# New Conditions for the Exponential Stability of Pseudolinear Difference Equations in Banach Spaces

Authors: Rigoberto Medina
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method.


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## Body

## 1. Introduction

The problem of stability and robustness of difference systems has been extensively studied in the last years [1–7]. However, mainly systems with linear leading parts were investigated. In fact, a few investigations have dealt with stability conditions for nonlinear difference or differential systems (see, e.g., [3, 8–14]). This itself highlights the importance of establishing new criteria to study exponential stability of nonlinear systems. Pseudolinear systems are an important class of nonlinear systems. The stability and robustness of pseudolinear differential equations are considered, for example, in [15–19].Martynyuk [19] derived new bounds for solutions of perturbed pseudolinear differential equations, basically using Gronwall-type inequalities. Dvirnyi and Slyn’ko [16, 17], constructing a piecewise differential Lyapunov function, established the stability of solutions to impulsive differential equations with impulsive action in the pseudolinear form. Banks et al. [15], 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 pseudolinear form. In summary, in the existing literature there are many results concerning the stability or asymptotic behavior of pseudolinear differential equations; however, in general, the assumptions are difficult to check or conservative.Our results compare favorably with the abovementioned works in the following sense:(a)
We established a local stability theory of discrete evolution equations.(b)
The Lipschitz assumptions are local in the state and general in the time.(c)
Explicit estimates to the norm of the associated evolution semigroups are established.In this paper we consider evolution difference systems defined in infinite-dimensional Banach spaces, with bounded operators on the right-hand side represented in the pseudolinear form. New estimates for the norms of solutions are derived giving us explicit stability and boundedness conditions. Our approach is based on the generalization of the freezing method to abstract difference systems. 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.Although the freezing method appears to be often utilized in practice in the control of linear time-varying systems, not much is currently known regarding the stability or asymptotic behavior of pseudolinear difference systems of the form considered here. In fact, we will develop a local stability theory to evolution pseudolinear difference systems.The remainder of this paper is organized as follows: In Section2, we introduce some notations, the statement of the problem, and a fundamental result concerning the exponential stability of abstract difference equations. In Section 3, sufficient conditions for the exponential stability of pseudolinear difference systems are given. In Section 4 pseudolinear difference systems in Hilbert spaces are studied. Finally, Section 5 is devoted to the discussion of our results: We highlight the main conclusions.


## 2. Statement of the Problem

LetX be a Banach space with an arbitrary norm ·,  ρA denotes the spectral radius of an operator A, and(1)ΩR=x∈X:x≤R,R≤∞.Consider in a Banach spaceX the equation(2)Δxk=Bxk,kxk;x0=x0,k=0,1,…,where Δ is the difference operator defined by Δuk=uk+1-uk and Bz,k are bounded linear operators in X, continuously depending on z∈ΩR; that is,(3)Bz,k:DBz,k⊆X⟶Xis a linear bounded operator in X, for each z∈X,k≥0.Denote byω∞R the set of sequences with values in ΩR:(4)ω∞R=hj∈ΩR:j=0,1,….Additionally, assume that Bz,k satisfies the conditions(5)Bh1,k-Bh,kω≤γRh1-hω,(6)Bh,k-Bh,sω≤μRk-sω,h1,h,ω∈ΩR;k,s≥0,where γR and μR are nonnegative constants independent of k,s, and ω.Remark 1.
Here we will consider system (2) as a perturbation of a fixed operator Bx0,k0. Thus, our results can be applied to robust stability; in fact, we can regard(7)Δxk=Bx0,k0xkas the nominal system and(8)Δxk=Bxk,kxkas a system with state-dependent parametric perturbations.
In order to establish the stability properties of (2), we will formulate a fundamental theorem concerning the exponential stability of the following nonlinear system of difference equations.
Consider inX the equation(9)xk+1=Akxk+fxk,k;k=0,1,…,where Akk=0∞ is a family of linear bounded operators in X with a common domain DAk≔D⊆X and f·,k:ΩR→X is continuous for each k≥0. In addition, there are nonnegative constants μ=μR and q=qR, such that(10)fz,k≤μz,z∈ΩR;k=0,1,…,Ak-Asω≤qk-sω,k,s≥0;ω∈ΩR,with q inpendent of k,s, and ω.Definition 2.
The zero solution of system (9) isexponentially stable with respect to a ball Ωη if there are constants M>0 and c0∈0,1 such that any solution xk of (9) with initial condition x0=x0∈Ωη satisfies the inequality(11)xk,x0≤Mc0kx0,∀k=0,1,….Denote by Ψ=Akjk≥j≥0 the discrete evolution family generated by the operator Aj, for j≥0.Theorem 3 (see [20]).
Under conditions (10), assume that(12)M=supk≥0⁡supl≥0⁡Akl<∞,θ0R=∑k=0∞qk+μsupl≥0⁡Akl<1.Then the zero solution of (9) is exponentially stable with respect to a ball ΩR. Moreover, any solution of (9) with initial vector x0∈ΩR satisfies the inequality(13)supk≥1⁡xk≤Mx01-θ0,provided that(14)Mx01-θ0-1<R.


