Well-posedness results for non-autonomous dissipative complementarity systems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Well-posedness results for non-autonomous dissipative complementarity systems Bernard Brogliato — Lionel Thibault. N° 5931 June 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5929--FR+ENG. Thème NUM.
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(175) ·Ø·ÏOÀ;¹»Ì\Òuݼ½¾À;êÀfÌ·Àf½OÒ.Îbáw½OÒ.ÝÀ 2¼½zÔ . ). f+. u(·) ≡ 0. Ò.Ì·Õ. F = 0. G. Ò.Ì·Õ 2ÀØ·ÏEÀÂOÊ·¹»ÏJÂEÀ½¾Ý¹»Ì·¼Î»¼ê.à ¹ ÂOÊ,ÏO¼ÝÀ. ĦÅÆ ÄÈÇ. ¢.
(176)
(177) "!#! $%. * V\eRVUjV¦p7jqV\kwTWvkm|kweqa"yscch!T5VZeqR!yze. 7. mk cukwgeVUsj.yz¨!wV7¤bysg!{ eR!ye h(·) kmcLy,{dkw¶BVUpsTWp7jqvR!kcqT¦jqp7T Rn kg7ep Rn ¢JQLRV>cqV |p7g!{dkweqkpsg!cysjqV cqVU¡sV>jqV7¤bRpV>¡sV>jeRVUayzjV'cyekcf)!V9{"V9ys|.ReqkT5V'eRV
(178) ceqpsj.yz7VZ¦h!g!|ekwp7g V (·) kc+y lh!ys{jyzeqkm|+¦hg0|eqkpsgkweqR|Upsg!ceysg7e +V9cqcqkmyzgª¢ * Vp7¨deyskwg z(t) ˙ = . . ∂2V ∂x2. ∂h T −1 (z(t))) ∂x (x)a(h. (x). +. 21. ∂h T −1 (z(t)))ζ(t) ∂x (x)b(h. +. A M0¢ MaB. ∂h T −1 (z(t)), u(t)) ∂x (x)e(h. 0 ≤ ζ(t) ⊥ w(t) = c(h−1 (z(t))) + g(u(t)) ≥ 0.. YZcqkgW¨!yscqk|'|Upsgb¡sVU¬"yzg!yswadcqkcLuV'jqV>jqkweqV A M0¢ MaBLyscueqR!V kwg!|Uwh0ckpsg ∂h T −1 (z(t))) ∂x (x)a(h. −z(t) ˙ + ∈. ªVUe\h!cZgp}yscch!T5V'eR!ye. sVUe. −z(t) ˙ +. _bVUeeqkg. ∂h T −1 (z(t)))u(t) ∂x (x)e(h. ∂h T −1 (z(t)))∂ψ(R+ )m (c(h−1 (z(t))) ∂x (x)b(h. b(x) = B. ∈. +. . ∂2V ∂x2. kc\y|p7g!cfe.yzge. n×m. ∂h T −1 (z(t))) ∂x (x)a(h. (x). − 12. ∂c T ∂x. +. T yzeqjk¬2¢+YZcqkg A M!¢ ZB+ysg!{. ∂h ∂x (x). ∂ψ(R+ )m (c(h−1 (z(t))) + g(u(t))).. S(t) := {z | c(h−1 (z)) + g(u(t)) ≥ 0} ◦ Φt .. ysg!{. =. . ∂2V ∂x2. (x). 12. uV. ∂h T −1 (z(t)), u(t)) ∂x (x)e(h. Φt (z) := c ◦ h−1 (z) + g(u(t)),. QLRkmcLV>lh!yswkwefaV>ys{!cLh!cueqpy7cqcqhTWVeqR!yzeeqRV>jqV'V¬dkmcfe.ccp7TWV'|Upsg!ceysge. ψ(R+ )m. + g(u(t))).. ρ>0. uVcV>VWeqR!yze. ch0|.R"eR!ye¦p7j+yz. ρBRm ⊂ B T Λ(x)(BRn ) + (R+ )m ,. ψS(t) =. x ∈ Rn. A M0¢ hB. RV>jqV B {dVUg!pzeqV9cueqRV oh!|k{V>yzg|UwpcV9{"hg!ke+¨!ysw kwg Rn |UVUgeqV>jqV9{yzeeqRV p7jqkskgª¢ QLRkmc
