the considered norm, a logarithmic normalways has the following properties (see [7]). Lemma 5. Let B be a square matrix. Then (i) μ(B) = limε→0+((ln ‖eεB‖)/ε), μ(B) = min{λ : ‖eBt‖ ≤ eλt, t ≥ 0}, and ‖eBt‖ ≤ 1 if and only if μ(B) ≤ 0; (ii) for any norm, we have − ‖B‖ ≤ −μ (−B) ≤ Reσ (B) ≤ μ (B) ≤ ‖B‖ ; (7) (iii) B is a stable matrix if and only if there exists a logarithmic norm μ such that μ(B) < 0. For the 1-norm ‖x‖1 = ∑i=1 |xi|, the induced matrix measure μ1 is given by μ1 (B) = max j {{ bjj + n ∑ i ̸ =j 󵄨󵄨󵄨󵄨󵄨bij󵄨󵄨󵄨󵄨󵄨 }} . (8) For the ∞-norm ‖x‖∞ = max1≤i≤n|xi|, the induced logarithmic norm is given by μ∞ (B) = max i {{ bii +∑ j ̸ =i 󵄨󵄨󵄨󵄨󵄨bij󵄨󵄨󵄨󵄨󵄨 }} . (9) Remark 6. Although logarithmic norm is only defined for constant fixedmatrices, it can be applied to anymatrix, 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]).


Introduction
The stability and robustness of differential systems have been widely investigated over the past decades; see, for example, [1][2][3][4][5][6][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  > 0.
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][9][10][11][12][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][15][16].In contrast, many alternative methods to Lyapunov's functions have been successfully applied to the stability analysis of differential systems, for example, Ngoc [11], 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. [17], using the concept of Lyapunov exponents and Bohl exponents, discuss the problem of stabilization for linear time-varying systems with bounded matrices.Coppel [15], 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 [18] and Gil and Medina [19].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 [12].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 [15], and the oscillatory behavior of retarded functional differential equations [22].
Pseudo-linear systems are an important class of nonlinear systems.The stability and robustness of pseudo-linear differential equations are considered, for example, in [8,10,[23][24][25].
Banks et al. [8] and Martynyuk [25] 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. [8], 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 pseudolinear 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 us consider a system described by the following equation in the Euclidean space   :   =  ()  () +  (,  ()) ,  ≥ 0, where () is a matrix-valued function, continuous and uniformly bounded on [0, ∞), and  : [0, ∞) ×   →   is a nonlinear and continuous vector function.The existence of solutions is assumed.For a number  ∈ (0, ∞], put We will use ‖ ⋅ ‖ to denote norms in   and  × , respectively. Definition 1.The zero solution of system ( 1) is exponentially stable with respect to a ball Ω() if there are constants ,  > 0, such that for any solution (,  0 ) of ( 1), with initial condition ( 0 ,  0 ) =  0 ∈ Ω(), the following inequality holds: The stability analysis with respect to a ball has been considered by many researchers (see, e.g., Furuta and Kim [26] and Hsiao et al. [27]).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 Ω() can be explicitly calculated in terms of known quantities.
(2) We want to point out that, considering solutions with initial functions into the region Ω(), 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  > 0, if () = [exp()]() is a bounded function, with () a solution of (1).In this case, the parameter  is called the convergence rate.
The logarithmic norm of a square matrix  is defined by This 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.
For the 1-norm ‖‖ 1 = ∑  =1 |  |, the induced matrix measure  1 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]).

Main Results
The results described here are based upon the following Coppel's inequality ( [14]):             ≤  () , ∀ ≥ 0, 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): (H 1 ) There is a positive real number  such that (H 2 ) For any logarithmic norm , the matrix () satisfies Remark 7. Condition (12) works on matrices (), for every fixed  ≥ 0.
Remarks 2. (a) Notice that this theorem is valid for an arbitrary logarithmic norm.For the specific case of  1 , Theorem 8 extends some results given by [3,10,25].(b) Theorem 8 asserts that any initial condition  0 ∈ Ω(), satisfying the condition belongs to the region of attraction.
Rewrite system (35) in the vector form of (1): where In addition, let Then the zero solution of system ( 36) is exponentially stable with respect to a ball Ω( 0 ), with  0 = (1 −  0 ), provided that ‖(0)‖ < (1 −  0 ), where Proof.Define the matrix norm by Hence, for small  > 0, 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 V = (V 1 , V 2 )  ∈ Ω(), and the coefficients (),  ≥ 0 are slowly varying; that is, there exists  > 0 such that By (39), simple calculations show that On the other hand, by (37), it follows that Hence, by Theorem 8 the zero solution of system (35) is exponentially stable.

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.

Conclusions
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).