An anti-diffusive scheme for viability problems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. An anti-diffusive scheme for viability problems Olivier BOKANOWSKI — Sophie MARTIN — Remi MUNOS — Hasnaa ZIDANI. N° 5431 Décembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5431--FR+ENG. Thème NUM. apport de recherche.
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(8). Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30. *. .
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(21) /() $x*+ $gu K -gOe; 9ubs³yds³gfu¦ R w U fe; u¦s}yd s}gub R %± $¯gfu f : R × U −→ R g ¨x@|µu}a)9yd{ w g[ts}gUgub9u}w9 u |µxwrydy 9wdy |©¹sx9e'{}e;ª§q@± |§wÑw2uu}± a gs}g[~Qydgfª7¼-¨¸gs}ydd-9s}g%u a #u f |§sjªµtfªµª§qªµ|§ s}ad|©u ´fwQu}|§wrydy s n. "!$#. p. %'&$(*) &$+ &-,/.1032547628+. { wQq'e;gs}yd{ dª§gÂfwQu}{ ª7¼r¨¸g%tg[wdu g. yx (t). n. n. u adgÂu}{+#vgu { q@s}u}|sv¹qr|§wd. ½¹Q9Á ½¾Á. y˙ x = f (yx , u) u ∈ U (y). Ñw)®r|§gf¨. u adg9w svu { |µwQux|§®g[w_rq½7Á¼t¨¸gs}gubu}a g¹ª§ªµ#¨x|§wd;tu |µe'ª¯wQu {}9ª- {}9dªµg[e ¹9{ ª§ª t > 0, } (P ) min {0, ∃u(·), y (t) ∈ K ²»gtg ¿ wdgÂu}a gO®#ªµydg¹y w u |µ9w)9s}s}t|#u g['u ;u}ad|sbd{ dª§gfe Qq yx (0) = x. x. x. V(x) := Inf(Px ). ¨xa gf{ glu adg®#ª§ydg% V(x) |§sls³yd Qs³g_u 'g +∞ |©u}adgs³gfu \wsvu { |µwQu sb|sxg[eÒduvq±bj\9yd{+s³g ¨¦gOu}ary sxa¬®glu}a u V(x) |µ / K ±%jª§s} ¼t¨¦gÂa ¬®g = +∞ x ∈ |µ ∃u(·) ¼ ∀t > 0 ¼ y (t) ∈ K 0 u adgf{ ¨x|s³g V(x) = +∞ `ba gfw u}adg à |§d|µª§|µuv q 9Âgf{ wdg[ªÃ|§sb9|µ®9gfwQq à |§ (K) = {x, V(x) = 0} = {x, ∃u(·), y (t) ∈ K ∀t ≥ 0}. {xg+a T > 0 ¼t¨¸g%|§wQu}{ tty gÂwd#¨ u}adg¹9ªµª§#¨x|§wd9tu}|§e'ª¯fwQu}{ ªÃd{ ªµg[e ¹9{ ª§ª t ∈ [τ, T ]} (P ) min {0, y (t) ∈ K ¨xa gf{ g y |s ;s³9ªµytu |µ9w ½49Á y˙ = f (y , u) u ∈ U (y) ½7#Á y (τ ) = x x. x. τ,x. τ,x. τ,x. τ,x. τ,x. τ,x. >0. ý. >@(.
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(23) Ç ¾È9Ê Å Æ-Ç9ÊŵËfÏÌ. . w ¨¸g%tg ¿ wdgªs}u adg%®#ª§ydgO¹ydw u}|§w2s s³tf|§u}g[@u};u ad|§s d{}9dª§gfe rq V (T − τ, x) = min(Pτ,x ).. ²»gs}gfgOu}a u. |© u}a ∃u(·), gf{ ¨x|§s}g ∀t ∈ [0, T ], y (t) ∈ K ¸Ú ¢ #ÇË ¬ËfÇ0 x ∈ K
(24) V (T, x) Í ¬Ëf Ç QËÌ%,Æ
(25) ¥È#/Ç ¬Ì V(x) È¬Ì T → +∞ bÇ
(26) ²»g ¿ {+s³u{}g[e;{}°Uu}a u T → V (T, x) |§s%w wtÐÑtg[f{}gs}|µw ± Ñw dgfg[¯¼s}ydd-9s}g T < T ± × ¼¥¹|§w {@wQq fwQu}{ ª u(·) ¼¸u}adg[{}g)gf«r|s³u s ¼u}tad≤g[{}gT|§sjs³wdy+u aRad|µu w a '#u u} yd(t){}#®9g∈/± ' Kg[± w org|µw¨¸gg2a¬ª®s³g V (T, x) = ∞ ¼ ¦ ¨ g t + u ± × t≤T V (T , x) = ∞ |§wªµª fs}g[s[± V (T, x) = 0 V (T, x) ≤ V (T , x) hj#¨L|µ V(x) = ∞ ¼tu adgf{ gÂg«t|svu+s T > 0 s}y +au}a u y (T ) ∈/ K %± 'jgfw fg¼ V (T , x) = ∞ ¼ w u ad|s¸|§e;dª§|µgs¦u}a u V (T, x) → ∞ y s³|§wdu}adgÂwd9wtÐÑtg[f{}gs}|µwdd{}9g[{³uvq9± ×u adgf{ ¨x|§s}g V(x) = 0 ¼ u adgfw ∃u(·) ¼ ∀t ≥ 0 ¼ y (t) ∈ K ¼dw V (T, x) = 0 ¹9{ ª§ª T > 0 ± ' g[w gª§s} V (T, x) → 0 ± 2 ²»gÂu}adg[w)a ¬®gOu}a g¹ª§ªµ#¨x|§wd)½¹¨xad|+a)d{}r |sbªµgf u u}Òu}a g%{}gtg[{+Á ¸Ú ¢ ËÆ Ω := {x, V (T, x) = 0} ÎdË
(27) ¥ËÂÎdÈ ¬Ë
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(29) #ÇË ¬Ëf0Ç T ≥ T ≥ 0
