BSTJ 60: 10. December 1981: Criteria for the Response of Nonlinear Systems to be L-Asymptotically Periodic. (Sandberg, I.W.)

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Criteria for the Response of Nonlinear Systems
to be L-Asymptotically Periodic

LW. SANDBERG:
(acusesiet recived hpi 22, 1981)

We consider the behaniar of general type of system governed by.
dan input-outpud operator ( that maps each excitation x into @
formesponding response 7. Here excitations and reqponsee are 3°

tealued funcluons dejinnton a s01 7. To acermumodate Both continuous
time and discrete linn ruses, 1 is alinved to be either {9, =) oF
(Gol, =+-). We uddrone te following question, Under what ennai
tions on tnd 1 Gn it true thatthe response ris L-egmplotically
eriutic in the sense that r = p+ q where p is periodic witha ven
Deri, and q has finite enargy fhe, is sqrare stommable)? This type
9f question arises naturaly in meany applications. Phe min results
noun fchice include ce nerenoary al ffcien condition) ave bast
tally "oolehenrems,” To ilunrate hoe they ean be used, en example
Uaiscuseid involving an integral equation that is often eaceurtored
{nthe theory of fendback aystems,

|. INTRODUCTION

In thie paper we coms the behavior ofa general typeof system
governed by an input-outgny operator (that mapa esch excitation 2
inca a corresponding response F. Here excitons and yesponsea are
‘sled fonetions defined on a set T- (As usual, n is ap exbitrary
psiive ines anoommodate both continuous time and diarete
time eases, we allow T vo be wither (0) ur (01,2 ===). As to Ex
stabiey theors"* each x is draw frum w funy BUZ) of fonetions
‘whore truncations belong 1o a sel [uf Sniteenery (ie, square
‘Rummeble) fanetios (ne dette given Sn Section 2.3), and @ ie
fsesumed to map) E(E) into Fi)

‘We adres, and give in Section 3.2 revulla concerning, the following
‘question. Under what conditions on G wn xii trae thatthe response
is Zeuegmptotcally periodic inthe sexe thal Y= p+ q, here p ia
periodic with w given pero 7, al ¢ ha Gil enmergy2? Thi ype of
‘question arises naturally in many applications. Qurresulta are basicaly
“Noa theorems” which appear to be widely applicahle. An example is
riven in Section 14

IL. MOTIVATION AND BACKGROUND MATERIAL

‘To provide motivation for considering an sbstact input-output
‘operator Gand also to deseribe some canier results related to those
‘of Section II, wo hogin hy recalling that am important example of
type of equation that arises fn he sly of phyla (ch
feedback systems or networks containing Tnear pest aml/r drs
‘ted clemonts ns well ax memrvess,posly ime-warying, nalinear
lens) ie tegen eation

x0 a

+ [ Betton ats «

in which x and r take values in A" (hore elements we take to be
column vector), £ i enn Xn matri-valued fonction, aod y maps
BE x [0, 0) inuo R. In oq (0, typically takes ino account initial
‘onditions a wellainpule and io the output tLe, i the intermediate
‘or final put) corresponding tx (fee, for instance, Ret. 2, pp. 872-4
Tor a specie application) Disevetstime councerarts of e9. (1) (368
[Ref 7, pp. 410-1, for exainple) al aoe alten fa system alien.

‘Much i know about the propertie of ea. (I}, eg, S68 Refs 2 8
‘and 5. In particular, ifm = T/and the “cecle citation” of Hef 2
together with certain aazociated conditions concerning &, $y % adr
scribe in he reference, aro roti.) is puriudic if with some
perio nd x — y+ x, with Fadl pic al ih xp
bounded and such thee o(@)+ 0 ae ¢—~, then we have r— p+ qin
which ps periodic with pesiad r, and ql) 0 anf» (see Ret. 2,
‘Theorem 4)

‘This remit generated to avbitary nit proved in Ref 6 by fst
showing thal cher isa rpetiicp defined on (—e, ©} such chat, with
[nested periodically for nepstive values the aailiay equation

sto =ptt+ [R= of, oh, 1 —

ea

in cot, andar efor sump, et Gt op Dy in

set pn ign ok wel

[Bowie 2 iow with mich sey (hat? soapy perc r te

Shee pleted tun deal otesrae eran cr
"Te srr elated se tre Hah 194 1 ee eet eae.

