# New Conditions for the Exponential Stability of Nonlinear Differential Equations

Authors: Rigoberto Medina
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

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.


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## Body

## 1. Introduction

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 α, we say that the system is exponentially stable. The concept of α-stability is related to the exponential stability with a convergence rate α>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 [8–13] 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 [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, 23–25].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 Bx,t satisfies a jointly Lipchitz inequality in x and t, 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 Section2, 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.


## 2. Preliminaries

LetCa,b,Rn be the set of all Rn-valued continuous functions on a,b.Let us consider a system described by the following equation in the Euclidean spaceRn:(1)dxdt=Atxt+Ft,xt,t≥0,where At is a matrix-valued function, continuous and uniformly bounded on 0,∞, and F:0,∞×Rn→Rn is a nonlinear and continuous vector function. The existence of solutions is assumed.For a numberη∈0,∞, put(2)Ωη=h∈Rn:h≤η.We will use · to denote norms in Rn and Rn×n, respectively.Definition 1.
The zero solution of system (1) isexponentially stable with respect to a ballΩλ if there are constants β,M>0, such that for any solution xt,t0 of (1), with initial condition xt0,t0=x0∈Ωλ, the following inequality holds:(3)xt,t0≤Me-βt-t0x0,∀t≥t0≥0.Remarks 1.
(1) 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 Ωλ can be explicitly calculated in terms of known quantities.
(2) We want to point out that, considering solutions with initial functions into the regionΩλ, we will ensure reasonable dynamics, for example, exponential decay rates.Definition 2.
System (1) is said to have astability degreeα (or to be exponentially stable), with α>0, if zt=exp⁡αtxt is a bounded function, with xt a solution of (1). In this case, the parameter α is called theconvergence rate.
It is assumed that, for a positive numberr≤∞, there is a constant γ=γr, such that the growth condition(4)Ft,u≤γu,u∈Ωr,t≥0.Remark 3.
If, for somep>1, there is a constant γp>0 such that(5)Ft,u≤γpup,u∈Rn;t≥0then for any fixed r>0, condition (4) holds with γ=γr=γprp-1.
The logarithmic norm of a square matrixB is defined by(6)μB=limε→0+I+εB-1ε.
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.Remark 4.
Different norms inRn×n give rise to different logarithmic norms. However, independently of the considered norm, a logarithmic norm always has the following properties (see [7]).Lemma 5.
LetB be a square matrix. Then
(i)μB=limε→0+⁡(ln⁡eεB/ε), μB=min⁡λ:eBt≤eλt,t≥0, and eBt≤1 if and only if μB≤0;
(ii) for any norm, we have(7)-B≤-μ-B≤ReσB≤μB≤B;
(iii)B is a stable matrix if and only if there exists a logarithmic norm μ such that μB<0.
For the1-norm x1=∑i=1nxi, the induced matrix measure μ1 is given by(8)μ1B=maxjbjj+∑i≠jnbij.
For the∞-norm x∞=max1≤i≤n⁡xi, the induced logarithmic norm is given by(9)μ∞B=maxibii+∑j≠ibij.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 ([14]): (10)eBt≤eμBt,∀t≥0,where μ is a logarithmic norm of a square matrix B.To establish our main results we make two basic assumptions on the coefficients of system (1):(H1)
There is a positive real numberq such that(11)At-As≤qt-s,∀t,s≥0.(H2)
For any logarithmic normμ, the matrix At satisfies(12)ρ≔-supt≥0⁡μAt>0.Remark 7.
Condition (12) works on matrices At, for every fixed t≥0.Theorem 8.
