orcos.tuwien.ac.atorcos.tuwien.ac.at/fileadmin/t/orcos/Research_Reports/Res_Rep_till... · ! "# $%...

27
1 2 3 4 5 1 2 3 4 5

Transcript of orcos.tuwien.ac.atorcos.tuwien.ac.at/fileadmin/t/orcos/Research_Reports/Res_Rep_till... · ! "# $%...

  • ���������������������������������������������� ��������!���"#������� $�%���!�&���'� �(����)*�,+-(.0/ 1 132546/ 798:44�?@2�AB=>C�4D4�?�/ =>E�;GF

    HJILK@ILMLNOK@MQP�R@SUTVSXWZYBP�I6P�NO[LK\[L]^ILK\_a`bIDcedfP�[HJ[6P�NGg�I6P�SihjSXce] [DclkmILK@n�S

    o +0p@��q�� ����� 1 rts + uv+ o ��&��(w3��!�� 2 r px+�y,���$������!�z���� 3 r px+�{�� ��z������ 4 r "|+ y6+}*���$��� 51 ~�� $> :��� ���� ��Lt9$O:O�X O D ���O G'�� � v� >G�'  G''¡�¢£�£'¤G¥�¥¦��L' ¡�: §*¢GO:·¦C�/Å83Ü�Ù�;>Ü5E�8:4�Ý2�Ù�=l/ 7Ü�; Ð ÛG=>/ÅÛO4�=>C�4�ÞC�E�7�E�;:ßlE�;,ÞÜ�;>4�8=>/ Ñ 4OßtE'Ú Ð 7 Ð . Ð ;?LÛ Ð 72�4 Ð 8:/ Ñ 7�/ à�Û Ð 7�=eÒ�E'á=>/ Ó Ð =E�; Ð 8l.�48e=>C�4 Ð . Ð ;?/Å8 Ñ / Ó�4C Ð =Ò Ð ALE�;eÒ Ð AL7�E'=lC Ð Ó�4 Ð à�Õ�4�?Ò�E�7�4O= Ð ;>ALÛOE�Ò�Ü5E�7�4;>E�1�Ò�E�?�4/ E�7x8:Ù Ñ�Ñ 4�8=8¦C�E$.^=>E@Ò Ð 7 Ð'Ñ 4L=>C�4 Ð . Ð ;?E Ó'4/ Ò�4�×�ØXC�4 Ð 7 Ð 1 A�8:/Å8�/Å8e/ 7�=>44�8=>/ 7 Ñ 254�Û Ð Ù98:4�E'Ú Ð ÞÄÚ Ð 1Å8:4Oß�8=>4 Ð ?�A8= Ð =>4�=>C Ð =l/Å8Ð ? â Ð ÛO4E¿2�Ù�=�E�Ù�=8:/Å?�4D=>C�4 Ð ?�Ò�/Å8>8:/ 2�1 4D;>4 Ñ / E�7 Ð 79?Z.0C�/ÅÛC*E'=>C�4.0/Å8:4DC Ð 80=>C�4ã Ù Ð 1 / = Ð =>/ Ó�4Ü�;>E�Ü�4/ 4�8¦E'Ú Ð 8=>4 Ð ?�A|8= Ð =>4�ä9=>C�44 Ð ;>4@-(/ 7�à97�/ =>4; Ð â4�ÛGá=>E�;>/ 4�8�ÛOE�7�Ó�4E/ = Ð 79?*-(/ 7�à97�/ =>4; Ð â4�ÛG=>E�;>/ 4�8t8= Ð ;:=>/ 7 ÑÐ ;>2�/ =>; Ð ;>/ 1 AÛO1 E�8:40=>E/ =�×å�E�; Ð 1 1�/ 7�/ =>/ Ð 15ÛOE�79?�/ =>/ E�798t=>C�44 Ð ;>4v/ 7�à97�/ =>44�8lÚ(E�;=>C�4DE�Ü�=>/ Ò Ð 1,8:E�1 Ù�=>/ E�7@=>C Ð =¦Û Ð 7�7�E'=v2�4L4OÓ Ð 1 Ù Ð =>4�?#/ 7Z=>C�4L8= Ð 79? Ð ;?#. Ð A�×�æÁ4L;>4Oá8:E�1 Ó�4�=>C Ð =lÜ�;>E�2�1 4E$Ó/ 7 Ñ6Ð 7�4E�Ü�E�8:/ =>/ E�7ÛOE�79ÛO47�/ 7 Ñ =>C�4XÓ Ð 1 Ù�4�E'Ú�=>C�4

    è�é:ê�ëìí5îï:ðÅê�ñ�ï:éò>óé:ôñªôvñ�ì ©�õ¦¬ ® · ��>

  • ¸�¬�õXG '�º�¬ ¶5¬�¸eC�4¦ö Ð Ò�/ 1 =>E�7�/ Ð 7/Å8t7�E�7áÄ÷E�Ù Ñ Cj/ 79ÛO4/ Ó�4�8C�4 Ð 7 Ð 1 A�8:/Å8BÚ(E�ÛOÙ98:4�8@E�7aÒ�E�7�4O= Ð ;>AÛOE�Ò�Ü�4/ E�73Ý Ð 8X/ 7@=>C�4D4OþBÛO/ 44LE'=>C�41Å?�× ß

    ö¦4O;>4�.�4�4OÕ�Ü�1 E�;>4t=>C�4 ã Ù�4�8=>/ E�7E'Ú�C�E .�2�4�8=�=>E�Ò Ð 7 Ð'Ñ 4�Ò�E'=>/ Ó Ð =>/ E�7 Ð 1 Ð . Ð ;?�8.0C�E�8:4LÓ Ð 1 Ù�4L/Å8 Ð =¦1 4 Ð 8=¦/ 7|Ü Ð ;:=¦;>4/ E�7 Ð 1Ý97�E'=¦E�7�1 A@Ò�E�7�4O= Ð ;>A�×9å�E�;¦4OÕ Ð ÒáÜ�1 4�Ý,C�E .bE'Úª=>4C�4C Ð =�=>C�4�8:4 ã Ù�4�8=>/ E�798C Ð Ó'4@4�8>Û Ð Ü�4�? Ð 7 Ð 1 A�8:/Å82A4�ÛOE�7�E�Ò�/Å8=84E�Ò 8:Ò Ð 1 1tÛOE�Ò�Ü�Ù�= Ð =>/ E�7 Ð 1lÛOE�8=8FGÝ2�Ù�= Ð =¦=>C�48 Ð Ò�4D=>/ Ò�4�Ý Ð 80=>C�47�Ù�Ò2�4C�4DE�Ü�=>/ Ò Ð 137�Ù�Ò2�4/ 4O=ABE'Ú Ð . Ð ;?�8E@8:Ù�298=>/ =>Ù�=>47�E�7�Ò�E�7�4G= Ð ;>ABÚ(E�;¦Ò�E�7�4O= Ð ;>A@;>48:/ 2�/ 1 / =A=>C Ð = Ñ / Ó/ 7 Ñ6Ð . Ð ;?�8,=>EE6E'Úª=>44�?�Ù9ÛO4�=>C�44/ E�7Ú(E�;e4OÕ�ÛO1 Ù98:/ Ó�/ =A Ð 79?�ÝC�4C�4Ó Ð 1 Ù�4D=>EB=>C�4;>4�ÛO/ Ü�/ 4