2960 THE BELL SYSTEM TECINICAL JOURNAL, OCCEMOCR 1901
‘is atte ‘Phen, using og, (1) ancl efi thar en. (2) gives

pid [ me melptoh el, 20,
itis proved thac when x = 3 + my we have rl) — pith > Ane r=,
in which inthe zero eloment of A similar ra shows that i
f1= | flor the ake of simplifying w states of «resull) snd both 2
nd, where stt}={t | io} ie fv ¢> 0 hava fini energy, then under
the conditions indicated above, r— hax fine energy [ste Rel. 3,
Crary 1a he proota in Let 2 aro of w functional analytic mature.
For mncarial related in a general tens eoncurning systems of difer
‘oil enstona, and in which » Lyapau-unciom xppronch i sed

0 Ke. 12, pp 210-28, Concerning mare recent material, areal slong
the aamue linc: wx the ane denerbed in the preceding paragraph for
inlereonneried systems governed by © somewhat dilferoal less of
integral equations, i proved in Ret 18 ‘There, too, an muxiliry
‘aqui approach ie used

‘Under reesonnble conditions on & and ¥ (ace Section 3), (hese
RAL, previously deserved and defined in Seetion TM, contains ex
tty one eolinn rf eg, () foreach x © BCL), Wo now inure the
typically trivial restrecon thac only solutions of eg, (1) contains in
BL) ate of interes to we. Thus, under saaonable contin thre
ascocavel in natural way with oq. (0) a map GiB(L) +r BUF) such
that r= Ge for ach 4 C-K12). Of course, many other examples can be
fron in which much w map (arbi

‘Assuming that y(-. of ins {1) ie independant of 6 and thac
‘a, )~ &, nooo that the G associated sith oq. 11} has the property
that it ie fime invariant in the sul sense thal the mespotae to 4
Glelayed nnn ie the delayed reponse to che original input, (Por x
proce definivion of ine invariance, se Sovluun 38) This tpe of
property of rather then the concep of an aullary quien, ply
‘oentrol ene in our approach in Section TT.

{W. LASYMPTOTIC PERIODICITY. TIME INVARIANCE, AND
PPERIODIOALLY-VARYING SYSTEMS

12.1 Pretmmary notation and dfitions

‘Throughout the zemainder of the ype Uhe alow
deinitone are used.

LASYMPTOTICALLY PERIODIC RESHONSE 2961
“te eymbol denotes eithor [U9 or (8,1, +++}. lemons of R™
are taken be column veccon, denote the tunspows ofan acbieary
We RS and fstanda forthe ero element of 2" =

It T= [0 w}, then £ donates the aot of Lobesguc measurable
feunetiona v from 7 into A” auch chat

| wttin(tidt <0,

Alternatively, when = (0,1, 2, «+-) standa forthe ost of maps »
from Pinto R” euch that

Evin ex

‘Tho norm [ol of an arbitrary clement v of L is defined by

(|

il=(3 eon) it T= 10.2.2

eipetnat) F= 10,9,

od

With this norm, £ is a Banach space of finite energy (ho square
rare) Funes

Pure R® avd s C7, tae denotes che map from into A
defined by (0) = v(t for 2G Twith £26, and ott) = Bfor fe
such thar f > o. We use E(L) to denote the “extended set
[eeP RP defor CT), and ty standa for the 2aro element of
‘B(LA, (Nove that BAZ) is tho ea of ald mape v: D> 4" when T= (0,
Lad

‘We say that a map HPI) — BE) ie causal (sce Ret 2p. B88) iF
wchave (FO) = [Pin foreach « & RIP) and each =P

Farany v © (L) and each we 724. 1) denotes the element x
of BIL) dafined by tle) = ott = rte T

“The symbol «denotes fixed postive eloment of 7; and stands for
the se of periodic functions (v@ EiZ}lot + 2)= old for¢ eT)

‘A velco i played by dhe sal defined by $= (2 C L|there is
ni ©. with che property that

Eve ted v0

ie Kh where 6 meas convergent in re lo

TiS ae eat a in 8 ere

2962 THE BELL SYSTEM TECHNICAL JOURNAL. DECEMBER 1981
Finally, fir each w € 7, the “delay mep" DsE(L) > BUL) is defined
by (Dust) = vle—w) for t= wand LDnit) = for tw.

