BSTJ 40: 5. September 1961: A Nonlinear Integral Equation from the Theory of Servomechanisms. (Benes, V.E.)

3.6 MB·English

A Nonlinear Integral Equation from
the Theory of Servomechanisms

By VE. BENES
(Atom ebro Meo 3, May

‘he equation 2) — 0) — BAFLAAM, whereof) Ho given signet,
Cj sem nusiione uortion, KL) the tocpane af @ Fear myst, ant
‘detox cnralis, seis a enerat elas ef teseomeehenons. op
ice of a abtion } be ont by fig fod pot
iperiie taf jue apn vig Sats hear

A grea lt of soln erwncaane ea by te
wn wo [ema e, ce ere,

Pov a eon eatin 11 pet Pea

haere extn un dR) he ter

tion eoresponding to k=) Wessenrse et FC} attisies the ifort
Lipschite

ss that #10} = 0,
\ else meth for stoning aonlneaeaervomenbanims ike that
of Fig iso specify extetiy he nonlieae element #1), to secu tht

Ine mespunse Mis te Cea neion of a dierent opera of lot
de, tl to use somes phase-pane analy Tae mth hes two
onieal isudvuagss i lake gererlity, so, whe pple,
nde inv not formatin, hon is need; seni provides elaed
Kone alsst a tetrieted ele of ere
1s ths pr all ar stand that ae the opposite carats.
ics it polos tll som! ef ighly evar fseemacien bout &
100. mn ene prem meeHEst TOMKAL, REMEADEHE EL

lang eso ease, We sll esenpi’y the nso af Scher fsed point
“han fot spn olatiea= =} of [thaws ying oithes

or PU} ir tail Ne staltivh dednize openies of ai by Hiding &
“aod pint (oom hn elation of the veto} th x. epee
tet ofp fmcien ogee Sie the fei quetiea the et ean be ches

th many tors, eppcling fa pare oh what prapertion of 2(+ are of
terest sul aetTl ass bard Zor «wie els f probs, ‘The
theory nthe sql fs therefore resrieted tw sample results for the
funtion spew ff suaieentexrble Fein, gad i to be regarded
ful as eure eannple ofthe rethod described above

In mang sitios iis desirale that the convolution term

[ec

= FEW) d

foto she spat sgrl st). °NHecer in shit approximation see «+1
itis Tis shew reexotie ln srk the space dy of se, ear
incest tunetioin iy facta of Hite enon Accor, We
tobe tel {ned oo) ane Ley at fe suk to ed ch egy
soi co wT

Now the hictivns of fz camiot gesme wales apprevialy diferent
Iron aera ens srt of enbitsasy lage ene Hewre the may bo ceed
Piysiealy ae lwo. Hy tetreane ao} sae the elaine a=) 20 Ly
treme therfone dying ths renee of they of Bg. ta oan
pls of finite enon We sul fr sural neve fo Badge
Fwy mac of the enney of) Tse nulaide a given tine terse

"he norm sera | nett denote the sire rok ef te energy
ff ftotion Ma feof) ind

el - (fC ow ta),

su sequence of fein -} Of salto covery ta wfc
tion tte} i Fecin fr — x [pends wero with remarg

M6. 2(.) fa fsnetion of Fe le

Toto)

forsee
Sunuunuane isvmanin Beto Bu

aos ele =

CL

Speen dings of eevomeshaaon,

‘teote its Fourier ceusfornsfor8(-) © fa, we eve Ee pec os

Thiet = Keel
“The operon 1m Le dined by the condition
moc) = [0 aRCaGe tu

hermes KC
tion of Le it Sol

em Fou 21) C Ly the Lisweisr option on #-) yi
[Fe 2 alls yso that Pst) = Ly

Te iow iow sul a se
onvolutint ef two by luelines lange a gs Men fff Als,
ty the Paes isons

s bout ,

ste elt = [ria #9) — TP

wp Kiet PPh — Fa IF

Rerthe op
hic eos that Hf ination
‘Now lot wt) vem unscmgalive futon of ky, ad Ite the
sera =) ind ote th
(2) Swit, slot everywhere, fey
evwnot 25 slowed and conver.

