Uniform asymptotic stability of time-varying damped harmonic oscillators

By Kazuki Ishihara and Jitsuro Sugie

Abstract

This paper presents sufficient conditions which guarantee that the equilibrium of the damped harmonic oscillator

is uniformly asymptotically stable, where is locally integrable. These conditions work to suppress the rapid growth of the frictional force expressed by the integral amount of the damping coefficient . The obtained sufficient conditions are compared with known conditions for uniform asymptotic stability. Two diagrams are included to facilitate understanding of the conditions. By giving a concrete example, remaining problems are pointed out.

1. Introduction

We consider the second-order linear differential equation

where the prime denotes , the coefficient is a nonnegative and locally integrable function on , and the number is a positive constant. The only equilibrium of Equation 1 is the origin . Equation Equation 1 is often called the damped harmonic oscillator when is a positive constant. Although the damped harmonic oscillator has a very simple form, there are extremely wide applications in science and engineering. Equation Equation 1 has been studied as one of the important physical phenomenon models by many researchers.

In the qualitative theory of differential equations, the study of asymptotic stability and uniform asymptotic stability occupy very important positions. The purpose of this paper is to present some growth condition about for the equilibrium of Equation 1 to be uniformly asymptotically stable and to clarify the relationship between these conditions. Before advancing to the main subject, it is useful to briefly describe the history of study of the asymptotic stability of Equation 1 and the results obtained.

Since equation Equation 1 is linear, if the equilibrium is attractive, then it is stable. Hence, we need only show that each solution of Equation 1 and its derivative tend to zero as in order to prove that the equilibrium is asymptotically stable. Many efforts have been made to find sufficient (also necessary and sufficient) conditions which guarantee that the equilibrium of Equation 1 is asymptotically stable (for example, see Reference 1Reference 3Reference 4Reference 9Reference 10Reference 11Reference 12Reference 13Reference 16Reference 18Reference 19Reference 21Reference 24Reference 25Reference 26Reference 27Reference 31). Among them, we should mention especially the criterion given by Smith Reference 21, Theorems 1 and 2. Let

Then the following result holds.

Theorem A.

Suppose that

Then the equilibrium of Equation 1 is asymptotically stable if and only if

When satisfies condition Equation 2, namely, has a positive lower bound , it is often called large damping. Smith’s condition Equation S is satisfied when has an upper bound or . On the other hand, condition Equation S is not satisfied when (for the proof, see Reference 11). From these facts, we see that condition Equation S prohibits rapid growth of the damping coefficient . Since condition Equation S is necessary and sufficient for the asymptotic stability of Equation 1, it is not too much to say that condition Equation S is very excellent. However, the weak point is that it is hard to check whether condition Equation S is satisfied or not. Although Artstein and Infante Reference 1 did not point out this fact, they gave another growth condition that guarantees the asymptotic stability of Equation 1 as follows.

Theorem B.

Suppose that condition Equation 2 holds. Then

implies that the equilibrium of Equation 1 is asymptotically stable.

Artstein and Infante’s condition Equation A requires that has to grow more slowly than . Artstein and Infante Reference 1 also showed that the exponent 2 of is best possible in the sense that it cannot be replaced by for any . Of course, condition Equation A is not as sharp as condition Equation S. For example, consider . Then it is clear that is unbounded. Hence, condition Equation A is not satisfied. However, by means of Ballieu and Peiffer’s result Reference 3, Corollary 7, we can verify that the equilibrium of Equation 1 is asymptotically stable in this example.

The advantage of condition Equation A is that it is easy to check. When an indefinite integral of is found, we may judge whether condition Equation A is satisfied or not. By numerical computation, it may be easy to check condition Equation A. However, it is very hard to confirm condition Equation S even with numerical computation.

Since condition Equation A is merely a sufficient condition for the asymptotic stability of Equation 1, it follows from condition Equation S. From another viewpoint, Hatvani et al. Reference 11 verified that condition Equation A implies condition Equation S. They proved that condition Equation S is equivalent to the discrete growth condition

where

provided that diverges to as (see Reference 11, Theorem 1.1). They also showed that condition Equation A implies condition Equation D in the proof of Corollary 3.7 in Reference 11 (see also Reference 1). Moreover, they gave another growth condition,

for any fixed natural number (their original form is slightly different) and clarified that condition Equation H implies condition Equation D under certain conditions including condition Equation 2 (see Reference 11, Corollary 3.6). We can show that condition Equation A implies condition Equation H by using Artstein and Infante’s result Reference 1, Lemma (for the proof, see Appendix). To sum up, we have the following diagram.

