The spectral mapping property of delay semigroups

Transcription

1 The spectral apping property of delay seigroups András Bátkai, Tanja Eisner and Yuri Latushkin To the eory of G. S. Litvinchuk Abstract. We offer a new way of proving spectral apping properties of delay seigroups in L p -history spaces with finitely any rationally depending delays based on a an explicit construction of approxiate eigenvectors. This allows us to provide proper generalizations of the existing spectral apping theores. Matheatics Subject Classification (2. Priary 47D6; Secondary 34K5, 34K2. Keywords. C -seigroups, spectral apping theore, delay equations. 1. Introduction Partial differential equations with delay play an iportant role in science and engineering and have been studied for any years and by any different ethods. Hale [6] and Webb [11] were aong the first to apply seigroup theory to delay equations, and we refer to [1], Diekan et al. [4], or Wu [12] for ore recent references on differential equations with delay. By now, it is well-known that fairly general classes of abstract delay equations generate strongly continuous seigroups on appropriate function spaces. However, spectral properties of the delay seigroup are not well understood. In particular, the following basic question still reains open, in general: If the spectral apping property ( σ T (t \ {} = e tσ(a, t >, (1.1 holds for the spectru σ( of the delay seigroup (T (t t and its generator A. The spectral apping property (1.1 is widely discussed in the literature on abstract strongly continuous seigroups, see e.g. the corresponding chapters in Nagel et al. [9], Engel and Nagel [5], van Neerven [1], or [3]. This property

2 2 András Bátkai, Tanja Eisner and Yuri Latushkin is obviously of great iportance because in concrete probles we often have a good characterization of the generator but no explicit knowledge of the seigroup itself. The spectral apping property always holds for any known classes of abstract seigroups, e.g., for eventually nor continuous, in particular, analytic or eventually copact seigroups. In any applications, for exaple, in the study of invariant anifolds for nonlinear evolution equations, even a ore restrictive result, T σ(t (t = T e tσ(a, (1.2 where T := { λ = 1} is the unit circle, is also of great iportance. Indeed, it tells us that the hyperbolicity of the seigroup (T (t t can be characterized via the spectru of A. Moreover, in the theory of strongly continuous seigroups one is often interested in a consequence of (1.2, that is, in conditions for the equality s(a = ω (A, (1.3 log T (t where s(a := sup{reλ : λ σ(a} and ω (A = li t t is the spectral bound of the generator and the growth bound of the seigroup, respectively. Dealing with delay seigroups, results of type (1.1, (1.2 or (1.3 are currently known only in the situations where the delay syste is finite diensional, or under soe restrictive copactness assuptions leading to the fact that the delay seigroup becoes eventually nor continuous. Thus, in these situations, properties (1.1, (1.2 and (1.3 follow fro general results on strongly continuous seigroups. However, there are any striking exaples of abstract seigroups for which the spectral apping properties fail. Using these exaples, we show below that, in general, the spectral apping properties fail for delay seigroups as well. In this paper we offer a new approach to properties (1.1, (1.2 and (1.3 that does not require that the delay seigroup is eventually nor continuous and give natural sufficient conditions for which these properties hold. More precisely, we study the spectral apping property of the seigroup associated to the abstract delay equation of the for u (t = Bu(t + Φu t, t, u( = x, (DE u = f, in a Banach space X, where (B, D(B is the (unbounded generator of a strongly continuous seigroup of linear operators on X, u t ( = u(t + on [ 1, ], and the delay operator Φ is assued to have the for Φ(f := C jf( j j=1, f : [ 1, ] X, (1.4 with given bounded operators C j L(X, j = 1,...,, N. Notice that this setting in fact covers a uch broader class of delays than (1.4, naely the case of finitely any rationally depending delays, see reark 2.4. As a first step one has to choose an appropriate state space for the solution u. One of the possibilities is to work in the space of continuous X-valued functions.

