Elsevier

Systems & Control Letters

Volume 96, October 2016, Pages 1-6
Systems & Control Letters

A version of a theorem of R. Datko for stability in average

https://doi.org/10.1016/j.sysconle.2016.06.015Get rights and content

Abstract

In this note we obtain a version of the well-known theorem of R. Datko for the notion of the exponential stability in average. We consider both cocycles over flows as well as cocycles over maps.

Introduction

In the process of extending the Lyapunov operator equation to the case of autonomous systems x=Ax when the operator A is unbounded, Datko  [1] established his famous theorem which asserts that the trajectories of a C0-semigroup {T(t)}t0 on a Hilbert space X exhibit an exponential decay if and only if they stay in L2(R+,X). Since then this theorem became one of the pillars of the modern control theory and has inspired numerous extensions and generalizations. In particular, Pazy  [2] proved that the conclusion of Datko’s theorem holds if L2(R+,X) is replaced with any Lp(R+,X) with p[1,). Furthermore, Datko  [3] obtained the version of his theorem which deals with the exponential stability of evolution families {T(t,s)}ts0 which describe solutions of the variety of differential equations. More precisely, he proved the following result.

Theorem 1

Let {T(t,s)}ts0 be an evolution family on a Banach space X. The following statements are equivalent:

  • (1)

    there exist D,λ>0 such thatT(t,s)Deλ(ts)for   ts0;

  • (2)

    there exists p[1,) such thatsups0sT(t,s)xp<for each   xX.

The first results related to discrete-time evolution families are due to Zabczyk  [4].

A major improvement of this ideas is due to Rolewicz  [5] who characterized exponential stability of evolution families in terms of the existence of appropriate functions N of two real variables (see  [6] for details and further discussion). This approach unified and extended many of the previously known results. The most recent contributions  [6], [7] deal with obtaining the version of Datko’s theorem for the notion of nonuniform exponential stability which was introduced by Barreira and Valls (see  [8]). Moreover, in  [9] the authors have obtained a certain ergodic version of Datko’s theorem.

The main purpose of the present paper is to obtain a version of Datko’s theorem for the notion of an exponential stability in average which is a particular case of a more general notion of an exponential dichotomy in average introduced in  [10], [11] for discrete and continuous time respectively. This notion essentially corresponds to assuming the existence of uniform contraction and uniform expansion along complementary directions but now in average, with respect to a given probability measure. We emphasize that this notion includes the classical concepts of uniform exponential dichotomy (and thus also of uniform exponential stability) as particular cases.

The paper is organized as follows. In Section  2 we recall some basic notions and the concept of an exponential stability in average. In Section  3 we prove the version of Datko’s theorem for cocycles over semiflows. Then, in Section  4 we do the same but for cocycles over maps. Finally, in Section  5 we imply those results to the study of the persistence of the notion of the exponential stability in average under small linear perturbations.

Section snippets

Preliminaries

We begin by recalling some well-known notions. Let Ω=(Ω,B,μ) be a probability space. A measurable map φ:R0+×ΩΩ is said to be a semiflow on Ω if:

  • (1)

    φ(0,ω)=ω for ωΩ;

  • (2)

    φ(t+s,ω)=φ(t,φ(s,ω)) for t,s0 and ωΩ.

For each t0 we can consider the map φt:ΩΩ given by φt(x)=φ(t,x), xΩ. Moreover, let X be a Banach space and let L(X) denote the set of all invertible bounded linear operators acting on  X. A strongly measurable map Φ:R0+×ΩL(X) (this means that (t,ω)Φ(t,ω)x is Bochner measurable for each xX)

Main result

The following result gives a complete characterization of the notion of an exponential stability in average. It can be regarded as a version of the classical Datko–Pazy results for this notion.

Theorem 2

The cocycle Φ is exponentially stable in average if and only if there exist C,p>0 such that(t0(ΩΦω(τ,t0)z(ω)dμ(ω))pdτ)1/pCΩz(ω)dμ(ω),for every t00 and zF.

Proof

Assume that the cocycle Φ is exponentially stable in average and take an arbitrary p>0. It follows from  (2) that t0(ΩΦω(τ,t0)z(ω)dμ(ω))

Cocycles over maps

In this section we obtain a discrete time version of Theorem 2. Let (Ω,B,μ) be a probability space and let f:ΩΩ be a measurable map. A measurable map A:N0×ΩL(X) is said to be a cocycle over f if A(0,ω)=Id and A(n+m,ω)=A(n,fm(ω))A(m,ω) for m,nN0 and ωΩ. We write Aω(m,n)=A(m,ω)A(n,ω)1. We also consider the map A=A(1,):ΩL(X). Clearly, A(m,ω)={A(fm1(ω))A(ω),m>0,Id ,m=0. We say that the cocycle A is exponentially stable in average if there exists D,λ>0 such that ΩAω(m,n)z(ω)dμ(ω)Deλ(mn

Robustness of the stability in average

In this section we use Theorem 3 to establish the persistence of the notion of an exponential stability in average for cocycles over maps under sufficiently small linear perturbations. In  [10] we have established similar property for the notion of the exponential dichotomy in average. Although this notion is more general then the notion of stability in average we emphasize that the robustness property of dichotomy does not imply the robustness property for stability.

Theorem 4

Let A and B be a cocycles

Acknowledgments

The author would like to express his gratitude to his collaborators Luis Barreira and Claudia Valls for many useful discussions regarding the joint works. Furthermore, the author wishes to thank referees for useful comments that improved the quality of the manuscript.

The author was supported by an Australian Research Council Discovery Project DP150100017 and in part by the Croatian Science Foundation under the project IP-2014-09-2285.

References (16)

There are more references available in the full text version of this article.

Cited by (8)

View all citing articles on Scopus
View full text