Compatible Versus Regular Well-Posed Linear Systems

Transcription

1 Compaible Verss eglar Well-Posed Linear Sysems Olof J. Saffans Deparmen of Mahemaics Åbo Akademi Universiy FIN-25 Åbo, Finland hp:// saffans/ George Weiss Dep. of Elecr. & Elecronic Eng. Imperial College of Sci. & Technol. Ehibiion oad, London SW7 2BT Unied Kingdom Keywords: Compaible sysem, reglar sysem, feedback, flow-inversion, ime-inversion, ime-flow-inversion. Absrac We discss differen reglariy noions for L p -well-posed linear sysems wih 1 p <, namely (from he weakes o he sronges) compaibiliy in he sense of Helon (1976), weak reglariy in he sense of Weiss and Weiss (1997) or Saffans and Weiss (2), srong reglariy in he sense of Weiss (1994a), and niform reglariy in he sense of Helon (1976) and Ober and W (1996). We frher invesigae how compaibiliy and reglariy are preserved nder varios ransformaions on he sysem, sch as he daliy ransformaion, feedback, flow-inversion, ime-inversion, and imeflow-inversion. Compaibiliy is he minimal assmpion nder which i is possible o describe he inp-sae-op behavior of a sysem wih iniial ime zero, iniial sae, inp, sae rajecory and op y in he sandard way () A() + B(), y() C() + D(),, (). Here A is he generaor of a srongly coninos semigrop A, B is he (possibly nbonded) conrol operaor, C is a (possibly nbonded, possibly eended) observaion operaor, and D is a bonded feedhrogh operaor. Also he sandard formla for he ransfer fncion, (1) D(z) C(zI A) 1 B + D, z > ω A, (2) where ω A is he growh bond of A, is based on compaibiliy. The operaors A and B are always niqe. The operaors C and D are niqe in he differen reglar cases, b no in he general compaible case. We give formlas for he operaors A, B, C and D of he ransformed sysems lised above in many of he cases where compaibiliy is preserved. 1 Well-Posed Linear Sysems Many infinie-dimensional sysems can be described by he eqaions (1) on a riple of Banach spaces, namely, he inp space U, he sae space X, and he op space Y. We have () U, () X and y() Y. The operaors A, B, and C are ofen nbonded whereas D is bonded. However, i is no known if all well-posed linear sysems can be wrien in his form, and i is herefore (and also for oher reasons) ofen more convenien o se an inegral represenaion of he sysem, which consiss of he for operaors from he iniial sae and he inp fncion o he final sae () and he op fncion y: () A + B,, y C + D. Here, A is he semigrop which maps he iniial sae ino he final sae (), B is he map from he inp (resriced o he inerval (, )) o he final sae (), C is he map from he iniial sae o he op y, and D is he inp-op map from (resriced o +, )) o y. We regard L p loc (+ ; U) as a Fréche space, wih he seqence of seminorms n Lp (,n;u). We regard L p loc (+ ; U) as a sbspace of L p loc (; U) and on he laer we define he bilaeral lef shif by, denoed τ, by (τ )(s) (s + ), < s, <. The well-posedness assmpion is ha (3) behaves well in an L p -seing, where 1 p <, i.e., () X and y L p loc (+ ; Y ) depend coninosly on X and on L p loc (+ ; U). If his is he case, we call he operaors C D a well-posed linear sysem, where B lim B τ, D lim τ D τ, each defined for hose L p loc (; U) for which he respecive limi eiss. I is possible o define a well-posed linear sysem Σ C D wiho any reference o he sysem of eqaions (1). See, for eample (alphabeically) Arov (3)

2 and Ndelman (1996), Crain and Weiss (1989), Salamon (1987, 1989), Saffans (1997, 1998a,c,b, 1999, 2), Weiss (1989a,b,c, 1991, 1994a,b), Weiss and Weiss (1997) (and he references herein). Mos of he available lierare deals wih Hilber spaces and p 2. Each well-posed linear sysem indces operaors A, B, and C corresponding o he operaors in (1). Here A is he generaor of he semigrop A. Before inrodcing he operaors B and C we need wo ailiary spaces X 1 and X 1. Choose any γ in he resolven se of A. We le X 1 be he domain of A, wih he norm X1 (γi A) X, and X 1 is he compleion of X wih he norm X 1 (γi A) 1 X. The semigrop A can be eended o a srongly coninos semigrop on X 1, which we denoe by