## 3. Main Results

Now, returning to system (2), we write it in the form(15)Δxk=Bhs,sxk+Bxk,k-Bhs,sxk,k=0,1,…,with a fixed integer s.Put(16)MR=suph∈ΩR,k≥0⁡Bh,kh,q0R=γRMR+μR.Assume that(17)pR,k=suph∈ΩR,s≥0⁡B^kh,s<∞,where B^h,k=I+Bh,k, with I the identity operator, and(18)χR≔supk≥0⁡pR,k<∞.Theorem 4.
For a positiveR<∞, let conditions (5), (6), (16)–(18), and(19)θ~0R=q0R∑k=0∞kpR,k<1hold. Then the zero solution of (2) is exponentially stable. Moreover, the solution xk of (2), with initial condition x0∈ΩR, is subject to the estimate(20)supk≥1⁡xk≤χRx01-θ~0R-1,provided that(21)χRx01-θ~0R-1<R.Proof.
Let us introduce the equation(22)Δyhk=Bhk,kyhk,k=0,1,…,where h∈ω∞R. If hk=xk is a solution of (2), then (22) and (2) coincide. If we put Ak=Bhk,k, with the function hk, then (22) takes the form(23)Δxk=Akxk,k=0,1,….By (5) and (6) we have(24)Bhk,k-Bhs,s≤Bhk,k-Bhs,k+Bhs,k-Bhs,s.On the other hand, by (16), we have Δxk≤MR, so it follows that xk-xs≤MRk-s,∀k≥s≥0.
Consequently(25)Bhk,k-Bhs,s≤γRxk-xs+μRk-s≤γRMR+μRk-s=q0Rk-s,∀k≥s≥0.Thus, applying Theorem 3 to (23) with Ak=B^hk,k and f≡0, the result follows.Remark 5.
Theorem4 remains true if the linear system(26)Δxk=B0,kxk,k=0,1,…satisfies conditions (17) and (18), and in this case condition (19) takes the form (27)θ~0R=μR∑k=0∞kpR,k<1.


