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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.

The stability and robustness of differential systems have been widely investigated over the past decades; see, for example, [

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 [

Pseudo-linear systems are an important class of nonlinear systems. The stability and robustness of pseudo-linear differential equations are considered, for example, in [

Banks et al. [

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

Let

Let us consider a system described by the following equation in the Euclidean space

For a number

The zero solution of system (

(1) The stability analysis with respect to a ball has been considered by many researchers (see, e.g., Furuta and Kim [

(2) We want to point out that, considering solutions with initial functions into the region

System (

It is assumed that, for a positive number

If, for some

The logarithmic norm of a square matrix

This logarithmic norm is often used as measure of stability and asymptotic decay in analytic and numerical studies concerning to ordinary differential equations (see [

Different norms in

Let

(i)

(ii) for any norm, we have

(iii)

For the

For the

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 [

The results described here are based upon the following Coppel’s inequality ([

To establish our main results we make two basic assumptions on the coefficients of system (

There is a positive real number

For any logarithmic norm

Condition (

Suppose that conditions

Then the zero solution of (

Let us take an initial value

Rewrite system (

There are two cases to consider:

Hence

Proceeding in a similar way, we have

By (

But

The right-hand side of (

Bound (

Now, let

To establish the exponential stability with respect to the ball

(a) Notice that this theorem is valid for an arbitrary logarithmic norm. For the specific case of

(b) Theorem

(c) If

Let us illustrate the obtained results by the following example:

Consider the nonlinear system in

Rewrite system (

Assume that there exist constants

Define the matrix norm by

To prove the exponential stability of the zero solution of (

Pseudo-linear systems are an important class of nonlinear systems. Theorem

Consider in

It is assumed that there are constants

Assume that, for any logarithmic norm

Here we will consider system (

For a positive

Let us introduce the linear equation

Theorem

Consider the following approximation to (

For a positive

The proof follows directly from Theorem

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 (

The author declares that there are no conflicts of interest regarding the publication of this paper.

This research was supported by Direccion de Investigacion under Grant NU 06/16 (Chile).