  • �/ Ñ / E�Ù98Ð . Ð ;?�8eÛ Ð 7Ò�E'=>/ Ó Ð =>4X4OÖ5E�;:= Ð 1 1�E�Ù�=tE'Ú�Ü�;>E�Ü�E�;:=>/ E�7L=>E�=>C�4EÁÛOE�Ò�Ü54O=>4BÚ(E�;=>C�4-� Ð 1Å?�;>/ Ñ 4 � . Ð ;?5×lØXC Ð =/Å87�E'==>EÁ8 Ð AÁ=>C Ð =.0/ 7�7�/ 7 Ñ C�E�7�E�; Ð ;>A Ð . Ð ;?�8t?�E4�8t7�E'=�C Ð Ó'4v;>4 Ð 1�4�ÛOE�7�E�Ò�/ÅÛ0Ó Ð 1 Ù�4�×�öv4/ÅÛ>Ô�8 Ð 79?' / 7 Ñ C Ð 1Ý ÿ �����ÄÝ5à979?x=>C Ð =6=>C�4B8=>E�ÛÔ|Ò Ð ;>Ô'4O=D;>4 Ð ÛG=8�Ü�E�8:/ =>/ Ó'4E ã Ù Ð 1 / =A Ð . Ð ;?Ð 7�7�E�Ù�79ÛO4E�Ò / 79?�4C Ð =BÓ Ð 1 Ù�4Z/Å8Ò�4�?�/ Ð =>4�?#=>C�;>E�Ù Ñ CB=C�4 Ð . Ð ;?32 8�;>4/ E�73×4 Ù�;6Ò�E�?�44�8:Ù�1 =8 Ð ;>4#7�E'=�?�;>/ Ó�44

  • µ ¸�¬�õXG '�º�¬ ¶5¬�¸e8Ú(;>4 ã Ù�41 A�Ýl8:C�E�Ù�1Å?É=C Ð =¿/ 7Ó�4�8=>Ò�4C�4�8:4 ã Ù�4�8=>/ E�798?�/ Ö544C�46Ò�E�7�4O= Ð ;>A¿ÛOE�Ò�Ü�E�7�4/ E�798,.0C�E�8:4 Ð 798:.�4O;8 Ð ;>4t7�E'=�E�2Ó/ E�Ù98Ð Ü�;>/ E�;>/×

    = {L���|���3����' Ù�Ü�Ü�E�8:4#=>C Ð =�.�E�;>Ô'4EÁ4OÕ�4; Ð 4OÖ5E�;:=E�;7�E'=*-(2�/ 7 Ð ;>AÉÛC�E�/ÅÛO4$FG×Î|Ð 7 Ð'Ñ 4E¦Ò Ð Õ�/ Ò�/ ÷C�4�?�/Å8>ÛOE�Ù�7�=>4�?L8:Ù�Ò E'Ú�=>C�4�7Ù�ÒL254C Ð =#/ =8#ÛOE�7�=>;>E�10Ó Ð ;>/ Ð 2�1 4 Ñ E Ó'4798=>C�4|Ü�;>E�Ü5E�;:=>/ E�7 E'Ú6Ü54; Ð4OÖ5E�;:=.0C�E Ð ;>4B;>4Ô�44#444�?É/ 7�=>E Ð1 E'=:=>4A -�ÛGÚ× Ý�� Ð ÷��(FGÝ Ð 79? Ð ; Ð 79?�E�Ò 8 Ð Ò�Ü�1 4 Ð ;>4 Ñ / Ó�4C�4 Ð . Ð ;? Ð =Ð 7A Ñ / Ó�4/ Ò�4�× F � 7�E'=>C�4O;L. Ð A|=>E@=>C�/ 7�Ô Ð 2�E�Ù�=6=>C�/Å8D/Å8 Ð 8�=>C�4¿; Ð =>4 Ð =L.0C�/ÅÛC?�4�8:4Ó/ 7 Ñ Ü544�;>4E�2�1 4Ô�4Ô�44�ÒA�E�Ü�/ÅÛ'×�ØXC�44�Ò Ð 7AD.�E�;>Ô�4C�4C�4C Ð =t=>C�4¦?�/Å8:Ù�=>/ 1 / =A�E'Ú�4OÕ�4/ 7 Ñ=>C�44OÕ=>; Ð 4OÖ5E�;:=¦/Å8¦C�4O=>4E Ñ 480/Å7�?�/ Ó�/Å?�Ù Ð 1ª8v.0/ =>CÁÛOÙ�ÒÙ�1 Ð =>/ Ó'4?�/Å8=>;>/ á2�Ù�=>/ E�7

    Φ(w)×�æ%/ =>C

    N?�4C Ð 8:/ Ò�Ü�1 4và97 Ð 79ÛO/ Ð 1

    ÛOE�Ò�Ü�E�7�4CvF Ð 79? Ð Ü98:A�ÛC�/ÅÛZÛOE�Ò�Ü�E�7�4/ 7 Ñ

    =>C�4B7Ù�Ò2�44�?ÁÜ Ð A�áE'Öç/ ÚX=>C�4B.�E�;>Ô�4O;D4OÕ�4; Ð 4OÖ5E�;:=L/ª8DC�4C�4B7�Ù�Ò2�4Ô�4464/ Ò�4 Ð ; Ñ Ù�Ò�44#=>C�4*8= Ð =>4 Ð 79?

    ÛOE�7�=>;>E�1XÓ Ð ;>/ Ð 2�1 4�84�8:Ü�4�ÛG=>/ Ó�4/ Ò Ð 1¦ÛGE�7�=>;>E�10Ò�E�?�4ÒÙ�1 Ð =>/ E�73×�vE'=>4�Ý�C�E .�4C�4L=>/ Ò�4 Ð ; Ñ Ù�Ò�4O7�=¦.0/ 1 1�Ò�E�8=>1 AZ2�4E�Ò�/ =:=>4�?*=>EB44 Ð ? Ð 2�/ 1 / =A�×

    ØXC�46?�A7 Ð Ò�/ÅÛC�4 Ð . Ð ;?32 8L;>4/ E�7�?�4�Û Ð A�84Òi/ 7a-@�$FL/Å8L=>C�4#; Ð =>4 Ð =.0C�/ÅÛCA Æ����� Á��>:

  • �C�4

    ;>4/ 7 Ñ Ú(;>E�Ò 4GÕ�=>; Ð .�E�;>Ô@Ò�/ 7Ù98¦=>C�4�ÛOE�8=8¦E'Úl=C�4 Ð . Ð ;?�8 Ñ / Ó�4C�4/ÅÛ Ð 15=>4Ò¿8C�4B?�/Å8>ÛOE�Ù�7�=L; Ð =>4�Ý

    ρ/Å8�=>C�4¿/ 79ÛO;>4E@=>C�4

    4C�4

    Ð . Ð ;?�Ý,=>C�44OÚ(E�;>4c ≥ v ÒÙ98=�C�E�1Å?�×1$�1 4 Ð ;>1 A�Ý,=>C�4ZÛOE�7�=;>E�1�C Ð 8L=>Eû8 Ð =>/Å8Ú(AÁ=>C�4ÛOE�798=>; Ð / 7�=

    u(t) ≥ 0 Ú(E�;04 Ð ÛC t ≥ 0.