he same symbol. We denoe he space of bonded linear operaors from U o Y by L(U; Y ), and le L p ω( + ; U) represen he space of fncions : + U for which e ω () belongs o L p ( + ; U). Proposiion 1.1. Le Σ C D be a well-posed linear sysem on U, X and Y. Denoe he growh bond of A by ω A. (i) Σ C D has a niqe conrol operaor B L(U; X 1 ), deermined by he fac ha he inp erm B in (3) is given by he sandard variaion of consans formla (he fncion inside he inegral akes i vales in X 1, b he final resl belongs o X) B A s B(s) ds, +, L p loc (+ ; U). (ii) Σ has a niqe observaion operaor C L(X 1 ; Y ), deermined by he fac ha he op erm C in (3) is given by (for almos all + ) (C )() CA, X 1. (iii) Σ has a niqe analyic L(U; Y )-valed ransfer fncion D defined (a leas) on z > ω A, deermined by he fac ha he Laplace ransform D of he inp-op erm D in (3) is given by D D(z)û(z), z > ω A, L p ω A ( + ; U), where û is he Laplace ransform of. The eisence of B and C is proved in Salamon (1989), Weiss (1989a), Weiss (1989b), and he eisence of a ransfer fncion is proved in Crain and Weiss (1989) and Weiss (1991). (See also Salamon (1989) and (Weiss, 1994a, emark 2.4).) The conrol operaor B is said o be bonded if he range of B lies in X, in which case B L(U; X). The observaion operaor C is said o be bonded if i is coninos wih respec o he norm of X, i.e., if i can be eended o an operaor in L(X; Y ). The operaor C in (1) is an eension of C, as will be eplained in Secion 2. To ge a ime-domain represenaion for he op y of a well-posed linear sysem similar o he second eqaion in (1) we inrodce he sbspace V of X U defined by V { X U A + B X }. (4) Also his space is a Banach space wih he norm V { 2 X + 2 U + A + B 2 X} 1/2. If X and U are Hilber spaces, hen so is V. Proposiion 1.2. Le Σ C D be a well-posed linear sysem on U, X and Y. Denoe he growh bond of A by ω A. We define he operaor C&D L(V ; Y ) by C&D C (αi A) 1 B + D(α), (5) where α C wih α > ω A can be chosen in an arbirary way (i.e., he he resl is independen of α as long as α > ω A ). We call C&D he combined observaion/feedhrogh operaor of Σ. (i) The op y C + D of Σ defined in (3) is given for all by () A y() C&D C&D + B, (6) () () for all X and all W 1,p loc (+ ; U) saisfying () () V. In pariclar, () V for all. (ii) The ransfer fncion D of Σ is given by D(z) C&D (zi A) 1 B I, z > ω A. (7) For he proof, see Arov and Ndelman (1996), Crain and Weiss (1989), Salamon (1987, 1989), Saffans (2), or Weiss (1989a,b). We call S C&D : V X U he sysem operaor of he sysem Σ, and we refer o A, B, and C&D as he generaing operaors of Σ. In Arov and Ndelman (1996), C&D is denoed by N. 2 Compaible Sysems I is sally no possible o spli C&D direcly ino C&D C + D, (8) where C C&D, X 1, D C&D, U, (9)

3 becase of he fac ha {} U is no conained in he domain V of C&D in general (his is re only when B is bonded). (The relaionship beween C and C&D epressed in (9) is always valid.) Ths, o spli C&D in his way we ms firs eend C&D o a larger domain conaining {} U as a sbspace. The smalles possible eended domain is Z U, where Z is defined as follows. We choose any γ in he resolven se of A, and le Z { z X z (γi A) 1 ( + B) for some X and U }. This is a Banach space wih he norm z Z ( inf 2 X + 2 ) 1/2, (γi A) 1 U (+B)z (1) saisfying X 1 Z X, and i is a Hilber space if boh X and U are Hilber spaces. I is easy o see ha V Z U, b he embedding V Z U need no be dense. Definiion 2.1. The well-posed linear sysem Σ C D is compaible if is combined observaion/feedhrogh operaor C&D can be eended o an operaor C&D L(Z U; Y ). We define he corresponding eended observaion operaor C L(Z; Y ) and feedhrogh operaor D L(U; Y ) by C C&D, Z, (11) D C&D, U. The eension of C&D o Z U need no be niqe, since V may no be dense in Z U. This means ha C and D need no be niqe eiher. However, here is a one-o-one correspondence beween C&D, C and D, i.e., any one of hese hree operaors deermines he oher wo niqely. This can be seen as follows. Clearly (11) defines C and D in erms of C&D, and conversely, if we know