This note studies the exponential asymptotic stability of the zero solution of the

linear Volterra equation

x˙ (t) = Ax(t) +  t

0

K(t − s)x(s) ds

by extending results in the paper of Murakami “Exponential Asymptotic Stability

for scalar linear Volterra Equations”, Differential and Integral Equations, 4, 1991.

In particular, when K isi ntegrable and has entries which do not change sign, and

the equation has a uniformly asymptotically stable solution, exponential asymptotic

stability can be identified by an exponential decay condition on the entries of K

Appleby, John A.D.

This note studies the exponential asymptotic stability of the zero solution of the\ud
linear Volterra equation\ud
x˙ (t) = Ax(t) +  t\ud
0\ud
K(t − s)x(s) ds\ud
by extending results in the paper of Murakami “Exponential Asymptotic Stability\ud
for scalar linear Volterra Equations”, Differential and Integral Equations, 4, 1991.\ud
In particular, when K isi ntegrable and has entries which do not change sign, and\ud
the equation has a uniformly asymptotically stable solution, exponential asymptotic\ud
stability can be identified by an exponential decay condition on the entries of K

Name not available

EXPONENTIAL ASYMPTOTIC STABILITY FOR LINEAR VOLTERRAEQUATIONSBy John A.D. ApplebySchool of Mathematical Sciences, Dublin City University[Received 4 August 2000.]AbstractThis note studies the exponential asymptotic stability of the zero solution of thelinear Volterra equationx˙(t) = Ax(t) +∫ t0K(t− s)x(s) dsby extending results in the paper of Murakami “Exponential Asymptotic Stabilityfor scalar linear Volterra Equations”, Diﬀerential and Integral Equations, 4, 1991.In particular, when K is integrable and has entries which do not change sign, andthe equation has a uniformly asymptotically stable solution, exponential asymptoticstability can be identiﬁed by an exponential decay condition on the entries of K.1. IntroductionIn this note, we study the exponential asymptotic stability of the zero solutionof the linear Volterra equationx˙(t) = Ax(t) +∫ t0K(t− s)x(s) ds, t ≥ 0 (1.1)where the solution vector x is in Rn, A is a constant n × n matrix, and K isan n × n matrix function with integrable entries. The question as to whether theexponential asymptotic stability of the zero solution of (1.1) is necessarily impliedby its uniform asymptotic stability was posed in the survey of Corduneanu andLakshmikantham [2]. As remarked in that article, the question is a reasonable oneas the right hand side of (1.1) is time homogeneous, and in view of the fact thatuniform asymptotic stability and exponential asymptotic stability are equivalentconcepts for linear functional diﬀerential equations with bounded delay.The paper of Murakami [6], on which our study is based, throws considerablelight on the nature of the relationship between uniform asymptotic stability and ex-ponential asymptotic stability of the zero solution of (1.1), detailing the connectionbetween the two types of stability for scalar equations of type (1.1).The ﬁrst theorem of Murakami’s paper [Theorem 1] neatly characterises suﬃ-cient conditions under which uniform asymptotic stability of the zero solution willimply its exponential asymptotic stability. Speciﬁcally, he shows that when the ker-nel satisﬁes an exponential decay condition, the two concepts coincide. Indeed, theresult can be readily generalised to Volterra equations in an arbitrary number ofﬁnite dimensions.Mathematical Proceedings of the Royal Irish Academy, ? (?), 1–7 (2000) c© Royal Irish Academy2 Mathematical Proceedings of the Royal Irish AcademyThe second result