3.2 L-Asymetote periodicity

‘We shall ute the following hypothe:
112: Ge a map from F(Z) into E(L) auch that for any v C EL), wo
have (GDa}t) = (D-Go)(t) for 0@ F wath ¢ +

"This hypothesis slated whenever (is a causal map of Bi) into
iteelf that is either time invariant or periodically varving with period
+ {see Secon 3.8). Our main result is he following:
‘Theorem 1: Assume that U1 holds. Let x © BL), and tet r denote
Gx, Then r has the form p~ q with p © P and q €L if and only if
(Gr— DES.
Proof: Suppose fist thut (Gx ~ GD,2) = for some v €S, Let v* €
be wuch that

Sot ean er

as K+, We shall une the proposition that
AD Ee HAO, o
hich follows from the inequality

Bet tte ta el
arctan Been

Sea

w

for 2 2,and the fae: that dhe righ side of inequality (4) approaches
ano an Ko,

Lt py danola + + 0°, which ie clearly an element of E¢L). Since
re) =(GD,2iI0 + ole) or C= Ti by HI. weave rfl — ree) +20)
fort = 1. Thersfore, fr fT, plt =) (eh) eerle tA = ele)
Fulet 7) 4 2°(e- 1). On che ober band, using oy (3), (0+ 009)
See ha =r = ol) = ple) for all CE Tit T
(0.1.2---). and for almost all #2 TMT = (0, =). Therefore,
wth p the element of I afin by 0) = pal) for €€ [0, 2) 07.
wwe have put) ~ ple) = for all © Tit T~ (0, 1,2, -»-} and for al
thet all ETH T=[0, 0), and clearly r~ p+ (pe —p— vin
which (po =p ~ 08) €T

Sunpose now thal r~ ps4 g-with pC P and q © Land let =
(Ge GD.) Fore 2, ul) = r= rl 9 = pO) + gil) —
Pit 7) ~ g(t 7} = ght) — g(t 11 which together with w © FUE),
Shows that we J

Let u"(.) in Zhe defined by

ASYMPTOTICALLY PERIODIC RESPONSE 2968
ute) = ¥ ale +e)

for ¢€ T and any positive integer K, und lot J be an intuper such that
o> K Using ult + he) = gle he) ~ gft-+ (B= Dr] for B= Land
TET, we have forte 7,

we) — WU) = Ewes be) Fale

= 5, wet as)

t+ Je) ~ alt +e

Taos fu? = wx lg + J) + [gle + ROL Sinoe [ye + KY
Oa KS, (Wt CL isu Cauchy sequence, and, by the
‘completonese of L, there a a” & Ze mveh that [x uf + Oa
+, This concludes he proot

92.1 Comments

“The following example shows Unit 8 is proper subwe of Z. Tet
G be the identicy opersiur um FUE}, lake m= Ip and lat 2 C RAE}
bbe defined by ale) = In 2 for 2 © (0, 2.1 F, and ate) = In ¢ for
fe G0) T. Lec =~ 1. Then, (Gx ~ GD.xiGl = rit) rie— 1)
Inde 1D] for # © [8 2) Using the inoquaity In(1 +o) =
valid for 920, wo aoe that Info — 17] @— 1) fore e[3, 0) 07,
and therefore that v, defined by v(t) = (Gx — GD-x\(0) for ¢ © T,
belongs to L. Sines here Gx eannct be waitton as p+ q with pc P and
.¢ Z, it folowa from the theorem that 8 ¢ 8.

Tis moc ifficle to verify chat che prouf given of the theoreca can be
‘modified to show that. canbe replaced with the following somewhat
‘weaker hypothesia.