    B Ì !����DC3�����æ%/ =>C

    V (R) = v + 1/R,-6��F

    E�Ù�;tE�Ü�=>/ Ò Ð 1�ÛOE�7�=>;>E�1�Ü�;>E�2�1 44-RF Ð 79?�E�7�4¦ÛOE�7�=>;>E�1,-

    uFl/Å8 Ñ / Ó'4/ÅÛ

    Ú(Ù�79ÛG=>/ E�73×��7j=>C�/Å8@Û Ð 8:4�Ý�C�E .�4C�44�?.0C�/ 1 4B=>C�4 ã Ù Ð 1 / = Ð =/ Ó'4B2�4C�4Ò�E�?�4C Ð 1 / 7�4 Ð ;tÚ(Ù�79ÛG=>/ E�7

    F×�öv44OÚ(4E Ð 7 Ð 1 A÷C�4¦Ò�E�?�4C

    F (w) = dw,- � F

    2�4�Û Ð Ù98:40.�4v?�ED7�E'=�1 E�8:4X/ 7�Ú(E�;>Ò Ð =>/ E�7 Ð 79?=>C�4vÜ�;>4�8:4/ E�7.0/ 798l=>; Ð 798:Ü Ð ;>446.�4 Ñ E=>C�;>E�Ù Ñ C#=>C�4 Ð 7 Ð 1 A�8:/Å8�E'Ú�E�Ù�;XÒ�E�?�4/ 4FE9A

    8:Ù�Ò�Ò Ð ;>/ ÷C�/Å8L8:4�ÛG=>/ E�73× 4 Ù�;DÜ�;>E�2�1 4C�4Z8:4O=

    Ω = {(R, u) ∈ � 2|R > 0, u ≥ 0}. ü¦8:/ 7 Ñ /eE�7�=>;>A Ð'Ñ / 72 8 Î|Ð Õ�/ ÒLÙ�Ò/t;>/ 79ÛO/ Ü�1 4-�8:4C*E�7�4�ÞÄ=>;>Ù�4OßD8=>4 Ð ?�A@8= Ð =>4

    (R̂, û)Ð 79?ZE�7�4�ÞÄÚ Ð 1Å8:4Oß

    8=>4 Ð ?�A8= Ð =>4(0, 0).

    æÁ4vÙ98:4X=>C�4v4OÕ�Ü�;>4�8>8:/ E�7ÞÄÚ Ð 1Å8:4OßX8=>4 Ð ?�A8= Ð =>4�Ý�2�4�Û Ð Ù98:4(0, 0)/Å8¦7�E'= Ð Ü�E�/ 7�=�E'Ú

    ΩÝ�2�Ù�=�/ =�C Ð 8v=>C�4 ã Ù Ð 1 / = Ð =>/ Ó'4LÜ�;>E�Ü54/ 4�80E'Ú Ð 8=>4 Ð ?�AZ8= Ð =>4��=>C�44 Ð ;>4*-(/ 7�à97�/ =>4; Ð â4�ÛG=>E�;>/ 4�86ÛOE�7Ó�4E

    (0, 0)Ý Ð 79?x=>C�44 Ð ;>4

  • ² ¸�¬�õXG '�º�¬ ¶5¬�¸e4; Ð â4�ÛG=>E�;>/ 4�8¦8= Ð ;:=>/ 7 Ñ@Ð ;>2�/ =>; Ð ;>/ 1 AZÛO1 E�8:4L=>E(0, 0)

    ×3ØXC�48=>4 Ð ?�AZ8= Ð =>4

    (R̂, û)/Å8 Ð 8 Ð ?�?�1 4LÜ�E�/ 7�=vÚ(E�;¦4 Ð ÛC|8:4O=¦E'Ú Ð ?�Ò�/Å8>8:/ 2�1 4DÜ Ð ; Ð Ò�4O=>44�8¦Ú(E�;6=>C�4�E�Ü�=>/ Ò Ð 1l8:E�1ÅÙ=/ E�7ÁÛOE�798:/Å8=8�E'Ú�=>C�;>48:4�8E'Ú,=; Ð â4�ÛG=>E�;>/ 4�8

    ��×0ØXC�46=>; Ð â4�ÛG=>E�;>/ 4�8�E�7#=>C�4L8= Ð 2�1 46Ò Ð 7�/ Ú(E�1Å?@E'Ú,=>C�4L8 Ð ?�?�1 46Ü5E�/ 7�=(R̂, û)

    ×��×0ØXC�4v8:4O=tE'Ú�=>; Ð â4�ÛG=>E�;>/ 4�8 Ñ E�/ 7 Ñ =>E

    (0, 0)Ý�8:/ =>Ù Ð =>4�?2�4C�4v8= Ð 2�1 40Ò Ð 7�/ Ú(E�1Å?

    E'Ú(R̂, û)

    Ð 79? Ð 2�E Ó'4�=>C�4DC�E�;>/ ÷E�;>A¿E�7#=>C�4

    Rá Ð Õ�/Å8XÛOE�7Ó�4E�=>C�4DE�;>/ Ñ / 73Ý�Ú(E�;0.0C�/ÅÛC

    u = 0.

    Î E�;>4Ü�;>4�ÛO/Å8:4;>A Ð'Ñ / 72 8 Î|Ð Õ�/ ÒÙ�ÒH/t;>/ 79ÛO/ Ü�1 4�×

    a(R0)/Å8t=>C�4¦Ó Ð 1 Ù�4vE'Ú�=>C�4�ÛOE�7�=>;>E�1

    uE�7�=>C�4

    8= Ð 2�1 4XÒ Ð 7�/ Ú(E�1Å?�ÛOE�;>;>4�8:Ü�E�79?�/ 7 Ñ =>ER0.