The equilibrium of Equation 1 is said to be asymptotically stable [AS] if

for every solution of Equation 1. As is well known, the equilibrium is not necessarily uniformly asymptotically stable even if it is [AS]. We need to check that each solution of Equation 1 and its derivative converge to zero with the speed of the same level in order to prove that the equilibrium is uniformly asymptotically stable. In this sense, we need to pay close attention to the analysis of uniform asymptotic stability.

To strictly describe definitions, we give some notation. Let and , and let be any suitable norm. We denote the solution of Equation 1 through by . The uniqueness of solutions of Equation 1 is guaranteed for the initial value problem.

The equilibrium is said to be eventually uniformly stable [EvUS] if for any , there exist an and a such that and imply that for all . If we can choose , the equilibrium is said to be uniformly stable [US]. The equilibrium is said to be eventually uniformly attractive [EvUA] if there exist an and a , and if for every there is a such that and imply that for all . If we can choose , the equilibrium is said to be uniformly attractive [UA]. The equilibrium is eventually uniformly asymptotically stable [EvUAS] if it is [EvUS] and [EvUA]. The equilibrium is uniformly asymptotically stable [UAS] if it is [US] and [UA]. With respect to the various definitions of stability, the reader may refer to the books Reference 2Reference 5Reference 6Reference 15Reference 20Reference 32 for example.

It is well known that the equilibrium of Equation 1 is uniformly asymptotically stable if and only if it is exponentially asymptotically stable [ExpAS]; namely, there exists a and, for any , there exists a such that and imply that for all . If the equilibrium of Equation 1 is [ExpAS], then the existence of a good Lyapunov function that satisfies the following conditions is guaranteed:

(i)

,

(ii)

  or   ,

(iii)

on , where , and are continuous increasing and positive definite functions, is a positive constant and is a positive suitable function (this is called a converse theorem on [UAS]). However, if the zero solution of Equation 1 is merely only asymptotically stable, such a good Lyapunov function does not necessarily exist. This is a big difference between [UAS] and [AS]. By using the converse theorem on [UAS], we can show that the uniform asymptotic stability is maintained even if a small perturbation term is added to equation Equation 1. Small errors cannot be ignored in model design. For this reason, it is necessary to consider the perturbation problem in actual phenomena analysis. From this point of view, the study of [UAS] is very important.

In this paper, we use the terminology “uniformly with respect to ”. This means the following: Let be a family of functions parametrized by . We say that uniformly with respect to if, and only if, for any there exists a such that and imply . We also use the symbol to mean the greatest integer that is less than or equal to a real number .

Sugie and Onitsuka Reference 30, Theorem 1.1 gave the following result.

Theorem C.

Suppose that

If

then the equilibrium of Equation 1 is uniformly asymptotically stable.

When condition Equation 3 holds, the damping coefficient is said to be integrally positive. The concept of the integral positivity was introduced by Matrosov Reference 14 (see also Reference 7Reference 8Reference 9Reference 17Reference 24Reference 28Reference 29). It is obvious that condition Equation 2 implies condition Equation 3. However, the converse is not always true. Integrally positive functions are allowed to have an infinite number of zeros. A typical example of integrally positive functions is .

Let us compare condition Equation 4 with condition Equation S. We notice that both are double integrals of . Condition Equation 4 requires that this double integral diverges uniformly with respect to . On the other hand, condition Equation S only needs to diverge when . Hence, we may say that Equation 4 is a uniform divergence condition. A growth condition similar to condition Equation 4 was first presented by Hatvani Reference 9, Theorem 2.5 as a sufficient condition for the zero solution of a certain two-dimensional linear system to become asymptotically stable.

By the same method as in the proof of Theorem C, we can obtain the following result.

Theorem D.

Suppose that condition Equation 3 holds. If

then the equilibrium of Equation 1 is eventually uniformly asymptotically stable.

For nonlinear differential equations, the concept of [EvUAS] is different from that of [UAS]. Of course, the equilibrium is uniformly asymptotically stable, so it is eventually uniformly asymptotically stable; namely, [UAS] implies [EvUAS]. Strauss and Yorke Reference 22, Lemma 2.7 gave a necessary and sufficient condition for the converse to be true as follows (see also Reference 23).

