Complex-Analytic Theory of the -Function*

Transcription

1 Jural f Mathematical Aalysis ad Applicatis 237, Article ID jmaa , available lie at Cmplex-Aalytic Thery f the -Fucti* Edmd A. Jckheere ad Nai-Pig Ke Departmet f Electrical EgieerigSystems, Ui ersity f Suther Califria, Ls Ageles, Califria jckhee@eudxus.usc.edu Submitted by Ala Schumitzky Received February 24, 1998 I this paper, we csider the determiat f the multivariable retur differece Nyquist map, crucial i defiig the cmplex -fucti, as a hlmrphic fucti defied a plydisk f ucertaity. The key prperty f hlmrphic fuctis f several cmplex variables that is crucial i ur argumet is that it is a pe mappig. Frm this sigle result ly, we shw that, i the diagal perturbati case, all preimage pits f the budary f the Hrwitz template are icluded i the distiguished budary f the plydisk. I the blck-diagal perturbati case, where each blck is rm-buded by e, a preimage f the budary is shw t be a uitary matrix i each blck. Fially, sme algebraic gemetry, tgether with the Weierstrass preparati therem, allws us t shw that the defrmati f the crssver uder Ž hlmrphic. variatis f certai parameters is ctiuus Academic Press 1. INTRODUCTION Arud the tur f this cetury, i a very crdial exchage f crrespdece betwee Picare ad Bruwer, the issue f the budary behavir f maps, triggered by the pieeerig wrk f Picare hlmrphic maps f several cmplex variables, became a fficial field f mathematwx 2. Bruwer s deep isight it the aximatic fudati f ical edeavr tplgy led him t frmulate his celebrated therem the ivariace f dmai, sayig that the hmemrphic image f a pe set is pe, i ther wrds, that a hmemrphism is a pe mappig. This therem, alg with the ivariace f the dimesi ad the JrdaBruwer *This is a cmpai paper t Real versus cmplex rbustess margi ctiuity as a smth versus hlmrphic sigularity prblem, J. Math. Aal. Appl Xr99 $3. Cpyright 1999 by Academic Press All rights f reprducti i ay frm reserved.

2 22 JONCKHEERE AND KE separati therem, put tplgy its fudati. Ather leadig mathematicia f this cetury, C. Carathedry, specialized the budary behavir prblem t cfrmal maps, culmiatig i the celebrated Carathedry prime ed therem w22 x. I this paper, we shw that, by relyig slely the thery f hlmrwx 6, e ca rederive i a phic fuctis f several cmplex variables self-ctaied maer the key features f the cmplex -fucti aalysis, derive strger results, ad remve the diagal, multiliear, eve lumped parameter assumptis, avidig ay kid f prgrammig argumet. I a certai sese, we develp a mre aalytical thery f the cmplex -fucw26 x. The budary behavir f the Nyquist map w13x i the case f a diagal ti as suggested by Zames perturbati is easily dispsed f. The budary behavir i the case f blck-diagal perturbati relies crucially the existece f cmplexaalytic sets embedded i the budary f the set f buded matrices. The latter prblem, which ca be als traced back t Picare, is a fudametal prblem f CR gemetry Žwhere CR stads fr either CauchyRiema r Cmplex-Real. w5, 6 x. We the tur ur atteti t the sigularity aalysis f the retur differece map. We itrduce the hlmrphic Jacbia ad defie the geericity f the Nyquist map. Ctiuity f the -fucti relative t prblem data w12, 14, 19x is apprached usig ccepts frm set-valued aalysis ad is shw t reduce t the prblem f the structural stability f the crssverthe preimage f q juder hlmrphic perturbati w1, 1, 17, 25 x. Ctrary t the real case where q j beig a critical value ca make the crssver badly behaved uder perturbati, i the cmplex case the crssver remais structurally stable, eve thugh q j is a critical value. The key igrediet i this case is the s-called Weierstrass preparati therem. 2. NYQUIST MAP FOR DIAGONAL AND OTHER PERTURBATIONS The DyleSafvAthas multiliearrž blck-. diagal perturbati frmulati f the multivariable gai margi fr the Ž pe-lp stable. lp matrix Ls is shw i Figure 1. See w3, 7, 8, 13, 18, 23, 24x fr relevat backgrud ifrmati. It is well kw that this margi prblem ivlves the Nyquist mappig f : = q Ž z, s. detž I q LŽ s. Ž z...

3 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 23 FIGURE 1 Stadard amg ur tati are the pe uit disk f the cmplex plae, its clsure, its budary, the uit circle, ad its amplified versi k s z g : < z < F k 4 i i. Whe the right half-plae is meat t q iclude a small strip arud the imagiary axis, we dete it as s z g : s ) y, ) 4. The ifiity rm is defied as 5z5 s max < z <4 i i. The Gai Margi r Žfr structured, diagal, multiliear perturbati. is defied as q km s sif z : fž z, s. s q j, s g 4 Ž 1. q ½ ž / 5 s if k: f Ž k., s q j, s g Ž q s if z : z, s g f q j, s g 3 q s if½k: Ž k. = l f Ž q j. / 5. Ž 4. It has bee quite ppular t d the abve at fixed frequecy, 1 k j s sif 5z 5 M : fž z, j. s q j4 Ž 5. Ž j. ½ ž / 5 s if k: f Ž k., j q j Ž s if z : z g f Ž q j. Ž 7. ½ 5 s if k: Ž k. l f Ž q j. / Ž 8.