    å�E�; Ð 7AL/ 7�/ =>/ Ð 1�Ó Ð 1 Ù�4XE'Ú9=>C�40;>4/ E�73ÝR0

    Ý5.�4=>C�44OÚ(E�;>4C Ð Ó�4/ 7�à97�/ =>44�8vÚ(E�;¦=C�4�E�Ü�=>/ Ò Ð 1e8:E�1 Ù�=>/ E�73×� 8:Ü�4�ÛO/ Ð 13Û Ð 79?�/Å? Ð =>46/Å8

    u = 0Ý�/× 4�×�=>E¿?�E�7�E'=>C�/ 7 Ñ ×

    �XA Ð 7 Ð 1 A�=>/ÅÛ Ð 1 Ð ; Ñ Ù�Ò�4C Ð =%=>C�4 =>; Ð â4�ÛG=>E�;>A 8= Ð ;:=>/ 7 Ñ Ð =(R0, a(R0))

    -(.0C�/ÅÛC/Å8 Ð Ü�E�/ 7�=lE�7=>C�408= Ð 2�1 4XÒ Ð 7�/ Ú(E�1Å?LE'Ú(R̂, û)

    F,/Å8�=>C�4XE�Ü�=>/ Ò Ð 18:E�1 Ù�=>/ E�73×E�ÛO42�/ =>; Ð ;>A

    u0 ∈ [0, a (R0)]Ý�.�4�ÛOE�Ò�Ü�Ù�=>4

    =>C�4Ù�=>/ 1 / =AZÚ(Ù�79ÛG=>/ E�7 Ð 1�Ú(E�;¦=>C�4=>; Ð â4�ÛG=>E�;>A*8= Ð ;>/ 7 Ñ#Ð =(R0, u0)

    Ð 79?|ÛOE�Ò�Ü Ð ;>4L/ =.0/ =>C@=>C�46Ù�=>/ 1 / =ABÚ(Ù�79Û=/ E�7 Ð 1�E'Ú,=>C�4�=>; Ð â4�ÛG=>E�;>A¿E�7#=>C�4D8= Ð 2�1 46Ò Ð 7�/ Ú(E�1Å?@8= Ð ;:=>/ 7 ÑÐ =

    (R0, a(R0)).Ø,EÛOE�Ò�Ü�Ù�=>4v=>C�4¦Ù�=>/ 1 / =A�Ú(Ù�79ÛG=>/ E�7 Ð 19Ú(E�;�=>C�4v=>; Ð â4�ÛG=>E�;>A�8= Ð ;:=>/ 7 Ñ

    E�7|=>C�4¿8= Ð 2�1 4Ò Ð 7�/ Ú(E�1Å?x.�4�Ù98:4 Ð .�4E�Ü5E�8:/ =>/ E�73Ý�.0C�/ÅÛ>CÁ.�4�Û Ð 7�7�E'=Ð Ü�Ü�1 AÉÚ(E�;=>C�4Z=>; Ð â4�ÛG=>E�;>A�8= Ð ;:=>/ 7 ÑûÐ =

    (R0, u0)Ð 79? Ñ E�/ 7 Ñ =>E

    (0, 0)Ýl2�4�Û Ð Ù98:4

    =>C�4�1 / Ò�/ =X=>E/ 7�à97�/ =A¿E'Ú3=>C�46ö Ð Ò�/ 1 =>E�7�/ Ð 7 Ð 1 E�7 Ñ =>C�/Å8�=>; Ð â4�ÛG=>E�;>A/Å8�7�E'=X4 ã Ù Ð 19=>E÷C�4#ö Ð Ò�/ 1 á=>E�7�/ Ð 7 Ð 1 E�7 Ñ =>C�4=>; Ð â4�ÛG=>E�;>A@/Å8và97�/ =>4�×�ØXC�4ÛOE�Ò�Ü�Ù�= Ð =>/ E�7|E'Út=>C�/Å8�1 / Ò�/ =�Ú(E�;¦=>C�4=>; Ð â4�ÛG=>E�;>/ 4�8�Ú(;>E�Ò =>C�4¦8:4�ÛOE�79?�ÛO1 Ð 8>8lE'Ú3Û Ð 79?�/Å? Ð =>4�8t/Å8 Ð 7 Ð 1 A=>/ÅÛ Ð 16�$Ú(;>E�Ò =>C�4 4 ( ú8:A�8=>4C�4C�4D8 Ð Ò�461 /ÅÒ�/ =0=>E�/ 7�à97�/ =AÁ-�8:44E�7�/ Ð 7H

    Ú(E�;E�Ù�;;>4/ E�7ÉÒ�E�?�4C=>C�4LÚ(Ù�79ÛG=>/ E�798

    V (R)Ð 79?

    F (w)Ð 8¦8:Ü�4�ÛO/ à94�?*/ 7�-6��F Ð 79?É- � FGÝ�;>4�8:Ü�4�ÛG=>/ Ó�44λÝ�C Ð 8�=>C�46Ú(E�;>Ò

    H(R, u, λ) = (ρ− cu+ λu)duV (R)− δλR.

  • �C�/Å8¦;>4/ E�73Ý�Ú(;>E�Òi-�>�× 2�FÐ 79?Á-�>�× Û�F�.�4D?�4/ Ó�4 Ð 7 4 ( ú 8:A�8=>4CZ/Å8 Ñ /ÅÓ'4/Å8à94�?�Ý�8:/ 79ÛO4H

    /Å8X/Å7 Ñ 4C�4*4 ã Ù�/ 1 / 2�;>/ Ð E'Ú�=>C�4 4 ( ú 8:A�8=>4

  • à ¸�¬�õXG '�º�¬ ¶5¬�¸eC�46Û Ð 7�E�7�á/ÅÛ Ð 158:A�8=>44%�¦.�4 Ñ / Ó�4 Ð Ü�C Ð 8:4�?�/ Ð'Ñ ; Ð ÒbE�7¿=>C�4¦Ú(4 Ð 8:/ 2�1 4

    ;>4 Ñ / E�7Ω.ØXC�4*/Å8:E�ÛO1 / 7�4�8�E'Ú¦=>C�4|8:A�8=>4C�468:/ Ñ 798�E'ÚṘÐ 79?

    u̇/ 7�=>C�4XÚ(E�Ù�;t/Å8:E�8:4�ÛG=>E�;8e?�4O=>4Ò�/ 7�4�?�2AL=>C�4v/Å8:E�ÛO1 / 7�4�8

    IRÐ 79?

