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Perturbation of Linear Quadratic Systems with Jump Parameters and Hybrid Controls

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(1)Perturbation of Linear Quadratic Systems with Jump Parameters and Hybrid Controls Rachid El-Azouzi, Mohammed Abbad, Eitan Altman. To cite this version: Rachid El-Azouzi, Mohammed Abbad, Eitan Altman. Perturbation of Linear Quadratic Systems with Jump Parameters and Hybrid Controls. RR-3816, INRIA. 1999. �inria-00072842�. HAL Id: inria-00072842 https://hal.inria.fr/inria-00072842 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Perturbation of Linear Quadratic Systems with Jump Parameters and Hybrid Controls Rachid El Azouzi — Mohammed Abbad — Eitan Altman. N° 3816 Novembre 1999. ISSN 0249-6399. ISRN INRIA/RR--3816--FR+ENG. THÈME 1. apport de recherche.

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(102) AO  + ,)' ! xrdŽisu—‰~C–Ši\p„ixe~u¬iÈ°MtgsCx{§®~Šgp·¶§®x¬`~u©ª©®~C³)p5x{dgsCxÈ·Ž§®¬,  Lg}r~Šh >. Bb<=. {. ; B. L. UPFZ . ™. . U7 Z h { . U7 ZDC 4 . 6·¶xrdŽiq .  R. . ¸ jq‘Šº. 7 ( 1I. & $ O. ƒw§®gŒix{dŽi$iq°Mtgsˆxr§ª~u ¸ jȚuº¦su‹Žh˜§®x{p\s7tŽŽ§¨°MtŽi$‡‰~Mp„§®xr§ª–ui$p„~Š©®tŽxr§ª~uÁ¬`~Š} ( U &_$ )Cö·Ax{dŽi{ U7 Z O &Ž· 7 ( 1I†i x 7 { U7 Zxrdgi R ; ; { U 7 Z O T Bb<= ; B { UFZCIced¶tgp    ;<=. X { FZ U & I LŽ}{~uh ¸ jq‘Šº ·Q§®x ¬`~Š©®©ª~C³)p7xrdgsˆ x E U 7 Z h { BWV ; U7 ZDC 4 &»sˆg‹x{dŽi T Bb<= ; B . ( { U 78Z &4¬`~u})su©®]© 7 15à ¿¬ ,  R   R 6·w³5ilœŠi x ¸ pr§ªgŒ i { U7 Z  &Šº.·. ¸ ,j PMº > { UPFZ  & I  " ; B. Bb<= ¿¬ ,  R  R 6·¶xrdgi ¸ Nj Ruº > { UF Z  U 7 Z h {  " U7 Z C 4 I #4 ; B. Bb<= Lg}r~Šh ¸ j PMºesˆ‰‹ miqh&sˆ}{µju·w³5i~u—wxsˆ§ªO E { U Z  {  " U7 ZQ¬`~ŠJ } 7 ( sˆg‹ {  " U7 Z  T < = U7 Z {  " U ZQ¬`~u6} 7 ( oI•)~C³\·  ". iÈ°MtgsCx{§®~Š ¸ Nj Ruº5Œqsˆ —‰i\³}{§´xrxriq‚sŠ,p E T Bb<= ; B { ]4 UF Z  U 7 Z h {  " U ZDC 4u· 7 ( IA€‚tŽ©®xr§ª‡Ž©®yM¯ §ªŽœ˜xrdgi\©ªsŠpzxiq°Mtgsˆxr§ª~u —¶y U7 ZU¬`~u}iÈsuŒ d 7 ( su g‹‚p„tŽh$h$§ªŽœ$~C–ui} ·wx{~$~u—wxsˆ§ªO E . K. 0. . . . 7  7. L. . . . . . . . . 7. 0. . . 7  7. L. 0. . . . . L. . . 7. . . I. 7  7. L. . 0. . . . L. . . . . . . . . .

(103) . . >. >. Bb<=  ;<  . U 78Z.   . L. ; B.  '. . X. . . L.  . . .   . . . . . . . . . . . .   . . {. UF Z. #4 .  . U  Z h {  . . . ". U   ZDC 4 $. ³dgi}{i U Z  T ;< U7 Z U 7 ZNz &ŽÃ ec d¶tgpq· U Z h {  " U Z C4  &Wsˆ‰‹ {  " U Z  & I. UPFZ cedgi}{i ¬`~u}{i{ U 78Z &g·#7 ( 13sˆg‹ T { &Ž·-7 ( 15Ãl

(104) 5yµtg pr§®Žœ x{dŽiÇprsuh$i  Bb<= ; B ]4 ‡g}r~wŒ iȋwtŽ}{iQ¬`~Š }exrdŽ i" Œqsupri ³dŽi}{i R , 6 w · 5 ³ l i Š ~ w — { x u s ® §  E R ¸ jq“Šº { U7 Z  I IPI  {ut  # U 78Z  & 7 ( 1$ v . . .    . . . . Ø Ø õ`gßa „ï . . . . . . . L. . . . . . . *. . . . . .