boh C and D hen we know C&D, since, by (11), C&D C + D, Z, U. (12) B here is also a one-o-one correspondence beween C and D since, by (7), D(z) C(zI A) 1 B + D, z > ω A. (13) This defines D as a fncion of D and C, and conversely, i defines he vales of C on he range of (zi A) 1 B as a fncion of D and D (he vales of C on X 1 are deermined by he original conrol operaor C, so hey are he same for all eensions). Ths we have shown he following: Proposiion 2.2. The well-posed linear sysem Σ is compaible if and only if C can be eended o an operaor C L(Z; Y ), and (11) is a one-o-one correspondence beween he eended C and he eended C&D. Ofen Z is a Hilber space and X 1 is a closed sbspace of Z, and hen i is clear from he above ha Σ is compaible. Someimes he eension of C comes narally, as in he case where C is bonded and has a (niqe) eension o an operaor in L(X; Y ). We can hen le his eension deermine he vales of C on Z. On he oher hand, in some cases (sch as bondary conrol sysems) i is possible o specify D arbirarily, i.e., o every bonded operaor D i may be possible o find an eended operaor C sch ha D is he feedhrogh operaor corresponding o C (see Saffans (2)). One possible choice in hese cases is o ake D. (If he sysem is a bondary conrol sysem wih a bonded observaion operaor, hen he wo mehods described above prodce, in general, wo differen eensions.) A hird mehod will be described laer: if he sysem is he resl of some ransformaion performed on some oher sysem, hen we may wan o choose C and D in sch a way ha he hey can be comped by a simple formla from he corresponding operaors for he original sysem. In spie of he possible non-niqeness of he eended observaion operaor C and he corresponding feedhrogh operaor D, independenly of how hese operaors are chosen, i is sill re ha he op eqaion (6) simplifies ino y() C() + D(),, (14) and we observed above ha (7) simplifies ino (13). In he seqel, raher han wriing C all he ime, we wrie C insead of C, b we sill hink of C as he (non-niqe) eended observaion operaor in L(Z; Y ). Whenever boh C and D appear in he same formla we reqire D o be he feedhrogh operaor indced by C. For a compaible sysem, we call a qadrple of operaors C A D B L(Z U; X 1 Y ) consrced as described above a se of generaors or generaing operaors of his sysem. As Helon (1976) commens, mos physically moivaed sysems seem o be compaible. This applies, in pariclar, o all sysems where B or C is bonded, o all bondary conrol sysems, and o all sysems wih finie-dimensional inp space U. All reglar linear sysems (see he ne secion) are also compaible. The eisence of a non-compaible wellposed linear sysem in he Hilber space cone is sill an open problem. A reasonably complee heory for compaible sysems is presened in Saffans (2). 3 eglar Sysems The non-niqeness of he eended observaion operaor C and he corresponding feedhrogh operaor D in a compaible sysem is slighly disrbing. I is possible o ge rid of his non-niqeness by sing a sronger reglariy assmpion on he sysem, and by sing a specific eension of C insead of an arbirary eension. Definiion 3.1. The well-posed linear sysem Σ C D is weakly, srongly, or niformly reglar if is ransfer fncion

4 D(z) has a weak, srong, or niform limi D lim D(z) (15) z + (where he limi is aken along he posiive real ais). As shown by Weiss (1994a) and Saffans and Weiss (2), every weakly reglar sysem is compaible and i is possible o eend C o he space Z in sch a way ha he corresponding feedhrogh operaor is D. I is even possible o eend C frher so ha he op formla y() C() + D() becoms valid for almos all nder he minimal assmpions ha X and L p loc (+ ; U). According o Saffans and Weiss (1998), every L 1 -wellposed sysem is weakly reglar, hence compaible. 