4 24 JONCKHEERE AND KE Ž. where f z s f z, j, ad the d the frequecy sweep k s if k j. 9 M M Althugh the abve equality has bee widely used, it is rted i a fudametal prperty f the fucti fž z, s. that has apparetly t yet bee ppularized Ž see Secti 4.. A few cmmets related t these defiitis are i rder: Ž. 1, Ž. 5 are frmulatis f the basis idea f fidig the smallest destabilizig perturbati. Ž. 2, Ž. 6 are just a rewritig f Ž. 1, Ž. 5 alg the lie f the quatitative feedback thery, where the idea is t fid the miimum gai such that the template itercepts q j. Ž. 3, Ž. 7, Ž. 4, Ž. 8 are merely rewritig f Ž.Ž.Ž.Ž. 1, 5, 2, 6, respectively, i the ucertaity space. Ž.Ž. 3, 7 are i the spirit f algebraic gemetry. Ž.Ž. 4, 8 ivlve sme kid f ctact Ž i a sese that will be made precise i Secti 6. betwee the stratified Ž q. maifld k = ad the algebraic variety f Ž q j. Ž j. ad are very much i the spirit f the thery f stratified spaces ad CR gemetry. The fixed frequecy Nyquist Ž retur differece. mappig f the cmplex -fucti i the frmulati Ž. 7 is a example f a hlmrphic fucti f several cmplex variables defied a plydisk: f : Ž k. ;Z z 1... det I q LŽ j. Ž z. z z1. Ž z. s... z Fr the techical reas that a hlmrphic fucti is defied ver a pe set, it is assumed that the fucti f is defied ver a pe set Z ctaiig the clsed plydisk Ž Fig. 2.. DEFINITION 1. The ctiuusly differetiable fucti f : Z,

5 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 25 FIG. 2. Ope dmai Z ctaiig, where s z g : < z< - 14 ad s j. where Z : is a pe, cected, simply cected subset f,is said t be a hlmrphic fucti f several cmplex variables if either ž / f 1 s qj f s z 2 x y i i i Ž zis xiq jy i, zis xiy jyi. r f Ž z, z,..., z. 1 2 is a hlmrphic fuctif e cmplex variablei each variable, separately. The first frmulati f hlmrphy, q j Ž f q j f. s, ž x y / i i is clearly equivalet t the usual CauchyRiema cditis, ƒ f y f s xi yi CauchyRiema cditis. f q f s x y i i A hlmrphic fucti f several cmplex variables eed t be a multiliear fucti; it eve eed t be a ratial fucti. Accrdigly, thse results f the -fucti relyig the hlmrphic prperty f the

6 26 JONCKHEERE AND KE Nyquist map d t eed the multiliearrž blck-. diagal perturbati assumpti; they d t eve eed the assumpti that the perturbati is ratial. Csequetly, we will csider the geeralized situati where the ly restricti is that all perturbatis zi eter the lp matrix i a hlmrphic fashi, typically, fž s, z, z,..., z. s det I q LŽ s, z,..., z This smewhat mre geeral frmulati allws us t csider pe-lp ustable systems. I this case, hwever, it is ecessary t assume that the clsed-lp system is stable fr z s ad that the umber f pe-lp ustable ples remais cstat as z g Ž k. fr the rage f variati f k beig csidered. See w13, Chap. 2, Therem 2.2 x. Observe that the hlmrphic aalysis ca be carried ver t the full Nyquist map f Ž istead f the fixed frequecy map f. by redefiig the basic map as F : = 1 q z, det I q L z,. 1 y ž ž / / 3. BOUNDARY BEHAVIOR Ituitively, k Ž j. i frmulati Ž 6. M is achieved whe the budary f the template itercepts q j. The atural questi is, what is the preimage f this situati i ucertaity space? Figure 3 attempts t depict this situati. This secti addresses these budary behavir issues Set-Valued Aalysis T prve that k Ž j. M is achieved the budary, we eed sme set-valued aalysis ccepts. DEFINITION 2. Let A be a subset f ad z be a pit f. The distace betwee the pit z ad the set A is defied as dž z, A. s if5z y a 5: a g A 4. DEFINITION 3. f A: Let A, B be subsets f. Defie the eighbrhd N s z g : d z, A -. 4 A

7 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 27 FIGURE 3 The the Hausdrff distace betwee A ad B is defied as 4 dž A, B. s if : A : N Ž. ad B : N Ž.. LEMMA 1. The cmpact set- alued mappig B ž / k f k is ctiuus fr the Hausdrff metric. Prf. This fact is implicitly ctaied i wx 4. T be self-ctaied, we sketch a simple prf. We must shw that, ), such that X < < X ž ž / ž // k y k - d f k, f k -. A It is claimed that the apprpriate is fud by ivkig ctiuity f f, viz., X 5 5 X z y z - f z y f z -.

8 28 JONCKHEERE AND KE X < X < Let k - k ad k y k -. Clearly, frm which it fllws that By ctiuity f f, we have which implies that Ž X. z zgž k. k : B, X ž / z zgž k. f Ž k. : f B Ž.. Ž 1. f B Ž. ; N Ž., z fž z. fž B z. : N fžž k zgž k. Cmbiig 1 ad 11 yields X ž / fžž k.. f Ž k. : N Ž.. Ž. X fžž k.. The prf f f k : N is trivial. The therem is prved. Q.E.D. THEOREM 2. k j is achie ed the budary, iz., M ž Ž M. / q j g f k. Ž Ž.. M Prf. Ideed, if q j g It f k, the by ctiuity f the set-valued mappig, there exists a ) such that ž ž Ž Ž M.. // q j g It f k y thereby ctradictig the ptimality f k. Q.E.D. M 3.2. Ope Mappig Therem The bulk f this secti deals with, amg ther thigs, a strg versi f a result f Dyle w7, Lemma 1 x, amely, f Ž q j. : Ž km.. ŽDyle w7, Lemma 1x prves that there exists at least e preimage pit i Ž k., while here we prve that all preimage pits are i Ž k... It M M

9 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 29 Ž. fllws that 7 ca be simplified t j 1 ke ~. k j s if k: det I q L j. M. s ke J ƒ fr sme F - 2. i The practical csequece f the abve result is that t fid, it suffices t sweep the subset Ž k. f Ž k.. Oe f ur claims is that this result is a crllary f the result f Picare, datig back t 19, which led Bruwer t develp the fudati f tplgy wx 2. This result f Picare is the s-called pe mappig prperty f hlmrphic fuctis f several cmplex variables w6, 1 x. The fudametal result is the fllwig: THEOREM 3 Ope Mappig Therem. A cstat hlmrphic fucti f se eral cmplex ariables, is a pe mappigthat is, f : Z z fž z 1, z 2,..., z., f Ž O. is pe i whee er O is pe i Z :. Prf. I the e variable s 1 case, take z g O ad let w s fž z.. We must shw that fr ay w clse eugh t w we ca slve the equati fž z. s w. Let be the rder f the first vaishig derivative, viz., d fž z. d fž z. X fž z. s w, f Ž z. s,..., s, /. dz dz