## 4. Pseudolinear Systems in Hilbert Spaces

As the previous theorems show, the extension of the freezing method to evolution equations is based on norm estimates for relevant semigroups. However, obtaining these estimates is usually not an easy task. Because of this, we restrict ourselves by equations in a separable Hilbert space with Hilbert-Schmidt coefficient operators.In the finite-dimensional case, the spectrum of a linear operator consists of its eigenvalues. The spectral theory of bounded linear operators on infinite-dimensional spaces is an important but challenging area of research. For example, an operator may have a continuous spectrum in addition to, or instead of, a point spectrum of eigenvalues. A particularly simple and important case is that of compact, self-adjoint operators. Compact operators may be approximated by finite-dimensional operators, and their spectral theory is close to that of finite-dimensional operators. We will assume that coefficient operators of (2) are not necessarily compact or self-adjoint. See, for example, [21, 22].To formulate the next results, let us introduce the following notations and definitions: LetH be a separable Hilbert space and A a linear compact operator in H. If ekk=1∞ is an orthogonal basis in H and the series ∑k=1∞Aek,ek converges, then the sum of the series is called the trace of the operator A and is denoted by(28)TraceA=Tr⁡A=∑k=1∞Aek,ek.Definition 6.
An operatorA satisfying the relation(29)Tr⁡A∗A<∞is said to be aHilbert-Schmidt operator, where A∗ is the adjoint operator of A.
The norm(30)N2A=NA=TrA∗Ais called theHilbert-Schmidt norm of A.Definition 7.
A bounded linear operatorA is said to bequasi-Hermitian if its imaginary component(31)AI=A-A∗2iis a Hilbert-Schmidt operator, where A∗ is the adjoint operator of A.Theorem 8 (see [22], p. 118).
LetA be a bounded linear operator acting on a separable Hilbert space and satisfying (31). Then(32)Am≤∑k=0mm!ρAm-kgIkAm-k!k!3/2for any integer m≥1, where gIA=2N22AI-∑k=1∞Im⁡λkA21/2, and λkA are the eigenvalues of A, including their multiplicities, and ρA is the spectral radius of A. Assume that(33)v^0=supk≥0⁡gIAk<∞,ρ^0=supk≥0⁡ρAk<1.Denote(34)M^0=supm≥1⁡∑k=0mm!ρ^0m-kv^0km-k!k!3/2.Theorem 9 (see [23]).
Under conditions (10) and (33), assume that(i)
Akk=0,1,… are Hilbert-Schmidt operators satisfying condition (31),(ii)
(35)SA·,f=∑j=0∞v^0jj!j+1q1-ρ^0j+2+μ1-ρ^0j+1<1.Then the zero solution of (9) is exponentially stable with respect to a ball ΩR. Moreover, any solution xk of (9) with initial condition x0∈ΩR satisfies the inequality(36)supk≥1⁡xk≤M^0x01-SA·,f-1,provided that(37)M^0x01-SA·,f-1<R.Now, in order to apply Theorem9 to solve the stability problem of the pseudolinear system (2), let us suppose that(38)ρ0R=suph∈ΩR,k≥0⁡ρB^h,k<1,v0R=suph∈ΩR,k≥0⁡gIB^h,k<∞.Put(39)χ~R=supk≥1⁡∑j=0kk!ρ0k-jRv0jRk-j!j!3/2.Theorem 10.
Under conditions (5), (6), and (38), assume that(a)
B^z,kk=0,1,… are Hilbert-Schmidt operators satisfying condition (5), with z∈ΩR,(b)
(40)SB^·=∑k=0∞v0kRk!k+1q0R1-ρ0Rk+2<1.Then the zero solution of (2) is exponentially stable with respect to a ball ΩR. Moreover, any solution xk of (2) with initial condition x0∈ΩR satisfies the inequality(41)supk≥1⁡xk≤χ~Rx01-SB^·-1,provided that(42)χ~Rx01-SB^·-1<R.Proof.
By Theorem8, we obtain(43)B^h,sk≤∑j=0kk!ρB^h,sk-jgIjB^h,sk-j!j!3/2,h∈ΩR;k,s≥0.This yields(44)suph∈ΩR,s≥0⁡B^h,sk≤suph∈ΩR,s≥0⁡∑j=0kk!ρB^h,sk-jgIjB^h,sk-j!j!3/2.Hence, by (38), we obtain(45)pR,k≤∑j=0kk!ρ0Rk-jv0Rjk-j!j!3/2,∀k≥1.It follows that(46)χR=supk≥0⁡pR,k≤supk≥1⁡∑j=0kk!ρ0Rk-jv0Rjk-j!j!3/2.Proceeding in a similsr way, we obtain that(47)q0R∑k=0∞kpR,k≤∑k=0∞v0Rkk!k+1q0R1-ρ0Rk+2.Hence, from Theorem 9, the result follows.Consider a perturbed system of autonomous pseudolinear equations(48)Δxk=Bxkxk,x0=x0;k=0,1,…,where Bz is a linear and continuous operator in ω∞R.Let suppose that(49)ρ~0R=suph∈ΩR⁡ρB^h<1,v~0R=suph∈ΩR⁡gIB^h<∞.Put(50)χ~R=supk≥1⁡∑j=0kk!ρ~0k-jRv~0jRk-j!j!3/2.In this particular case μR=0 because system (48) is autonomous.Corollary 11.
Under conditions (5) and (49) assume that(a)
B^z is a Hilbert-Schmidt operator, with z∈ΩR,(b)
