Stability and Sensitivity Analysis for Optimal Control Problems with a First-order State Constraint having (nonessential) Touch Points

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(1)Stability and Sensitivity Analysis for Optimal Control Problems with a First-order State Constraint having (nonessential) Touch Points J. Frederic Bonnans, Audrey Hermant. To cite this version: J. Frederic Bonnans, Audrey Hermant. Stability and Sensitivity Analysis for Optimal Control Problems with a First-order State Constraint having (nonessential) Touch Points. [Research Report] 2006, pp.34. �inria-00087573v2�. HAL Id: inria-00087573 https://hal.inria.fr/inria-00087573v2 Submitted on 26 Jul 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. Stability and Sensitivity Analysis for Optimal Control Problems with a First-order State Constraint having (nonessential) Touch Points J. Frédéric Bonnans — Audrey Hermant. N° ???? Juillet 2006 Thème NUM. N 0249-6399. apport de recherche.

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(33)

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(36) !"$#%&('*) +( -,$%.&/% 01&/% 2+ I. ¹ZV¼ ¹Z~ T¼ [p1 (τex )] = 0, τex ∈ Tex ¹Z¼ [p1 (τto )] = −ντ gyµ (y(τto )), τto ∈ Tto . Éh}±V~V¸_m>qvq, $ÂqÃ¸ÁV p1o}$}±°>q,}moq 9 % !% %. ½W¸<m>qv V>~>} q $«hm> F V [0, T ] \ (I ∪ T ) ¹ZV¼ g µ (y(t)) < 0 b to V Int I ¹Z ¼ η˙ 1 (t) ≤ 0 b ¹Z9 L¼ + − = η1 (τen ), τen ∈ Ten ; η1 (τex ) = 0, τex ∈ Tex ντ1 ¹ZV¼ ντ = 0 τ ∈ Tto . m>}2v~ |~p~}~½~q oqTFmoqÃ}q,Woq,|W{ ½ ¾¿q}}±opqªm>4> Voq,p >Äm>$Æm>q1$q }$}±°>q,},½_(P) «¯m p¯ , η¯ > T¯ }A}}±~ q, pFooq, }Æ m>q¯q >(¯u$,y¯ÂV)q ¸ÁV p1o$>1opq,},½~moqA1¸Á1$«o¡}}±>po>G} F ¢ s5o¸Á p }± o¡\ |Âq È~¯{¿¸moq Qvp¯o «´ T´ T´\moq W ªÂ$ oqV½2µ´ q´Çmoq, q q È~} } α > 0 ½o}± m*m>$ ¸Á 5 uˆ ∈ R > t ∈ [0, T ]. ¹Z I¼ ˜ µ (ˆ u, y¯(t), p¯ (t± ), η¯ (t± )) ≥ α, H [p1 (τen )]. = −ντ1en gyµ (y(τen )),. τen ∈ Ten. to. en to. 0. 1. uu. 1. ¢ s5o¸Á p q ¡> {v¸~moq?}±$qª >}± WoqT Åmoq? W 2}±q ,½$Z´ qV´½Hmoq qÃq È~}±} }± mm>$ ¸Á 5>´ >´ t, dist(t; I(gµ (y))) ≤ ε > uˆ ∈ R. |(g µ )(1) (ˆ u, y¯(t))| ≥ β,. . . 0. β, ε > 0. ¹ZKV¼. 0. u. ¢