    Iu×�ØXC�/Å8lÚ Ð ÛG=

    Ð 79?-/t;>E�Ü�E�8:/ =>/ E�7N��� Ð 1 1 E . Ù98v=>E@ÛC Ð ; Ð ÛG=>4/ ÷C�4�8= Ð 2�1 4Ò Ð 7�/ Ú(E�1Å?*/ 7 Ð 8:Ò Ð 1 17�4C�EE�?BE'Ú�=>C�4L8=>4 Ð ?�A#8= Ð =>4

    (R̂, û)Ð 8�Ú(E�1 1 E .v8C�4ÁE'=>C�4E�ÒU/ 7�à97�/ =A

    Ð 79? Ñ E�/ 7 Ñ =>E(R̂, û)

    Ý�;>44/ E�7D/Å83Ó'4A Ñ EE�?�äG=>C�4 Ð . Ð ;?�/Å83E'Ú/ Ò Ð 1�ÛOE�7�=>;>E�1/Å8,Ó�4AD8:Ò Ð 1 1×$ØXCÙ98/ Ð 1 1 A6/Å8�Ó'4A6Ü�;>4�8=>/ Ñ / E�Ù98E¿?�4C�4D;>4/ E�7#=>EE ã Ù�/ÅÛÔ1 A�×

  • �/ Ò Ð 1�8:E�1 Ù�=>/ E�73Ý�/× 4�×9=>C�48= Ð 2�1Å4DÒ Ð 7�/ Ú(E�1Å?*E'Úe=>C�48 Ð ?�?�1 4Ü�E�/ 7�=L4 ã Ù�/ 1 / 2�;>/ Ù�Ò (R̂, û) ×,övE .�4C�42�4E6=>C�;>4E�;8E�;¦2�4O=.�4C�4L/ 798= Ð 2�1 4LÒ Ð 7�/ Ú(E�1Å?*E'Ú

    (R̂, û)/Å8 Ð Ü Ð ; Ð 2�E�1 /ÅÛ68:4�ÛG=>E�;L-ªÚ(E�;¦8:E'áÛ Ð 1 1 4�?#7�E�7�CAÜ�42�E�1 /ÅÛDÛO;>/ =>/ÅÛ Ð 1�Ü5E�/ 7�=8v/ 7:� 2 Ý98:4E�7�E Ó¿4O= Ð 1× Ý5ÿ ���ÝE�;�/e4Ô'E9Ý5ÿ ����(FG×C�4uá Ð Õ�/Å8�=>C�446/Å8 Ð 7�E'=>C�4E�;�×�æÁ4DÛ Ð 7

    8 Ð A=>C Ð =�=>C�4wE9E .jC�44¦C Ð 8�=>C�468:C Ð Ü�4�E'Ú Ð ã Ù Ð ;:=>4C�4C�4X=; Ð â4�ÛG=>E�;>A8= Ð ;:=>/ 7 ÑûÐ =

    (R0, us(R0))

    Ñ E�/ 7 Ñ =>EÁ=>C�4|8=>4 Ð ?�Aç8= Ð =>4(R̂, û)

    Ð 1 E�7 Ñ / =8¿8= Ð 2�1 4Ò Ð 7�/ Ú(E�1Å?#/ª8 Ð 7�E'=>C�48:/ 2�1 4D8:E�1 Ù�=>/ E�73×

    Ø�ED?�4O=>4Ò�/ 7�40=>C�40E�Ü�=>/ Ò Ð 198:E�1 Ù�=>/ E�73Ý'.�4vÛOE�Ò�Ü�Ù�=>4X=>C�4vÙ�=>/ 1 / =AÚ(Ù�79ÛG=>/ E�7 Ð 1�Ú(E�;4 Ð ÛC�Û Ð 79?�/Å? Ð =>4 Ð 79?Á=>C�44�=>C�4#;>4�8:Ù�1 =8C�/Å8 Ð / ÒZÝ,.�4B;>4�Û Ð 1 1t=>C�4?�4Oà97�/ =>/ E�7ZE'Ú,=>C�4DÒ Ð Õ�/ Ò�/ ÷E�7�/ Ð 73Ý

    H0(R, u)

    H0(R, u) = maxλ

    H(R, u, λ),

    Ð 79?#.�4D/ 7�=>;>E�?�Ù9ÛO4�=>C�4D7�E'= Ð =>/ E�7

    U(R, u) = ρF (uV (R))− cuF (uV (R)) .�7#E�;?�4E Ñ / Ó�4 Ð Ú(E�;>ÒÙ�1 Ð Ú(E�;�=>C�4¦Ù�=>/Å1 / =A�Ú(Ù�79ÛG=>/ E�7 Ð 19Ú(E�; Ð =>; Ð â4E�;>A.0/ =>C

    Ð 7@/ 7�à97�/ =>4D=>/ Ò�4DC�E�;>/ ÷

  • ¤G£ ¸�¬�õXG '�º�¬ ¶5¬�¸eEx=>C�4B=>; Ð â4�ÛG=>E�;>/ 4�8D=>C Ð =¿8 Ð =>/Å8Ú(AÁ=>C�4@7�4�ÛO4�8>8 Ð ;>A

    ÛOE�79?�/ =>/ E�7986E'Ú[/�E�7�=>;>A Ð'Ñ / 72 8 Î|Ð Õ�/ ÒLÙ�Ò/t;>/ 79ÛO/ Ü�1 4�.�4�C�4;>A Ð'Ñ / 72 8 Î|Ð Õ�/ ÒÙ�Ò/t;>/ 79ÛO/ Ü�1 4 Ð 7�? Ð Ü�Ü�;>E Ð ÛC

    (0, 0)Ð 8

    t→ +∞.ØXC�4LÛOÙ�;>;>4E�7�/ Ð 7Z/Å8

    H(R, u, λ) = (ρ− cu+ λu)du(v +

    1

    R

    )− δλR.

    ØXC�4|Ò Ð Õ�/ Ò�/ ÷/ E�7ýÚ(E�;#/ 7�=>4/ E�;#8:E�1 Ù�=>/ E�798C�44OÚ(E�;>4λ − c ≤ 0 / 7 Ω. �vE'=>4=>C Ð =L8:/ 79ÛO4 λ − c ≤ 0 Ý5=>C�4ö Ð Ò�/ 1 =>E�7�/ Ð 7

    H/Å80ÛOE�79Û Ð Ó'46/ 7

    uÝ�2�4�Û Ð Ù98:4

    ∂2H

    ∂u2= 2(λ− c)d

    (v +

    1

    R

    ).

    ü¦8:/ 7 Ñ =>C�/Å80/ 7�Ú(E�;>Ò Ð =>/ E�7@.�4 Ñ 4O=X=C�4DÒ Ð Õ�/ Ò�/Å÷O4�?Zö Ð Ò�/ 1 =>E�7�/ Ð 7

    H0(R, λ) = − ρ2 d

    4(λ− c)

    (v +

    1

    R

    )− δλR.