10 21 JONCKHEERE AND KE By the Weierstrass preparati therem, the equati f z s w is lcally equivalet t z q a1ž w. z q qaž w. s, where a Ž w.,...,a Ž w. 1 are hlmrphic fuctis. By the fudametal therem f algebra, the abve equati has a sluti prvided w is clse eugh t w w2, 21 x. The geeral case G 1 is prvided as fllws. Take z g O. Csider the e-variable hlmrphic fucti g : fž z q k., g, k g. Clearly there exists a such that g is t a cstat fucti Žfr therwise f wuld be a cstat fucti.. Take k small eugh such that z q k: g 4 : O. By the e-variable case gž. is pe. Furthermre, fž z. g g Ž. : f Ž O.. Therefre z g O, fž z. has a pe eighbrhd ctaied i fž O., ad hece fž O. is pe. ŽFr a geeralizati f this result t fiite m hlmrphic maps it, m G 1, see Grauert ad Remmert w1, p. 7 x.. Q.E.D. Remark 1. I the e variable case, z fž z., if f X Ž z. /, it fllws frm the Cmplex Implicit Fucti Therem that f is, lcally arud z fž z., a hmemrphism. The pe mappig prperty therefre fllws frm the Bruwer dmai ivariace, which says that the hmemrphic image f a pe set is pe. COROLLARY 4. Let f : Z z fž z 1, z 2,..., z. be a hlmrphic fucti f se eral cmplex ariables. Let ; K / Z

11 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 211 be cmpact ad let The fž K. s H. f Ž H. : K. Prf. By ctradicti, assume z g itž K. such that fžz. g H. Sice z g it K, pe eighbrhd O f z such that z g O K. z z It fllws that fž O z. : H. By the Ope Mappig Therem, fo Ž. is pe ad it fllws that Ž. Defie s s f z fž z. g fž O z. : itž H.. z Ope ad bserve the fllwig, s g fž O. : itž H., by precedig argumet z Ope s g itž H. s g H s H _itž H., by ctradictig hypthesis ½, s itž H. a ctradicti. Q.E.D.

12 212 JONCKHEERE AND KE 3.3. Diagal Perturbati COROLLARY 5. Let f : Ž k. Ž z,..., z. fž z,..., z. 1 1 be a hlmrphic fucti defied er the plydisk k. Fr example, Let fž z,..., z. s det I q LŽ j, z,..., z H s f ž Ž k. / be the Hrwitz template. The f H : k. Ž budary f the plydisk.. Remark. k s k k, fr G 2, is the distiguished

13 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 213 Prf. j We must shw that z g f H ; we have z g k s ke : g w, 2.4 i. Csider the partial Nyquist Mappig, Clearly, Defie the pit Clearly, f i: zi fž z1,..., zi, z i, ziq1,..., z. Hi s f z 1,..., z i, k, z iq1,..., z. H : H. i s s fiž z1,..., zi, zi, ziq1,..., z.. s g H ; s g H. i Ad the fllwig strig shuld be bvius: s g H _it H s g H s g H i: Hi s itž H. s itž H. ½ ½ s g H. i By the Ope Mappig Therem f the e-variable case, zi g fi Ž s. : k. It fllws that z g k, as claimed. i i Q.E.D.

14 214 JONCKHEERE AND KE 3.4. Blck-Diagal Perturbati We w exted the results f the previus secti t the case f a blck-diagally perturbed lp; fr example, z1 ƒ z 2... z11 z12 z1m s, z21 z22 z2 m... zm1 zm2 zmm... where ad z g k i z z z m z z z... z z z m m=m m1 m2 mm g kb, m= m where B detes the Ž clsed. uit ball f m = m matrices, that is, 4 m= m m=m B s A g : 5 A5s Ž A. F 1. Frm here we set k s 1 t simplify the tati. The space m= m is tplgized by the distace 5 A y B5s Ž A y B. max. The mai result pertais t a retur differece Nyquist map f the frm max m=m f : iž. = B Ž z, z,...,z,... det I q LŽ j., 1 2

15 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 215 where ad z 1 z 2. s.. Z z z z m z21 z22 z2 m Z s.... z z z m1 m2 mm As befre, we defie the Hrwitz template, m=m ž i / H s f = B, ad the prblem is t lcate f Ž H.. The targeted result is that Ž z1, z2,...,z,... g f Ž H. zi g ½ Z g UŽ m., where Um detes the uitary grup f cmplex m = m matrices. As i the previus secti, these results are prved usig hlmrphic fucti thery. We first dispse f the diagal perturbati terms. THEOREM 6. Prf. Ž z, z,...,z,... g f Ž H. 1 2 zi g s Csider the partial Nyquist mappig f d : Ž z 1, z 2,.... fž z 1, z 2,...,Z,... H s f,,...,z,.... d It is a hlmrphic fucti f several cmplex variables defied the plydisk Ž.. Ž. Ž Take z, z,...,z,... g f H. Clearly f z, z, d 1 2 g H Ž. Ž f z, z,... it H ; sice H : H, we get f z, z,... itž H.. d 1 2 d d 1 2 d

16 216 JONCKHEERE AND KE Ž. Ž The last exclusi tgether with f z, z,... g H yields f z, z,... d 1 2 d d 1 2 Ž. g H. Therefre z, z,... g f Ž H. d 1 2 d d. As a csequece f the pe mappig therem fr hlmrphic fuctis defied a plydisk, it fllws that Ž z 1, z 2,.... g iž.. Hece the prf. Q.E.D. We w fcus the Nyquist mappig relevat t the blck-diagal perturbati term, f b: B m= m Hbs fž z1, z2,...,b m= m,... Z fž z1, z2,...,z,..., where z g. Our majr result is that f Ž H.: Um. i b We first prve the fllwig weaker frm f the targeted result. THEOREM 7. f Ž H. : B m= m. b b Prf. It suffices t csider fb as a hlmrphic fucti f a great may cmplex variables z 11, z 12,..., z 1m, z 21,.... Take s g Hb ad Z Ž m= g f s. Clearly, Z it B m. b, sice the cverge wuld be a vilati f the pe mappig therem. Therefre Z g B m= m. Q.E.D. At this stage, we have t lk mre carefully at B m= m. LEMMA 8. 4 B m= m s A g B m= m : iž A. s 1 fr sme i Prf. Obvius. Q.E.D. Clearly, UŽ m. s A g m= m : iž A. s 1,i 4, s that UŽ m. B m= m, m ) 1. The fllwig lemma is the crerste f this part f the paper. LEMMA 9. Z g B m= m _UŽ m., there exists a parameterized cmplex aalytic set S ctaiig Z ad embedm= m ded i B _UŽ m,. iz., Z : S : B m= m _UŽ m.. m= m T be specific, S is a cmplex-aalytic set embedded i B _UŽ m. ad

