We study the set of $T$-periodic solutions of a class of $T$-periodically
perturbed Differential-Algebraic Equations, allowing the perturbation to
contain a distributed and possibly infinite delay. Under suitable assumptions,
the perturbed equations are equivalent to Retarded Functional (Ordinary)
Differential Equations on a manifold. Our study is based on known results about
the latter class of equations.Comment: 13 pages. Revision: Incorporate changes suggested by readers.
  Corrected a few typos across the paper, definition of BU added, revised the
  (previously incorrect) definition of solution of RFDAE, made slight changes
  in the Introduction. Replacement of Dec. 6, 2012: introduced further changes
  suggested by referee, bundled addendum/erratum containing a corrected version
  of Lemma 5.5 and Corollary 5.

Bisconti, Luca

Spadini, Marco

English

arXiv

Spadini, M.

Repository of the Academy's Library

Electronic Journal of Qualitative Theory of Differential Equations2012, No. 97, 1-5; http://www.math.u-szeged.hu/ejqtde/CORRIGENDUM TO ON A CLASS OFDIFFERENTIAL-ALGEBRAICEQUATIONS WITH INFINITE DELAYLuca Bisconti∗ Marco Spadini†AbstractThis paper serves as a corrigendum to the paper titled On a class ofdifferential-algebraic equations with infinite delay appearing in EJQTDEno. 81, 2011. We present here a corrected version of Lemma 5.5 andCorollary 5.7.1 IntroductionIn Section 5 of [1] we investigated examples of applications of that paper’sresults to a particular class of implicit differential equations. For so doing weused a technical lemma from linear algebra that, unfortunately, turns out tobe flawed. As briefly discussed below this affects only marginally our paper’sresults (just a corollary in Section 5 of [1]).The simple example below shows that there is something wrong withLemma 5.5 in [1]. In the next section we provide an amended version of thisresult.Example 1.1. Consider the matricesE =(0 10 0), C =(1 00 0).2000 Mathematics Subject Classifications : 34A09, 34K13, 34C40Keywords and phrases : differential-algebraic equations, retarded functional equations,periodic solution, periodic perturbation, infinite delay∗Dipartimento di Sistemi e Informatica, Universita` di Firenze, Via Santa Marta 3,50139 Firenze, Italy, e-mail: luca.bisconti@unifi.it†Dipartimento di Sistemi e Informatica, Universita` di Firenze, Via Santa Marta 3,50139 Firenze, Italy, e-mail: marco.spadini@unifi.itEJQTDE, 2012 No. 97, p. 1Clearly, kerCT = kerET = span{( 01 )} for all t ∈ R. The matricesP =(1 00 1), Q =(0 11 0),realize a singular value decomposition for E. NeverthelessP TCQ =(0 10 0)which is not the form expected from Lemma 5.5 in [1]. The problem, as itturns out, is that kerC 6= kerE.Luckily, the impact of the wrong statement of [1, Lemma 5.5] on [1]is minor: all results and examples (besides Lemma 5.5, of course) remaincorrect, with the exception of Corollary 5.7 where it is necessary to assumethe following further hypothesis:ker C(t) = ker E, ∀ t ∈ R.