(105) jNK. AN,$.

(106) = VA\+9

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(108). {. U 78Z. . *. H-,  &. . >. ;  B. L. {'=UPFZ. ¸ jq™Šº. &_I . Bb<=  ,  { UPFZ & 7 ( 1 I T Bb<= L ;  B " I  R    +& ,  { UFZ U 7 Z h {   * U7 Z C4 7 ( 1  R T Bb<= L ;  B 4   * {   U7 Z & -H ,  { UF Z & T Bb<= L ;  B 4   * *  , +& R  {   U 7 Z IPI IPIPI { " U 7 Z & $ 7 ( 1$    *. ¿¬ ·¶xrdŽiq Ž· ¿ ¬ Ž·¶xrdgi gà ¿¬³5iŒ~uMxr§ª¶tŽi³§´x{d xrdŽip{sˆh$i‡Ž}{~¶Œiq‹wtg}ri¦¬`~u} sug‹ ,. ¸ šKŠº. >. ^5sŠp„iš EU€. ¿¬. u· 5ÃU¿xe¬`~u©ª©®~C³)p5x{dgsCx g·w§´x¬`~Š©®©ª~C³)p5xrd‰sCx E. ¨§ pe~w‹Ž‹'à 6·¶xrdgi. Bb<=. L. { PU FZ & ;  B \ . .  R. >. Bb<=. ;  B L. <. .  . {'UFZ. &_$ . & $ $fIPIPI IP$ I. . R I. ¸ šwjȺ I. ¸ šušuº. 7 ( 1I. ¿¬, SR  +&g·x{dŽi T Bb<= ; B { " UPFZ  U7 Z {  # U 7 Z 4 $7 ( 1Icedgi}{i ¬`~u}{ i {  # U7 Z  &7sˆg‹ { UPFZ &Ž·7 ( 15Ã

(109) 5y tgpr§ªŽœ&xrdgi4p{sˆh$*i‡Ž}{~wŒ iq‹ŽtŽ}risŠp§ªÁxrdŽi˜ŒsŠp„i\³dŽiq* }ri˜€ §ªp T Bb<= ; B " iq–uiq­·w³Åi~Š—wx{su§®‚sˆgsu©®~Šœu~Š tgp5}riÈp„tŽ©®x{pqà ¿¬ ,  &Ž·g³Åiœui x E U  R  U 78Z8ZG{'!U 7 Z  x9U 7 Z R U 7 Z h {u'=U7 ZDC 4  > ? ; B {u'=UPFZ  > ; B { UFZC$ 7 ( 1 I ¸ šˆ‘Šº . . . I L. L. . . . . . . . . . Lg}r~Šh. ¸ šŽjȺ5~u} ¸ šušuº ·Ž³5ild‰s^–ui¦xrdgsˆx E >. Bb<=. . Bb<=. L. ;  B. {'UFZ . &_$. Bb<=. 7 ( 1I. L. . . ¸ šPMº Ö`×Ø­ÖÚÙ.

(110)  

(111) 

(112) 

(113) !

(114) "$#&%(')*,+-'$./10324 +567

(115) 8

(116) +9,*)'-

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(118) . . . . . >. >. Bb<=  ;<  . >. . . >.  '. Bb< ;<. .   . ¥ iÁ§ªŠx{}r~w‹wt‰Œ i xrdŽi‚¬`~Š©®©ª~C³§ªŽœÀŽ~ˆxsCxr§ª~u‰p,E  U Z  U Z U U ZZ *# U U7 Z U7 ZZ *# sˆ‰‹2?   T ;< ced¶tgpq·¶xrdgi\iq°MtgsCx{§®~Š ¸ š=TuºÊ—AiqŒ~uh$iqp E . . . . . . U. . . R. .   . U   ZZ {  ' U   Z . . Z R. . . < =.   . . . x  U  [. X. > . U7 Z.  .     . . U7 Z@?. {. UPFZ . .  . ; B.   . < =.  . UFZ { '!U    ZDI. . Bb<.  > .  . >. ;< Bb<. U 7 Z  U 7 Zr· [ x  U  Z   U 7 Z   ?   ; B T.    . ¸ šˆ“Šº I. .  >. . . T ;<   T ;<. U  Z h {  ' U  Z C 4. . ; B. .

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(121) j^š. AN,$.

(122) = VA\+9

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(125)  

(126) 

(127) 

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(129) "$#&%(')*,+-'$./10324 +567

(130) 8

(131) +9,*)'-

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(133) & S04AJ('E./0

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(135) &lxU7 Zqz & 57 ( 1"!)#O0   S

(136) =+['9

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(138) !+;1'. =. 104")(5 AQ)S

(139) A 5 ( 8

(140) "V")")

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