Theorem E.

Suppose that the equilibrium is eventually uniformly asymptotically stable. Then it is uniformly asymptotically stable if and only if the zero function is a unique solution defined on the interval

Since equation Equation 1 is linear, from Theorem E it turns out that [EvUAS] is equivalent to [UAS]. Hence, condition Equation SU is a growth condition on uniform asymptotic stability. In this paper, based on the uniform divergence condition Equation SU, we intend to present other growth conditions on uniform asymptotic stability and give a correlation diagram showing their relation.

2. Conditions for suppressing the rapid growth of

As mentioned in Section 1, Hatvani et al. Reference 11 presented the discrete growth condition for the asymptotic stability of Equation 1, which is equivalent to Smith’s condition Equation S. Inspired by this result, Sugie and Onitsuka Reference 30, Theorem 4.2 gave the discrete growth condition

for uniform asymptotic stability of Equation 1 and proved that condition Equation 5 implies the uniform divergence condition Equation 4 under the assumption Equation 3. Using the same idea, we can show that

implies condition Equation SU. We may regard condition Equation DU as a discrete version of Equation D.

Unfortunately, in general, it is not so easy to check whether a given satisfies conditions Equation SU and Equation DU. In this section, we propose other growth conditions corresponding to conditions Equation A and Equation H given by Artstein and Infante Reference 1 and Hatvani et al. Reference 11, respectively. We also reveal implications between conditions Equation SU, Equation DU and these new growth conditions.

Theorem 1.

Suppose that If

then condition Equation DU holds.

Remark 1.

If condition Equation 3 is satisfied, then . Condition Equation AU is a uniform convergence version of condition Equation A of Artstein and Infante Reference 1.

Theorem 2.

Suppose that

If condition Equation AU is satisfied, then

Remark 2.

Condition Equation 3 implies condition Equation 6. Condition Equation HU is a uniform divergence version of condition Equation H of Hatvani et al. Reference 11.

Theorem 3.

Suppose that

Then condition Equation HU yields condition Equation SU.

Combining Theorems 1, 2 and 3 with Theorems D and E, we can give the diagram for [UAS] shown in Figure 2:

Figure 2.

The marks ”, ”, and [UAS] mean “addition to”, “implies”, “if and only if” and the uniform asymptotic stability of Equation 1, respectively.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{SVG} \let\eqref\fakeeqref\begin{align*} &\hspace{-3mm}\protect\eqref{eq:3}\Longrightarrow\protect\eqref{eq:6} \\[-1mm] &\hspace{9.5mm}\downarrow\hspace{15.4mm}\protect\eqref{eq:2}\Longrightarrow\protect\eqref{eq:3} \\[-1.5mm] &{\rm(AU)}\Longrightarrow\!\!\:\protect\eqref{eq:HU}\hspace{5.7mm}\Uparrow\\[0mm] \lim_{t\to\infty}H(t) = \infty\longrightarrow\!\!\!&\hspace{1mm}\Downarrow\hspace{14mm}\Downarrow\;\longleftarrow\protect\eqref{eq:7} \\[-1mm] \Uparrow\hspace{14mm}&\protect\eqref{eq:DU}\Longrightarrow\!\protect\eqref{eq:SU}\!\Longrightarrow\![\rm{UAS}] \\[-1mm] \protect\eqref{eq:3}\hspace{12.7mm}&\hspace{9.5mm}\uparrow\hspace{12.3mm}\uparrow\\[-1mm] &\hspace{9mm}\protect\eqref{eq:3}\hspace{9.75mm}\protect\eqref{eq:3} \\[-5.5mm] \end{align*} \end{SVG}

3. Proofs

Proof of Theorem 1.

Let and for all . Since , the sequence is increasing and diverges to as . Hence, we can choose an so that

Define . In order to show condition Equation DU, it suffices to show that for any there exists an such that implies that

for any . From Equation AU it follows that

For any , let

Suppose that there exists an with such that

Then, since is an increasing function on and for all , we see that

Since is an increasing sequence, we see that . Using Equation 8 with and , we obtain

Hence, we have

This contradicts Equation 9. We therefore conclude that

We can rewrite Equation 8 as

Using Equation 10 and Equation 11, we get

Hence, we have

that is,

By means of the Cauchy-Bunyakovski-Schwarz inequality, we have

for any . Hence, it follows from Equation 12 that