(A corrected statement of Corollary 5.7 of [1] can be found in the next section,Corollary 2.2.)2 Corrected Lemma and its consequencesWe present here a corrected version of Lemma 5.5 in [1].Lemma 2.1. Let E ∈ Rn×n and C ∈ C(R,Rn×n)be respectively a matrixand a matrix-valued function such thatker CT (t) = ker ET , ∀ t ∈ R, and dimker ET > 0, (2.1)Put r = rank E, and let P,Q ∈ Rn×n be orthogonal matrices that realize asingular value decomposition for E. Then it follows thatP TC(t)Q =(C˜11(t) C˜12(t)0 0), ∀t ∈ R, (2.2)with C˜11 ∈ C(R,Rr×r)and C˜12 ∈ C(R,Rr×n).If, furthermore,ker C(t) = ker E, ∀ t ∈ R, (2.3)then C˜12(t) ≡ 0. Namely, in this case,P TC(t)Q =(C˜11(t) 00 0), ∀t ∈ R, (2.4)with C˜11(t) nonsingular for all t ∈ R.EJQTDE, 2012 No. 97, p. 2Proof. Our proof is essentially a singular value decomposition (see, e.g., [2])argument, based on a technical result from [3].Observe that (2.1) imply rankE = rankC(t) = r > 0 for all t ∈ R. Infact,rankE = rankET = n− dimkerET == n− dimkerC(t)T = rankC(t)T = rankC(t).Since rankC(t) is constantly equal to r > 0, by inspection of the proof ofTheorem 3.9 of [3, Chapter 3, §1] we get the existence of orthogonal matrix-valued functions U, V ∈ C(R,Rn×n)and Cr ∈ C(R,Rr×r) such that, for allt ∈ R, detCr(t) 6= 0 andUT (t)C(t)V (t) =(Cr(t) 00 0). (2.5)Let Ur, Vr ∈ C(R,Rn×r)and U0, V0 ∈ C(R,Rn×(n−r)) be matrix-valuedfunctions formed, respectively, by the first r and n − r columns of U andV . An argument involving Equation (2.5) shows that, for all t ∈ R, thespace imC(t) is spanned by the columns of Ur(t). Also, (2.5) imply that thecolumns of V0(t), t ∈ R, belong to kerC(t) for all t ∈ R. A dimensional argu-ment shows that they constitute a basis kerC(t). Analogously, transposing(2.5), we see that the columns of Vr(t) and U0(t) are bases of imC(t)T andkerC(t)T respectively.1Let now Pr, Qr and P0, Q0 be the matrices formed taking the first r andn−r columns of P and Q, respectively. Since P and Q realize a singular valuedecomposition of E, proceeding as above one can check that the columns ofPr, Qr, P0 and Q0 span imE, imET , kerET , and kerE, respectively.We claim that P T0 Ur(t) is constantly the null matrix in R(n−r)×r. To provethis, it is enough to show that for all t ∈ R, the columns of P0 are orthogonalto those of Ur(t). Let v and u(t), t ∈ R, be any column of P0 and of Ur(t),respectively. Since for all t ∈ R the columns of Ur(t) are in imC(t), there isa vector w(t) ∈ Rn with the property that u(t) = C(t)w(t), and〈v, u(t)〉 = 〈v, C(t)w(t)〉 = 〈C(t)Tv, w(t)〉 = 0, ∀t ∈ R,because v ∈ kerET = kerC(t)T for all t ∈ R. This proves the claim. Asimilar argument shows that P TrU0(t) is identically zero as well.1In fact, the orthogonality of the matrices V (t) and U(t) for all t ∈ R, imply thatthe columns of Ur(t), Vr(t), U0(t) and V0(t) are