    � 7%4 ã Ù�/ Ó Ð 1 4Ò E'Ú H0 Û Ð 7ý2�4x?�4/ Ó�4�? Ð 7 Ð 1 E Ñ E�Ù�8:1 A�2A 8:Ù�298=>/ =>Ù�=>/ 7 Ñ λ/ 798=>4 Ð ?#E'ÚuÚ(;>E�Ò =>C�4Lö Ð Ò�/ 1 =>E�7�/ Ð 7@Ò Ð Õ�/ÅÒ�/ ÷/ E�73Ý�A/ 4C�4 � Ü�Ü544@=>C�4Z1 / Ò�/ =|-@����FGÝ.0C�/ÅÛCj.�4x7�4C�4xÙ�=>/ 1 / =A Ú(Ù�79ÛG=>/ E�7 Ð 1¦Ú(E�;#=>C�4|=>; Ð â4�ÛG=>E�;>/ 4�8Ñ E�/ 7 Ñ =>E

    (0, 0)Ú(;>E�Ò/t;>E�Ü�E�8:/ =>/ E�7:� � ×

    u6� �9�������(������! B� d�y(R, u)

    l�e�SZ�e6UL\�Se(R0, u0) ∈

    Ωl#Se6U]l�1P�leDOC�4@8:Ü�4�ÛO/ Ð 1�Û Ð 8:4

    c = 0/ =/Å8Ü�E�8>8:/ 2�1 4¿=>E|E�2�= Ð / 7�4OÕ�Ü�1 /ÅÛO/ =>1 A

    =>C�44GÕ�Ü�;>4�8>8:/ E�798XÚ(E�;v=>C�48:E�1 Ù�=>/ E�7(R, u)

    8= Ð ;:=/ 7 ÑBÐ =(R0, u0)

    Ð 79?@=>E#ÛOE�Ò�Ü�Ù�=>4?�/ ;>4�ÛG=>1 A¿=>C�4D1 / Ò�/ =�×

  • �C�=>C Ð =t=>C�44v/Å8 Ð =>; Ð â4�ÛG=>E�;>A

    (R, u)8= Ð ;:=>/ 7 Ñ

    Ð =(R0, u0)

    Ð 79? Ñ E�/ 7 Ñ =>E(0, 0).

    ØXC�4EBÛOE�Ò�Ü Ð ;>4¦=>C�4LÓ Ð 1 Ù�4�8XE'Ú�=>C�4LÙ�=>/ 1 / =ABÚ(Ù�79ÛG=>/ E�7 Ð 1Å8XÚ(E�;0=>C�4�8:46=.�E�=>; Ð áâ4�ÛG=>E�;>/ 4�8C�4¦Ù�=>/ 1 / =AÚ(Ù�79ÛGá

    =>/ E�7 Ð 1�/Å8�-(2�A Ð .�4 Ð Ô@Ú(E�;>Ò E'Ú/t;>E�Ü5E�8:/ =>/ E�7.� � Ý5ÛGÚ×�å�4C�=>/ 7 Ñ 41Ý�ÿ >���F4 ã Ù Ð 15=>E

    1

    rH0(R∗(0), u∗(0)) =

    1

    r

    2

    (dus(R0)

    R0+

    δR0us(R0)

    + dvus(R0)

    )− cδR0

    ].

    -@����Få9E�;¦=>C�4=>; Ð â4�ÛG=>E�;>A

    (R, u)8= Ð ;:=>/ 7 Ñ#Ð =

    (R0, u0)Ý Ð ÛE/t;>E�Ü�E�8:/ =>/ E�7

    � � Ð 79?��>=>C�4DÙ�=>/ 1 / =A#Ú(Ù�79ÛG=>/ E�7Z/Å8X4 ã Ù Ð 15=>E

    1

    rH0(R0, u0) −

    1

    rlim

    t→+∞e−rtH0(R(t), u(t)) =

    =1

    r

    2

    (du0R0

    + dvu0

    )− cδR0

    ].

    -@����F

    $�E�Ò�Ü�Ù�=>/ 7 Ñ =>C�4Á?�/ Ö544/ E�7 Ð 1¦Ó Ð 1 Ù�4�8x-@����F Ð 79?-@����FGÝ�.�4 Ñ 4O=

    ρ

    2r

    [(us(R0)− u0) d

    (v +

    1

    R0

    )+

    δR0us(R0)

    ].

    å�;>E�Òb-@���$FOÝus(R0)−u0 > 0

    Ý�8:EX=>C�/Å8,?�/ Ö544/ Ó�4�×4/ E�7

    R0=>C�4¦8:E�1 Ù�=>/ E�7¿.0/ =>C�=C�4¦/ 7�/ =>/ Ð 1�ÛOE�7�=>;>E�1

    us(R0)/Å8t2�4O=:=>4C Ð 7¿=>C�4

    8:E�1 Ù�=>/ E�7%.0/ =>C%=>C�4x/ 7�/ =>/ Ð 16ÛOE�7�=>;>E�1u0

    ×[�7jE'=>C�44/ E�;l=>; Ð â4�ÛG=>E�;>/ 4�8tÛOE�7�Ó�4E

    (0, 0)Ð 8lE�Ü�=>/ Ò Ð 1�8:E�1 Ù�=>/ E�798C�4D8=>4 Ð ?�A¿8= Ð =>4�.0/ =>C#=>C�462�E�;?�4; Ð â4�ÛG=>E�;>A¿E�7#=>C�4

    Rá Ð Õ�/Å8/ E�7ûÜ�;>E�2�1 4C Ð =L=>C�4BÙ�=>/ 1 / =AÚ(Ù�79ÛG=>/ E�7 Ð 15Ú(E�;�=>C�4�=>; Ð â4�ÛG=>E�;>A

    (R∗, u∗)8= Ð ;:=>/ 7 Ñ�Ð =

    (R0, us(R0))

    E�7B=>C�4D8= Ð 2�1 4Ò Ð 7�/ Ú(E�1Å?#E'Ú

    (R̂, û)/Å8XÜ�E�8:/ =>/ Ó�4�Ý.0C�/ 1 4�=>C�4DÙ�=>/ 1 / =A¿Ú(Ù�79ÛG=>/ E�7 Ð 1�ÚªE�;X=>C�4�=>; Ð â4�ÛG=>E�;>A

    8= Ð ;:=>/ 7 Ñ¿Ð =(R0, 0)

    /Å8X÷

  • ¤O ¸�¬�õXG '�º�¬ ¶5¬�¸eC�4Ù�=>/ 1 / =A@Ú(Ù�79ÛG=>/ E�7*Ú(E�;0=>C�4L=>; Ð â4�ÛG=>E�;>A#E�7Z=>C�4Rá Ð Õ�/Å80/Å8v/ 7*Ú Ð ÛG=¦÷/ 7 Ñ¿Ð =

    (R0, us(R0))

    Ý�=>C�4DÙ�=>/ 1 / =ABÚ(Ù�79ÛG=>/ E�7 Ð 13/Å8 Ñ / Ó'44Ò�/ 7�4D.0C�4O=>C�4C�4D1 Ð 8=04OÕ�Ü�;>4�8>8:/ E�7#/ 7É-@����F�/Å80Ü�E�8:/ =>/ Ó�46.�4L?�4Oà97�4

    Rc :=dρ2v +

    √d2ρ4v2 + 4dρ2c2δ

    2c2δÐ 79?B=.�EÚ(Ù�79ÛG=>/ E�798

    c1Ýc2 : [Rc,+∞)→

    � +

    Ñ / Ó�4C(R, u) ∈ Ω .0C�/ÅÛCZ/Å8X7�E'=v8:/ =>Ù Ð =>4�?#2�4O=.�4C�46Ú(E�1 1 E .0/ 7 Ñ 1 4C�4�à9;8=eÜ Ð ;:=eE'Ú�=>C�/Å8�1 4E6ÛOE�Ò�Ü�Ù�=>4�=>C�40?�4/ Ó Ð á=>/ Ó�4�80E'Ú

    c1Ð 79?