(179) RysccqhTWvdeqkpsg A M!¢» B
(180) yswp7g&keR eqRVyscchTWvdekwp7g psg eqR!V"jVUsh!ysjqkwefapz V (·) VUg!cqhjV eqR!yze
(181) eqR!V"|p7g¡7V¬ A R!VUg!|UVe.yzg7VUgeqkmyza jqV>shmyzjB'¦hg0|eqkpsg ψ ysg!{ eRV7y7|p7¨kysg´ye z(t) pz ψ ¦h,)!LeqRVjV>lhkjqV>TWVUge ¦psj\eqR!V,V9l7h0yzkefa¦psjT h!y"kwg -w¤2QLRVUp7jqV>T (R 9) ¢ 1;yzg!{RVUg!|UV eRV5V>lh!yzkwefa ∂c◦h S(t) ∂c ∂h −1 yswp+c+h!c\ep = ∂x ∂z ∂x eqj.yzg0cmyeV\eRV'y7cfeZ{k¶BVUjVUgeqkmyz2kg!|Uwh!cqkpsgkwgeqp m. + m. −1. −z(t) ˙ +. ∂h T ∂h T (x)a(h−1 (z(t))) + (x)e(h−1 (z(t)), u(t)) ∈ ∂ψS(t) (z(t)), ∂x ∂x. A M!¢ hB. A B¢Jx+V>|Uyswkg eR!ye RV>jqV {dVUg!pzeqV9c A cqVU V -·? 1 BoeqRV'kwTWkweqkg ch¨B{dkw¶<V>jqV>geqkmyz2pzeqRV¦hg0|eqkpsg eqR!V\kwTW∂ψ kweqkgS(t) c(z(t)) h!¨<{dkw¶BVUjVUgeqkmyzBpz eqRVkwg0{dk|>yepsj¦hg!|ekwp7g ψ pzy5cVUe Q kcugpseqRkg5VUmcVψS(t) ¨hdeLke.ckT5kweqkgWgp7jqT yz |p7gV N (Q; ·), uVT y·ajVUjkweqV\eqRV kg!|h!cqkwp7g&kwg A M!¢ Z BuTWpsQjV'|UpsTWv!y7|eqayscu¦p7wp+c> A M!¢é B. ˜ −z(t) ˙ + h(z(t)) + e˜(z(t), u(t)) ∈ N (S(t); z(t)).. «ZcuJ y )0jceuceqV>v"kgeqR!Vysjq7hTWVUgecopz QLRV>psjVUT M!¢» 'uVZV>ceys¨kcqR eqR!VZ¦p7wpkwg! VUTWT y¢q3¥gkwecuvjppsªV\h!cqV eqR!V TWyskwg km{dV>ypzeqRV{VU¡sV>wp7vTWVUge'pzueqRV cqh|kVUg!|Uav!yzjqe pzLQLRVUp7jqV>T ¢ M7kwg -·?1o¨!hde
(182) V {dV9yz-kweqR eqRV kgV>lh!yswkwefaW¨0V>wp kgysw0eqRV'cqv!ys|UV Rn . A Z|Upshj.cV7¤seRVkwgV9lh!yzkefa |>yzg¨<VcqVUV>g"yscuy%=Z,h8 7TWVUeqjk|ZjqV>shmyzjkefaIB¢. ´ " Q 9h U 67
(183) i @j ] 5n0\Ib J U JZ< >= <9Yb n m :` U −1 ! >9 zˆ ∈ Rn \ U cb67 <
(184) i y 7→ d(ˆz, k−1 ((R )m − y)) k;: R $_→ R9 5 U k(z). o :=
(185) c 6 ◦ h6a.:`(z) U 1/ρ + 5 U %?9 67e6a H Rn .. o® . . . . Äm¹»Ï 2¼½¦ÂOÊ Ì9¼.ÂO¹»Ì·êLÂEÊÒ Ý+ÒàZÌ·¼. zÀJÒuÁf¼ÌUÓÀ¥í'ÏEÀ¥ÂqÃzÏE¼uÂOÊ·ÒÂÂEÊ·ÀJÌ·¼.ÂE¹»¼Ì·Ï¼.á!ÏOØ sÕ9¹ 7Àf½¾ÀfÌUÂE¹éÒ.ÎÒ.Ì·Õ'Ì·¼½¾Ý+Ò.ÎbÁf¼Ì·ÀJ¼.á!Áf¼ÌUÓÀ¥í Ò.ÌÒ.Î à>ÏO¹»ÏªØ·ÏEÀÕ\¹»Ì\ÂEÊ·À;º·½¾À¥Ó9¹»¼Ø·S(t) ÏÏOÀfÁ¥ÂO¹»¼ÌÊÒÓÀ-ÂE¼ zÀ;À¥í>ÂEÀfÌ·Õ·ÀfÕ'ÂO¼Ý¼½¾À-êÀfÌ·Àf½EÒ.ÎÌ·¼Ì9Á¼ÌÓÀ¥í ¼9¿¾ÀÁ¥ÂEÏqÔ. Æ Æ Ì ÑìÙ9ç.