respective orthogonal bases of the spacesimC(t), imC(t)T , kerC(t)T and kerC(t).EJQTDE, 2012 No. 97, p. 3Since for all t ∈ RP TU(t) =(P TrUr(t) 00 P T0 U0(t))is nonsingular, we deduce in particular that so is P TrUr(t).Let us compute the matrix product P TC(t)Q for all t ∈ R. We omit here,for the sake of simplicity, the explicit dependence on t.P TCQ = P TUUTCV V TQ =(P TrUr 00 P T0 U0)(Cr 00 0)(V TrQr VTrQ0V T0 Qr VT0 Q0)=(P TrUrCrVTrQr PTrUrCrVTrQ00 0),which proves (2.2).Let us now assume that also (2.3) holds. We claim that in this case V T0 Qris identically zero. To see this we proceed as done above for the productsP T0 Ur and PTrU0. Let v(t), t ∈ R, be any column of V0(t), hence a vector ofkerC(t) for all t ∈ R, and let q be a column of Qr(t). Since the columns ofQr lie in imET , there is a vector ` ∈ Rn with the property that q = ET `,and〈v(t), q〉 = 〈v(t), ET `〉 = 〈Ev(t), `〉 = 0, ∀t ∈ R,because v(t) ∈ kerC(t) = kerE for all t ∈ R. This proves the claim. Asimilar argument shows that V TrQ0(t) is identically zero as well. Hence,V (t)TQ =(Vr(t)TQr 00 V0(t)TQ0)thus V Tr(t)Q0, and VT0 (t)Qr are nonsingular. Also, plugging VT0 Qr = 0 in theabove expression for P TCQ one gets (we omit again the explicit dependenceon t)P TCQ =(P TrUrCrVTrQr 00 0). (2.6)Which proves the assertion because P TrUr, Cr, and VTrQr are nonsingular.In view of the corrected version of the above lemma, the statement ofCorollary 5.7 of [1] can be rewritten as follows:Corollary 2.2. Consider EquationEx˙(t) = F(x(t))+ λC(t)S(xt), (2.7)EJQTDE, 2012 No. 97, p. 4where the maps C : R → Rn×n and S : BU((−∞, 0],Rn)→ Rn are contin-uous, E is a (constant) n × n matrix, F is locally Lipschitz and S verifiescondition (K) in [1]. Suppose also that C and E satisfy (2.1) and (2.3), andthat C is T -periodic. Let r > 0 be the rank of E and assume that there existsan orthogonal basis of Rn ' Rr × Rn−r such that E has the formE '(E11 E120 0), with E11 ∈ Rr×r invertible and E12 ∈ Rr×(n−r).Assume also that, relatively to this decomposition of Rn, ∂2F2(ξ, η) is invert-ible for all x = (ξ, η) ∈ Rr × Rn−r.Let Ω be an open subset of [0,+∞)×CT (Rn) and suppose that deg(F ,Ω∩Rn) is well-defined and nonzero. Then, there exists a connected subset Γ ofnontrivial T -periodic pairs for (2.7) whose closure in Ω is noncompact andmeets the set{(0,p) ∈ Ω : F(p) = 0}.This result follows as in [1] taking into account the modified version ofthe lemma.References[1] L. Bisconti and M. Spadini, On a class of differential-algebraic equa-tions with infinite delay, Electronic Journal of Qualitative Theory ofDifferential Equations 2011, No. 81, 1-21.[2] G. H. Golub and C. F. Van Loan, Matrix computations, 3rd edition, J.Hopkins Univ. Press, Baltimore 1996.[3] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysisand Numerical Solution, EMS Textbooks in Mathematics, 2006.(Received November 22, 2012)EJQTDE, 2012 No. 97, p. 5