(30) (i) Ω ⊂ Ω ⊂ K (ii) ∩ Ω = Viab à j h # ¨ ¦ ¨ % g d ²»gOu}ary sbª§r°Ò{}¹9{ Qs³w glu}@ df{}¬e;«td|§e'ytu #g u}|§Ωw@¹¹{ { VT(·,ª§T{}9)gO|§w g[wdKy ±²a¯¼dg°rw wd@#¨u @u a d# ub{}u}¬a «tgl|§e;¹y w u}gu |µ9|w V s Q#qu}|s Ω¿ g[s ± w '+ g[~Qy #u |µ9w u adg¹ª§ª§#¨x|µwd¹{ e ½ QÁ V − min (f (x, u) · ∇ V ) = 0, t > 0, x ∈ K; ½ Á V (0, x) = 0, x ∈ K; ½ Á V (t, x) = +∞, t > 0, x ∈ / K. ×w gOu adgÂuv¨¸Ò¹9ªµª§#¨x|µw ;9s}s}yde;tu |µ9w sb{ gÂs #u |§s ¿ g ¼ ¼ ¼ (i) ∃α > 0 ∀x ∈ ∂K ∃u ∈ U η · f (x, u) < −α d{ ªµª x ∈ K ¼ A(x) := {u, y ∈ K} 6= ∅ (ii) u adgfw V |§sl@9w9u |µwryd9y sx¿®r|s}f9s}|©uvqs³9ªµytu |µ9w) ½ QÁ± ' g[{}g9¼ à ad#¨¦gf®9gf{¼ru}adgs s³yde;tu |µ9w s (i) { (ii) {}gxw u¸w g[gs}s { |§ªµqÒs}u}|s gü9s}|§w g |µu¸¨¦y ª§;|§e;dªµqÒu a #u | = K ± 'jgfw fgj|§w@g[wdgf{+ªdu}adg ¹ry q w u { |µ9wdw °V#¨ |s}sx°#wd|§wujÓï¬9Ö×w9± u |µÑwrw ydu}9ady |s[sx
(31) ± ×ul|§g[sx{[s}¼tad¨¦#g¨xdw{}u}9;Q-s³ggl;u}'s³9fªµyte;u |µd9w ytu gÂb(K) u} ½ad9gÂÁ¦¹|§ydw2w;u} |§w {}u}V|ydrªq@{xy s}s}gf|µw w s}g%u ad|§g®g[s}w [ª§ªµg }i ªµu}{+#Ð×-gfg Âs}+adg[e;gÒÓ ¼¬Ö¯¹{bu adgt|s}f{}gfu}|s}u}|§w x½ 9Á± `badgd|§s { gu |§s #u |µ9w_g[~Qy u}|§w¶½ QÁ¥¨x|§ª§ª¯gs³u}yt|µg|µw2s}g[u |µ9w ± V (T, x) =. . 0 +∞. τ,x. 0. 0. x. 0. 0. 1. x. 1. 1. T →∞. T →∞. x. T. T0. 0. T. T →+∞. T. K. T. t. K. x. u∈U. x. x. F. ý ý ð3 ü ú " . T.
(32) bÅ ¾ËfÇ%# "!. #
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(35) È¬Ì È9È . (,4 2 )( &$6. $gu\u adgbu {}9gu |§w ¢d¿ wd
(36) |µu}gxu |µe;g -Cg¹9-{}gjgxÂ-9s}yds s} |µs}dgª§uWqª§g[ ¬R®r|§wd±\ `bKadg Qs}qÒyd #uWs}gª§u\g[9svHuW|µw 9|©wdu g¦|§u}ª{+s³#u v#g[u g[u}s9{}xq ∈y K(·)s³y|sW+afu ªµa ª§g[#u u}Ca g |§sW£{}¢gC + ad¡gÚ |µw rdy w g_)u}tadg[gwds³gfu u³g[Ð×2®¬¦ªµy g[t·u e'(C) ± ¨xa |§+aRf|§w |tgs¨x|µu}a 9ytu s}|tg ¼\g[~Qy ª§su} $guysC|§wQu}{ K |µws³|tg à C w g~9yªs@u u}a g»fwr®gf« arydFª§ªl à {0} ∪ F (x) w ∂C ± F× K |s) C{ gf-gfª§ªµg[{@¹{ 0 tu ± (C) = | (K) ±&ju}adg[{}¨x|s³g¨¸ga ¬®g|µwLg[wdgf{+ª F à |§½¹ |7± g9± (K)|= ¦(K) du =(C)∅Á;∪u à adgf|§w ¦(K) {%y {O yd{}-9s}g¼Ãª§gu T > 0 ± $¯gfu χ g;u adg'f{ 9u}g[{}|s³u}|%¹ydw u}|§w¶ C ¼|7± g9±Òtg ¿ wdgKrq |© w χ (x) = +∞ u}a gf{ ¨x|§s}g±-$¯gfu ϑ (t, x) w ϑˆ (t, x) -gtg ¿ wdg[ rq χ (x) := 0 x ∈ C w y (s) ∈ K ¹{ s ∈ [t, τ ]}, ϑ (t, x) := min {χ (y (τ )), y˙ (s) ∈ F (y (s)) w w y (s) ∈ K ¹{ ªµª s ∈ [t, T ]}. ϑˆ (t, x) := min{χ (y (T )); y˙ (s) ∈ F (y (s)), hj#¨ ¨¦gtg ¿ w glu adg £¢ ¡ Ú ¢d
(37) Ú ×Þ ¡ Ú Ú T Qq ¦tu (C; T ) := {x ∈ K, ϑ (0, x) = 0}. Ñw {}u}|y ª§{[¼ T → ¦tu (C; T ) |§sO|§w { g[9s³|§wd'¹9{lu adgÒ|µw fªµys³|§w¯¼¯w Kª§s}_¨¦g ¿ w Uu adgÒy s³yª [tu yd{ gOs}|µw)s T → ∞ ¦tu (C; T ) = ¦tu (C). lim ×u;|§swdu;t| @ydªµuÒu}»tu+|§w ½¾9s}s}yde;|µw |skU{ +ay ü\w |§w {³u |§fydª§{u a #u F (x, U ) fwr®gf«'¹{ ªµª xÁ¼du adg%|tgfwQu}|µuvq@ u adg%d{ gf®r|µ9Fy s¦®#ª§ydgO¹ydw u |µ9w s n. x. F. C. F. FC. F. FC. F. F. C. C. C. T. τ ∈[t,T ]. T. C. C. T. t,x. t,x. t,x. t,x. t,x. C. T. t,x. t,x. t,x. T. F. F. F. T →∞. 'jgfw fg¨¸g%a ¬®9gOª§s}. F. ϑT (t, x) = ϑˆT (t, x).. ¦ tu (C; T ) := {x ∈ K, ϑˆ (0, x) = 0}. `ba gj¹ydwu}|§w ϑˆ |§sb®#ª§ydg ¹ydwu}|§wCw@9tu}|§e'ªÃ9w9u {}9ª {}9dªµg[eá½¹e;{ gjd{ g[f|§s}gfª§q¼t Ë ËfÉ #ÄrÌxd{ dª§gfe¨x|µu}a¶svu+#u}g;9w s³u}{+|§w9u+s Á± Ñw» {³u |§fydª§{j¨¦gÒ[wKs³u u}g; dqQwe;|§d{ {+e;e;|§wd {}|§w |§dª§gO¹{ ϑˆ ½¹¹{ t + ∆t ≤ T Á ½ 9Á ϑˆ (t, x) = min ϑˆ (t + ∆t, y (t + ∆t)), w ½ ¬Á ϑˆ (T, x) = χ (x). ¨xa gf{ g y { gs³9ªµytu |µ9w sb y˙ ∈ F (y ) 9w [t, t + ∆t] ±¦i w tg[{ls}e;gÂu}g+adwd|fª¯s}yde;tu}|§w s[¼ ¿ ϑ ¦ˆ s}tu u}|s gs%ªs³¼r)u}adg%w ª's³+u g[~Qgy ~Qy u}#|§u w |µ9sÂw½|§w·9ÁÑ Ð+½¬9gfwdÁ¸g[¨x{ |§ªµªµª¯|§´f-ggKts}|gfs}w f{}s}gfg2u}|§½¾´fs}ggf_g)|§Óïw¬Ö¹Áu}±adg%Ñw»wdgf«r{+ujtgfs³g{Ou}u}U|§wd± d{ ¬«t|µe'#u g T. F. T. yt,x. T. T. t,x. t,x. C. t,x. C. t,x. F (C; T ). >0. ý. >@(.