17 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 217 passig thrugh Z if there exists a plydisk r ad a cstat i i hlmrphic map, such that S s hž r. i i i. ŽThe map h: r i i h: iri B m= m _UŽ m. m= m O Z, hž. is hlmrphic if all cmpets h ij: iri are hlmrphic fuctis f se eral cmplex ariables Žsee wx.. 5. m= m Prf. Take a pit Z g B _UŽ m.. This implies that Z has a sigular value decmpsiti f the frm max s 1 ƒ... l s 1 Z s UL U R, - 1 l - m. Clearly, the mappig lq M h: m Ž 1 y. B m= m _UŽ m. islq1 i Ž lq1,...,m. 1 s 1 ƒ... U L s 1 l lq1 q lq1... q Ž m defies a cmplex-aalytic set, S s h Ž 1 y.. islq1 i, passig thrugh m= m Z ad embedded i B _UŽ m.. Q.E.D. m m U R

18 218 JONCKHEERE AND KE Nw, we are i a psiti t frmulate the fllwig: THEOREM 1. f Ž H. : UŽ m.. b b Prf. Assume by ctradicti that there exists a Z g f Ž H. b b such m=m that Z g B _UŽ m.. By the previus lemma, there exists a cmplex-aalytic set S defied by h: iri S : B m= m _UŽ m. O Z m=m passig thrugh Z ad ctaied i B _U m Fig. 4. Csider the hlmrphic fucti f several cmplex variables, fb h: iri fbž S. : Hb O s s f Ž Z. g H. Clearly, O g Ž f h. Ž s. b. The fact that s lies at the budary while the preimage O des t ctradicts the pe mappig prperty f fb h. Q.E.D. b b FIGURE 4

19 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 219 Hlmrphic fucti thery des t allw us t arrw dw the preimages f the budary mre accurately tha withi Um, fr the fllwig reas: PROPOSITION 11. bedded i UŽ m.. There are parameterized cmplex-aalytic sets em- Prf. Assume, by ctradicti, that such a parameterized cmplexaalytic set exists. This parameterized cmplex aalytic set itself ctais hlmrphic curves. ŽBy defiiti, a hlmrphic curve i the cstat hlmrphic image f.. Therefre, uder the ctradictig hypthesis, there wuld exist a cstat hlmrphic mappig, h: UŽ m. h11ž. h1mž... h Ž. h Ž. m1 Sice h maps it the uitary grup, we have h11ž. hm1ž. h11ž. h1mž..... h h Ž. h Ž. 1mŽ. hmmž. m1 mm s I, g, Ž 12. where ad mm hijž. s Ý aij, k ks k hijž. s Ý aij, k. ks Nw, bserve that if s q j, r s 1 Žr y jž r..ž y j. s. Therefre, takig the hlmrphic derivative f Ž relative t yields h X h X 11 1mŽ. hž... s. X X h Ž. h Ž. m1 mm k

20 22 JONCKHEERE AND KE Sice h is uitary, it fllws that h X h X 11 1mŽ... s. X X h Ž. h Ž. m1 Sice the disk is cected, it fllws that hž. is a cstat fucti. A ctradicti. Q.E.D. Remark 2. The same argumet Žsee w6, Crllary 3, p. 15x. shws, fr example, that there are parameterized cmplex aalytic sets i a sphere. The existece f parameterized cmplex aalytic sets i ther bjects is the mai pit f Chapter 3 f wx 6. mm 4. FREQUENCY SWEEP Ž. We quickly prve the kw frequecy sweep fact 9 with a vel prf that relies the budary behavir f hlmrphic fuctis. Ž. THEOREM 12. Fr the pe-lp stable f s, z s det I q Ls z frmulati, we ha e Prf. Frm Ž. 2 it fllws that km s if kmž j.. q M ½ ž / 5 k s if k: f k, q j. Therefre, csiderig the set-valued mappig it fllws that q ž / k f k,, q ž Ž M. / q j g f k,. Ž. Ž Ž q.. M Sice z, s f z, s is hlmrphic, z, s g f f k,,we have Therefre q s g O s. 13 ž Ž M. / q j g f k,

21 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 221 ad k s if k: f k, q j M Ž. 4 ½ ž / 5 ½ ž / 5 s if k: f k, j q j s if if k: f k, j q j s if km Ž j.. Q.E.D. Fr the pe-lp ustable case, there are sme techicalities t be wrked ut. THEOREM 13. Csider the frmulati fž s, z. s detži q Ls, Ž z.., where Ls, Ž z. is mermrphic as a fucti f s i the right half-plae with sigularities p 4 that d t deped z. The Prf. i km s if kmž j.. I the pe-lp ustable case we have q M ½ ž i4/ 5 k s if k: f k, _ p q j. The prf fllws the same lies as the precedig, except fr the crucial differece that istead f 13 we have Frm this, it fllws that where s g q _ p 4 s j p 4. i i ½ 5 k s mi if k Ž j., k Ž p., M M M i 4 k Ž p. s if 5z 5 : fž z, p. s M i i Because f the behavir f the ples, fž z, p. i s, z, ad therefre k Ž p. s, ad the result fllws. Q.E.D. M i 5. SINGULARITY AND GENERICITY T avid pathlgies, we defie a geeric -prblem. As we will state mre precisely later, a prblem is geeric if it keeps the same structure uder data perturbati wx 9. Geericity is a ccept relevat t the sigularity structure f the map.