(186) Y V= gi $ S<U i8 67e. · ²ª² ( psg!cqk{VUjkwg,eqR!V
(187) cVUeF¡·yswh!V>{TWysvvkg . . M : Rn ⇒ Rm. keR. M (z) := −k(z) + (R+ )m ,. V'R!y·¡7V. d(ˆ z , M −1 (y)) = d(ˆ z , k −1 ((R+ )m − y)) =: ϕ(y) ∈ R+ ∪ {+∞}.. kw¬£yzgba&jqV9yzgbhT ¨<VUj. yzg!{eZy +sV,ysgbacV9l7h!VUg!|UV (y ) |Upsgb¡sV>jq7kwg!Weqp y yzg!{Écqyzeqkmcf¦abkg ϕ(y ) ≤ α. ( Rpbp7cqV α n k e R QLRVcqV>lhV>g!|nV (z kmc-¨0p7hg!{dV9{,ysg!{ R!VUg!|UVukweqR!pshde;p7nccps07VUgV>jyswkwefa zn ∈ M −1 (yn ) ϕ(yn ) = kˆ z −zn k. n )n uV T y·aÉcqhvv<p7cqV eqR0ye kwe,|p7gb¡sVUjsV9ceqpcqpsTWV z kg Rn . \¨b¡bkpsh!cqwa kˆz − zk ≤ α ysg!{ kwe kc V>y7ca£eqp£cqVUVWeR!ye QLR!VUjV¦psjVs¤ yzg!{"eqR!kcabkVUm{cueqR0ye kmcpV>jcV>T5km|p7geqkgh!psh!c>¢ −1 z ∈ M kw¬g(y). p§yzgba (¯y, y∗ ) kwϕ(y) g£eqRVW≤sαj.yzv!RpzJeRV ji>|.RVUecqhϕ¨<{k¶BVUjVUgeqkmyz-ps ϕ A cqVUV5Vs¢ !¢-·?1-¦p7jZeRV {dV)!g!kekwp7g B ysg!{ |.Rpbp7cqV z¯ ∈ M −1 (¯y). QLR!VUgɦpsj V>y7|.R ε > 0 eqRV>jqV VU¬bkmcec'cp7TWV5g!VUksRb¨0p7jqR!ppd{ Y ps y¯ cqh!|.RÉeqR!yze ¦psj
(188) yz y∈Y. R!k|.RTWV>yzg0cu¦psj+yz. hy ∗ , y − y¯i ≤ ϕ(y) − ϕ(¯ y ) + εky − y¯k, z ∈ Rn. h0, z − z¯i + hy ∗ , y − y¯i ≤ kz − zˆk + ψgph M (z, y) − k¯ z − zˆk − ψgph M (¯ z , y¯) + εky − y¯k.. kc5y ! jqi9|.RVe5ch¨!sj.ys{dkVUge pseqRV"¦hg!|ekwp7g (z, y) 7→ kz − zˆk + ψ ye RV>jqV {dVUgpseqV9c eqR!V7jysvR pzeqR!VcqVeqO¡yzhV9{ TWysvvkg gphM.M (z, QLRy) Vefup(¯¦zh,g0y¯|),eqkpsg0c kg eRV
(189) cqhT¨<VUkg jqV>shmyzjysg!{eRV )!j.cfe+p7gV'¨0V>kwg!|psgb¡7V¬"|p7geqkgh!psh!c>¤dch¨B{dkw¶<V>jqV>geqkmyz cqhT jhV'vjqp¡bkm{dV>cLcqpsTWV cqh!|.R&eqR0ye ∗, y∗ ) kmcZy !jqi9|.RVeZg!psjTWys2eqp gph M ye (¯z, y¯). nJhdeqeqkg p¯ = y¯ + k(¯z) ∈ (R )m , kwe\kmc zg!∗pze+ ∈{dB + k R|Uhwe+eqp5eqj.yzg0c(z myeV\eRV myeeVUjkgeqp ∗ yzg!{ ∗ ∗ m Y\ckg z ∗ ∈ B yzg!{£eqR!V5kg!|Uwh!cqkpsg A M!y¢» B∈¤ªN VW((R p7¨d+eys)kwg£; p¦¯ps)j
(190) yzgba zq ∈= By ◦ ∇k(¯ cqpsTWzV ).q0 ∈ B yzg0{ p ∈ (R )m R R R + cqh!|.ReqR!yze QLRbh!c>¤. (0, y ∗ ) gph M := {(z, y) : y ∈ M (z)} n. n. m. n. ρhy ∗ , bi = hy ∗ , ∇k(¯ z )(q 0 ) + pi ≤ hy ∗ ◦ ∇k(¯ z ), q 0 i = hz ∗ , q 0 i ≤ 1,. R!k|.RV>geyzkmc ky∗k ≤ 1/ρ. «|U|Upsj.{dkgWeqp"eqRkmc !jqi9|.RVe'ch!¨<{dkw¶BVUjVUgeqkmyz¨0p7hg!{dV9{dgV>ccyzg!{&ep eqR!V,pV>jZcqVUTWk |Upsgeqkgbhkwefapz ϕ, eqRV
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