Suppose that conditions(H1), (H2), and (4) are satisfied. In addition, let (13)GA·,F=qρ2+γρ<1.
Then the zero solution of (1) is exponentially stable with respect to a ball Ωλ, with λ=r1-GA·,F, provided that(14)x0<r1-GA·,F.Proof.
Let us take an initial valuex0∈Ωλ, satisfying inequality (14).
Rewrite system (1) in the form(15)dxdt=Aτxt+At-Aτxt+Ft,xt,regarding an arbitrary τ≥0 as fixed. If xt is a solution of system (15), then(16)xt=exp⁡Aτtx0+∫0texp⁡Aτt-sAs-Aτxs+Fs,xsds.
There are two cases to consider:r=∞ and r<∞, respectively. First, assume that (4) is valid with r=∞; then taking norms of both sides and using conditions (H1), (4), and (12), we obtain the relation(17)xt≤eμAτtx0+∫0teμAτt-sqs-τ+γxsds.Taking t=τ, we get(18)xτ≤eμAττx0+∫0τeμAττ-sqs-τ+γxsds.Hence, for any finite T, with 0≤τ≤T, we have(19)sup0≤τ≤T⁡xτ≤sup0≤τ≤TeμAττx0+qsup0≤τ≤Txτ∫0TeμATT-sT-sds+γsup0≤τ≤Txτ∫0TeμATT-sds.The last inequalities yield(20)supt≤Txt≤supt≥0e-ρtx0+qsups≤T⁡xs∫0TeμATT-sT-sds+γsup0≤s≤T⁡xs∫0TeμATT-sds.But (21)∫0teμAtt-st-sds≤∫0te-ρt-st-sds≤∫0∞e-ρzzdz.
Hence(22)∫0teμAtt-st-sds≤1ρ2.
Proceeding in a similar way, we have(23)∫0teμAtt-sds≤∫0te-ρt-sds≤∫0∞e-ρzdz=1ρ.
By (17), (22), and (23), we get(24)supt≤Txt≤x0+sups≤Txsqρ2+γρ.
But(25)qρ2+γρ=GA·,F.Consequently,(26)supt≤Txt≤1-GA·,F-1x0.
The right-hand side of (26) does not depend on T. Hence, condition x0<λ, with λ=r1-GA·,F, ensures this bound for all T≥0. Thus, it follows that(27)supt≥0xt≤1-GA·,F-1x0.
Bound (27) proves the Lyapunov stability with respect to the ball Ωλ, with λ=r1-GA·,F.
Now, letr<∞. Then by a simple application of the Urysohn’s lemma [28, p.15]. and taking into account the relation(28)1-GA·,F-1x0<r,we get the Lyapunov stability in this case.
To establish the exponential stability with respect to the ballΩλ of the zero solution of system (1), let us define a new variable(29)xαt=xteαt,with a small enough α>0. Here xt is a solution of (1). Substituting (29) in (1), we obtain(30)dxαdt=Aαtxαt+F1t,xαt,t≥0,where Aαt=αI+At, and(31)F1t,u=eαtFt,e-αtu;u∈Ωλ;t≥0.The growth condition (4) yields(32)F1t,u≤eαtFt,e-αtu;u∈Ωγ,t≥0.Applying our reasoning above to (30), we get according to inequality (27) that xαt 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 ofμ1, Theorem 8 extends some results given by [3, 10, 25].
(b) Theorem8 asserts that any initial condition x0∈Ωλ, satisfying the condition(33)x0<r1-GA·,F,belongs to the region of attraction.
(c) IfGA·,F=0, then any solution of (1) with initial value(34)x0∈u∈Ωλ:u≤ris bounded.Example 9.
Let us illustrate the obtained results by the following example:
Consider the nonlinear system inR2 given by(35)dx1dt=-a1+d1tx1t+d2tx2t+γ1x1te-α1x1tdx2dt=-a2+d2tx2t+d1tx1t+γ2x2te-α2x2t,where aii=1,2 and αi>0 and γi>0i=1,2 are constants. diti=1,2 are bounded nonnegative functions.
Rewrite system (35) in the vector form of (1):(36)dxdt=Atxt+fxt,where At=-a1+d1td2td1t-a2+d2t, xt=x1t,x2tT, fxt=f1,f2T, and(37)fixit=γixite-αixit,i=1,2.Define(38)q=max⁡2q1,2q2,qi>0are  constants  fori=1,2.Theorem 10.
Assume that there exist constantsqi>0i=1,2 such that(39)dit-dis≤qit-s,∀t,s≥0i=1,2.In addition, let(40)min⁡a1,a2>max⁡γ1,γ2,qmin⁡a1,a22-max⁡γ1,γ2min⁡a1,a2<1.Then the zero solution of system (36) is exponentially stable with respect to a ball Ωλ0, with λ0=r1-S0, provided that x0<r1-S0, where(41)S0=qmin⁡a1,a22-max⁡γ1,γ2min⁡a1,a2.Proof.
Define the matrix norm byB=max1≤j≤2⁡∑i=12‍bij; then(42)μB=max1≤j≤2bjj+∑k=1,k≠j2bkj.Hence, for small ε>0, we have (43)I+εAt=1-εa1+d1tεd2tεd1t1-εa2+d2t=max⁡1-εa1,1-εa2=1-εmin⁡a1,a2.Thus,(44)μAt=-min⁡a1,a2;t≥0.