(38) È Æ É ÄrÌ #ËÌÍ+ÎdËfÏ'Ë
(39) Ç ¾È9Ê Å Æ-Ç9ÊŵËfÏÌ . +* $O d(x! C ²»gls}a ªµª-d{ g[s}gfwQuWu}a gOij¿ ª©u { Ð7-gfg½4i xÁ¸s}+adg[e;gx¹{¸u}adg '2+ g[~Qy #u |µ9wK½QÁ\|§w_s} fgjt|§eÒg[w s}|µ9w t¼ ®9g[u |µ9w'|µwK[ü9u}adg[w@|§w ürw ¿yw s³|§wdª§%q@u |§adw){}g[gl@s³¹u}9g[{b u}s ad Cg ¨¦'g + {+svu¦g~Qdy {}#gs³u g[|µ9wQw¯uW± u}addgO{xi { 9s +u}ad|fgfe;ªHgxd¹yd9{ {¥ª§Q|µwds³g9g¼ru}{¦adg t+∞ ®#ª§ydg%fw-g%{}g[dª§9g@rq d ¼ . w µ | ) w. ³ {. u § | f d y ª ¦ { } u d a g f . w d © |. u µ | 9 » w ½. [ b Á [ w % g. { f g. § ª 9 [ g r q +1 |µ x ∈/ K. ½¾ÕQÁ V (t, x) = +1 (5 ÌfÍ+ÎdËfÏ; Ë
(40) # Ç 'Å Ë+È#ÇÈ #Ë+ÍÆ 1 ²»g9w s³|tg[{bu}adgt|s { gu}|s #u}|§w ½ 9Á v + f (x) v = 0, t > 0, x ∈ R v(0, x) = v (x) ¨xa gf{ g x → f (x) |§sª§|§ s}+a |©u ´ÐÑwQu |µwd9y s[¼w Ku}a g'|µwd|µu}|ª¸wt|©u |µ9w v |§s9s}s}yde;g[|§w L (R) ± s³y+a'u}a u x − x = ∆x w t = n∆t glydwd|µ¹{ e s³ 9gjw Òu}|§e;glt|s { gu}|s #u}|§wsf¼ ) ¨x$ga u gf{ (xg ∆x, { gbu}adgjeÒgs³a@s}|§´fg[s[-± $¯gu V tgfwdu}gsW%wryde;gf{ |§[ªddd{ ¬«r|§e'#u |µ9wu Âu}adgOs³9ªµytu |µ9w ∆t. ¼ b ` d a % g i Ls}+a gfe;gO¹{½ QÁ¸u °g[s¦u adgO¹ª§ªµ#¨x|§wdÒ¹{ e v(t , x ) V −V V −V ½v9Á + f (x ) = 0, ∆t ∆x ¨x|µu}au adg%|µw |©u |§ªµ|§´[u}|§w Z ½v9Á 1 V := v (x)dx, ∆x ¨xa gf{ g x = x + -± ' g[{}g V w V { g¦wQy eÒg[{}|fª ¦Ä QË̯u a #u¥¨x|§ª§ªdg dg ¿ wdgÒg[ªµ#¨%± ²»g%¨x{ |©u g;½³[QÁ¥|§wu}adg%g~Qyd|µ®#ªµg[w9uxwd9wtÐÑws³g[{}®##u |µ®9g ¹9{}e ½³Á =V −ν V V −V , ¨xa gf{ g ∆t ν := f (x ) |s ³ªµtfª * -$ %wQy e-gf{±\²g9s}s}yde;gOu}a u |ν∆x| ≤ 1 ∀j ± Ñw u adgf9s³g ν = 0 ¼d¨¦gOu}ary s fw s}|tgf{ ¼dw u}ad g yd«rgs V wdgfg w uxu};-gtg ¿ wdg[ñ =V V ²»g ¿ {+svu s}gu |© ν > 0, ( b := max(V , V ) + (V − max(V , V )), ½³[9Á t. x. 0. 1 loc. 0. j. n. j+1. j. n n j. j. n+1 j. n j. n,L j+ 21. j. n,R j− 21. xj+ 1. 0 j. 2. 0. xj− 1 2. j. j+ 21. n,L j+ 21. ∆x 2. n+1 j. n,R j+ 21. n j. n,L j+ 21. j. j. n,R j− 21. j. j. n+1 j. + j Bj+. j. |© ν < 0, j. ý ý ð3 ü ú " . j. n,R/L j+ 21. n j. (. :=. n n j j−1 n min(Vjn , Vj−1 ). 1 νj + ν1j. n j (Vjn. −. n n j j−1 n min(Vjn , Vj−1 )).. 1 n n n n n b− j := max(Vj , Vj+1 ) + |νj | (Vj − max(Vj , Vj+1 )), − 1 n n n n n Bj := min(Vj , Vj+1 ) + |νj | (Vj − min(Vj , Vj+1 )),. ½³fQÁ.
