8/11/2019 1410983%283%29

1/5

Pure and Applied Mathematics Journal2014; X(X): XX-XXPublished online MM DD 2014 (http://www.sciencepublishingg oup.co!/"/p#!")doi: 10.11$4%/".XXXX.2014XXXX.XX&'' : 2 2$-*+*0 (P int); &'' : 2 2$-*%12 (,nline)

On the construction of regions of stabilityLuciano Miguel Lugo 1, Juan Eduardo Npoles Valds 1, 2 , Samuel Ivn Noya 2

1 #cult#d de ienci#s #ct#s 3 . 5ibe t#d 6640 ( 400) o ientes 378 9& 32 #cult#d 7egion#l 7esistenci# 9 ench 414 ( 600) 7esistenci# 378 9& 3

Email address:l!l!b #hoo.co!.# (5. M. 5ugo) "n#polese #.unne.edu.# (#ld?s) s#!ueli #nno # g!#il.co! ('. &. o #)@

o ci!e !"is ar!icle:5uci#no Miguel 5ugo #ld?s '#!uel & =n o #. ,n the onst uction oA 7egions oA 't#bilit . Pure and Applied

Mathematics Journal. >ol. X o. X 2014 pp. XX-XX. doi: 10.11$4%/".XXXX.2014XXXX.XX

#$s!rac!: &n this p#pe we built # st#bilit egion # ound the o igin Ao the 5i?n# d eBu#tion (4) to ensu e st#bilit #nd boundedness oA solutions oA this eBu#tion without !#Cing use oA the cl#ssic#l 'econd Method oA 5 #puno . e co!p# eou esult with so!e othe s p oposed b diAAe ent #utho s.

%ey&ords: 5 #puno 9 #"ecto ies 3s !ptotic Builib iu!

1' In!roduc!ion

9he te ! Est#bilit F o igin#tes in Mech#nic toch# #cte iGe the eBuilib iu! oA # igid bod . 'o theeBuilib iu! is c#lled st#ble iA the bod etu ns to its o igin#l

position h# ing been Edistu bedF b being !o ed slightlA o! its position oA est. &A the bod #Ate # slightdispl#ce!ent tends tow# d # new position its eBuilib iu! isc#lled unst#ble.

9he 'econd Method oA 5 #puno h#s been est#blished #sthe !ost gene #l !ethod to stud the st#bilit oA eBuilib iu! positions oA s ste!s desc ibed b diAAe enti#ldiAAe ences o Aunction#l eBu#tions (o s ste!s). 9his!ethod w#s Aound in cl#ssic#l !e!o oA 3le #nde Mi"#Hlo ich 5 #puno 1 published in 7ussi#n in 1%*2t #nsl#ted into ench in 1*0+ ( ep inted 40 e# s l#te 2)

1

Io n on

8/11/2019 1410983%283%29

2/5

2 5uci#no Miguel 5ugo et al. : ,n the onst uction oA 7egions oA 't#bilit

dete !ine the st#bilit #nd #s !ptotic st#bilit oA # s ste!without e plicitl integ #ting the nonline# diAAe enti#leBu#tion) is the en#!ed te t JoshiG#w# oA the si t 4. &n theBu#lit#ti e stud oA # nonline# s ste! whethe #utono!ous

K ( ); 7 n (1)

o non-#utono!ous

K ( t); 7 n (2)

suppose th#t (t ) is continuous in (t ) on & D whe e D is# connected open set in 7 n & denote the inte #l 0 tN #nd7 n denote the uclide#n n-sp#ce with the no ! . &n theBu#lit#ti e theo so!e oA the !ost studied Bu#lit#ti e

p ope ties # e st#bilit #s !ptotic st#bilit #nd the boundedness (#lso c#lled continu#bilit ) 6:

9he solution O(t) oA (2) is stable in the Lyapunov sense iA Ao #n 0 #nd #n t 0 & the e e ists (t0 ) 0

such th#t iA ( ) #s .&A >R is not identic#ll Ge o #long #n solution othe th#n

the o igin then the s ste! (1) is co!pletel st#ble +.9he !#in diAAicult in using 9heo e! I oAten is th#t one

c#n const uct # 5 #puno Aunction s#tisA ing the th eeeBui e!ents. Sence it is !uch e#sie to stud the

boundedness oA the solutions #s # sep# #te p oble! A o!which # ises the need to build #pp op i#te egions whe e

we c#n ensu e the boundedness.Iuilding the st#bilit egion oA # gi en eBu#tion is

#nothe w# to stud the p oble! oA con e gence #s t tendsto inAinit oA #ll solutions oA this eBu#tion. 9his p oble! isoA # p# #!ount ele #nce in the Bu#lit#ti e theo .

9he pu pose oA this note is to const uct # new st#bilitegion Ao eBu#tion ( ) using # diAAe ent #pp o#ch oA e# lie esults #nd without !#Cing uses oA co!!on conditions.i st we su!!# iGe Cnow esults we p esent b illust #tionthe p ooA oA Ai st #nd l#te we p esent ou esults.