22 222 JONCKHEERE AND KE U DEFINITION 4 Critical Pit. A critical pit z f a hlmrphic map f: Z defied ver a cmplex aalytic maifld Z is a pit where the iduced liear map defied ver the hlmrphic taget space, d U f : H U z z Z, is t surjective, that is, dim d U fž H UZ Ž z z Observe that if Z is a pe set cverig, the Z is a cmplex aalytic maifld.. Usig lcal crdiates t chart the cmplex aalytic maifld Z yields a mre ituitive defiiti: DEFINITION 5 Critical Pit. A critical pit f the hlmrphic mappig f: Z is a pit where the hlmrphic partial derivatives with respect t all lcal crdiates vaish, ž / f 1 f f s yj s, k s 1,...,, z 2 x y k k k r equivaletly, the rak, ver the grud field, f the Jacbia represetati f d U f, z ž 1 2 / f f f J U fs z, z z z is - 1. The hlmrphic sigularity set is give by the simultaeus slutis t fr z s, i. Usig ly e sigle cstrait yields i 5 f Vi s ½z g : s. z i V is a cmplex aalytic variety Žw6, p. 19 x, w25 x. i. The sigularity set ivi is als a cmplex aalytic variety. ŽBy defiiti wx 6, a cmplex aalytic variety V is a subset f such that z g V, there exists a eighbrhd Oz f z i V such that Ozl V is the set f slutis t fiitely may hlmrphic equatis.. DEFINITION 6. The prblem f cmputig the sigularity set is said t be geeric, r the V i s are said t be i geeral psiti, r t itersect trasverally iff ž / 2 f rak s, z z i j U z

23 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 223 where the matrix f partial derivatives is evaluated at a arbitrary critical pit z U. T justify the statemet that a matrix f secd-rder derivatives f full rak is the geeric case, we eed the fllwig therem: THEOREM 14. Ay hlmrphic fucti f : ca be apprximated arbitrarily clsely with a fucti f the frm f q azq 1 1 qaz that has sigular matrix f secd-rder deri ati es at e ery critical pit. Prf. This is a cmplex-aalytic versi f the s-called Mrse apprxw11x f the real, smth case. T prve the therem it imati lemma suffices t shw that, except fr a set f Ž a, a,...,a. 1 2 s f zer measure, we have 2 Ž f q azq 1 1 a2z2q qaz. rak s, Ž 14. z z ž / i j z where the abve matrix f partial derivatives is evaluated at the sluti z f fž z. q a s. T prve the latter statemet, csider the mappig f q Ý az i i : ž / is1 Ž f q Ýis1az i i. z 1... z Ž f q Ýis1az i i. The Jacbia f that mappig is ž 2 Ž f q Ý is1az i i. 2 f J Ž f q iaz i i. s s. z z z z / ž / i j i j Csider a pit Ž a, a,..., a. at which Ž fails. The matrix f secdrder derivatives is evaluated at the sluti t fž z. q a s. If Ž 14. fails, it fllws that ž / ž / 2 Ž f q Ýaizi. 2 f rak s rak -, z z z z i j z i j z

24 224 JONCKHEERE AND KE s that z is a critical pit f the mappig f, ad the crrespdig critical value is fž z. sya. By the cmplex Sard therem wx 6 the set f critical values has zer measure. Hece the set f Ž a, a,...,a. 1 2 where Ž 14. fails has zer measure. Q.E.D. Nw, we ca state the fllwig: THEOREM 15. I the geeric case, the sigularity set f a hlmrphic fucti f: csists f at mst islated pits. Prf. Take a critical pit z U f a geeric f. It is a zer pit f the mappig f : f z 1. z.. f z By geericity hypthesis, J U z f is sigular, s that the map f is lcally arud z U a hmemrphism. Hece z U is a islated zer pit f f ad hece a islated critical pit f f. Q.E.D. EXAMPLE 1. As a example f a geeric case, it suffices t csider f s z1z3q z2z 3, because ideed, ž / 2 f 1 rak s rak 1 s 2-3. i j 1 1 z z ž / Because this example is t geeric, its critical set is mre tha a set f pits. Ideed, f s z 3 Ž 15. z 1 f s z 3 Ž 16. z 2 f s z 1 q z 2, Ž 17. z 3 3 s that the critical set is the liear variety z, z, g : z q z s

25 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 225 We ca fix this example by either f the fllwig tw methds: 1. f s z z q z z q z, where / The perturbed critical set is give by f s z 3 q Ž 18. z 1 f s z 3 Ž 19. z 2 f s z 1 q z 2. Ž 2. z 3 Clearly, there are critical pits i this case. Hece, the perturbed case is geeric. 2. f s z z q z z q z z, where / The perturbed critical set is give by f s z 3 q z 2 Ž 21. z 1 f s z 3 q z 1 Ž 22. z 2 f s z 1 q z 2. Ž 23. z 3 Clearly, the ly critical pit i this case is Ž,,., ad the assciated matrix f secd-rder derivatives is ž / 2 f 1 rak s rak 1 s 3. i j 1 1 z z ž / Hece this secd perturbed case is als geeric. EXAMPLE 2. It is very easy t shw that a affie Nyquist map f e cmplex ucertaity has critical pits. A multiaffie map f tw cmplex ucertaities, with vaishig leadig cefficiet, has ly e

26 226 JONCKHEERE AND KE critical pit, ad this ca be shw as fllws. Csider fž z 1, z2. s a12z1z2q azq 1 1 a2z2q a, where a 12 /, a 1, a 2, a g. Takig hlmrphic partial derivatives yields f Ž z 1, z 2. s a 12 z 2 q a 1 s z 1 f Ž z 1, z 2. s a 12 z 1 q a 2 s. z 2 Therefre, the ly critical pit is a2 a1 Ž z 1, z2. s y, y, where a 12/. a a ž / EXAMPLE 3. A multiaffie map f three variables, with vaishig leadig cefficiet, has tw critical pits. Ideed, take fž z 1, z 2, z3. s a123z1z2z3q a12z1z2q a23z2z3q a13z1z3 qa1zq 1 a2z2q a3z3q a, a 123/ Takig hlmrphic partial derivatives yields f s a 123 z 2 z 3 q a 12 z 2 q a 13 z 3 q a 1 s z 1 f s a 123 z 1 z 3 q a 12 z 1 q a 23 z 3 q a 2 s z 2 f s a 123 z 1 z 2 q a 23 z 2 q a 13 z 1 q a 3 s. z 3 Frm the last tw equatis, we derive a2q a12z1 a3q a13z1 z3sy, z2sy. a z q a a z q a Substitutig the right-had sides f the abve fr z3 ad z2 i the first equati yields a3q a13z1 a2q a12z1 a3q a13z1 a2q a12z1 a123 y a12 y a13 a123 z1 q a23 a123 z1 q a23 a123 z1 q a23 a123 z1 q a23 q a23 s.