3 SMT for delay 3 However, the state space E := X L p ([ 1, ], X turns out to be a better choice with regards to certain applications (e.g., in control theory, see Bensoussan et. al. [2], in nuerical ethods, see Kappel [7], and will be used in this paper. To forulate the ain result of the current paper, for any λ C and τ [ 1, ] we let ɛ λ (τ := e λτ and define the operator, Φ λ L(X, as follows: Φ λ x := Φ(ɛ λ Idx = Φ(e λ( x for x X. (1.5 Theore 1.1. Assue that B and Φ are such that for each λ C the strongly continuous seigroup generated by the operator B + Φ λ has the following spectral apping property: for all rational t >, ( σ e t(b+φ λ \ {} = e tσ(b+φλ. (1.6 Then the operator with the doain A := ( B Φ d dσ (1.7 D(A := {( x f D(B W 1,p ([ 1, ], X : f( = x } (1.8 generates a strongly continuous seigroup (T (t t on the Banach space E := X L p ([ 1, ], X such that the spectral apping property (1.1 is satisfied for rational t >, and, oreover, equality (1.2 holds for all t. Corollary 1.2. Under condition (1.6, the delay seigroup (T (t t satisfies equality (1.3. Condition (1.6 is satisfied in the case where (B, D(B generates an iediately nor continuous or copact seigroup. As we have entioned, outside of this class, (1.6 is a nontrivial assuption. We stress again that, in general, neither of the spectral apping properties (1.1, (1.2 or (1.3 holds for the delay seigroup, see Reark 2.7 below. 2. The delay seigroup Let us suarize soe results fro onograph [1], which will be needed later, on the seigroup approach to linear partial differential equations with delay. Consider the general delay equation u (t = Bu(t + Φu t, t, (GDE u( = x, u = f, where x X, X is a Banach space, B : D(B X X is a linear, closed, and densely defined operator, f L p ([ 1, ], X, p 1,

4 4 András Bátkai, Tanja Eisner and Yuri Latushkin Φ : W 1,p ([ 1, ], X X is a linear, bounded operator, having the for Φ(f := 1 dηf (2.1 with a given function η : [ 1, ] L(X of bounded variation, u : [ 1, X and u t : [ 1, ] X is defined by u t (τ := u(t + τ. Definition 2.1. We say that a function u : [ 1, X is a (classical solution of (GDE if (i u C([ 1,, X C 1 ([,, X, (ii u(t D(B and u t W 1,p ([ 1, ], X for all t, and (iii u satisfies (DE for all t. To solve (GDE by seigroup ethods, we introduce the Banach space E := X L p ([ 1, ], X and the linear operator A described in (1.7 and (1.8. Consider now the abstract Cauchy proble { v (t = A v(t, t, (ACP v( = v, associated to the operator atrix (A, D(A on the Banach space E with initial value v := ( x f. There is a natural correspondence between the solutions of the two probles, (GDE and (ACP. Lea 2.2. [1, Theore 3.12] (i If u is a solution of (GDE, then t ( u(t u t is a solution of the Cauchy proble (ACP. (ii If t ( u(t v(t is a solution of (ACP, then v(t = ut for all t, and u is a solution of (GDE. As an easy consequence of Lea 2.2, we have that if (A, D(A generates a strongly continuous seigroup (T (t t, then the solutions u of equation (GDE are given by the first coponent of the function t T (t ( x ( f for xf D(A: { x + B t u(t = u(s ds + Φ t u s ds for t, (2.2 f(t for a.e. t [ 1,. By eans of the perturbation theore of Miyadera-Voigt, (see [5, Corollary III.3.16], one can forulate the following sufficient condition of the well-posedness of (GDE. Let us assue that (B, D(B is the generator of a strongly continuous seigroup (S(t t on X, (T (t t is the nilpotent left shift seigroup on L p ([ 1, ], X, and S t : X L p ([ 1, ], X is defined by { S(t + τx, t < τ, (S t x(τ :=, 1 τ t.