FL": GEIL) > FUL) ia map such tat for any 0 BL), there is an
$€ Such thee (GD.0}t) ~ (D.Gei0) + sie) far C= TAL, 2)

‘The simple example: n — 1, (Go}(e) = o() +e" for € © Pand each

© BU) one for which HY, but nt ILI, i met

32.2 Corotarie (the use of weighing factions)

{In this section, and in the Appendix, w denotes any function from T
into R such that there isa constant > 0 for which w(0) = (L+ fe)®
when ¢ © 7, and such that w is measurable on T and bounded on
ounded subsets of Tif T= [0, x). By we, where ve KUL), we mean
the element of FUE) defined by (wov(t) ~ weet) for ¢= 7.

Corollary 1: Suppose that ILL is met, that x © E(L), and that

2964 THE BELL SYSTEM TECHNICAL JOURNAL, OECEMBER 1981
w(Gs — GD.x} © L. Then Gz = p + @ for some pC B and some
gel.

Proof: Lel = wiGr ~ GD.) and lets denote (Gx — GD, 2). Olnerve
that's © L. Forny nitive integers and K with J> K,

Batre — Sot +40] -[) 3 a v4
2,3 le +H
E14 ager that + a

= 3 a base ia,

which shows that

Lectin Saran] oo
ts of and K approach infinity. By the completeness of F, wo have
1S and ube orollany follow

Ta Corollary 2, hein (= Fist + 9) —-)] denotes Ue element
of BiZ) whose values ste wit « ilale+ 7) ~ 310)
Coroldury 2: Araume that 1LL 18 met, and Oho! there is a positive
fonstant p uch thot

[tc = ery] = sete eal 6

for anal w in RAD) and ws & Tf x © KIL) ts such that whe + 7)
“fale #1) — ale) Fy then Ux — p+ 9 for some p & P end some
gel.

Proof Wehavellass ~ GU-2h\ Sol u(r D,xa,|for oc Tand
fang a © BU). When w(- + lla: + 1) — 2097 © F,3 follows chat
Sioes [tet ~ D xh l= a: hence, wie — GD-2} CD. By Corallary
.Gr= p 1 qwith p and gas indicated

2.29 Comments

‘Tae condition chae m6 + aie = =) — atl] © Li met if
sujerfote- Yl} 20 and x = p+ @ with pe =P and wn =F,
Ind af couree supyenlut + n/wid] = o is atisied if, for example,
tele) efor €or wll) = (1 + AN" for C= 7, with Xn positive
constant. Inpot-auepu: stability theory techniques can frequently be
hel to abr, n specific casos, that cg), with an appropriate wis

"Regarding che caso in which T= (0, 1,2, +++} inte yeuuld hase
‘been taken to be unity, Theorem 1 and Corllaries 1 nel 2 provide
conditions under which ris L-usymplotally convdant in Uve sense
that r= e+ g with g © Land c& C, whine Ci the set of constant R
valued fonctions {0 © Pol) = wu for 6 T and wee WER
(Corresponding results for 7 [Uae given ia the Append,

3.3 Time invariance and peviodicalyarying ystems

Hypothesis 1 plays a prominent zola in Section 2.2. Here wo give

Asfnitions which mule precise the eeuamally sl-vvident.propusition

hat HLL bs me i ca mip of CP) ia el ha be,

the weal sense, time invariant or periodically varying with petod »
“Tat H he an arbitrary causal map of £(Z) into (E}

Definition 1: Ha ime invariant i) there san element» of B” euch

hat (210(e) = »for ce 7. and (i) for any x © EZ), we have

DK) =» rEmanT
OHM), FE tent

for neh ws (P= (Oh.
Definition 2 Is periodically varying with period +i (@) He =» (or
fnmne y © P, and (i) fo each x © £(L) and any positive inveger &
Uda = vit feQeinT
DaHN, 8G Tr),
[Notice chat £1 is “periodically varying” with period r if His time
Invariant. Arete definition is the follwing

Definition 2: H's periodiealy varying with period vf) H@x ~ for
tome v © P, and (i forany x © EWL). we have

UD sie) = re, tefonnr
DFM, Efe.