    IuÐ 79?#=>EB8:C�E .a=>C Ð =0=>C�4467�4 Ñ�Ð =>/ Ó�4 Ð 79?@Ü�E�8:/ =>/ Ó�4�Ý�;>4�8:Ü�4�ÛGá

    =>/ Ó�4; Ð / Ñ C�=:Ú(E�;>. Ð ;?�× *× ú × (×�Ä=@;>4Eç8:C�E .V/Å8�=>C Ð =B=>C�4|Ó Ð 1 Ù�4*E'Ú�=>C�4xÙ=/ 1 / =A Ú(Ù�79ÛG=>/ E�7 Ð 10Ú(E�;B=>C�4

    8:E�1 Ù�=>/ E�7@8= Ð ;:=>/ 7 ÑÐ =(R0, u

    s(R0))E�7B=>C�4D8= Ð 2�1 4�Ò Ð 7�/ Ú(E�1Å?B/Å8�Ü�E�8:/ =>/ Ó'4�×��ÄÚ

    R0 ≥Rc

    Ý5=>C�4C�4¿Ò�E�7�E'=>E�7�/ÅÛO/ =A*E'Úus

    Ý5=>C�4¿;>4/ E�7&��× 2xÚ(;>E�Òq/t;>E�Ü5E�8:/ =>/ E�7&���9Ý=>C�4L8:4�ÛOE�79? Ð 79?B=>C�46à9;8=0Ü Ð ;:=vE'Ú�34

  • �EL4C ÐDÑ / Ó�44/ E�73×�ØXC�44D/Å807�E¿=>C�;>4�8:C�E�1Å?Z8:4/ 7 Ñ / 7�/ =>/ Ð 1,;>4/ E�798v8:Ù9ÛC@=>C Ð =¦C�/ Ñ C�áÜ�;>4�8=>/ Ñ 4�;>444�8=>/ Ñ / E�Ù98 Ð 79?Á=>C�E�8:4¿2�4C Ð =D=>C�;>4�8:C�E�1Å?Á7�44�Ý$=>C�44v/Å8 Ð Ò�E�7�E'=>E�7�/ÅÛ0;>4/ E�798:C�/ Ü254O=.�4;>44/ E�7ÉE'ÚX=>C�4 Ð . Ð ;? Ð 79?Á=>C�4¿Ú(;>4 ã Ù�4C�.0C�/ÅÛ>C�/ =8:C�E�Ù�1Å?û2�4 Ñ / Ó'4C Ð =Ò�E�Ò�44¿Ü�;>4�8=>/ Ñ / E�Ù98�=>C�4 Ð . Ð ;?ÁÛOÙ�;>;>41 Ax/Å8C�4B1 4�8>8LE'Úª=>4E Ð ? âÙ98=�=>C�4;>4/ E�7*Ú(;>E�ÒV/ =8�/ 7�/ =>/ Ð 1Ó Ð 1 Ù�4�Ýt2A Ð 1 =4O;>/Å7 Ñ =>C�4*Ú(;>4 ã Ù�4C%.0C�/ÅÛ>C%/ =#/Å8 Ñ / Ó�4C Ð = Ð ? âÙ�8=Ò�4O7�=8:C�E�Ù�1Å?û2�4BÒ Ð ?�4#8:Ò�E�E'=>C�1 A�× � 8L1 E�7 ÑxÐ 86=>C�4#Ü Ð ; Ð Ò�4O=>44¿C�44 Ð ?�A8= Ð =>46C Ð 8 Ð 7 Ð 7 Ð 1 A�=>/ÅÛ Ð 154OÕ�Ü�;>4�8>8:/ E�7

    (R̂, û) =

    (√d

    δ

    ρr

    2c(r + δ),

    ρr

    2c(r + δ)

    ).

    �7@=>C�/Å8vÛ Ð 8:4�Ý�=>C�4D1 E�7 Ñ áÄ;>Ù�7@Ü�E�1 /ÅÛOABÜ�;>4�8>ÛO;>/ Ü�=>/ E�7@/Å808:/ Ò�Ü�1 4�×9ØXC�4�Ú(;>4 ã Ù�4C.0C�/ÅÛC*=>C�4 Ð . Ð ;?Z/Å8 Ñ / Ó'44 Ð ?�A*8= Ð =>4/Å8v=>C�4Ü�;>E�?�Ù9ÛG=�E'Ú¦-@�$F0=>C�4Ó Ð 1 Ù�4E'ÚC Ð Ó/ 7 Ñ|Ð .�E�;>Ô�44/ Ó�4�=>E*=>C�4Á- Ð ?�Ò�/ 7�/Å8=>; Ð =>/ Ó�4�FDÛOE�8=E'Ú/Å8>8:Ù�/ 7 ÑLÐ 7 Ð . Ð ;? Ð 79?Z-6��F,=>C�4¦?/ª8>ÛGE�Ù�7�=t; Ð =>4v?�/ Ó/Å?�4�?2AL=>C�4¦?�/Å8>ÛOE�Ù�7�=t; Ð =>40Ü�1 Ù98=>C�4�; Ð =>4 Ð =X.0C�/ÅÛCB=>C�46?�/ 1 Ù�=>/ E�7#E'Ú3=>C�4 Ð . Ð ;?32 8t;>4/ E�7#?�4�Û Ð A�8�E Ó'4/ Ò�4

    δ.ØXC�4và9;8=t=>4Ò /Å8tE�2Ó/ E�Ù98C�4�8:4�ÛOE�79?�/Å8�/ 7�=>44�8=>/ 7 Ñ ×�æ%C�4C Ð =¦/ =8;>4/ E�7@/Å80C�/ Ñ C3×�æ%C�44/ E�7