in the paper is a very striking one. The author proves [Theo-rem 2, [6]] that when the kernel is integrable and does not change sign on R+, andthe zero solution is exponentially asymptotically stable, then the kernel satisﬁesan exponential decay condition. Thus, for scalar equations of type (1.1), exponen-tial asymptotic stability will only arise if the kernel decays exponentially [Theorem3, [6]].As is mentioned at the end of Murakami’s paper, it is reasonable to conjecturethat this result can be extended to systems with an arbitrary number of ﬁnitedimensions. The proof of this conjecture is the subject of the following note. Wemirror Theorem 3 in [6] by showing for a system in which the entries of the matrixkernel K do not change sign that the exponential and uniform asymptotic stabilityof the zero solution are equivalent if and only if all the entries of K satisfy anexponential decay criterion.2. Preliminary MaterialIn this section, we introduce some notation, give deﬁnitions and allude to somekey results employed in this article.Let R = (−∞,∞) and R+ = [0,∞). Let Mn(R) be the space of n×n matriceswith real entries. We say that the function K : R+ → Mn(R) is in L1(R+) ifeach of its entries is a scalar Lebesgue integrable function. Denote by C(J ;Rn)the space of continuous Rn-valued functions on the compact interval J ⊂ R. Forφ ∈ C([0, t0];Rn), deﬁne |φ|t0 = sup{‖φ(t)‖1 : 0 ≤ t ≤ t0}, where ‖x‖1 is the sumof the absolute values of the entries of the vector x ∈ Rn. For K = (kij) ∈Mn(R),we deﬁne ‖K‖N =∑ni=1∑nj=1 |kij |. Note that this deﬁnes a norm on a ﬁnitedimensional linear space, so that all other norms on this space are equivalent to‖ · ‖N . We denote by C the set of complex numbers, and the real part of s ∈ C bys. If f : R+ →Mn(R), we can deﬁne the Laplace transform of f at s ∈ C to befˆ(s) =∫ ∞0f(t)e−st dt.If α ∈ R and ∫∞0 ‖f(t)‖Ne−αt dt < ∞, then fˆ(s) exists and is continuous in sfor s ≥ α, and analytic on s > −α. See for example, Churchill, p.171 [1], orWidder [7].Consider the linear Volterra equationx˙(t) = Ax(t) +∫ t0K(t− s)x(s) ds, t ≥ 0 (2.1)where A, K(t) ∈ Mn(R), K is continuous on R and K ∈ L1(R+). For any t0 ≥ 0and φ ∈ C([0, t0],Rn), there is a unique Rn-valued function x(t), which satisﬁes(2.1) on [t0,∞) and for which x(t) = φ(t) for t ∈ [0, t0]. We denote such a solutionby x(t; t0, φ). The function x(t) ≡ 0 is a solution of (2.1) and is called the zerosolution of (2.1).Appleby—Exponential Asymptotic Stability for Linear Volterra Equations 3Consider now the matrical equationX ′(t) = AX(t) +∫ t0K(t− s)X(s) ds, t ≥ 0 (2.2)with X(0) = In, where In is the identity matrix in Mn(R). The unique X(t) ∈Mn(R) which satisﬁes (2.2) is called the resolvent, or principal matrix solution for(2.1).We further recall the various standard notions of stability of the zero solution re-quired for our analysis. The zero solution of (2.1) is said to be uniformly stable (US),if, for every ε > 0, there exists δ(ε) > 0 such that t0 ∈ R+ and φ ∈ C([0, t0],Rn)with |φ|t0 < δ(ε) implies ‖x(t; t0, φ)‖1 < ε for all t ≥ t0. The zero solution is saidto be uniformly asymptotically stable (UAS) if is US and there exists δ > 0 withthe following property: for each ε > 0 there exists a T (ε) > 0 such that t0 ∈ R+and φ ∈ C([0, t0],Rn) with |φ|t0 < δ implies ‖x(t, t0, φ)‖1 < ε for all t ≥ t0 + T (ε).The zero solution of (2.1) is said to be exponentially asymptotically stable (Ex AS),if there exists C,α > 0 such that‖x(t; t0, φ)‖1 ≤ Ce−α(t−t0)|φ|t0 , t ≥ t0 ≥ 0for any φ ∈ C([0, t0],Rn).These deﬁnitions are standard: the reader may refer to Miller [5]. Existence anduniqueness