To prove the exponential stability of the zero solution of (36), it suffices that condition (4) of Theorem 8 holds for any vector v=v1,v2T∈Ωr, and the coefficients At, t≥0 are slowly varying; that is, there exists q>0 such that(45)At-As≤qt-s,∀t,s≥0.By (39), simple calculations show that(46)At-As≤max2q1,2q2t-s,∀t,s≥0.On the other hand, by (37), it follows that (47)fx≤diagγ1,γ2x1e-α1x1x2e-α2x2≤diagγ1,γ2x=max⁡γ1,γ2x,x∈Ωr.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. Theorem8 will be a fundamental theorem to establish the stability and robustness of this kind of differential systems.Consider inRn the equation(48)dxdt=Bxt,txt,t≥0,where Bz,t continuously depends on z∈Ωr and t≥0. Everywhere below, it is assumed that Bz,t are bounded linear operators in Rn for each z∈Ωr, t≥0. The existence of solutions of (48) is assumed.It is assumed that there are constantsηr and vr such that(49)Bh1,t-Bh,t≤vrh1-h,Bh,t-Bh,s≤ηrt-s,for h,h1∈Ωr; t,s≥0.(50)Mr=suph∈Ωr,t≥0Bh,th,and q0r=vrMr+ηr.Assume that, for any logarithmic normμ, the matrix Bht,t satisfies μBht,t<0 for every fixed t≥0. Namely,(51)ρ~=-suph∈Ωr,s≥0μBh,s>0.Remark 11.
Here we will consider system (48) as a perturbation of a fixed operator Bx0,t0. Thus, our results can be applied to robust stability; in fact, we can regard(52)dxdt=Bx0,t0xt,t≥0as the nominal system and(53)dxdt=Bx,txas a system with state-dependent parametric perturbations.Theorem 12.
For a positiver<∞, assume conditions (49)–(51) and(54)q0rρ~2<1.Then the zero solution of system (48) is exponentially stable with respect to a ball Ωλ with λ=r1-q0r/ρ~2, provided that(55)x0<r1-q0rρ~2.Proof.
Let us introduce the linear equation(56)dyhdt=Bht,tyh,t≥0,where h:0,∞→Rn is a differentiable function. If ht=xt is a solution of (48), then (48) and (56) coincide. Further, the continuous dependence of solution on initial data implies that, under condition x0<r, there is t0, such that(57)xt∈Ωrfor0≤t≤t0for a solution xt of (48). If we put At=Bht,t with some function ht, then (48) takes the form(58)dxdt=Atx.Due to (49), we have (59)Bht,t-Bhs,s≤Bht,t-Bhs,t+Bhs,t-Bhs,s≤vrht-hs+ηrt-s;t,s≤t0.Hence, (60)Bxt,t-Bxs,s≤vrxt-xs+ηrt-s≤vrsupt≤t0dxdtt-s+ηrt-s;t,s≤t0.But according to (48),(61)dxdt≤Mr,t≤t0.Thus,(62)Bxt,t-Bxs,s≤q0rt-s;t,s≤t0.Applying Theorem 8, with At=Bxt,t to (48), we have the bound(63)supt≤t0xt≤1-q0rρ~2x0,t≤t0.But condition (55) allows us to extend this bound to all t≥0. Hence, estimate (63) is proved for all t≥0. It yields the Lyapunov stability. Following the lines of Theorem 8, the exponential stability is proven.Remark 13.
Theorem12 remains true if the linear system(64)dxdt=B0,tx,t≥0satisfies (50). In this case condition (51) takes the form ρ^=-supt≥0⁡μB0,t>0, and (54) becomes ηr/ρ^2<1.


## 5. Application

Consider the following approximation to (1):(65)dxdt=Atx,xt0=x0.Denote by Ut,t0 the fundamental matrix of this system. We assume that(66)supt≥t0Ut,t0≤m1t0,supt≥t0U-1t,t0≤m2t0.Make the change of variables xt=Ut,t0yt in (1). Hence, (1) reduces to the form(67)dydt=U-1t,t0Ft,Ut,t0yt,yt0=y0=x0.Assume that the right-hand side of (67) reduces to the form(68)U-1t,t0Ft,Ut,t0yt=B~y,ty.We additionally assume that B~y,t satisfies conditions (49)-(50) applied to (67); that is,(69)M~r=suph∈Ωr,t≥t0B~h,th,q~0r=vrM~r+η~r,ρ^=-suph∈Ωr,s≥0μB~h,s>0.Theorem 14.
For a positiver<∞, assume conditions (49) and (69) hold. In addition, assume that(70)q~0rρ^2-1<1.Then the zero solution of system (65) is exponentially stable with respect to a ball Ωλ^, with λ^=r1-q~0rρ^2-1, provided that(71)x0<r1-q~0rρ^2-1.Proof.
The proof follows directly from Theorem12.


## 6. 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).



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