5 SMT for delay 5 Theore 2.3. [1, Theore 3.29] Let (B, D(B be the generator of a strongly continuous seigroup on X and assue that the delay operator Φ is of for (2.1. Then the operator (A, D(A is the generator of a strongly continuous seigroup on E. Thus, (GDE is well-posed. Iportant special cases are operators Φ defined by Φ(f := n k= C kδ hk (f, f W 1,p ([ 1, ], X, (2.3 where δ hk (f = f(h k are the δ-easures supported at h k, the operators C k L(X are given, and h k [ 1, ] for k =,..., n. In particular, Theore 2.3 holds for the operators Φ given in (1.4. Reark 2.4. This see to be the appropriate point to ake a coent on tie easureent and the special for (1.4 of the delay in (DE. Assue that we are given an equation of the for v (t = Bv(t + k j=1 C jv(t h j, t, (DE-1 Then, denoting h := ax{h 1,..., h k }, we can rescale the tie aking the change of variables t = hs and u(s = v(hs. Hence we obtain a delay equation where the largest delay equals 1. Further, if all the delays were rationally dependent, eaning that h j αq for soe α R and all j = 1,..., k, then the new delays in the rescaled equation will becoe rational nubers. Hence, there exists N such that h j = nj for suitable n j N. Introducing sufficiently any zero operators C j, we see that (DE-1 could be rewritten as u (s = Bu(t + j=1 C j u(t j, t. (DE-2 Hence, the delay operator in (1.4 actually covers the case of finitely any rationally dependent delays. Notice that the transforation above does not affect the asyptotic behaviour of the solutions (for exaple, unifor exponential stability or hyperbolicity, but it ay affect the speed of the exponential convergence. Finally, we characterize the resolvent set and the resolvent operator of A (see [1, Section 3.2]. Let ɛ λ (t := e λt and Φ λ L(X be defined by Φ λ x := Φ(ɛ λ Idx = Φ(e λ( x for x X. Further, let (A, D(A be the generator of the nilpotent left shift seigroup (T (t t in L p ([ 1, ], X. We use notation ρ(, σ(, σ p (, σ a (, σ r (, and σ ess (, respectively, for the resolvent set, spectru, point, approxiate point, residual, and essential spectru of an operator (. Lea 2.5. Let X be a Banach space, (B, D(B be a linear, closed and densely defined operator, and Φ : W 1,p ([ 1, ], X X be linear and bounded. Let

6 6 András Bátkai, Tanja Eisner and Yuri Latushkin (A, D(A be the operator atrix defined in (1.7 and (1.8. Then λ σ(a if and only if λ σ(b + Φ λ, λ σ p (A if and only if λ σ p (B + Φ λ, λ σ a (A if and only if λ σ a (B + Φ λ, λ σ r (A if and only if λ σ r (B + Φ λ, λ σ ess (A if and only if λ σ ess (B + Φ λ. Moreover, for λ ρ(a the resolvent R(λ, A is given by ( R(λ, B + Φλ R(λ, B + Φ λ ΦR(λ, A. (2.4 ɛ λ R(λ, B + Φ λ [ɛ λ R(λ, B + Φ λ Φ + Id]R(λ, A Using a perturbation arguent, one can show the following regularity results for the delay seigroup, see [1, Proposition 4.3], [8]. Theore 2.6. Consider the atrix operator (A, D(A defined by (1.7 and (1.8. (a If (B, D(B generates an iediately nor continuous seigroup, then the seigroup generated by (A, D(A is nor continuous for t > 1. (b If (B, D(B generates an iediately copact seigroup, then the seigroup generated by (A, D(A is copact for t > 1. Hence, in these so called regular cases the spectral apping property for the delay seigroup does hold. It is also known, however, that if (B, D(B generates an eventually nor continuous or eventually copact seigroup, then this property can be destroyed by the delay ter. Reark 2.7. Forula (2.4 shows that one should not expect the spectral apping properties (1.1, (1.2 or (1.3 to hold, in general, for delay seigroups. Indeed, consider Φ as in (1.4 with = 1. Choose B and C = C 1 such that the spectral apping property (1.6 does not hold, see exaples for such operators in [3, Exaples 2.1.5] or in Engel and Nagel [5, Section IV.3.a]. Specifically, let us assue that 2kπi ρ(b + C for all k Z, but (B + C 2kπi 1 as k. Then Φ 2kπi = C e 2kπi = C, and hence R(2kπi, B + Φ 2kπi = (B + C 2kπi 1, as k. (2.5 By (2.4, we know that 2kπi ρ(a for all k Z, but also by (2.4 and (2.5, we have that R(2kπi, A as k, and thus 1 σ(t (1. This shows that assuption (1.6 is a natural assuption on (GDE if one expects the spectral apping property for (T (t t. 3. Proof of the ain result We consider fro now on delay operators with finitely any rational depending delays, i.e., operators of for (1.4, Φ := Σ j=1c j δ j,