are covered in [3].The properties of the resolvent X are deeply linked to the stability of the zerosolution of (2.1). It is shown in [5] that the zero solution of (2.1) is UAS if andonly if X ∈ L1(R+). Moreover, in [4], it is shown that X ∈ L1(R+) if and only ifH(s) = sIn −A− Kˆ(s) (2.3)satisﬁes detH(s) = 0 for s ≥ 0.3. ResultsIf the zero solution of (2.1) is UAS, X ∈ L1(R+), so Xˆ(s) exists and is con-tinuous in s for s ≥ 0. Taking Laplace transforms both sides of (2.2), we getH(s)Xˆ(s) = In for s > 0. By the continuity of Xˆ, H , we haveH(s)Xˆ(s) = In s ≥ 0. (3.1)Thus Xˆ(s) = H−1(s), for s ≥ 0, as detH(s) = 0 for s ≥ 0.Theorem 1 in [6] says that if∫ ∞0|K(t)|eγt dt <∞, for some γ > 0 (3.2)and the zero solution of the scalar equation (2.1) is uniformly asymptotic stable,then the zero solution is also Ex AS.It is mentioned in [6] that the proof can be generalised, with minor modiﬁcations,to establish a corresponding result for systems.4 Mathematical Proceedings of the Royal Irish AcademyTheorem 3.1 (Murakami). If the zero solution of (2.1) is UAS, and the kernel Ksatisfies ∫ ∞0‖K(t)‖Neγt dt <∞ for some γ > 0then the zero solution is Ex AS.We now turn to a partial converse of Theorem 3.1, which is strongly motivatedby Theorem 2 in [6], and whose proof it follows very closely. The hypotheses of thattheorem can be extracted from the following by setting the dimension n = 1 andconsidering the scalar case. Indeed, the proof of our Proposition 3.2 is exactly thatof Theorem 2 in [6] with n = 1. However, Murakami did not succeed in obtaining theextension to ﬁnite dimensions. In view of this gap, the nuances involved in passingfrom the scalar to the general ﬁnite dimensional case, and to achieve a coherentexposition, we show the details here.Theorem 3.2. Let K ∈ L1(R+) and suppose that none of the entries of K(t) ∈Mn(R) change sign on R+. If ‖X(t)‖N ≤ Ce−βt, for some C, β > 0, then thereexists γ > 0 such that ∫ ∞0‖K(t)‖Neγt dt <∞. (3.3)Proof. Let 0 < α < β. Clearly, by hypothesis, Xˆ(s) exists and is analytic in s fors > −α and continuous for s ≥ −α. Patently, (3.1) holds. Since the zero solutionis UAS, detH(0) = 0, so det Xˆ(0) = 0. Since Xˆ(s) is continuous at s = 0, and thedeterminant of a matrix is a continuous function of its entries, s → det Xˆ(s) iscontinuous at s = 0. Therefore, there exists an open neighbourhood of 0 (U ′, say)such that det Xˆ(s) = 0 for s ∈ U ′: thus Xˆ−1(s) exists on U ′. Since the entries ofXˆ(s) are analytic on U := U ′ ∩ {s : s > −α}, so must be the entries of Xˆ−1(s).ThusF (s) = sIn −A− Xˆ−1(s)is analytic on U , and satisﬁesKˆ(s) = F (s), s ≥ 0. (3.4)The proof of the result now follows by a sequence of contradictions: we posit thattnK(t) ∈ L1(R+) for n = 1, 2, . . . in turn, and show that each of these hypothesesintroduces a contradiction. The existence of the “moments” of K, together withthe analyticity of F and (3.4) enables us to prove (3.3).We claim tK(t) ∈ L1(R+). If this is false, there exists T > 1 such that∫ T0‖K(t)‖N(t− 1) dt > Mwhere M = ‖F ′(0)‖N . Next, it is possible to construct δT = min(1, e−T ), so that ifAppleby—Exponential Asymptotic Stability for Linear Volterra Equations 50 < h < δT then1− e−hth≥ t− 1, t ∈ [0, T ]. (3.5)Thus for 0 < h < δT , we have∥∥∥∥F (h)− F (0)h∥∥∥∥N=∥∥∥∥∥Kˆ(h)− Kˆ(0)h∥∥∥∥∥N=∥∥∥∥∫ ∞0K(t)1− e−hthdt∥∥∥∥N=n∑i=1n∑j=1∣∣∣∣∫ ∞0kij(t)1 − e−hthdt∣∣∣∣=n∑i=1n∑j=1∫ ∞0|kij(t)|1− e−hthdt,using the fact that kij(t) has the same sign on R+ at the last step. From (3.5), for0 < h < δT , we thus have∥∥∥∥F (h)− F (0)h∥∥∥∥N≥n∑i=1n∑j=1∫ T0|kij(t)|(t− 1) dt =∫ T0‖K(t)‖N(t− 1) dt.Using the continuity of