(41) Õ. bÅ ¾ËfÇ%#. #
(42) dÎ ¾Ë
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(44) È¬Ì È9È . hj#¨%¼t¨¸g%tg ¿ wdgÂu adg yd«rgs V w V 9s¦¹ª§ªµ#¨ s½4s³g[g'Ó ¬Ö¹Á × u}adg[w)dg ¿ wdg V := min(max(V , b ), B ) • ν >0 × u}adg[w)dg ¿ wdg ± • ν <0 , b ), B ) • × ν ≤ 0 w ν ≥ V0 ¼du}adg[w):=dg ¿min(max(V wdg w V := V . ½³Á V := V × ¼ru adgfwUtg ¿ wdg V := V ½¹|µ ν > 0Á¸{ V := V ½¹|µ ν < 0Á± • ν ν >0 {xsvu+d|§ª§|©uvq@w _fwr®g[{}9gfw fgxd{}9g[{³u |µgs¦¯u ad|§sbs +adgfe;g9¼Q¨¦gO{}gf¹gf{¸u}Ó Ö7±Whju}glu}a#ub|µw u}adg [s}g ν > 0 ∀j ¼¨¦g;a ¬®g V = V w Uu ary sf¼Ctgfw u}|§wd V = V ¼u}adg@s +adg[eÒg2½³Á u+°9g[s¸u}adg%e;9{}g%s}|µe;dª§g¹{ e n,L j+ 21 n,L j+1/2 n,R j−1/2. j j j. + j − j. n j+1 n j−1. + j − j. j+1. n,R j+ 12. n,L j+ 21. n j. j. n,L j+ 21. n,R j+ 21. j. n,L j+ 21. n j+1. n,R j+ 21. j j+1. (5. n,R j+ 21. n,L j+ 21. n,R j+ 12. n j+ 21. n,L j+ 21. j+1. n n Vjn+1 = Vjn − νj Vj+ . 1 − V 1 j−. ÌfÍ+ÎdËfÏ;Ë
(45) #Ç 'Å Ë+È#ÇÈ #Ë+ÍÆ1 hj#¨ ¨¦gÂfw s}|tgf{bu adg%g[~Qy u}|§w 2. 2. ½v 9Á ½v Á. vt + f1 (x, y)vx + f2 (x, y)vy = 0.. ²»g%9w s³|tg[{b;[{}u}g[s}|w_e;gs³a (x , y ) ¨x|µu}a2fw s³u w9ube;g[s}as}|§´fg[s x ¼ w 9s}s}yde;gOu}adg*-$·fw t|µu}|§w y = ∆y v(0, x, y) = v0 (x, y). j. k. i+1. − xi = ∆x. w. yj+1 −. j. ½vÕ9Á `ba gi s +adgfe;g_½³9Áx|sjgf«Qu gfw dg[)u ½v 9Ábrq)y s}|µw _s³|§e;dª§q) `{ u}u}g[{ls}dªµ|µu³u |µw ½¾s}gfg Ó Ö Á±l`badg |§wd|µu}|ª§|µ´#u |µ9ws³u}g[|s Z ½v 9Á 1 V = v (x, y) dx dy ∆x∆y ¨xa gf{ g w ±O`ba gfwK¨¦g ¿ {+s³uO9e;dytu}g (V ) ¹{ e (VI =) r[xq_s}−ª§®r|µw ,Òx9wd+gOu}|§e;]gsvu gfJC=u}ad[ygi − s +ad,gfye;+gÂ|µwu ad] g x t|§{}gu}|§w ¹{ ∆x ∆y max max |f1 (xi , yj )| ≤ 1. , |f2 (xi , yj )| i,j ∆t ∆t. 0 i,j. i n i,j. i. ∆x 2. i. ∆x 2. 0. Ii ×Jj. j. ¹9{bg[+a_|§®gfw y = y ±%C|µw ªµq'¨¦gOtu+|§w i s +adgfe;g%|§w_u adg y t|§{ g[u |µ9w_¹9{. j. ∆y 2. ¹{}9e. vt + f1 (x, y)vx = 0. j. n+1 ) (Vi,j. n,1 i,j. ∆y 2. j. n,1 (Vi,j ). rq@s³9ªµ®r|§wd9wdgOu}|§eÒg%s³u}g[_ u}adg. ¹9{Wg[9+a;|§®gfw x = x ±`ba gO*-$2wt|©u |µ9wU½vÕ9ÁC|s\w #u yd{+ªd-g[[y s}gxadgf{ gx¨¦g ws³|tgf{¸%`{ u}u}gf{ s}½4Õ9dÁª§|©± u}u}|§wd ± ªs³¼Ã¹{O9ydw d{}qUfw d|©u |µ9w sO¨¸g;+adr9s}gu adg;®¬ªµy g V = 1 |© (x , y ) ∈/ K sO|µw vt + f2 (x, y)vy = 0. i. n i,j. i. j. >0. ý. >@(.