(51)S~B^·=γRMR∑k=0∞v~0kRk!1-ρ~0Rk+2<1.Then the zero solution of (48) is exponentilly stable. Moreover, any solution xk of (48), with initial condition x0∈ΩR, satisfies(52)supk≥1⁡xk≤χ~Rx01-S~B^·-1,provided that(53)χ~Rx01-S~B^·-1<R.Proof.
The leading part of (48), Bx, is invariant with respect to the second argument; then μR=0. Besides, formulae (49) are independent of time. Consequently, this corollary is a consequence of Theorem 10.Example 12.
We present an example that illustrates Theorem10. In a finite-dimensional Hilbert space E2,·∞ (E2 the 2-dimensional Euclidean space) and x∞=maxx1,x2, with an appropriate inner product, we consider a system of difference equations(54)Δukvk=01-ξuk,vk,k-fuk,vk,kukvk,assuming that the functions f=fu,v,k and ξ=ξu,v,k are positive, bounded, and continuous with respect to the arguments u,v, under the conditions(55)k≥0,u2+v2≤R2with a fixed positive number R<∞. Thus,(56)Buk,vk,k=biju,v,ki,j=1,2,where b11u,v,k=0, b12u,v,k=1, b21u,v,k=-ξ, and b22u,v,k=-f.
Hence, using the inequality(57)g2A≤N2A-TraceA2,the Hilbert-Schmidt norm of a matrix A, we obtain(58)v^0R=2+supk≥0,u2+v2≤R2⁡ξu,v,k,provided that the supremum is finite. Consider(59)q0R=supk≥0,u2+v2≤R2⁡fu,v,k+1-fu,v,k,ξu,v,k+1-ξu,v,k,ρ0R=infk≥0,u2+v2≤R2⁡fu,v,k2-fu,v,k4-ξu,v,k1/2.
If the inequality(60)SR=q~0R11-ρ0R2+2v^0R1-ρ0R3<1holds, then, by Theorem 10, the zero solution of (54) is exponentially stable with respect to the ball u2+v2≤R2. Moreover, any solution xk=uk,vk of (54), with initial value x0=u0,v0∈E2, satisfies the inequality(61)supk≥1⁡xk≤χ~Rx01-SR-1,provided that(62)χ~Rx01-SR-1<R,where(63)χ~R=supk≥1⁡ρ0kR+kv^0Rρ0k-1R.Example 13.
LetH=l2C, where l2≔l2C is the Hilbert space of complex sequences, equipped with the norm(64)ul2=∑j=1∞uj21/2,u=uj∈l2.Consider the infinite-dimensional triangular system, defined on l2:(65)Δxk=Bxkxk,x0=x0∈ΩR,where Bz is a linear and continuous operator, represented by the infinite-dimensional matrix(66)Bz=bijz,1≤i≤j;z∈l2.Additionally, assume that Bz satisfies the following conditions:(a)
There exists a nonnegativeγR such that(67)Bh1-Bhω≤γRh1-hω,whereγR is independent of ω∈ΩR.(b)
ρ~0R=suph∈ΩR⁡ebiih<1,  i=1,2,….(c)
v~0R≤JB^·<∞, where(68)JB^·=12suph∈ΩR⁡∑j=1∞∑k=1,k≠j∞bjkh-b¯jkh21/2(d)
χ~R≤χ~1R, where(69)χ~1R=supk≥1⁡∑j=0kρ~0Rk-jJjB^·k-j!j!3/2.(e)
(70)S~1B^·=γRMR∑k=0∞JkB^·k!1-ρ~0Rk+2<1.Then the zero solution of (65) is exponentially stable. Moreover, any solution xk of (65), with the initial condition x0∈ΩR, satisfies the inequality (71)supk≥1⁡xk≤χ~1Rx01-S~1B^·-1,provided that(72)χ~1Rx01-S~1B^·-1<R.In fact,(73)ρ~0R=suph∈ΩR⁡ρB^·=suph∈ΩR⁡1+biih,i=1,2,…≤suph∈ΩR⁡ebiih,i=1,2,….In many cases, gIA is difficult to calculate. But using the inequality(74)gI2A≤12N2AI,we obtain a new estimation for gIA; namely,(75)gIA≤12∑j=1∞∑k=1,k≠j∞ajk-a¯jk21/2.It follows that(76)v~0R=suph∈ΩR⁡gIB^h≤12suph∈ΩR⁡∑j=1∞∑k=1,k≠j∞bjkh-b¯jkh21/2≔JB^·<∞.Consequently, B^h is a Hilbert-Schmidt operator for h∈ΩR. On the other hand, (77)χ~R=supk≥1⁡∑j=0∞ρ~0Rk-jv~0Rjk-j!j!3/2≤supk≥1⁡∑j=0∞ρ~0Rk-jJjB^·k-j!j!3/2≔χ~1R.Finally,(78)S~B^·=γRMR∑k=0∞v~0Rkk!1-ρ~0Rk+2≤γRMR∑k=0∞JkB^·k!1-ρ~0Rk+2.Then, by Corollary 11, the zero solution of (65) is exponentially stable. Moreover, any solution xk of (65), with initial condition x0∈ΩR, satisfies the inequality(79)supk≥1⁡xk≤χ~1Rx01-S~1B^·-1,provided that(80)χ~1Rx01-S~1B^·-1<R.


## 5. Conclusions

New conditions for the exponential stability of a class of infinite-dimensional nonlinear difference systems are derived. We establish the robustness of the exponential stability, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. Unlike the classic method of stability analysis, we do not use the technique of Lyapunov function in the process of construction of the stability results. The proofs are carried out using the semigroup theory combined with the freezing method. 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 two examples which show how this approach brings out different aspects of the stability problem of pseudolinear equations.



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