4 Daliy In his secion we sppose ha U, X and Y are refleive, and ha 1 < p <. Then he (casal) dal of he well-posed sysem Σ C D is given by Σ d A C, (16) B D where is he reflecion operaor defined by ( )() ( ) for all and for all L p loc (). The sysem operaor of he dal is he adjoin of he sysem operaor of he original sysem: A d B d (C&D) d. (17) C&D In pariclar, A d A, B d C and C d B. The qesion of wheher or no D d D in he compaible case is more difficl. By Saffans and Weiss (2) or Saffans (2), weak and niform reglariy are preserved nder he daliy ransformaion and D d D in his case. By Saffans (2), if X 1 is dense in Z hen compaibiliy is preserved nder he daliy ransformaion and again D d D. I is no known o wha een his is re when X 1 is no dense in Z. According o Saffans and Weiss (2), srong reglariy is no preserved nder daliy in general (b, of corse, he dal of a srongly reglar sysem is weakly reglar, hence compaible). 5 Feedback To ge a sandard saic op feedback connecion for he sysem Σ we choose some operaor K L(Y ; U) and replace in (3) by Ky + v, where v L p loc (+ ; U) is he new inp fncion. Then i is easy o see ha I KC I KD v. (18) This eqaion defines and niqely and coninosly in erms of and v if and only if I KC I KD is inverible in X L p loc (+ ; U), or eqivalenly, if and only if I KD is inverible in L p loc (+ ; U). Under his assmpion we ge a well-posed closed-loop sysem Σ K whose inp-sae-op relaionship is given by (A K ) (B K ) () y C K D K v A B I C D KC I KD 1. v (19) The corresponding closed-loop sysem operaor is given by A K B K (C&D) K M 1, (2) C&D where I M. (21) I K(C&D) In pariclar, M maps he space V ono he corresponding closed-loop space V K. In he classical finie-dimensional case he above feedback sysem is well-posed (i.e., I KD is inverible) if and only if I DK is inverible, or eqivalenly, if and only if I KD is inverible. When his is he case we can wrie he closedloop generaors in he form A K B K C K D K A + BEKC BE (I DK) 1 C DE, (22) where E (I KD) 1. In he infinie-dimensional wellposed case he siaion is more complicaed, and he inveribiliy of (I DK) is neiher necessary nor sfficien for he closed-loop sysem o be well-posed. (For eample, in a bondary conrol sysem D can be chosen in an arbirary way, hence I KD will be inverible for some D b no for all D, independenly of wheher he closed-loop sysem is well-posed or no). Le s assme ha he closed-loop sysem is well-posed. Then lef-inveribliy of I DK is sfficien o imply ha he closed-loop sysem is compaible, and if I DK is inverible (from boh sides), hen (2) holds, as shown by Mikkola (2) and Saffans (2). (In pariclar, in a bondary conrol sysem we can make I KD inverible by aking D.) Uniform reglariy is always preserved nder feedback (and I DK is inverible in he niformly reglar case). I is no known if srong reglariy is always preserved (i is perserved if I DK is inverible; see Weiss (1994b)), b srong reglariy implies ha I DK is lef-inverible, hence he closed-loop sysem is compaible. According o Saffans and Weiss (2), weak reglariy is no preserved in general nder feedback, even in he case where I DK is inverible. 6 Flow-Inversion The idea behind flow-inversion is o keep he relaionships beween he sae and he signals and y in (1) inac, b o reinerpre y as he inp and as he op. By (3), I C D y. (23)

5 This eqaion defines and niqely and coninosly in erms of and y if and only if I C D is inverible as an operaor from X L p loc (+ ; U) o X L p loc (+ ; Y ), or eqivalenly, if and only if D is inverible as an operaor from L p loc (+ ; U) o L p loc (+ ; Y ). We hen ge a wellposed flow-invered sysem Σ, whose inp-sae-op relaionship is given by () (A ) (B ) C D y A B 1 (24) I. I C D y The corresponding flow-invered sysem operaor is A B 1 I (C&D). (25) I C&D In pariclar, I C&D maps he space V ono he corresponding flow-invered space V. In he classical finie-dimensional case flow-inversion is possible if and only if D is inverible, and in ha case A B A BD C D 1 C BD 1 D 1 C D 1. (26) Acally, flow-inversion can be inerpreed as a special case of feedback: if we (for simpliciy and wiho loss of generaliy) sppose ha U Y, hen we ge he flow-invered sysem by firs replacing y by y (meaning ha we replace D by D I) and hen sing negaive indeniy feedback. Ths, everyhing ha we said in he feedback secion applies o flow-inversion if we replace I KD by D, (I KD) by D, (2) by (25), and (22) by (26). In pariclar, if D is inverible, hen flow-inversion preserves compaibiliy and srong and niform reglariy. However, weak reglariy is no preserved nder flow-inversion. For more deails, see Saffans and Weiss (2) and Saffans (2). 