27 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 227 Multiplyig by Ž a. 123 z1 q a23 2 yields a quadratic equati i z1 that has tw slutis. Hece there are tw critical pits. THEOREM 16. A multiaffie map f ariables, with aishig leadig cefficiet, has y 1 critical pits. Prf. Defie Ý Ý f z s a z z, i1 id i1 id ds i jg1,..., 4, i j-i jq1 where, fr cveiece a s 1. It suffices t shw that, i, Ž f r z. 12 i is t icluded i Ý Ž f r z., where Ž f r z. j/ i j j detes the pricipal ideal f wz, z,..., z x 1 2 geerated by the plymial fr z j. The prf is by iducti. Clearly, the therem has bee prved fr s 1, 2, 3 i Examples 2 ad 3. Let the asserti f the therem be valid fr y 1, ad we prve, by ctradicti, that it shuld als hld fr. Write fž z 1,..., z. s fž z 1,..., z. q z gž z 1,..., z.. Assume by ctradicti that This implies that f z f g. z ž / Ý j/1 1 j ž / f g f g z q z z g Ý z qz z q Ž gž z 1,..., z j/1, j/ j j This wuld imply that g z g g, z Ý ž / 1 j/1, j/ j which ctradicts the iducti hypthesis. Q.E.D. 6. OPTIMALITY Nw that we have idetified the preimage f the budary f the template t be withi the distiguished budary f the plydisk, we prceed t characterize the ptimal preimage pit, that is, the preimage f the budary pit f H that first itercepts q j as k icreases.

28 228 JONCKHEERE AND KE The situati q j g H, traced back t the dmai f defiiti, yields f q j : f Ž H.: Ž k. M, which meas that there is sme ctact betwee the cmplex-aalytic variety f Ž q j. ad the distiguished budary Ž k. M f the plydisk km. Observe that we are dealig with a ctact betwee tw differet structuresf Ž q j. is a cmplex-aalytic variety Ževe a cmplex-aalytic maifld if q j is t a critical value., while Ž k. M des t have the full cmplex-aalytic structure because its defiig equatis zi zis k 2 d t satisfy the CauchyRiema cditis: M z z s z /. i i i z i 2 4 w x Such a bject as z g : zi zis km is called a real hypersurface 5, 6. It is a particular case f a CR maifld w 5, 6 x. Ctacts betwee cmplex-aawx 6. lytic ad CR structures are the cetral theme f CR gemetry It turs ut that ifk: q j g fžž k..4 ca be frmulated as a trasversality prblem. First we itrduce sme tati. T f Ž q j. z detes the real ta- get space t f q j at the pit z that is, the taget space fr the uderlyig real structure f f Ž q j. i terms f the real variables x 1, y 1,..., x, y, where zis xiq jy i. The same defiiti applies t T Ž k.. Clearly, the defiig equatis f f Ž q j. are fž xq jy. z s, fž xq jy. s, ad the defiig equatis f Ž k. are g i xi 2 q yi 2 s k 2. Frm these bservatis, the fllwig lemma is easily prved: LEMMA 17. f f f x1 y z f f f x y y T f q j s ker T k s ker 1 1 z J z f x y 1 1 x2 y2 z... x y z J z g

29 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 229 The fllwig therem is the crucial ctact cditi: DEFINITION 7. The smth Ž real. maiflds Ž k. ad f Ž q j. are said t itersect trasversally if, z g Ž k. l f Ž q j./, we have z T Ž k. q Tf Ž q j. s, z where T detes the real taget space at z. z THEOREM 18. At ptimality Ž k. ad f Ž q j. M d t itersect tras ersally, that is z g Ž k. l f Ž q j.,such that M Ž z M z. dim T k q T f q j - 2. Prf. Assume by ctradicti that Ž k. ad T f Ž q j. M z itersect trasversally. The crucial fact is that trasversality is a pe prperty that is, fr ay perturbati f the maiflds cfied t sufficietly small tubular eighbrhds, the perturbed maiflds still itersect trasversally. Therefre, fr sme ), ŽŽ k y.. ad f Ž q j. M itersect trasversally ad km culd t be the miimum. Q.E.D. I terms f Jacbias, the crucial trasversality cditi ca be rewritte, successively, dim ker J g q ker J Ž z z f. - 2 H H Ž. Ž Ž z z.. dim Rw J g q Rw J f - 2 Ž. H dim Rw J g l Rw J Ž z. Ž z f. - 2 dim Rw J g l Rw J f G 1. z z I ther wrds, the rw spaces f the tw Jacbias must have empty itersecti, which meas that the system f equatis T T Ž. Ž z z. J g s J f must have a trivial real sluti. Still, i ther wrds, the crucial trasversality cditi ca be rewritte T < T Ž z. z ker J g J f /. Ž T Clearly, the cmpsite matrix J < T g J f. is 2 = Ž q 2. z z. Therefre, if q 2 F 2, that is, G 2, the crucial trasversality cditi reduces t T < T Ž z. z rak J g J f - q 2.