(46) È Æ É ÄrÌ #ËÌÍ+ÎdËfÏ'Ë
(47) Ç ¾È9Ê Å Æ-Ç9ÊŵËfÏÌ. . ¦± $9ytu |µc[{}gxÓ #Örd{ #®gju adg¥®9gf{ qj|§wQu}gf{ g[s³u}|§wd {}9g[{³uvqlu}a uu adg¸i »s +adg[eÒg¸t®9g[u+s¯gf«t9u ªµq {³u |§fydª§{Ofª§9s}slbsvu gfK¹ydw u |µ9w s[¼¯|µwKu}adg'[s}g¸9w svu+wQu%t®9g[u |µ9w¯±d9{Â|µw s³u w g9¼H¹{¬Ð ds} |µe;9gfg ws³|§ w ª¸ {}9dªµg[e'sf¼\ª§gu u s³y+a u}a#u V |§wd|©u |§ªµ|§´fgR9s|§wL½³ 9Ág[ªµ9wd9su}u}a g_¹9ªµª§#¨x|µw S b) ∈ {0, 1, 2}, V =V }. ¸9w s}|§tg[{;u}adgUi s S+ad:=gfe;g{(V¹{ ),v ∀(a, x ¨ d a [ g } { g |s@·9w s³u wQu , f ) = const a9t¬®®ggu |µ9w»®9w g[u}{ R ±U`ba gfw¯¼\+s s}ydfe;·|µ∇v wdUu}=adg0* -$Lwft|©u =|µ9w (fmax(|f ¼C¨¸g | , |f | ) ≤ 1 ∀i, j n≥0 Z 0 ij. 0. i,j. 3i+a,3j+b. t. 2. 3i,3j. 1. n Vi,j =. 1 ∆x∆y. 2. ∆t 2 ∆y. ∆t 1 ∆x. v(t, x) dx. ¨xa gf{ g v(t, x) = v (x − f t) |s¦u}adg%gf«t9u s}ª§ytu}|§w¶½¾s}gfg%ªs} Ó #Ö¯¹9{beÒ9{}gÂ9gfwdg[{ ª¹y w u |µ9w sbu}a#u {}gOg«d9u}ª§q_9t®gu}g Á± u ad×ugj|sÃi u Rad|s sÃ+gad«dgfe;uÃg¼u}{+¨xadwd|§-+a'{}e;u u}u}|§|§®¬wOdu}g{ s-y gfs\{}¹uv9q{\¼¨xy s³ad|§|§wd+%a%|µfu¥|§{ w{}g¹s³{ -wQuWw ds-{ u} 9w fu}|§wQwu}|dt{}|9s}s}d|µª§gf#e'u}|§s\®s³g y¯+-a;gfa9s ¬®Q½|§QÁ{ ± -×uxg%|®sbg[{}ªq@s³ÒfªµwrQyds³e;gOu gf;{ |§[¹ª§ydªµq;wu}|§s³g[w2{}®9s}g[ ;fu g%a #s³uxy +|µa) Vs Strg[ u}s¦gfwd{ ub¹-g[¨gfª§u w |µe;gu s³u}u gfadg%sf±ª²»s g%s S{}gf¼r¹gfu}{badg[u}w 'V lz[s}du}{ g[c[w sxds¦wu} $ 9ytu |µc[{}g9Ó Ö¯¹9{xu}a gf{x|§w9u gf{ g[s³u}|§wd;d{ -gf{}u}|§g[sb u}adgi s +adgfe;g9± /xÉ (5TÌfÍ Î ËÏ'Ë ²»g%wd#¨Lfw s}|tgf{¦u}a gt|§s { gu |§s #u |µ9w Cw '+ g[~Qy #u |µ9w u adg¹{ e ½49Á v − min (f (x, y, u) · ∇v) = 0, t > 0, (x, y) ∈ K. ²»gO9s}s}yde;g u adg* $»9w t|µu}|§wK½³[ÕQÁ±\`badgO|µwd|µu}|ª§|§´[#u |µ9w@ V |s¸t9wdgÂs¸|µw»½v QÁ± ¦u¥u |µe;g ¼ ¹9{l'9|µ®9gfw ¨¦g9w s}|§tg[{ '9|µ®9gfw2t|s}f{}gfu}|§´[#u |µ9w)\u}a gte;|s}s}|§dªµg t=t s}gu U (x , y ) ±l²gÒt(xgfwd, u}yg)rq V (u)(u u ad) gi Ts +adg[eÒgtu+|§wdg[)¹{ e (V ) rqy s}|µwd_u}adg 9t®gu |µ9w ¼d|7± g±§¼twdgÂu |µe;gs³u}gfCu}a gi s +adg[eÒgO¹9{ Ii ×Jj. 0. 0 i,j. t. n i,j. u∈U (x,y). 0 i,j. n. i. i. j. f (·, u). j. k k=1,...,N. n+1,U B i,j. `ba gfw u}adg'+ Ls}+adg[e;gO|sb|§®g[w rq n+1 Vi,j. n i,j. vt − f (x, y, u) · ∇v = 0.. = min. . n+1,U B Vi,j (uk ). . ½7tÁ. .. ²»g~Qg2y #{ u g|µ¹9gfw {;sWu ¨x |µu}Ó aÖ t¹|§s { 9¿ wQ{ u}s³|§wdu@y ds¸ |§ªµwd|f|©u |§u}|§ªÃdw #s;u+dl±`bu adadg gO}s i +adªµu}g[{+eÒ#Ð×gOs³g[g[ggf)e's s¸+ad¨¸g[g[eÒªµªHgu d¶tu}u}adgg2;{ u}g[s}u}{ ª§ytg[u}|§u¦wt|s wQ'2u +|µwt Ð y y sÂs}ª§ytu}|§w s%w |µw» {}u}|y ª§{l¨xadg[w»u}adg;®#ª§ydgÒ¹ydw u}|§w»u °g[sO9wdªµq)uv¨¦)®#ª§ydg[s;½ 0 w 1Á± 'j#¨¸g[®g[{[¼Qd{ g[s}gfwQu}ª§q@¨¦gÂd'wduxa ¬®9gÂ;fwr®g[{}9gfw fgjd{ r u ad g '2+ ¸Ð×i s}+a gfe;gO¹{½49Á± /xÉ (5TÌfÍ Î ËÏ' Ë
(48) #ÇƹÎdËÒ Í #2Ï -ÄtÆÑÈ#Æ 1
(49) È Í+ È -Æ7ÄtÇ}ËÊ+Ȭ0Ì Ê Ë
(50) #Ç ËÆ Ï'Ë T u+{ `bgfadu g'C¼ ªµ9|§s%{ u}|©adu adg;e ¹9ªµ¹ª§#{%¨x|µw 9eÒ |§w»ytu}u}|§adwdg_2 ¶[s³gftu³u u}yd|§wd{ g ± 9' s}gfs}{ |§w¶g'¨¦¦g;d9u s}s}(C; yde;gÒTu})a -ug¹x9{}gÒ∈ u R|µe;g±Tl¼Hyd¹{{%|§e ||§s®g[u}w uk. F. ý ý ð3 ü ú " . 2.