the norm ‖ · ‖N , we obtainM = ‖F ′(0)‖N = limh→0+∥∥∥∥F (h)− F (0)h∥∥∥∥N≥∫ T0‖K(t)‖N(t− 1) dt > M,a contradiction. Thus tK(t) ∈ L1(R+).Next, since 1− x ≤ e−x for x ≥ 0, for h > 0, s ≥ 0 and t ∈ R+, we have∣∣∣∣1− e−hth e−stkij(t)∣∣∣∣ = t∣∣∣∣1− e−htht∣∣∣∣ |e−st||kij(t)| ≤ t|kij(t)|. (3.6)Thus, using the Dominated Convergence Theorem, in conjunction with (3.4) and(3.6), we getlimh→0+Fij(s+ h)− Fij(s)h= limh→0+kˆij(s+ h)− kˆij(s)h= limh→0+∫ ∞0e−ht − 1he−stkij(t) dt=∫ ∞0−te−stkij(t) dt,so Fij(s) = −∫∞0 te−stkij(t) dt for s ≥ 0. Therefore F ′(s) = −∫∞0 te−stK(t) dt.By applying the argument for K(t) in the foregoing paragraphs to tK(t), we6 Mathematical Proceedings of the Royal Irish Academyobtaint2K(t) ∈ L1(R+), F ′′(s) =∫ ∞0t2e−stK(t) dt, s ≥ 0,and repeating this procedure givestnK(t) ∈ L1(R+), F (n)(s) = (−1)n∫ ∞0tne−stK(t) dt, s ≥ 0 (3.7)for n = 1, 2, . . .. Since F is analytic on U , we have that∞∑m=0F(m)ij (0)m!smis absolutely convergent on the disc D = {z ∈ C : |z| < γ}, for some γ > 0 and alli, j = 1, . . . n. Thus, by (3.7), we have∞ >∞∑m=0|F (m)ij (0)|m!γm =∞∑m=0γmm!∫ ∞0tm|kij(t)| dt, (3.8)and so, using Fatou’s Lemma∫ ∞0eγt‖K(t)‖N dt =∫ ∞0eγtn∑i=1n∑j=1|kij(t)| dt =n∑i=1n∑j=1∫ ∞0eγt|kij(t)| dt=n∑i=1n∑j=1∫ ∞0∞∑m=0(γt)mm!|kij(t)| dt=n∑i=1n∑j=1[ ∞∑m=0γmm!∫ ∞0tm|kij(t)| dt]<∞,by (3.8), and the proof is complete.We now have all the ingredients to prove Theorem 3.3.Theorem 3.3. Suppose that K ∈ L1(R+) and that the entries of K do not changesign on R+. If the zero solution of (1.1) is uniformly asymptotically stable, thenthe following are equivalent:(i) The zero solution of (1.1) is exponentially asymptotically stable,(ii) The principal matrix solution of (1.1), X(t), satisfies ‖X(t)‖ ≤ Ce−αtfor some C,α > 0,(iii) There exists γ > 0 such that∫ ∞0‖K(t)‖eγt dt <∞,where ‖ · ‖ is any matrix norm.Appleby—Exponential Asymptotic Stability for Linear Volterra Equations 7Proof. Since Mn(R) is a ﬁnite dimensional linear space, norm equivalence in thisspace means that the result need only be established for the special choice of norm‖ · ‖N .Clearly, (i) implies (ii) trivially. That (ii) implies (iii) is the subject of Theorem3.2. (iii) implies (i) is simply Theorem 3.1. Hence (i)-(iii) are equivalent.The proof of Theorem 3.3 shows how Theorem 3 in [6] can be extended to thenon-scalar case, a problem which was proposed in [6].AcknowledgementsIt is a pleasure to acknowledge David Reynolds for his careful reading andinsightful criticism of this paper, and Kieran Murphy for several interesting discus-sions on the subject matter.References[1] R. V. Churchill, 1958 Operational Mathematics. New York. McGraw-Hill.[2] C. Corduneanu and V. Lakshmikantham, 1980 Equations with unbounced delay: a survey.Nonlinear Anal. 4, 831-877.[3] R. D. Driver, 1962 Existence and stability of solutions of a delay diﬀerential system. Arch.Rational Mech. Anal. 10, 401-426.[4] S. I. Grossman and R. K. Miller, 1973 Nonlinear Volterra integrodiﬀerential equations. J.Diﬀerential Equations 13, 551-566.[5] R. K. Miller, 1971 Asymptotic Stability Properties of Linear Volterra Integrodiﬀerential Equa-tions. J. Diﬀerential Equations 10, 485-506.[6] S. Murakami, 1991 Exponential asymptotic stability of scalar linear Volterra equations. Dif-ferential Integral Equations 4:3, 519-525.[7] D. V. Widder, 1946 The Laplace Transform. Princeton, N.J. Princeton University Press.

Exponential asymptotic stability for linear volterra equations

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Exponential asymptotic stability for linear volterra equations

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