Corrigendum to On a class of differential-algebraic equations with infinite delay

Florence Research

On a class of differential-algebraic equations with infinite delay

Bisconti Luca

Spadini Marco

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2012, No. 97, 1-5; http://www.math.u-szeged.hu/ejqtde/CORRIGENDUM TO ON A CLASS OFDIFFERENTIAL-ALGEBRAICEQUATIONS WITH INFINITE DELAYLuca Bisconti∗ Marco Spadini†AbstractThis paper serves as a corrigendum to the paper titled On a class ofdifferential-algebraic equations with infinite delay appearing in EJQTDEno. 81, 2011. We present here a corrected version of Lemma 5.5 andCorollary 5.7.1 IntroductionIn Section 5 of [1] we investigated examples of applications of that paper’sresults to a particular class of implicit differential equations. For so doing weused a technical lemma from linear algebra that, unfortunately, turns out tobe flawed. As briefly discussed below this affects only marginally our paper’sresults (just a corollary in Section 5 of [1]).The simple example below shows that there is something wrong withLemma 5.5 in [1]. In the next section we provide an amended version of thisresult.Example 1.1. Consider the matricesE =(0 10 0), C =(1 00 0).2000 Mathematics Subject Classifications : 34A09, 34K13, 34C40Keywords and phrases : differential-algebraic equations, retarded functional equations,periodic solution, periodic perturbation, infinite delay∗Dipartimento di Sistemi e Informatica, Università di Firenze, Via Santa Marta 3,50139 Firenze, Italy, e-mail: luca.bisconti@unifi.it†Dipartimento di Sistemi e Informatica, Università di Firenze, Via Santa Marta 3,50139 Firenze, Italy, e-mail: marco.spadini@unifi.itEJQTDE, 2012 No. 97, p. 1Clearly, ker CT = ker ET = span{( 01 )} for all t ∈ R. The matricesP =(1 00 1), Q =(0 11 0),realize a singular value decomposition for E. NeverthelessP T CQ =(0 10 0)which is not the form expected from Lemma 5.5 in [1]. The problem, as itturns out, is that ker C 6= ker E.Luckily, the impact of the wrong statement of [1, Lemma 5.5] on [1]is minor: all results and examples (besides Lemma 5.5, of course) remaincorrect, with the exception of Corollary 5.7 where it is necessary to assumethe following further hypothesis:ker C(t) = ker E, ∀ t ∈ R.(A corrected statement of Corollary 5.7 of [1] can be found in the next section,Corollary 2.2.)2 Corrected Lemma and its consequencesWe present here a corrected version of Lemma 5.5 in [1].Lemma 2.1. Let E ∈ Rn×n and C ∈ C(R, Rn×n)be respectively a matrixand a matrix-valued function such thatker CT (t) = ker ET , ∀ t ∈ R, and dim ker ET > 0, (2.1)Put r = rank E, and let P, Q ∈ Rn×n be orthogonal matrices that realize asingular value decomposition for E. Then it follows thatP T C(t)Q =(C̃11(t) C̃12(t)0 0), ∀t ∈ R, (2.2)with C̃11 ∈ C(R, Rr×r)and C̃12 ∈ C(R, Rr×n).If, furthermore,ker C(t) = ker E, ∀ t ∈ R, (2.3)then C̃12(t) ≡ 0. Namely, in this case,P TC(t)Q =(C̃11(t) 00 0), ∀t ∈ R, (2.4)with C̃11(t) nonsingular for all t ∈ R.EJQTDE, 2012 No. 97, p. 2Proof. Our proof is essentially a singular value decomposition (see, e.g., [2])argument, based on a technical result from [3].Observe that (2.1) imply rankE = rankC(t) = r > 0 for all t ∈ R. Infact,rank E = rankET = n − dim ker ET == n − dim ker C(t)T = rankC(t)T = rankC(t).Since rank C(t) is constantly equal to r > 0, by inspection of the proof ofTheorem 3.9 of [3, Chapter 3, §1] we get the existence of orthogonal matrix-valued functions U, V ∈ C(R, Rn×n)and Cr ∈ C(R, Rr×r) such that, for allt ∈ R, det Cr(t) 6= 0 andUT (t)C(t)V (t) =(Cr(t) 00 0). (2.5)Let Ur, Vr ∈ C(R, Rn×r)and U0, V0 ∈ C(R, Rn×(n−r)) be matrix-valuedfunctions formed, respectively, by the first r and n − r columns of U andV . An argument involving Equation (2.5) shows that, for all t ∈ R, thespace im C(t) is spanned by the columns of Ur(t). Also, (2.5) imply that thecolumns of V0(t), t ∈ R, belong to ker C(t) for all t ∈ R. A dimensional argu-ment shows that they constitute a basis ker C(t). Analogously, transposing(2.5), we see that the columns of Vr(t) and U0(t) are bases of im C(t)T andker C(t)T respectively.1Let now Pr, Qr and P0, Q0 be the matrices formed taking the first r andn−r columns of P and Q, respectively. Since P and Q realize a singular valuedecomposition of E, proceeding as above one can check that the columns ofPr, Qr, P0 and Q0 span im