30 23 JONCKHEERE AND KE I ther wrds, all Ž q 2. = Ž q 2. submatrices f the cmpsite matrix Ž T J < T g J f. must cacel. z z 7. CONTINUOUS DEFORMATION OF CROSSOVER AND CONTINUITY T cpe with the ctiuity issues, we itrduce a perturbed mappig, f : = X Ž z,. fž z,., where s is the mial value, that is, fž z,. s fž z.. The perturbed mappig is cmplex-aalytic i bth variables z,. Ata level mre fudametal tha the ctiuity prblem, the issue is the uderstadig f hw the sluti set 4 f Ž q j. s z: fž z,. s depeds. It will be shw that f Ž q j. sustais a ctiuus defrmati as varies Ž Fig. 5.. FIGURE 5

31 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 231 Mre clsely related t the ctiuity prblem, the issue is whether the cmpact set-valued mappig f q j l r is ctiuus i the Hausdrff metric fr all r ). Ideed, the fllwig result hlds: THEOREM 19. If f q j l r is ctiuus fr the Haus- drff metric, the 5 5 ½ 5 k Ž. s if z : z g f Ž q j. l Ž r. M, r is ctiuus i. Prf. This result is implicitly ctaied i wx 4. T be cmplete, we sketch a simple prf. Let F s f q j l r r ay cmpact set fr that matter. We must shw that e ), ) such that X < < X M, r M, r y - k y k - e. It is claimed that it suffices t take such that < X X y < - d FŽ., FŽ. - e. ŽExistece f this is guarateed by ctiuity f FŽ... Let z g FŽ. be a pit such that 5z5 s k Ž.. Sice džfž., FŽ X.. M, r - e, it fllws Ž Ž X X that d z, F.. - e. Let z g FŽ X. be such that 5 z y z X 5 - e. By the triagle iequality, it fllws that Therefre 5 X X z F z q z y z - z q e. k Ž X. - k Ž. q e. M, r Iterchagig the rle f, X, ad repeatig the same argumet yields Therefre M, r k Ž. - k Ž X. q e. M, r M, r X M, r M, r k y k - e, ad the therem is prved. Q.E.D.

32 232 JONCKHEERE AND KE THEOREM 2. If k is ctiuus r ), the M, r is ctiuus k Ž. s if z : z g f Ž q j. M Prf. T prve that k Ž. M is ctiuus, we have t shw that, fr Ž. - a - b, k a, b is pe. If a, b are fiite, peess f k ŽŽ a, b.. M M fllws trivially frm ctiuity f k M, b. Hece it remais t prve that k ŽŽ a,.. is pe. This is de as fllws: M k Ž a,. s k Ž a, b. M M b) a s k Ž a, b. b) M s b) k M, bž Ž a, b... Sice k ŽŽ a, b.. M, b is pe by ctiuity f k M, b, its ui fr all b s is pe, ad therefre s is k ŽŽ a,... Q.E.D. M Clearly, if we ca prve that the crssver is ctiuusly defrmed uder the perturbati f the map, we will have prved that the cmplex k is ctiuus. M 7.1. Ctiuus Defrmati Take a pit z g f Ž q j.. If the hlmrphic Jacbia des t vaish, we ca select a variable, say z, such that Ž fr z.žz. 1 1 /. By the cmplex implicit fucti therem, we ca slve the equati Ž f z, z,..., z, s fr z i a eighbrhd f z, ; i ther wrds, there exists a hlmrphic fucti Ž z,..., z,. such that fž Ž z 2,.. Ž..., z,, z 2,..., z, s fr z 2,..., z i a eighbrhd f z 2,..., z. ad i a eighbrhd f. It fllws that the mappig Ž z,..., z,. 1 2 z 2. Ž z z g O, i z i i s 2,..., prvides the hlmrphic defrmati f the crssver uder hlmrphic defrmati f the Nyquist map. Nw assume that at z g f Ž q j. the hlmrphic Jacbia vaishes. Geerically this ccurs ly at islated pits. We take z t be such a represetative pit ad ivestigate the defrmati f the crssver arud that pit.

33 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 233 We first eed sme chage f variables t btai the crrect parameterizati f the prblem. LEMMA 21. There exists a liear, sigular chage f ariables, Ž z y z. s AŽ Zy Z., A g GLŽ,., such that, fr sme selected ariable, say Z, ad fr sme fiite k, if we 1 defie we ha e FŽ Z,. s fž z q AŽ Zy Z.,., F Z 1 FŽ Z1, Z2,...,Z,. s, Z 1, Z 2,...,Z, s, k k 1 2 Z 1 k F k Z 1 Z, Z,...,Z, s, Z 1, Z 2,...,Z, /. Prf. See Grauert ad Remmert w1 x. ŽObserve that the fiite-rder case is geeric.. Q.E.D. The defrmati f the crssver arud a sigular pit is described by the fllwig: Uder the ab e hy- THEOREM 22 Ž Weierstrass Preparati Therem.. ptheses, the crss er equati is equi alet t fž z,. s FŽ Z,. s where k k Ž Ž 2. Ž 1 1. Z y Z q r Z,...,Z, Z y Z q q rkž Z 2,...,Z,. s, Ž 25. r : = X i Ž are hlmrphic fuctis, defied i a eighbrhd f Z,...,Z,. 2, such that

34 234 JONCKHEERE AND KE riž Z2,...,Z,. s. Prf. This ca be viewed as a crllary f the Weierstrass divisi therem w6, 9, 16 x: Give a divisr FŽ Z,. satisfyig the abve cditis, give a dividat dz, Ž., there exist qutiet qz, Ž. ad remaider rž Z,. hlmrphic fuctis such that where ad dž Z,. s qž Z,. FŽ Z,. q rž Z,., qž Z,. / k Ý i is i rž Z,. sy r Ž Z,...,Z,.Ž Z y Z.. Ž. k Takig dz, s Z, Z yields the result. Q.E.D. 1 1 T uderstad the eed fr the chage f variable f Lemma 21, csider Fr i, fr k, d we have Actually, we have f z f z fž z 1, z2. s Ž z1y z1.ž z2y z2.. k f Ž z. /. k z i fž z1, z2. s 2 f s Ž z2y z2. s, s,... 2 z f s Ž z1y z1. s, s,... 2 z 2 2 We cat use the Weierstrass preparati therem directly the rigial variables. We have t destry the multiliear structure t get the crrect parameterizati f the prblem. Fr example, take frm which it fllws that ž / ž / 2 ž 2 2/ Z z1y z1 1 1 s, Z 1 z y z z y z z y z s Z Ž Z q Z. FŽ Z, Z