Proof Unt (+) |S be a soqurnee of funelions approcehing
In, Thon foe >

Dand gi} = Lebesgue mensure, "
ental lanate

aani> a.
1512 su wets srsnuac mEeasTesT toma, SuPvuamAE BOL

However,

[a [al — 10 | + 1
[xt — mid) | = wl.
Henne | 20) | — w(t) > ciple | 260) — zal) | > end
alte at) — anf) [> dB te [210 |= wl > ab
Letting appre nny the ew find Hct
leet) | wl > a
tas measuse sore fur cnch « > 0, Hence almost overyutere
[sw | 8 with

und 30 Sa elosed, The convey ef 33 vin
"Wee dole By tae ene of fetione 3+) of Ly whieh ane ban
Vise" te fountey ene. € B40 a copeeeemable

‘Vhs phys iterptaon of swacbemeip ia feof ease thet the
sinus velletione int whieh a taprtin is docomposed the
Fuaser seanstorn are rstited in Feguoncs to che nferal {88};
lag Peja? = Mor [>

"Te ipl aignal ane the e=poase (+) wil beusnsned to helene
lo BL we deli the operat J 0% he oy

thon the range of fis asubert of BT Jl tha any slition of (12,
fay y Hoe pt J il ely Bae Tong ast) and B=) do
je “bul ling ssitine i natal physraly, bere of Uae
Jesowt alienation at high Feguencescharsctotie of physical eeu
an i il have an anportane mataematira ein Sadig ed pints of
To particule ne tha £8 CB.

‘To aban «bond ot the smnant of energy that whation 2) bs
‘asda given interval call supp tha the no-aezative fusion
Wei) al ds, een th dfntnn 08 8, satis the itera eles

ates) a fae wt foe te swt wo

‘This innelity may bo thought of os defining ov wecate Tar
So!) does
ih oe Lact

rode wl fe ate ten na felon

"The anal fain s+ tes oem C9

souetant tof eter 18,

Leming 118 We FSO = 8, Aloe
8. "Phen

Panga Tet! hed

[ates fab Liebe oe

sei.
is pains ace sented
Leming £287 8 tin
Pegs Les B= ns ain of 801 ans (vainly nc
outa mah real

Cle

‘his atone ne salt

Yun

(8 famosas

= CVO).

wing irbynanent oa 3. Abo the

he tat boi 6 he rg

yu atae [| tad a
startin af

“@)

show tye the Hovde S11 Bane vata ome
Since ioe S nd Fane rhe sot, tins 40 prove Lamesa 4} 20
lune et S118 asecmaCallycompace Met xis = SVE be aa e32i-

nay equa taco, "Uhe scr} ane erly Sony! aed ne
Fron’ siontinnous, Lp eandaed sayovnl onguricat eng
cesar ess of ee real in, 6 eap slet wt seb a whi

Converge hese sean 4f- uniformly 097

rich oe We Fae

[efit t+ ate

i Pf tott nates Pate

soup casey ts tf [et fo,

Porch Sxl 6 he fiw ave ters rn the vig af the last inequality
pri woe om benemes eg alr hil term eat moe: ||? =
Poms ein Monee [aS Bend at de. Uso Miukow
inal 0 al

Pall marl
(emt)
Usain ee)

“Phe diese too othe ga a be td ie aly 8 hae
og sh of omiforzalin f° ¢ Eve, Cn br yes se
tees Hae the yt 4 sucremme tw et im Fe hie proves
Tema 4

1 Puaventas uaicuve ron cemMRAL FL)

Peoran 25 Let ef wad Bf} Wwhng Bit Ki} Sounded, et
2:0} = Oyo he fogon ownage tin, Then tere ests
fptoleien 05 (1) wi lnenee 8P1A,wth propetse

iy 2 mies tauden |S lon),

"Pela de
segciseas ermal navstas 1

Pray 44 sauiznoue toupsing of ths clase conve ct 8 ints ig
sume scbuek = FT, hy she “atone torn” af Sears theory
Ahr eines a piut 24) 08 MHL ageh ths eS Ve pret
Tint lene ae iene eonennrnnes 0 telonging to 22,

