Abstract and Applied Analysis
Volume 2016 (2016), Article ID 5098086, 7 pages
http://dx.doi.org/10.1155/2016/5098086

Research Article

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

Rigoberto Medina

Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933, 5290000 Osorno, Chile

Received 28 December 2015; Revised 18 March 2016; Accepted 24 March 2016

Academic Editor: Patricia J. Y.  Wong

Copyright © 2016 Rigoberto Medina. 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.

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.

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 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 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 Section 2, 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

Let be a Banach space with an arbitrary norm ,   denotes the spectral radius of an operator , and

Consider in a Banach space the equation

where is the difference operator defined by and are bounded linear operators in , continuously depending on ; that is,

is a linear bounded operator in , for each .

Denote by the set of sequences with values in :

Additionally, assume that satisfies the conditions

where and are nonnegative constants independent of , and .

Remark 1. Here we will consider system (2) as a perturbation of a fixed operator . Thus, our results can be applied to robust stability; in fact, we can regard

as the nominal system and

as 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 in the equation

where is a family of linear bounded operators in with a common domain and is continuous for each . In addition, there are nonnegative constants and , such that

with inpendent of , and .

Definition 2. The zero solution of system (9) is exponentially stable with respect to a ball if there are constants and such that any solution of (9) with initial condition satisfies the inequality

Denote by the discrete evolution family generated by the operator , for .

Theorem 3 (see [20]). Under conditions (10), assume that

Then the zero solution of (9) is exponentially stable with respect to a ball . Moreover, any solution of (9) with initial vector satisfies the inequality

provided that

3. Main Results

Now, returning to system (2), we write it in the form

with a fixed integer .

Put

Assume that

where , with the identity operator, and

Theorem 4. For a positive , let conditions (5), (6), (16)–(18), and

hold. Then the zero solution of (2) is exponentially stable. Moreover, the solution of (2), with initial condition , is subject to the estimate

provided that

Proof. Let us introduce the equation

where . If is a solution of (2), then (22) and (2) coincide. If we put , with the function , then (22) takes the form

By (5) and (6) we have

On the other hand, by (16), we have , so it follows that .
Consequently

Thus, applying Theorem 3 to (23) with and , the result follows.

Remark 5. Theorem 4 remains true if the linear system

satisfies conditions (17) and (18), and in this case condition (19) takes the form

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: Let be a separable Hilbert space and a linear compact operator in . If is an orthogonal basis in and the series converges, then the sum of the series is called the trace of the operator and is denoted by

Definition 6. An operator satisfying the relation

is said to be a Hilbert-Schmidt operator, where is the adjoint operator of .
The norm

is called the Hilbert-Schmidt norm of .

Definition 7. A bounded linear operator is said to be quasi-Hermitian if its imaginary component

is a Hilbert-Schmidt operator, where is the adjoint operator of .

Theorem 8 (see [22], p. 118). Let be a bounded linear operator acting on a separable Hilbert space and satisfying (31). Then

for any integer , where , and are the eigenvalues of , including their multiplicities, and is the spectral radius of . Assume that

Denote

Theorem 9 (see [23]). Under conditions (10) and (33), assume that

(i) are Hilbert-Schmidt operators satisfying condition (31),(ii)  

Then the zero solution of (9) is exponentially stable with respect to a ball . Moreover, any solution of (9) with initial condition satisfies the inequality

provided that

Now, in order to apply Theorem 9 to solve the stability problem of the pseudolinear system (2), let us suppose that

Put

Theorem 10. Under conditions (5), (6), and (38), assume that

(a) are Hilbert-Schmidt operators satisfying condition (5), with ,(b)  

Then the zero solution of (2) is exponentially stable with respect to a ball . Moreover, any solution of (2) with initial condition satisfies the inequality

provided that

Proof. By Theorem 8, we obtain

This yields

Hence, by (38), we obtain

It follows that

Proceeding in a similsr way, we obtain that

Hence, from Theorem 9, the result follows.

Consider a perturbed system of autonomous pseudolinear equations

where is a linear and continuous operator in

Let suppose that

Put

In this particular case because system (48) is autonomous.

Corollary 11. Under conditions (5) and (49) assume that

(a) is a Hilbert-Schmidt operator, with ,(b)  

Then the zero solution of (48) is exponentilly stable. Moreover, any solution of (48), with initial condition , satisfies

provided that

Proof. The leading part of (48), , is invariant with respect to the second argument; then . Besides, formulae (49) are independent of time. Consequently, this corollary is a consequence of Theorem 10.

Example 12. We present an example that illustrates Theorem 10. In a finite-dimensional Hilbert space ( the -dimensional Euclidean space) and , with an appropriate inner product, we consider a system of difference equations

assuming that the functions and are positive, bounded, and continuous with respect to the arguments , under the conditions

with a fixed positive number . Thus,

where , , , and .
Hence, using the inequality

the Hilbert-Schmidt norm of a matrix , we obtain

provided that the supremum is finite. Consider

If the inequality

holds, then, by Theorem 10, the zero solution of (54) is exponentially stable with respect to the ball . Moreover, any solution of (54), with initial value , satisfies the inequality

provided that

where

Example 13. Let , where is the Hilbert space of complex sequences, equipped with the norm

Consider the infinite-dimensional triangular system, defined on :

where is a linear and continuous operator, represented by the infinite-dimensional matrix

Additionally, assume that satisfies the following conditions:

(a)There exists a nonnegative such that

 where is independent of .(b),  .(c), where

(d), where

(e)  

Then the zero solution of (65) is exponentially stable. Moreover, any solution of (65), with the initial condition , satisfies the inequality

provided that

In fact,

In many cases, is difficult to calculate. But using the inequality

we obtain a new estimation for ; namely,

It follows that

Consequently, is a Hilbert-Schmidt operator for . On the other hand,

Finally,

Then, by Corollary 11, the zero solution of (65) is exponentially stable. Moreover, any solution of (65), with initial condition , satisfies the inequality

provided that

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.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research was supported by Fondecyt Chile under Grant no. 1130112.

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