35 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION 235 Nw, we have 2 F Ž. /. 2 Z 1 The Ž z y z.ž z y z situati ccurs mre easily tha e wuld imagie i ctrls. Ideed, it suffices t csider the case f a lp matrix becmig triagular at sme : L Ž j. L Ž j z1 det I q ž z / L 22Ž j. 2 ž ž / / ž /ž / s L11 L22 z1 q z2 q. L L T uderstad the Weierstrass therem, csider a high-degree rtlcus prblem that has a breakaway pit. At the breakaway the characteristic plymial has a duble rt that bifurcates as the gai is perturbed. The Weierstrass preparati therem says that, whatever the degree f the characteristic plymial, the lcal behavir f the lcus arud the breakway is give by a mic plymial f degree 2, with its cefficiets hlmrphically depedig the gai. EXAMPLE 4. Csider s 2 q s. Figure 6 shws the s secti thrugh the bifurcati f the zer set. FIGURE 6

36 236 JONCKHEERE AND KE The set f Z -slutis t Ž is a k-sheeted brached cverig surface ver a pe subset f the hyperplae Ž Z,...,Z,. wx 2 9. Write Ži. Z Ž Z,...,Z,., the sluti lyig the ith sheet. Clearly, the sluti 1 2 Ž Z,...,Z,. Z1 Ži. 2 Z 2. Ž Z is ctiuus, ad eve aalytic away frm the brach pits. Gluig tgether the slutis as prvided by Eqs. Ž 24. ad Ž 26., usig the affie trasfrmati f Lemma 21, prvides the ctiuus defrmati f the crssver relative t. Figure 7 attempts t describe this situati. Ž Ž1. Žk. Observe that the mappig Z,...,Z. 1 1 ca be made hlmrphic by mappig the rts t the symmetrized pwer f, where tw pits whse crdiates differ by mre tha a permutati are idetified w6, 13, 25 x. FIG. 7. Gluig tgether the sigular ad the sigular defrmati f the crssver.

37 COMPLEX-ANALYTIC THEORY OF THE -FUNCTION Ctiuity We w cme back t the result ecessary t prve ctiuity f k. M THEOREM 23. f q j l r is ctiuus fr the Hausdrff metric, r. Prf. We must shw that ), ) such that < < - ž / d f q j l r, f q j l r -. Frm ctiuity f Ž 26., we ca fid a Ž Z,...,Z.) such that Defie 2 Z2 Z2 y - Ž Z Z Ži. Ži. Z1 Z 2,...,Z, Z1 Z 2,...,Z, - Ž Z 2,...,Z.. s if Ž Z,...,Z.. Z,...,Z gk 2 2 Clearly, by cmpactess f r, we have ). It clearly fllws frm 27 that s that f q j l r : N f Žqj.l Ž r. Ž. f q j l r : N f Žqj.l Ž r. Ž. ž / d f q j l r, f q j l r -, ad the mappig is ctiuus. Q.E.D. 8. CONCLUDING REMARKS If we attack the case f real perturbati fllwig the guidelies develped i this paper, we will arrw dw sme specific discrepacies betwee the real-smth ad the cmplex-hlmrphic cases. I the real-smth case, the retur differece map is t always pe. I the real-smth case, the sigularity set is geerically a etwrk f curves

38 238 JONCKHEERE AND KE frmig a smth maifld, istead f a set f islated pits. Furthermre, i the real case, the image f a sigular curve crssig q j culd create lack f ctiuity, while i the cmplex case it des t. T aalyze the disctiuity f the real assciated with a bifurcati f the crssver i a eighbrhd f a sigular pit, we use the Malgrage preparati therem w9, 16x istead f the Weierstrass preparati thew15 x. rem. These issues are expaded up i a cmpai paper ACKNOWLEDGMENTS May thaks t F. Callier ad J. Wiki, Uiversity f Namur, Belgium, ad H. Zwart, Uiversity f Twete, the Netherlads, fr may helpful discussis. May thaks t J. P. D Agel, Uiversity f Illiis at Urbaa-Champaig, fr his critical cmmets a early versi f this paper. REFERENCES 1. V. I. Arld, S. M. Gusei-Zade, ad A. N. Varchek, Sigularities f Differetiable Maps, Vl. II, Birkhauser, Bst, P. S. Aleksadrv, Picare ad Tplgy, Russia Math. Sur eys 27 Ž 1972., M. F. Barrett, Cservatism with Rbustess Tests fr Liear Feedback Ctrl Systems, Ph.D. thesis, Uiversity f Miesta, Jue C. Berge, Tplgical Spaces, Dver, New Yrk, A. Bggess, CR Maiflds ad the Tagetial CauchyRiema Cmplex, CRC Press, Bca Rat, FL, J. P. D Agel, Several Cmplex Variables ad the Gemetry f Real Hypersurfaces, CRC Press, Bca Rat, FL, J. C. Dyle, Aalysis f feedback systems with structured ucertaity, IEE Prc. 129 Ž 1982., M. Fa ad A. Tits, Characterizati ad efficiet cmputati f the structured sigular value, IEEE Tras. Autmat. Ctrl AC-31 Ž 1986., M. Glubitsky ad V. Guillemi, Stable Mappigs ad Their Sigularities, Spriger- Verlag, New Yrk, H. Grauert ad R. Remmert, Cheret Aalytic Sheaves, Spriger-Verlag, New Yrk, V. Guillemi ad A. Pllack, Differetial Tplgy, Pretice-Hall, New Yrk, D. Hirichse ad A. Pritchard, A te sme differeces betwee real ad cmplex stability radii, Systems Ctrl Lett. 14 Ž 199., E. A. Jckheere, Algebraic ad Differetial Tplgy f Rbust Stability, Oxfrd Uiv. Press, OxfrdrNew Yrk, E. A. Jckheere ad N. P. Ke, Tplgical Thery f r Ambiguities i Rbust Ctrl, i Prceedigs, IEEE 36th Cferece Decisi ad Ctrl, Sa Dieg, Califria, 1997, pp E. A. Jckheere ad N. P. Ke, Real versus cmplex margi ctiuity as a smth versus hlmrphic sigularity prblem, J. Math. Aal. Appl., t appear.

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