‘To soe that Definitions 2 and 2 ary oonsslent, we observe the
following: If meets the conditions af Definition 2, then obviously HT
suites tn: eons a Patino 2 On the ner hand, iF st
the oonwiions of Definicion 2 and x © A) ix given, and If
Dae — 0, reloanor (6a)

A WWilioe, FELT (by

for some A, then, hy the condicons of Definition 2 with x replaced
wvith, Da

Davai) ~ 10, 1e1QNT
= (DFM, 1S ANT
Since HDs-r has the values given by wus. (a) ane (Bb, we ee thar
(Dy sVt) = 0, 1610, ae OP
= Prt, CE [+ IIT,
‘whic showa tht the conditions of Definition 2 uno ne.
"Notice ie our assumption that His caval is no explicitly used,

“That ussumaptins eesriets the class of overstors Hf that the defi
tions given above are propria nnd sacaral >

24 An example
Lot T= (0, and consider eq. 1} which is repeated below:

niearo=[ Memtehako, =o a

Assume the following, in which, donates these of functions from
10,0) t0.8" chat wre amable aver [D, ©)
AL 2 © E(D), ba measurable soil » Xn matria-valued function
“efined om (0 2) auch thst nach kbs hounded and belongs vo Ls, and
Vis a top from 2" % (0, =) inte RY with the properties that
$0.0) = Bfor # = O,and

(i) shore is a constant ¢ > 0 such that [$y 9 — (oO S
elu el forall u, 0 R" and all 0, in which |+| waome norm on
Sond

id) Yle(-h -]is meanurable on 0, =) whenever 2 © EZ

Since 2 € RUL) and each hy € LiL fllows that u defined by

sity ~ [Ae onaien olde, r= 0

fs an eloment nf KUL. Alo, singe wach dy ie bounded, there is
oneant such Ut [A(¢— afl, a) ~ glam a] =e 24~ zl for al
onnegetive cand such that > oy nod fo all zy and 2: in B*, Then
theo obwervations show that proof given by Tricomi (see Rel 1p,
42-7) can be modified to prove that KL) eantains a unique selution r
ofeq. O14

et @ be the map of B(E) into BL defined by the condition chat
for each 2 Bil), r-~ Gx ia the solution in FUL) af eq. (0). Since

‘uly owen gm ever” Alex Defias oot cere
Sieh dit pt finan cues

ETM hdae Pe KL a einae ad ao ely cay age

LASYMPTOTICALLY PERIODIC RESPONSE 2587
HO, #1 = 0 for = 0 it is eany to aon that TLL fs met when,
He, C41 = gla, for 2 O and 2&2

"Now consider four additonal esrumpcons
Aa: Yle.0) = Ya, 47 for ¢ Oand all 2 R
‘Ait Por aay’ 2 apd 2: in E(L), there is meseurable real n x n
‘matrixvalued function D defined on (0, = such that () each Di is
Pounded on on), fat. €1— yet, £] = Didf2.0 ~ auth} for
£20, and (i the velation

implice that we have ry © L whenever 5 © E(E} and 2c € F. (See Ret.

2 pp. STi for condone under whieh 3 hole when i hana cerkaln

import epeciti frm)

A: For cach dand jth € Ey orp = 1.24

AS: Conceming 2g. (1), X= U4 + us with us EP and (ur € Z for

01

We shall prove the following

Theorem 2: If Ad through AS hold, then E{L) contains a unique

scat of (i) and howe r= afr ome Pontos
eb.

Proof sinned cain 3 plies chat there «unig solution

toteq. (1) ie ULI, Let rand » depote Gx and GD.x,respeccivey, and

let D vatisty Ure), £1 ~ vfs) d= De trio) ~ st] 2 O wih D euch

hat (and (i) of A. Hold. Then, with A= r~ sand v= =~ Dia,

cw = 40+ fae -eversiene, ea,

[Note that v(t) ~ 21 ort © 10), ad wl) = ale) — ant — 9 fort
from shih it easily follows that (1 + "0 © Z for p= 0, 1,2.
Ry A, Se. In addition, observe that we have

+a = 0 naK)

+ [Ae ermion + aioe

Since thy € Ly for all jan j. ts given by

By hy wemean ot cur the tap am 02k we aha I.

2368 THC DELL SYSTEM TECHNICAL JOURNAL, OECEMBEN 1981