2' (reliminary )esul!shile the e # e so!e p e ious esults in the AiAties t he

Ai st esult oA this n#tu e w#s obt#ined b 5#'#lle in 1*$0when he showed th#t #ll solutions oA (4) # e st#ble #nd

bounded using #n #pp op i#te bounded egion.

2.1. Region 1 8

9heo e! 1. nde #ssu!ptions #) #nd b) iA we h# e

)( x F x

duu f 0

)( Q #s x

6

A. 9heo e!s 4 #nd 6 oA 5#'#lle (1*$0).7 ontinu#ble Ao us.

8 5#'#lle (1*$0).

8/11/2019 1410983%283%29

3/5

Pu e #nd 3pplied M#the!#tics K(4)( ) KQg( ) K T g( )Qg( )( - ( )) Tg ( ) ( ) 0.

'o we h# e >K (4)( ) 0 in #n subset oA . 5et be theegion deAinite b

= )(/)( 2 y x ! y x# with is #

positi e e#l nu!be such th#t . &t is cle# th#t the bette #lue oA is the lowest #lue between 8(#) #nd 8(T#).3g#in we h# e the no positi it oA >K (4)( ) we

gu# #ntee the boundedness #nd st#bilit oA solutionsst# ting in (7egion 2 in the Aigu e bellow).

Region 2

2.3. Our Region 12

hile the const uction oA the st#bilit egion c#n be #tti!es b the 'econd Method oA 5 #puno we sh#lldis eg# d it in the esults p esented below.10

'ee S#s#n #nd Vhu (200+).11

A. J#det# (201 ).12 3 p eli!in# e sion oA this esult w#s p esented #t the 3nnu#l Meeting oA

the M3 l#st e# . 'ee 5ugo =poles #nd o # (201 ).

8/11/2019 1410983%283%29

4/5

4 5uci#no Miguel 5ugo et al. : ,n the onst uction oA 7egions oA 't#bilit

e ne t gi e # st#bilit egion #llowing us to obt#insuAAicient conditions Ao the boundedness #nd st#bilit oA solutions oA the s ste! (4) #nd conseBuentl Ao theeBu#tion ( ).

9heo e! . &A in #ddition to conditions #) #nd b) we h# e

A g(7) {A (7)/A( )T g( ) 0 iA 0; A( )Q g( ) 0 iA

N0 }then #ll solutions oA (4) # e st#ble #nd bounded.

P ooA. 5et ( (t) (t)) be # solution oA (4) with initi#l#lue( (0) (0)) ( 0 0) #nd we t#Ce the egion:

+=$

% & x' G$ y( x ) ! & y * x' +G$ .

he e C is # positi e e#l nu!be s#tisA ing the eBu#tion

$ xG$ y

1)( 00 += ; #nd #nd # e the solutions oA

eBu#tion +$

% & x' G = (note th#t p oposed eBu#tion to Aind

the #lue oA C is eBui #lent to #sCing th#t the solutionEst# tF on the bound# oA the egion).

#lcul#ting the slopes oA the bound# we h# e:

L)(

)( x

x g xd yd

x$g == A o! he e we h# e th#t

$ x

1L= iA 0

L)(

)( x

x g xd yd

x$g == whe e

$

x 1

L = iA N0.

Our Region

'o iA we h# e$

x 1

L= iA 0 #nd$

x 1

L = iA N0

the t #"ecto ies th#t begin #t the bound# oA this egionEA#llF into the s#!e which ensu es the st#bilit #ndcontinu#bilit oA the solution conside ed (see ,u 7egion inthe p e ious Aigu e).

*' +inal )emar s7egion 1 h#s 4 points oA inte section in the gi en

bound# #nd those points c#n not spe#C oA de i ed .9he 7egion 2 does not h# e th#t p oble! but #s the#lue oA # is Ai ed in #d #nce the p oposed egion

c#n not co e #ll solutions oA the eBu#tion .7egion h#s the #d #nt#ge th#t it w#s not neededto deAine # 5 #puno Aunction (so in ce t#in sense

is # con e se theo e!) #nd #lso includes #nsolution to the eBu#tion since this egion isconst ucted when the initi#l condition . 9hedis#d #nt#ge is th#t it h#s 2 points oA inte section

between the gi en edges whe e no one c#n spe#C oA de i ed .3s #n #tte!pt to o e co!e the #bo e !entioned8uido iGGi const ucted # A#!il 5 #punoAunctions1 which we e tended 14 to non-#utono!ous5i?n# d eBu#tion RRQA( ) RQ#(t)g( ) 0 t#Cing #s #5 #puno Aunction Ao the A#!il

)()()(

1)( xG y x,

t a y xt +=

.

ith = x

ds s g xG0

)()( #nd

+=

)(

01

)( x F y

s

sds y x,

which #llowed us to deAine

#s the Aollowing sets:

7 2 iA0

W( ): ( )- -1 iA 0

W( ): N ( )- -1 iA N0.

9he esult obt#ined in the 9heo e! is consistentwith so!e p e ious esults oA the second #utho 16.

)e-erencesU1Y 3cost#

8/11/2019 1410983%283%29

5/5

Pu e #nd 3pplied M#the!#tics #ldes