(51) . bÅ ¾ËfÇ%#. #
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(54) È¬Ì È9È . d|§s { gu |§s}g¼¹{Â@9|µ®9gfw ¼-u adg¹y w u |µ9w V (t, x) := ϑˆ (T − t, x) ¨xadg[{}g `ba gO-y w d{ q_fw t|µu}|§Tw>|s 0 V (t, x) = 1, (x) ∈ / K, w _u}adg%|§wd|µu}|ªfw t|µu}|§w|s T. ϑˆT. 9g[qtsjg~QsÒ½Á±. 7| ± g±u}a g%®¬ªµy gl¹9w u |µ9w)|§s w 9wdª§q± ²»g ¿ {+svu'{ gfe'{ °»u}a u'wrq0 s}Cª§ytu}g[ªµq fwQu}|§wrydysÒs}ª§ytu |µ9wRlu}adg2t|µÀHgf{ gfwQu}|ªx|§w ª§y s}|µ9w y˙ ∈ 9{}{ g[s}-w dslªs³_u}) s}ª§ytu |µ9wK y˙ = f (y, u, v) ¹{% |§®g[w (u, v) ∈ L [0, T ], U × F (y) ¼t¨xadgf{ g {0, 1} |µ¹x{ ∈ K\C, w ¹{ uw ∈ U, f (x, u) vf (x, u) f (x, u, v) := u adgfx{ ¨x∈|∂C, s³g9± u ∈ U, v ∈ {0, 1}, 0 Ñw'9{ dgf{\u t|s}f{}gfu}|s³gbu adgltqrw e;|§ f ¼Q¨¸gltg ¿ w gj%{+t|§y s ρ := (∆x + ∆y ) ¨xadgf{ g ∆x w {}gu}adg;e;g[s}a¶s³|§´fgsf¼¯w ¼¨xadgf{ g d |slu}adg .Wy fªµ|t|w ∆y ≥ ρ} d|§s³u w g@½ C |s 's³y s³gfu CCÁ±b`b:=adg[{xw¨¦∈gK,fw s}d((x, |§dgf{b¹y),{ K\C) w v ∈ {0, 1} u}adg%¹9ªµª§#¨x|§wd u ∈ U (x) dd{ ¬«t|µe'#u g[ tqrw e;| f |© x ∈ K\C, f (x, u) |© x ∈ C\C , ½7Á vf (x, u) f (x, u, v) := © | 0 x∈C . hju glu a #u ¨¦g%tg[dy g¹{ en½ #QÁbÒdqQwe;|§Od{ 9{ e;eÒ|§wdd{}|§w f|µdª§g¹{ V ½49Á V (t + ∆t, x) = min V (t, y (t + ∆t)), w y˙ = f (y, u(s), v(s)) w [t, t + ∆t] ± Ñw x u ¨xada g¶gf{ fg s}yg»=¨¸gya ¬®g|§s_u}wdadª§gq s³9wdªµytg»u |µ9|§w ®g[w y(t) 9wQu}{ =ª (u(s), ¹{ s ∈ [t, t + ∆t] ¼lu}a g»s}ª§ytu}|§wL|§s v(s)) ¦ u. u µ | ; e g ± ×ux|§s¦u ary sxd {}¬«t|§e;u}g@rq;u}adgi V (t + ∆t, x) = V (t, y (t + ∆t)) su ad+adgg[eÒg[ªµgª¯Qfgfq)wQ;u}g[®#{}g_ª§yd|§gw dgfwdÁu}%± g['jgfVw fg%¨¸gÂd(u,{ -v)9ts}¼gl9u}tadu =gÂ|µ¹wdt9g+ªµª§#∆t¨x¹{}|µ9w e ;s u +adadgg[®#eÒg ªµyd gs V #u u |µe;g t = t ½¾|µw x ½7#QÁ (u , v) . V = min V ½¾¨xadhjgf{ g u}gO(u|§w_)¾9u¦u}a u¸|u sxad;|§sbs |§+®adg[w gfe;tgÂ|s 9{ g{}u}{ |§g[´[s}u}9|§w wds¥g «du}adgu}ª§s}qgu u Uu (xadg'2)Á+ ± ¸Ð×i s +adg[eÒg dªµ|§g[@u}u}adg ¹9ªµª§#¨x|§wd ¹{ e'ª 2'+ g~Qy #u |µ9w ½79Á v − min (f (x, u, v) · ∇v) = 0, t > 0, x ∈ K. V (0, x) = 1K\C (x),. C. (x) ∈ K,. ∞. C. C. 2. ρ. 2 1/2. ρ. ρ. ρ. ρ. ρ. ρ. (u,v)∈U ×{0,1}. t,x. t,x. C. t,x. n+1. n+1,U B i,j. n i,j. n. i,j. n+1 i,j. uk , v∈{0,1}. k k=1,...,N. t. n+1 i,j. k. i,j. u∈U (x), v∈{0,1}. ρ. >0. ý. >@(.
(55) È Æ É ÄrÌ #ËÌÍ+ÎdËfÏ'Ë
(56) Ç ¾È9Ê Å Æ-Ç9ÊŵËfÏÌ. 9. `ba |§sb|sx¨xarq@¨¦gÂs}a ªµª¯ª§s};{}gf¹gf{bu}Òu ad|§sjs}+adg[e;gÂ9sxw '2+ ¸Ð×i s}+a gfe;g± s -+&Æ ad g[ eÒ g9¼d @u adÍf0Çg Æ×dËf{ 0Ç |§¾w È |§ddª§9g{W|§u}s a ¿ gl{+sv9u e;u @dytgfu ®9u}ª§|§®g%w'|§w2u}%|§e;®rg|§dw|µª§2|µuvq°9e;gf{ dwdydgfu}ªg{¥s}e;fgdu}dydd{ { gx¬«t |µe's}|§#w@u |µy 9s}w |µwd u}VadgO(t,i x) ys³|§wd'u adg '+ ¸ÐÑi Tª§{ |µu}ade)¼ ½¾y s}|µwd;u}|§e;gs³u}g[ ∆t > 0 s #u |§s³¹qr|µwd;u adg;* -$ºwt|©u |µ9w [ÕQÁ± gs V|sld{ dg{ ¬wr«ryd|§e;e'gf#{ u |g[fª§ª§rq)qUf¦wr®tg[u {}9|µw ±l`ba ¹|§{s e;`ba ggfwKw sj¨¦|µgwUdg[{}|u}t|gydu}ª){jsvu u}a 2uju adu}a g;gs}f+adtg[e;u}y ¿ g{}g¨xadg[s}wU|µw»u ad¦g®#tu ªµyd(C) s}s³u}9e;dg -g[T >¨xad0gf± wÑu}wa gd{ ~99yu |§wQfu}g|µuv¼-q ¹{lu}adg {+s³uluv¨¦@u g[s³u sOWu adgP¹ª§ªµ#¨x|§wd s}g[u}|§w¯¼-s u #adu g'|§s ¿ i g[ s (C; s +adgfTe;) g|s |V − V | ||V − V || := ∆x∆y n i,j. F. n. n−1. F. L1. i,j. n i,j. n−1 i,j. ||V n − V n−1 ||L1 ≤ 10−4 .. ) b$¸*¶ xÃC Ñwu adg¹ª§ªµ#¨x|§wd;wryde;gf{ |§[ªHu g[s³u s[¼d¹{bu adg®r|d|§ªµ|µuvq_ªµ9{ |©u ade)¼t¨¸g%a ¬®9gÂy s³g u}adg% 9s³|®g[{ s}|§w 9sbd{ g[s}gfwQu}g_|§wÓ Õ#Ö×± ¢ Ú v£ Þ C Þ ¢d Þ C¡ Þ Ú ²»g;ws³|tgf{Ou adg {}9dªµg[eá¸fe;dytu |µw _u adg;®Q|#Ð |µª§|©uvq_°gf{ wdg[ªH¹ { ½7 99Á x(t) ˙ = x(t) − y(t), ½4 Á y(t) ˙ ∈ [−c, c], ¨xf|µw u}a s}ce;e'=#1/2 ±ºu ad`bg_a |§s'9{}f{ g[{ s}{}-gs³-w tw |§wd su}u |§»e;g ∈ [0, 3] u |µ9w¼¸ {}w 9 dªµu}g[ade gUfÓ Õt¼Cw ¬s³u}Ö7±{+' |§wQg[u w s x(t) g'adg[{}∈g K[0, 2]:= [0,w 2]y(t) . w × [0, 3] dgf-gfw dwQul '2+ d{ dª§gfe |s . Vt + max (−f (x, y, u) · ∇v) = 0,. ∀t > 0, ∀(x, y) ∈ K,. u=±c. V (0, x, y) = 0, V (t, x, y) = 1, x−y f (x, y, u) = u Ωt = {x, V (t, x) = 0}. ∀(x, y) ∈ K, ∀(x, y) ∈ / K, t ≥ 0,. ¨xa gf{ g ±Ò²»g'a ¬®9g{ gfdªfg[Ku}adg +∞ ®#ª§ydg;rq 1 ¹{fe;e;rd|©uvq9¼w s³u}|§ª§ª¯a ¬®g ± ²»ga ¬®9g%dªµu³u g[)|§w |§ ±O%u adg{ g[s}ydª©u+sx|§®g[w)Qq_u}adg®r|§d|µª§|µuvq ª§9{}|µu}a e w )rq_u}adg '+ ¸Ð e;gs³a¶w s³|§ ´fg)½ P x ={ g[s}Pgyu}=|§®g[50ªµq9¼w w »100s³u}9Ád± -dg[{Ou}u adadg@g@fi e;ds yt+u+ad#g[u eÒ|µ9gÒw¶¨¦guÂa u}|§¬eÒ®9gg yi s³gKs +u ad|µe;gfe;g@gs³¼Hu}g[¹9 {Âs ®¬∆t{}|§'ysl0.013 d{}¬«t|§e'#u}ª§q±2²»g_0.007 a¬®g_ª§s}2y s}g[ ½ Á±U`ba g_dª+°»ªµ|§wdgs t = n∆t = 5 c} dgfª§|µhje;|µuxu}gOu}adu a g%#-u¦u}{+adtgÂgf{x®r|du |§adªµ|µg%uvq@g«dª§9u {}s³|µu}9a ªµyteØu |µ9fw¯e;± dytu g[s¦®#ª§ydNg[s ={ 2 ± uÑw_∈9{−c, yd{xª§{ |µu}ade)¼Q¨¸gÂ9eÒ ytu}g ®#ªµydgsx¨xad|+a2{ g 0 { 1 ¼ {js}e;g%|§w9u gf{ e;g[t|{ q@®#ª§ydg±¦`b0adg|§1wQu}gf{ e;g[t|{ q@®#ª§ydgs { g% s}gf{ ®g u 'gª§¨¦¬qts¦w2 +¹{ wQu |µg[{ l¨xad|+a) w d¨x|§ru a2|§sj-ytuj9wdg%{buv¨¦'e;g[s}a2s³|§´fg9±¸`badgg[{}{ {b9w n. u. ý ý ð3 ü ú " .
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(60) È¬Ì È9È . u ad|sO¹{}9w9u |µg[{Â|§sÂwdut¿ |©ÀHy s}g[rq2u}adg's +adgfe;g2½ u u}adg'9wQu}{+{ q u )eÒQsvuÂwryde;gf{ |§[ªe;gu adtdss otgfe;|©,Ð ${+wd9|§w%{ wd|µu}g d|©ÀHgf{ gfw fgbeÒgfu}adtds+Á¼9dytu¥s³u ¬qtsC¨¸g[ªµªtª§tfªµ|s³g|§w'Os}e'ª§ªr w d¨x|§ru a¯± ®r|dÑw|§ª§|©uvCq@|§ °9±Øgf{ wdKgfª\½¾w { fu}adtgUu}y ¹{}ª§gªµ# ¨x9|§s³wd|§ w¼¦Á± u adgs}e'ª§ªjdª+° s}~Qy {}g2{ gf9|µ9w s;{}g[d{ g[s}gfwQu'u}adg»9eÒ ytu}g {lu}adg;i s +adgfe;g9¼u adgÒdª§9+°s ~Qy { g{ gf9|µ9w sj{}gs s³t|#u g[)¨x|µu}aKu}adgÒ9|µwQu+sl¨xa gf{ g 0 ≤ ¨x|µu}a ½ u adgl-|§wQu s¥¹{}9e ¨xa |§+a@¨¦gls}ad9ydª§@gOdª§gxu}{}g+a;u adg u+{ gu¸|µw'u |µe;g V ≤ ª§g[s s}gf{lu}a w9{O=g~Q10y ªCu} t Á u}a gÒ9{}g[q29|µwQu+sl{ gf {}gs³g[w9u+slu}adg;e;g[s}aK-¬«K¨x|µu}a¶wK|§wQu}g[{}e;g[d|§{}q ®#ªµydgO V -guv±W¨¦`bgfg[adw |§sj0½¾d{ { ªg[+s}°dÁ¥9w ;w u}'1 e;½¹¨xgs³ada|µu}g¬Á¬±W«tk2gsb9¨x{}gladgfd{ { gOg[u}|ads}ggfª§tq¼Q|s u}adgwQs³u gO|µwr-yd¬|µ«ruvq@gs¸|sx{}tgOgu { g[gfdu { g[g[Ãs}gf± wQu}g;|© ≤V ≤1− ¢ Ú WÚ ¡ Ú Þ C¡ Þ Ú