7 SMT for delay 7 for soe N. Note that soe operators C j ay be zero. Recall, see Reark 2.4, that we rescaled the tie so that the greatest tie delay is 1. We have the following necessary condition for λ C to be in the spectru of T ( 1. Proposition 3.1. Let Φ be as in (1.4, and pick a nonzero λ σ a (T ( 1. Then λ σ a (e 1 (B+Φµ for every µ satisfying e µ = λ. Proof. By assuption, there exists a sequence {( x n } fn E such that n=1 ( x n fn E = 1, (T ( 1 λ( x n fn E, n. (b Our ai is to show that the sequence (x n fors an approxiate eigenvector corresponding to the eigenvalue λ for the operator e 1 (B+Φµ. Let u be the ild solution of (DE given by forula (2.2. By Lea 2.2, assertion (b eans, in ters of u, that (a u( 1 λx n X, n, (3.1 u 1 n( L p ([ 1,],X, n. (3.2 Also by (b, we have, for j {1,..., + 1}, [T ( j j 1 λt ( ]( x n fn E T ( j 1 (T ( 1 λ( x n fn E, and therefore u j as n u( j j 1 λu( X, n, j = 1,...,, (3.3 ( λu j 1 ( L p ([ 1,],X, n, j = 1,...,. (3.4 Take any µ satisfying e µ = λ. By representation (2.2 we have for all t u(t = x n + B t t u(sds + Φ µ u(sds + g(t, where the function g is given by t t [ t t g(t := Φ u s ds Φ µ u(sds = Φ u s ds ɛ µ Then u (t = Bu(t + Φ µ u(t + g (t, t, u( = x n, and we obtain by the variation of constant forula u(t = e t(b+φµ x n + First we show that t ] u(sds, t. (3.5 e (t s(b+φµ g (sds, t. (3.6 g L p ([,1],X, n. (3.7

8 8 András Bátkai, Tanja Eisner and Yuri Latushkin By (3.5 and the for of the delay operator Φ in (1.4, we have: g (t = Φ [u t ɛ µ u(t] = C j [u(t j e j µ u(t] = j=1 λ j C j [λ j u(t j u(t]. j=1 We fix j {1,..., } and show that u( λ j u( j L p ([,1],X, as n. Indeed, u( λ j u( j L p ([,1],X = u(1 + λ j u(1 j + L p ([ 1,],X = u 1 λ j u 1 j L p ([ 1,],X u 1 λu 1 1 L p ([ 1,],X + λ u 1 1 λu 1 2 L p ([ 1,],X λ j 1 u 1 j 1 λu 1 j L p ([ 1,],X, and by (3.4 each suand converges to as n. This proves property (3.7. Next, we show that e 1 (B+Φµ x n λx n X as n. Indeed, e 1 (B+Φµ x n λx n X e 1 (B+Φµ x n u( 1 X + u( 1 λx n X. The first suand here converges to as n by applying the Hölder inequality in (3.6, and using (3.7, while the second by (3.1. Finally, we show that li inf n x n X >. Passing to a subsequence, we have fro (3.2 and (3.4 as n : λf n L p ([ 1,],X u 1 ( λf n L p ([ 1,],X + u 1 L p ([ 1,],X = u 1 L p ([ 1,],X + o(1 = u 1 ( 1 λ u 2 1 L p ([ 1,],X + 1 λ u 2 1 L p ([ 1,],X + o(1 = 1 λ u 2 1 L p ([ 1,],X + o(1 =... 1 = λ 1 u 1 1 L p ([ 1,],X + o(1 = λ 1 u L p ([,1],X + o(1. This iplies, by (3.7 and representation (3.6, that λ f n L p ([ 1,],X M x n X + o(1 for M := sup t [,1] e t(b+φµ. Thus, x n would contradict (a. Hence, {x n } n=1 is an asyptotic eigenvector for e 1 (B+Φµ corresponding to λ σ a (e 1 (B+Φµ. Reark 3.2. Let Φ be as in (1.4 and N N. Rewriting Φ as Φ = N j=1 C jδ j N by adding zero operators C j, when appropriate and applying Proposition 3.1 with