E, im ET , ker ET , and ker E, respectively.We claim that P T0 Ur(t) is constantly the null matrix in R(n−r)×r. To provethis, it is enough to show that for all t ∈ R, the columns of P0 are orthogonalto those of Ur(t). Let v and u(t), t ∈ R, be any column of P0 and of Ur(t),respectively. Since for all t ∈ R the columns of Ur(t) are in im C(t), there isa vector w(t) ∈ Rn with the property that u(t) = C(t)w(t), and〈v, u(t)〉 = 〈v, C(t)w(t)〉 = 〈C(t)T v, w(t)〉 = 0, ∀t ∈ R,because v ∈ ker ET = ker C(t)T for all t ∈ R. This proves the claim. Asimilar argument shows that P TrU0(t) is identically zero as well.1In fact, the orthogonality of the matrices V (t) and U(t) for all t ∈ R, imply thatthe columns of Ur(t), Vr(t), U0(t) and V0(t) are respective orthogonal bases of the spacesim C(t), imC(t)T , kerC(t)T and kerC(t).EJQTDE, 2012 No. 97, p. 3Since for all t ∈ RP TU(t) =(P TrUr(t) 00 P T0 U0(t))is nonsingular, we deduce in particular that so is P TrUr(t).Let us compute the matrix product P T C(t)Q for all t ∈ R. We omit here,for the sake of simplicity, the explicit dependence on t.P T CQ = P TUUT CV V T Q =(P TrUr 00 P T0 U0)(Cr 00 0) (V TrQr VTrQ0V T0 Qr VT0 Q0)=(P TrUrCrVTrQr PTrUrCrVTrQ00 0),which proves (2.2).Let us now assume that also (2.3) holds. We claim that in this case V T0 Qris identically zero. To see this we proceed as done above for the productsP T0 Ur and PTrU0. Let v(t), t ∈ R, be any column of V0(t), hence a vector ofker C(t) for all t ∈ R, and let q be a column of Qr(t). Since the columns ofQr lie in im ET , there is a vector ℓ ∈ Rn with the property that q = ET ℓ,and〈v(t), q〉 = 〈v(t), ET ℓ〉 = 〈Ev(t), ℓ〉 = 0, ∀t ∈ R,because v(t) ∈ ker C(t) = ker E for all t ∈ R. This proves the claim. Asimilar argument shows that V TrQ0(t) is identically zero as well. Hence,V (t)T Q =(Vr(t)T Qr 00 V0(t)T Q0)thus V Tr(t)Q0, and VT0 (t)Qr are nonsingular. Also, plugging VT0 Qr = 0 in theabove expression for P T CQ one gets (we omit again the explicit dependenceon t)P TCQ =(P TrUrCrVTrQr 00 0). (2.6)Which proves the assertion because P TrUr, Cr, and VTrQr are nonsingular.In view of the corrected version of the above lemma, the statement ofCorollary 5.7 of [1] can be rewritten as follows:Corollary 2.2. Consider EquationEẋ(t) = F(x(t))+ λC(t)S(xt), (2.7)EJQTDE, 2012 No. 97, p. 4where the maps C : R → Rn×n and S : BU((−∞, 0], Rn)→ Rn are contin-uous, E is a (constant) n × n matrix, F is locally Lipschitz and S verifiescondition (K) in [1]. Suppose also that C and E satisfy (2.1) and (2.3), andthat C is T -periodic. Let r > 0 be the rank of E and assume that there existsan orthogonal basis of Rn ≃ Rr × Rn−r such that E has the formE ≃(E11 E120 0), with E11 ∈ Rr×r invertible and E12 ∈ Rr×(n−r).Assume also that, relatively to this decomposition of Rn, ∂2F2(ξ, η) is invert-ible for all x = (ξ, η) ∈ Rr × Rn−r.Let Ω be an open subset of [0, +∞)×CT (Rn) and suppose that deg(F , Ω∩Rn) is well-defined and nonzero. Then, there exists a connected subset Γ ofnontrivial T -periodic pairs for (2.7) whose closure in Ω is noncompact andmeets the set{(0,p) ∈ Ω : F(p) = 0}.This result follows as in [1] taking into account the modified version ofthe lemma.References[1] L. Bisconti and M. Spadini, On a class of differential-algebraic equa-tions with infinite delay, Electronic Journal of Qualitative Theory ofDifferential Equations 2011, No. 81, 1-21.[2] G. H. Golub and C. F. Van Loan, Matrix computations, 3rd edition, J.Hopkins Univ. Press, Baltimore 1996.[3] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysisand Numerical Solution, EMS Textbooks in Mathematics, 2006.(Received November 22, 2012)EJQTDE, 2012 No. 97, p. 5

Corrigendum to on a class of differential-algebraic equations with infinite delay

Marco Spadini

Luca Bisconti

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Electronic journal of qualitative theory of differential equations

This paper serves as a corrigendum to the paper titled On a class of differential-algebraic equations with infinite delay appearing in EJQTDE no. 81, 2011. We present here a corrected version of Lemma 5.5 and Corollary 5.7

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On a class of differential-algebraic equations with infinite delay

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Florence Research

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