Ñw»u}ad|s%gf«te;dªµg9¼¯¨¦g@e;dydu}g;u}a g@fdu}yd{ g; s}|µw¹{ -gf{ e;gfª§¥{}9dªµg[e ½4 9Á x(t) ˙ = 1 − ay(t) + u cos(θ), ½7 #Á y(t) ˙ = u sin(θ) |§w u}adgUde'|§w (x, y) ∈ K := [−6, 2] × [−2, 2] ¼¦w º¹{@fwQu}{ ªs 0 ≤ u ≤ u := 0.44 ¼ ± θ ∈ [0, 2π[ `badg%®r|¼td¨x|§|µª§u}|©uva q_aªµ=90.1{ |©u ad±\en`bad½4gÂs³g[u g'{}Ó ¬9Ö gux|µw|§sxu}a +a |§s 9[s}gfws}gadÁ¦gf{ wg%_s¦u adu}ga gOi Lª§s}ª +Ca gfe;:=g%B(0, { gÂfr)e;¨x |µu}a{}g@r =|§w 0.44 |§ ±ït¼ ¨x|µu}a ± d9{lu adg;i s +adgfe;g9¼-¨¦gÒa ¬®gy s³g -|§wQu sÂw ± u adg`bs³u}adPg@ xdf=|µ|µ{+w ;Pª§g'fy{}t|µ=gfu}ª§g[|§{}eÒ100 ||µu ¨¦s%9u}sadg;||V-{+−tgfV{%¥u}||a g;u ≤{ 10gfu[¼¼dw ¨xKadu}|+ada)g'9d¬Nª®9+gl°K'=ªµs³|§wd20u}g[ s%d|µª§w s})u t|µgfe;ª§|µg e;t|µdtu'u}'a 7gÒ± 0.019 gf«du [tu yd{ g 9s³|§wR½¹¨¦ga¬®gfe;dytu g[2u adgÒªµ|§e;|©uOu}{+#vgu}9{}|§g[sjrqys³|§wd@u adgÒC9wQu}{ q99|µw2¥{}|§w |§dª§g¼ s}gfg hj¦{ qru}g_s}u}wa uw 2' 9Q2tÓ§¶Ö¹Ád± { gfª§|§eÒ|§w {}q»dd{ ¬«t|µe'u}|§w»|§sªs³U9tu |µw g[¶rqKu adg_i s +adg[eÒg'g[®g[w ¨xs |µe;u}a@gÂ%wrs}yde'e-ª§gfªd{xwrydef-gfwQ{¸u}{ Hªe;s[¼rg[s}w a' 9|µwQu s[-± d{¥|µw s³u w gj|µw C|µ± ¨¸gxa ¬®9gby s³g P = P = 20 ½¹¨x|µu}a Á± ∆t ' 0.27 ¢ Ú Ñw%u ad|§s g«deÒ ªµg9¼¨¦g¥fe;dytu gWu}a g [tu yd{ gW s}|§w ¹{ u}adg¥¹ª§ªµ#¨x|§wdj%{}u u}|§w ª dqQwe;|§ −10. n ij. n i,j. n. n i,j. . . 2. max. n. n−1. u. −4. L1. x. y. . 9wKu adg@t9e;|µw |7± g±§¼. f (x, y, u) =. K = [−1, 1]2. . y −x. . ±@`badgÒu {}9gu%|sOu}adg' ªµªWfgfwQu}g[{}gK|§w. (0.5, 0). w ¦{ 9t|µys 0.2 ¼. C := {(x, y) ∈ K, (x − 0.5)2 + y 2 ≤ 0.22 }.. hju g¥u a #u\|§wu adg¦dqQwe;|§ f u adgf{ g¦|§sCwdÂtgf-gfwtgfw fq#®g[{ O9w9u {}9ª u ad#¨¦gf®9gf{¼fu adgxi ·s +adgfe;g dQgÑsw y s} g |§ O± _dqQ¨¦wge;f|§e;¸u a #{}u¥g%tu g[adgÒg[®Qw |; |µHª§|©%uvq29w9ªµ9u {}9{ ª|©u vades|§w2w )½4u 9adÁ¦gÒ½¹adi g[ w s}gx+|§a w;gfe;d{ g9u uj|§fu}g¦|§e;¨¦ggxa ¬®9g N ½¾a =ªµ¸2Á ± u yd{ wÁ¼r¨x|µu}a P = P = 100 ½¾w ∆t = 0.02ÁWw P = P = 200 ½¾w ∆t = 0.01T Á=±C`bπa gls}e;ªµª f|µ{+ª§g%tgfª§|§eÒ|µu sxu}adgÂu+{ guxw_u adg%dª+°'ª§|µw g%{}g[d{}g[w s}gfwQu s¸u}a g%9{ tg[{x u}adg%gf«t9u s}ª§ytu}|§w¯± u. x. y. x. y. >0. ý. >@(.
(61) È Æ É ÄrÌ #ËÌÍ+ÎdËfÏ'Ë
(62) Ç ¾È9Ê Å Æ-Ç9ÊŵËfÏÌ. . Viability Algorithm. Viability Algorithm. 3. 3. 2. 2. 1. 1. 0. .. 0. 0 1. 2. .. 0. UB 3. 2. 2. 1. 1. .. 0. 0 1. 2. .. 0. 1. C{}|§9ydd{ ªµg[g'e b¸e; { |s³9w_u adg®r|§d|§ªµ|µuvq ªµ9{ |©u ade w_u adgi L s +adgfe;g¹9{xu}adgfw s}e;e'#u |µ9w P x = P y = 50. ý ý ð3 ü ú " . 2. UB. 3. 0. 1. P x = P y = 100. 2.
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(66) È¬Ì È9È . Viability Algorithm 2. UB. .. 2. 1. 1. 0. 0. −1. −1. −2. .. −2 −6. −5. −4. −3. −2. −1. 0. 1. 2. −6. −5. −4. −3. −2. C|§yd{ g d{}¬«t|§e'#u}|§w Cu}adg[tu yd{ gOs}|µw ¹{bu adg%´fg[{}e;gfª§;d{ dª§gfe)¼. −1. 0. 1. 2. Px = Py = 100. UB 2. 1. 0. −1. −2. .. −6. −5. C|µ9yd{ g . −4. −3. −2. −1. 0. -gf{ e;gfª§Ò {}9dªµg[e)¼ P . 1. x. 2. = Py = 20. >0. ý. >@(.
(67) È Æ É ÄrÌ #ËÌÍ+ÎdËfÏ'Ë
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