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

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 equationwhere 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 conditionswhere 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 regardas the nominal system andas 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 equationwhere 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 thatwith 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 inequalityDenote by the discrete evolution family generated by the operator , for .

Theorem 3 (see [20]). Under conditions (10), assume thatThen the zero solution of (9) is exponentially stable with respect to a ball . Moreover, any solution of (9) with initial vector satisfies the inequalityprovided that

3. Main Results

Now, returning to system (2), we write it in the formwith a fixed integer .

PutAssume thatwhere , with the identity operator, and

Theorem 4. For a positive , let conditions (5), (6), (16)–(18), andhold. Then the zero solution of (2) is exponentially stable. Moreover, the solution of (2), with initial condition , is subject to the estimateprovided that

Proof. Let us introduce the equationwhere . If is a solution of (2), then (22) and (2) coincide. If we put , with the function , then (22) takes the formBy (5) and (6) we haveOn the other hand, by (16), we have , so it follows that .
ConsequentlyThus, applying Theorem 3 to (23) with and , the result follows.

Remark 5. Theorem 4 remains true if the linear systemsatisfies 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 relationis said to be a Hilbert-Schmidt operator, where is the adjoint operator of .
The normis called the Hilbert-Schmidt norm of .

Definition 7. A bounded linear operator is said to be quasi-Hermitian if its imaginary componentis 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). Thenfor 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 inequalityprovided that

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

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 inequalityprovided that

Proof. By Theorem 8, we obtainThis yieldsHence, by (38), we obtainIt follows thatProceeding in a similsr way, we obtain thatHence, from Theorem 9, the result follows.

Consider a perturbed system of autonomous pseudolinear equations where is a linear and continuous operator in

Let suppose thatPutIn 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 , satisfiesprovided 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 equationsassuming that the functions and are positive, bounded, and continuous with respect to the arguments , under the conditionswith a fixed positive number . Thus,where , , , and .
Hence, using the inequalitythe Hilbert-Schmidt norm of a matrix , we obtainprovided that the supremum is finite. ConsiderIf the inequalityholds, 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 inequalityprovided thatwhere

Example 13. Let , where is the Hilbert space of complex sequences, equipped with the normConsider the infinite-dimensional triangular system, defined on :where is a linear and continuous operator, represented by the infinite-dimensional matrixAdditionally, 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 thatIn fact,In many cases, is difficult to calculate. But using the inequalitywe obtain a new estimation for ; namely,It follows thatConsequently, 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 inequalityprovided 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.