7 Time-Inversion The idea behind ime-inversion is o solve (1) backward in ime, i.e., we le in (1) be negaive. The se of eqaions (1) wih canno direcly be inerpeed as he eqaions describing he evolion of a well-posed linear sysem, since sch sysems evolve in he forward and no in he backward ime direcion, b his can easily be aken care of by an era reflecion of he ime ais. Ths, we replace by in (1). This changes he sign of he derivaive in he firs eqaion in (1), b i has no inflence on he second eqaion. Ths, if we are o obain a well-posed ime-invered sysem by arging in his way, hen he sysem operaor of he ime-invered sysem Σ ha we ge ms be given by A B. (27) (C&D) C&D By simply ignoring he inp and he op y in (1), we realize ha a necessary condiion for he ime-inveribiliy of a well-posed linear sysem Σ C D is ha A can be eended o a grop. As shown in Saffans and Weiss (2), his condiion is also sfficien, and (27) holds. By (27), he combined observaion/feedhrogh operaor of he ime-invered sysem is he same as he original one, hence, in pariclar, compaibiliy is preserved nder imeinversion. Moreover, since he ime-invered ransfer fncion D can be comped hrogh (7) from he original observaion/feedhrogh operaor C&D (defined in (5)), we ge ha for all z in he righ half-plane ( z) > ω A, (zi A) D ( z) C&D 1 B I (28) C (zi A) 1 (αi A) 1 B + D(α), where ω A is he growh bond of he semigrop generaed by A and α > ω A. Comparing his epression o (7), we realize ha D ( z) and D(z) are obained from he same formla, which in one case is valid for z > ω A and in he oher case for z < ω A. The fncion prodced by his formla is analyic in he resolven se of A. Ths, if he specrm of A does no separae he righ half-plane z > ω A from he lef half-plane z < ω A, hen D ( z) and D(z) are analyic coninaions of each oher. I is shown in Saffans and Weiss (2) wih a scalar conereample ha reglariy need no be preserved nder ime-inversion. (In he single-inp single-op case weak, srong, and niform reglariy are all eqivalen, and hence none of hem is preserved.) Moreover, even if boh he original and he ime-invered sysem are reglar, i is sill possible ha he forward eended observaion and feedhrogh operaors differ from he backward eended observaion and feedhrogh operaors, as can be seen from he simple eample D(z) 8 Time-Flow-Inversion e z 1 e z. To ge a ime-flow-invered sysem, we perform he wo operaions described in Secions 6 and 7 simlaneosly. Tha is, we inerpre y as he inp and as he op, and replace < in (1) by >. If Σ C D if flow-inverible, and if he flow-invered sysem is ime-inverible, hen by combining (25) and (27) we ge he epeced formla for he sysem operaor of he ime-flow-invered sysem Σ, namely A B (C&D) A B I 1 I, (29) C&D which in he compaible case wih an inverible operaor D shold become A B A + BD 1 C BD 1 D 1 C D 1. (3) C D In he finie-dimensional case a ime-flow-inverible sysem is necessarily boh ime-inverible and flow-inverible

6 and (3) holds. In pariclar, he dimensions of U and Y ms be he same. This is no re in he infinie-dimensional case, as will be shown below. I is qie simple o find a necessary and sfficien condiion for ime-flow-inveribiliy. Le >, and define (π (,) )(s) (s) if s (, ) and (π (,) )(s) oherwise. Then (3) can be rewrien in he form () A () + B π (,), π (,) y C + D π (,),, (31) where C π (,) C and D π (,) Dπ (,). Defining Σ A B C D, (32) we can wrie his as () Σ () π (,) y. (33) π (,) If he sysem is ime-flow-inverible, hen i ms be possible o epress () and π (,) in erms of () and π (,) y, and his implies ha Σ ms be inverible from X L 2 ((, ); U) o X L 2 ((, ); Y ), for all >. I is shown in Saffans and Weiss (2) ha his necessary condiion is also sfficien for he eisence of he ime-flow-invered sysem. Moreover, if Σ is inverible for one >, hen i is inverible for all >, and he ime-flow-invered sysem operaor is given by (29). As menioned earlier, in he finie-dimensional case, a sysem canno be ime-flow-inverible nless he dimensions of U and Y are he same. This is no re in he infiniedimensional case as he following conereample shows: ake U {}, Y C, le A be he lef-shif on X L 2 ( + ), and le C I. This sysem is conservaive, meaning ha he operaor Σ A is niary from X o X L 2 (, ) for all, hence inverible. The inverse of a niary operaor coincides wih is adjoin, and i can be shown ha he ime-flow-invered sysem coincides wih he dal sysem in his case. Ths, A A d A is he righshif on L 2 ( + ), (B ) (C T ), i.e., B y π (,) y, and he generaors of he ime-flow-invered sysem are A A and B C. The sysem described above is ime-flow-inverible b neiher ime-inverible nor flow-inverible. Boh he sysem iself and he ime-flow-invered sysem are reglar. The feedhrogh operaor D is no inverible, and he righ-hand side of (3) is no even well-defined (b of corse, (29) is sill re). The qesion wheher compaibiliy or reglariy are preserved nder ime-flow-inversion is sill wide open. eferences Arov, D. Z. and M. A. Ndelman (1996). Passive linear saionary dynamical scaering sysems wih coninos ime, Inegral Eqaions Operaor Theory, 24, pp C Crain,. F. and G. Weiss (1989). Well posedness of riples of operaors (in he sense of linear sysems heory), in Conrol and Opimizaion of Disribed Parameer Sysems, Birkhäser-Verlag, Basel, vol. 91 of Inernaional Series of Nmerical Mahemaics, pp Helon, J. W. (1976). Sysems wih infinie-dimensional sae space: he Hilber space approach, Proceedings of he IEEE, 64, pp Mikkola, K. (2). Infinie-dimensional H and H 2 reglaor problems and heir algebraic iccai eqaions wih applicaions o he Wiener class, Docoral disseraion, Helsinki Universiy of Technology. Ober,. and Y. W (1996). Infinie-dimensional coninos-ime linear sysems: sabiliy and srcre analysis, SIAM J. Conrol Opim., 34, pp Salamon, D. (1987). Infinie dimensional linear sysems wih nbonded conrol and observaion: a fncional analyic approach, Trans. Amer. Mah. Soc., 3, pp Salamon, D. (1989). ealizaion heory in Hilber space, Mah. Sysems Theory, 21, pp Saffans, O. J. (1997). Qadraic opimal conrol of sable well-posed linear sysems, Trans. Amer. Mah. Soc., 349, pp Saffans, O. J. (1998a). Coprime facorizaions and wellposed linear sysems, SIAM J. Conrol Opim., 36, pp Saffans, O. J. (1998b). Qadraic opimal conrol of wellposed linear sysems, SIAM J. Conrol Opim., 37, pp Saffans, O. J. (1998c). Feedback represenaions of criical conrols for well-posed linear sysems, Inerna. J. obs Nonlinear Conrol, 8, pp Saffans, O. J. (1999). Admissible facorizaions of Hankel operaors indce well-posed linear sysems, Sysems Conrol Le., 37. Saffans, O. J. (2). Well-Posed Linear Sysems, Book manscrip. Saffans, O. J. and G. Weiss (1998). Well-posed linear sysems in L 1 and L are reglar, in Mehods and Models in Aomaion and oboics, Proceedings of MMA98, Ags, Miedzyzdroje, Poland, pp Saffans, O. J. and G. Weiss (2). Transfer fncions of reglar linear sysems Par II: inversions and daliy, Manscrip. Weiss, G. (1989a). Admissibiliy of nbonded conrol operaors, SIAM J. Conrol Opim., 27, pp

7 Weiss, G. (1989b). Admissible observaion operaors for linear semigrops, Israel J. Mah., 65, pp Weiss, G. (1989c). The represenaion of reglar linear sysems on Hilber spaces, in Conrol and Opimizaion of Disribed Parameer Sysems, Birkhäser-Verlag, Basel, vol. 91 of Inernaional Series of Nmerical Mahemaics, pp Weiss, G. (1991). epresenaions of shif-invarian operaors on L 2 by H ransfer fncions: an elemenary proof, a generalizaion o L p, and a conereample for L, Mah. Conrol Signals Sysems, 4, pp Weiss, G. (1994a). Transfer fncions of reglar linear sysems. Par I: characerizaions of reglariy, Trans. Amer. Mah. Soc., 342, pp Weiss, G. (1994b). eglar linear sysems wih feedback, Mah. Conrol Signals Sysems, 7, pp Weiss, M. and G. Weiss (1997). Opimal conrol of sable weakly reglar linear sysems, Mah. Conrol Signals Sysems, 1, pp