VORCESUNG I02 . I

.

2017

Monotone AWA, exponential Sfabihtiit

Definition I.

1 ( Monotone AWA )

Die Iuuhtion fat , x ) geuiigteieorutlouofoueebedingy "

won net X Ct ) > 0 gilt :

- ( flt ,x ) - fct , y) ,

× . y ) 7 AH ) 11×-4112 tha, HEYKD ( ^ )

Setae X : = III NH . a

Motivation :

i• skalarer fall f Ct ,x)=g G) e R.

- (gcx ) - gcy) ,x - y ) = . ( gcxtgud ) ( x . y) z #-) ¥2

2,0

⇐ ) - GIGI z AH )also sekabenssagy

Kay ) negativ .

hzw . gwohofonfallend .

Da linear, vehtorwortiger Fall

of Ct,

x ) = Act ) × t b It ),

dofiirlaetet

CD :

- ( Actlx - Akly ,x -

y ) = ( - AA ) ( × - y ),

x . y ).

= ( × -YTGACH ) C x - y ) 7Mt ) 11×+42d. h

.

- AH ) istpontiv define ,also Act ) negatiudfiuittft

Definition

VI. 2 ( Expouentielle Sfabihfeit )

Eimegleobale Lousy UHI einar AWA heist a exponential

stabile"

,Wen es S

, x. A > 0 gift so dam gilt :

Zujedim Zeitpuht t*7 to mdjedan w*E Rdmet 11W

* 4 < S hat die gestorte AWA

vytkfctivctl ),

t3t*,

vCt*l=uCt*kw*

ebenfallseine glohale Lousy vct ) fiir die gilt

Hvlt ) - uct ) K E A Eat # A wall, t>t* *

A noIdee : =#

utt*kw* 'll1 D

to +*

Banerkngen :

a Gestate Ljsylanff exponential shall of

urspniyliche Losey z winds.

a Diet Z. B .

and der Untersady von Frxpuukthautououor System

f ( ue ) = O ⇒ n' Ct ) = of ( uctl ),

u(to)=ue=) act ) = he

D• Es gift winter Stabihtoitobeqntfe

" Kjeapuwov - sfobieifoit "

@EE:sEEee¥geIeaYEwua

.

• Nt) =fcuttl ) it > to,

UHOIEV

gilt a (f) € U fee allet >cto .

SatnI.3C6lobalerStabiGfEfssatrfAlleLEmyeeiueL-stetigehmdgleichmoiRgmouotonaAWAmitMouotouiekomdatedhabeneiueglobaleLsgundsindexpoueetiellsfabiCmtsbeliebig.x-dmdA-1.ImFallsEyfoHfCtc0HfcosindalleL5syegleidmoifhgCd.h.emobh.vout1besdrainkt.BeweisiCi1AunalneuCt1tOVtD6Liu4t1-fCf.uCtH-OVt7totsCiiH.fet.ucHhauttttnD-o@scikI.uk

) KUCHK'

) - (

fCt.uCtshuCtIHuCtIltYaOEimsdnfiaEwtmIadafEIuEtihatylEnuiHthEEudtHEatuctI.iiCtYAuCtsll@ElluCtH-CfCt.uk

)) - fao ) , @KI - o ) HUCHIM) =

Einsatrdr Monotoniebedaoy :( fft ,

01,

UCHHUCHIIY

HUCHH daflluktlltdlluttlltelluttl # HUHN - lfltieattfk ,o ),

uttto )E ( fft ,

o ),

utt ) ) E Hflt , ON VUCHHC is .

⇐ ddzlluttkex Hawk E Hflt ,0111 (2)

Cit Nehme on,

dam nfto )=O ,ultttosirttto

,n' ltlstetig

Fiirttto gilt gcaudiy- Schwart

daquan =

( a #'

n' ata

" null Hutt 'll= guy , , ,

nutty-

Mutty

AusstetigerDifferaniabarhetrouulf7folgtfdttauHsHlt.IalejmofdaanaHItHaliuttH.CiiiIglobaleBesdweiuhtheitdrLiswgiMultiplitioreC4mitexltEd.eNttddq11uctHtdeNttolyuwyeeNttdllfCtProdeletregeli@ddffeNt-tolyucts1ifEeMttosllfet.oill

Integral iifer [ to , to ] :

e' ' ' ttol HUCHH - Hutto ) 11 ± §iedlstd Hfcs,o÷

£ SIEGE, Hfcsiollfte's ⇐ told ,

⇐excs . toyalso : to

HUHSK ÷ e-'ktdyuctdy + FEEE.gl#' 0M¥ { 1 - E

' ' #toy

ET→ Hu # E EX # td

Half .)At ¥ TEE,⇒

Hfcsioll

gleidmoemgetbsdrawktheit falls sssufpoaFCS ,o)H< a

C iv ) exponential stability :

AWAS :

n' Ct ) = fct ,UCH ) ,

tzto,

ucfokuo

viltl = f ( t ,va ) , t7t*

,vCt*)=uH*Hw*

Setae wlt ) = vctl - uct ),

also W' A) = vyt ) - ul Ctt

DGL fir WHI :

w' Ct) - ( flt ,

vltl ) - fcteuat ) ) = 0

Shalane Multiplication mitwct ) :

( w' A) iwttl ) - ( f ( t.ve ) ) . flt ,

ultl ) ,was ) = 0

Eiuslnb :

Edatllwasltatzdat ,⇐wEtD=.Ff±¥HwetDwiHalso : ( was

,wilt ))

0 =

tzdftHwtHH2-CfCt.vCtH-fCt.uathwCtDsnzdatHwHlPtA11wHHtMnltiplizierennit2e4HtHliefatie2HttHdatllwHlpt2xe2HttHl1wct7ll2Pwduhbregeli@ddzfe2d4H11wcyHYE0inlegriereiiGrEt.t

]

f*tda*[e"" s #Hwcgy

']ds=e2'' ( t ' #Hw # 112 - Hw # HE 0

( ⇒ e2Ht#hwµH2E Hwc # 2154 e× 't #Hwa ) 11 ellwttxtk

⇐ |hokN±e*⇐t*'HwH#f cq

Kooollor I. 4

Awwendy auf die linear AWA.

Sei A K ) : [ to,

a ] → IRDH gleiohmafhg negahiv define

uudb Ct ) : [ to ,a ] → Rd besdmeubt

. Dam hat die

linear AWA

u' (f) = A A) ultltbcf ) ,

t >, to ,

who ) = no

nine eindeutige globule [ doing ,die besohraiuhtend

exponential sfalnk it.

Beweis :

a VL II ,Satan .

6 : Linear AWA → eiudentige , global Lsg .

Da Alt ) glen negativ defiant → AWA wonotonnnf A.

Dart folger net Satre I. 3 exponential Stability .

a §;pqH fft , oh =

sfyfo11 Alt ) 0 + but 11 = typo 11 but 11 < a

→ Besobianhtheif der Leong made Satt I. 3. Tg