"The Flowing light nection a Chess Liven no esp
io

etna I, nao: othe bypothose af ‘Ubrcre 1,

oan [Rie < hy mn

Ho Gt smi of Phannees |r be aed

MTS) gam ten

rei sa aff inion? reir)

son | Kia’ || Pe

Le | Soup [Ae |

shew thse J © GH, Siowo Une Lnmdngy i¢ Hous, QB ine
‘ese subset of che compel oo 8 8, io arpa. Tie wt
lions ium Seder’ Fee

Lesolar than eenger assum wor eoring be tase Fn te
{£;0) ane nosssniry Foe ave ttn seule hut mae the neng of

20s thar Ut ol 421.8 pasty ean a se ce lee
Piece, om 0s
an, #42 iNet eg

TMCESS i aa tei wee wane sategen 2) = 2)

rE em ts

’

ithe Mule = LCP LS WU, the see sorverging for |] =
vl
fel

ein,

app al that tere few aur 6, 1< a < py eth tha

ee res) on

hea f] <eimplios thet PE ae the aig off

in tls LN a
toe hn £4. fecepewente hy a ower serine 0 lle sg oe
ferme ue sowie 1 wage
ace we a comic 21 4 beard sown of CL abtaizad by
conning F121 = fw shall 3986 the hoot af Ue weaclting Tacariaed
puto: this fe tin deed i ite Rene “rae
yd =, THe
wah Te

uuilaly, che ohes-nyp ranster Eneior the Terr Toop ie the

rns tsar

Kis

Bea

Vhs coins il atti the thea

pro
Ty dit of our rene usps oa 2-1 cme ae Tea
invegealineqnaity from (1, We azure ine! hal tho ex ra
owrnegstive hve en vie © Fob at
win £6 fn ojo taes is
[yi this quality plang Mo role of (82, the meted sed tp
‘enarean 1 sate apoio: sien! slbont odbc ior
Tower sw gal net #1 enna ina 1 esha:
Aigesn ea gre feet eee ie allt. Ome wae
tel Tis I Fine a cvmangataye v1} = By that antes (3) sh
suai, fea soko ot the molcwr ee er
eA [60 Foe
en

{Well >how how he lsiee ou: neon principe oe cesptee met
morc eed a iad soliton 9) (8, @ Ried pone of Me
Stet soa eesenopiid be
Tena iW tr

sf snimces «

sve have for some > 0

we re) Hie =

Tne sighs, Here 111 ha tos tate > Ah an foe

sw Met) eup lai natn ie

‘hoe MY CV. Toshow hat ea east eel shat eonengene
‘Ze implies convergene mse, Le ty © gona to i Lo
that an

Helgi =i |2 a0
for one «> 0, Hauer, slings enyehis have
= [x0 a6) Sa S| ty | tens

sid 90

whens at} denotes Lesoaguo maaute. Letting «0, 6 Bnd
1M nm nts svemar mer seMaNnd, sk eReCEA 6
esprit) Sa a

coins &

131 i vonteseting on Yel a) and yl) be abitrny

‘)

To so
fuetigne in FT

date asp ga( flint mee

Hwee,

rey) ey [mire — 3,

jet es lew ae a"
an 0, dis La
Yate ~My) Seep [fel niztia = 9°"

bet 2iGu =
But tna — 2° <1, Ma pontine on
esnips, hy the enatrcerios peneile, that these exits
tuviguesolaion of) of (0 ince see YF, ceaizable ap the eit
acai wanes wiki a: any poh a

et matsorret wesoum #E) satea HIQaT A HH ones
To Re Tie otf sein be Ut say th cin

SIH] 8 600, almost everwhere,

vhere f+) 6h fain i the eet (8). ago Ean
2 shows that is eooed and conve, ad that of Letom 4 shows that
ROB isco.

heen 221 a > Wand M4) have the properties (5) and (6), and
Co) haf) both tg ba By wile oes) Daa Re) hy

ssn (8) holes
< any

‘thon a schstion o>) of (1) existe inJ th the propertice