9 SMT for delay 9 N instead of, we have the following: If a nonzero λ σ a (T ( 1 N, then λ σ a (e 1 N (B+Φµ for every µ satisfying e µ N = λ. We will now forulate the following characteristic equation for T ( 1. Proposition 3.3. Let Φ be as in (1.4 and assue that (1.6 holds for every λ C. Then the following relation holds for λ : λ σ(t ( 1 λ σ(e 1 (B+Φµ for all/soe µ with e 1 µ = λ. Proof. By Proposition 3.1 we only have to prove the iplication. Let λ σ(e 1 (B+Φµ for soe µ with e µ = λ. By assuption (1.6 there exists ν σ(b + Φ µ with λ = e ν. Since ν = µ + 2πik for soe k Z, we have Φ ν = j=1 C je j ν = j=1 C je j µ = Φµ. Therefore ν σ(b + Φ ν, which eans, by Lea 2.5, that ν σ(a. By the spectral inclusion theore for strongly continuous seigroups, this iplies λ = e ν e 1 σ(a σ(t ( 1. Proof of Theore 1.1. In view of Reark 3.2 and the spectral apping theore for polynoials, it is enough to prove (1.1 for t = 1. By the spectral inclusion theore for strongly continuous seigroups, we only have to prove in (1.1 the inclusion. Fix a nonzero λ σ(t ( 1. By the spectral apping theore for the residual spectru, we ay assue that λ σ a (T ( 1. Take any µ satisfying e µ = λ. By Proposition 3.1 and assuption (1.6, we have: λ σ(e 1 (B+Φµ = e 1 σ(b+φµ. Therefore, there exists ν σ(b + Φ µ such that e 1 ν = λ. As in the proof of Proposition 3.3, we have ν = µ + 2πik for soe k Z and Φ µ = Φ ν. Therefore, ν σ(b + Φ ν. By Lea 2.5, ν σ(a and we thus conclude λ e 1 σ(a. Finally, equality (1.2 follows by the fact that if the spectru of T (t for one t > does not intersect a circle centered at zero, then it will not intersect the correspondingly rescaled circle for any other t >, which can be seen by an appropriate odification of [5, Proposition V.1.15]. Acknowledgents This research was supported by the DAAD-PPP-Hungary Grant, project nuber D/5/1422. A. B. was supported by the János Bolyai Research Fellowship, the Bolyai-Kelly Scholarship and the OTKA Grant Nr. F T. E. was supported by the Studienstiftung des Deutschen Volkes. Y. L. was supported in part by the NSF grants and , by the CRDF grant UP OD-3, and by the Research Board and the Research Council of the University of Missouri; also, the support of Marie Curie Transfer of Knowledge project TODEQ MKDT-CT is gratefully acknowledged.

10 1 András Bátkai, Tanja Eisner and Yuri Latushkin References [1] Bátkai, A., Piazzera, S., Seigroups for Delay Equations, A K Peters, 25. [2] Bensoussan, A., Da Prato, G., Delfour, M. C., and Mitter, S. K., Representation and Control of Infinite Diensional Systes I-II., Birkhäuser, [3] Chicone, C., Latushkin, Y., Evolution Seigroups in Dynaical Systes and Differential Equations, Matheatical Surveys and Monographs 7, Aerican Matheatical Society, [4] Diekann, O.,van Gils, S. A., Verduyn Lunel, S. M., and Walther, H. O. Delay Equations, vol. 11, Appl. Math. Sci., Springer-Verlag, [5] Engel, K.-J., Nagel R., One-paraeter Seigroups for Linear Evolution Equations, Springer-Verlag, Graduate Texts in Matheatics 194, [6] Hale, J. K., Functional Differential Equations, Appl. Math. Sci., vol. 3, Springer- Verlag, [7] Kappel, F., Seigroups and delay equations, Seigroups, Theory and Applications, Vol. II., (Brezis, H., Crandall, M. G., Kappel, F. eds., Pitan Research Notes in Matheatics 152, Longan, 1986, pp [8] Mátrai, T., On eventually copact perturbations of partial differential equations with delay, Seigroup Foru 69 (24, [9] Nagel, R. (ed., One-paraeter Seigroups of Positive Operators, Springer-Verlag, Lecture Notes Math. 1184, [1] van Neerven, J. M. A. M., The Asyptotic Behaviour of Seigroups of Linear Operators, Operator Theory: Advances and Applications, Vol. 88, Birkhäuser, [11] Webb, G., Functional differential equations and nonlinear seigroups in L p -spaces, J. Diff. Eq. 29 (1976, [12] Wu, J., Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, Appl. Math. Sci. 119, András Bátkai ELTE TTK, Departent of Applied Analysisand Coputational Matheatics, 1117 Budapest, Pázány P. Sétány 1C, Hungary e-ail: batka@cs.elte.hu Tanja Eisner Matheatisches Institut, Universität Tübingen Auf der Morgenstelle 1, D-7276, Tübingen, Gerany e-ail: talo@fa.uni-tuebingen.de Yuri Latushkin Matheatics Departent, University of Missouri-Colubia, Colubia, MO 65211, USA e-ail: yuri@ath.issouri.edu