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(55) ).% . ªË. d2 − < 0, g(¯ y (t))|t=τen dt2. τen ∈ T¯en ;. d2 + < 0, g(¯ y (t))|t=τex dt2. τex ∈ T¯ex .. ¹µL¼. q moqÇoq, B qTÀ VW v>À}±qÇ}>V q,}*q¿ q,}qTÂq,{ V := L2(0, T ) Z := }*moq y| q Â }> q\¸F¸Áo>V>} 2 «¯m j«ªqT5 n ½Æ«moq q 1 H oq (0, Â$$T;ÂVRqv ) L2(0, T )H´ (0,q °>ToqA) moqW>o Æ V}±?¸Áo>V$Âq V × Z L (0, T ) . 1. J1 (v, z) :=. «m>q q. Z. T. ˜ (u,y),(u,y) (¯ H u, y¯, p¯1 , η¯1 )((v, z), (v, z))dt X ν¯τ1 z(τ )∗ gyy (¯ y (τ ))z(τ ), + z(T )∗ φyy (¯ y (T ))z(T ) + 0. ¹µV¼. τ ∈T¯en. ˜ H. }?moqo¡pq,Vq, tQ p>¿¹L¼½>*moq}q 5¸4 >}± W}GF [0, T ] ; z(0) = 0 z˙ = f (¯ u, y¯)z + f (¯ u, y¯)v y. u. g(u,y) (¯ u(t), y¯(t))(v(t), z(t)) = 0. τ ∈ T¯en t ∈ I¯b. gy (¯ y (τ ))z(τ ) ≤ 0. τ ∈ T¯to .. gy (¯ y (τ ))z(τ ) = 0 (1). ¹µ I¼ ¹µ9KV¼ ¹µ9 <V¼ ¹ V¼. ¾¿q V>}~q, ÃmoqA¸ÁV$«>¡}q, >~²ZV ~q, Ã >o¯V F ¸Á 5 (v, z) ∈ V × Z ½ (v, z) 6= 0 ½o}$}±¸Á{Wo¡Ä¹Z I¼³²¹»9<V¼´ ¹ T¼ J1 (v, z) > 0, 6 +(&7 ~´ ¹»¼F¾¿q 5|o$« |{ : o½ q pp o´ =ªm>1«Ãq* q È~o q,}}Am>qW>o ¬ W} ½ }±o¡ (¯p, η¯) ~q °>>q,¬|{ ¹Z <V¼>}±q,V¸ (¯p , η¯ ) ½|$ÂVq moqA}±>V qv¸ÅoqT 2B,q, q, q,} (v,Jz)1 1 1 }$}±¸Á{|>¡¹µ I$¼ ½o|{ F J1 (v, z) =. Z. T 0. H(u,y),(u,y) (¯ u, y¯, p¯)((v, z), (v, z))dt + z(T )∗ φyy (¯ y (T ))z(T ) Z T + z(t)∗ gyy (¯ y (t))z(t)d¯ η (t).. »¹ Ö¼ >o¯V\¹ T¼º}?}± o¡Vq m>m>qvo²Z¡W*}q, >~²ZV ~q, º}NM¬ q W? >o¯VÅ½~ m> q ² BT$V¸m>q}±qT |²ZV oq Ã¡ $«?m ~¯V¿¹µ}±q,qS: =Ö¼F ¸Á 5 (v, z) ∈ V × Z ½ (v, z) 6= 0 ½o}$}±¸Á{Wo¡Ä¹Z I¼³²¹ V¼ . ¹ ¼ J1 (v, z) > 0, 6 +(&7 o´ l?moqÆ}±qT V>|²³ ~q }N M¬ q WÃ >o¯V\¹ ,¼Q>}q,¬moqv}±o{>}q >}ÂW{ >{~}±}Å}«ªqT 5q, <mmoqªVoqº¡VÂVq : , =³½$«moq, qºmoqªq W {AVW4 >}± Wª¹» KV¼Å}4p±q,<´ l?m>qÆ~moV }ªo q,}q WF|opq pq mo~<½~>}q,*, _q,W>>},½W$«o¡ moqT! 5moq |q ÂW{¬¸4moqFW>V~ Æ¸Á p $Âq ?moqF}o>}> q~q °oq,|{K¹» I¼>K¹µ9 <V¼ ½>«mo mÄ¸2¸ Wq, qT}oo,$V>} ½|«moqAmoqJÂV1 q ¯°,$V*¸¹ ,¼Ã> V qÆ q p>}? q ´ 0. ÔÔ æY>ÜTÛOZà![ Ý\ Þ ß].

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