    1/R̂/Å834OÕ Ð ÛG=>1 A √

    δ/d?�/ Ó/Å?�4�?D2�A�=>C�4�8=>4 Ð ?�A68= Ð =>4 Ð . Ð ;?¦Ú(;>4 ã Ù�4C Ð =�C�/ Ñ CD; Ð =>4�8 δ E'ÚÚ(E�; Ñ 4O=:=>/ 7 Ñ Ü Ð 8= Ð . Ð ;?�/ 7 Ñ 8:C�E�Ù�1Å?/ 79ÛO;>4 Ð 8:4�=>C�4v4 ã Ù�/ 1 / 2�;>/ Ù�Ò ;>4/ E�7;>4/ Ó�4=>E=>C�4 Ð . Ð ;?¿Ú(;>4 ã Ù�48XE�2Ó/ E�Ù98 Ð Ü�;>/ E�;>/52�Ù�=v44 Ð 8:E�7 Ð 2�1 4/ 7 ;>4O=>;>E�8:Ü54�ÛG==>C Ð =�=>C�4|8 Ð Ò�4Z8:C�E�Ù�1Å?ç2�4Z=>;>Ù�4@Ú(E�; Ñ ;>4 Ð =>4/ 1 / =AçE'Ú�.�E�;>Ô-�8:Ò Ð 1 1 44 Ð =>44�8:/Å8= Ð 79ÛO4B;>4 ã Ù�/ ;>4�8 Ñ ;>4 Ð =>44C�4�8:46=.�E ã Ù Ð 7�=>/ =>/Å4

  • ¤µ ¸�¬�õXG '�º�¬ ¶5¬�¸eC�4LÛ Ð 8:4v > 0.

    � 1 13.�4DÔ7�E .a/Å8�=>C Ð =

    û =

    √δ

    d

    R̂√1 + vR̂

    .

    æÁ4�?�EL7�E'=�Ô7�E .ý.0C�4O=>C�4C�4¦4 ã Ù�/ 1 / 2�;>/ Ù�Ò ;>4/ E�7¿.0/ 1 1�2�4¦2�4O=:=>4E�Ò ÷C Ð =,.0C Ð =>4/ Ù�Ò;>4/ E�73Ý�=>C�4@ÛOE�;>;>4�8:Ü�E�79?�/ 7 Ñ Ú(;>4 ã Ù�4CÉ.0C�/ÅÛCû=>C�4 Ð . Ð ;?x/Å8 Ñ / Ó�48�=>C Ð 7�.0C�4C Ð =�8 Ð Ò�4X4 ã Ù�/ 1 / 2�;>/ Ù�Ò ;>4OÜ�Ù�= Ð =>/ÅE�7/Å8 Ð ÛC�/ 4E'= Ð 1�Ó Ð 1 Ù�4 E'Ú#4 Ð ;>7�/ 7 Ñ E�7�4 ;>4CÉ.�4�à979?û2�E'=Cû/ 7û=>C�4BE�2�â4�ÛG=>/ Ó�4�Ú(Ù�79ÛG=>/ E�7 Ð 1 Ð 79?x=>C�48= Ð =>404 ã Ù Ð =>/ E�73× 4 2Ó/ E�Ù98:1 A�Ý .0C Ð =tÒ Ð =:=>4C�4Ü Ð ; Ð Ò�4O=>4C�4C�4L;>44/ E�73Ý9/× 4�× Ð C�/ Ñ C|Ó Ð 1 Ù�4LE'Ú1/R

    Ý Ð 79?Z=>C�48 Ð Ò�4/Å8�=>;>Ù�46Ú(E�; Ð 2 Ð ?#;>4/ E�7@=>E Ñ 4O=>C�4C Ð C�/ Ñ CZÒ�E�7�4O= Ð ;>ABÛOE�Ò�Ü5E�7�4O7�=�×

    æ%C Ð =vÒ Ð Ô'4�8�=>C�/ 7 Ñ 8 Ð 1 / =:=>1 462�/ =¦ÛOE�Ò�Ü�1 /ÅÛ Ð =>4�?BC�44�/Å8�=>C�4�Ú Ð ÛG=X=>C Ð =v.0C�/ 1 4v- Ð 8:A�8=>44/ Ð 2�1 4�=>C Ð =�ÛC Ð 7 Ñ 4�8lE$Ó�4/ Ò�4�×�ØXC�/Å8�Ò�4 Ð 798v=>C Ð =¦=C�4; Ð =>/ E#2�4O=.�4C Ð 7 Ñ 4�8�ÛOE�7�=>/Å7�Ù�E�Ù98:1 A�×

    å�E�;�E�Ù�;X8:4/ Ó/ =A Ð 7 Ð 1 A�8:/Å8t.�4v=>C�44OÚ(E�;>4¦Ü�Ù�=v/ 7�=>E;>4/ E�7B.0/ =>CB=>C�4D8=>4 Ð ?A

    8= Ð =>46Ó Ð 1 Ù�46E'Ú,=>C�4D;>4/ E�73Ý1/R̂,

    .0C�/ÅÛ>CZ/Å80E'ÚeÛOE�Ù�;8:4 Ð ÛOE�798= Ð 7�=�×æû4�ÛOE�798:/Å?�4C�;>4CÁ;>4�8:Ü�4�ÛG=�=>E

    v�l-@�$F¦Ü�Ù�;>4C�E�7�E�;�-

    v = 0FGÝt-6��F

    / 7a8=>4 Ð ?Aý8= Ð =>4*=>C�4xC�E�7�E�; Ð 79?ýÒ�E�7�4O= Ð ;>A ÛOE�Ò�Ü�4/ E�7 Ð ;>4|4 ã Ù Ð 1 1 AçÓ Ð 1 Ù�áÐ 2�1 4É-

    v = 2.78FGÝ Ð 79? -b"× Ð áÄ2�F/ 7%8=>4 Ð ?�Aç8= Ð =>4@=>C�4*C�E�7�E�;¿/Å8�E�7�1 A�E�7�4Oá=>4C

    Ð 8@Ó Ð 1 Ù Ð 2�1 4 Ð 8#=>C�4ûÒ�E�7�4O= Ð ;>AjÛOE�Ò�Ü�E�7�4/ÅÛOÙ�1 Ð ;�ÝXå�/ Ñ áÙ�;>4:�*Ü�1 E'=8=>C�4Z8= Ð 2�1 4 Ð 79?ÉÙ�798= Ð 2�1 4@Ò Ð 7�/ Ú(E�1Å?�8LÚ(E�;=>C�4�8:4#=>C�;>4C Ð =�/ 79ÛO;>4 Ð 8:/ 7 Ñ =>C�4-�ÛOE�798= Ð 7�=GFlÒ�E�7�4O= Ð ;>A�ÛOE�Ò�Ü�E�7�44 Ð 8:4�8FG× ØXC Ð =lÒ Ð Ô'4�8l8:4C�404OÕ=>4C Ð =e=>C�40;>44/ E�73Ý�=>C Ð =X;>4/ E�7BÒLÙ98=X254¦Ü�;>4�8:4Ó�4�?�×�ØXC�4�Ò�E�;>4vÜ�;>E�Ò�/ 7�4C�4à97 Ð 79ÛO/ Ð 1lÛOE�Ò�Ü�E�7�4C�4�1 4�8>8�?�44

  • �C�44/ E�7ZE'Úe=>C�4E�Ü�=>/ Ò Ð 1�ÛOE�7�=>;>E�1u∗

    / 7Z=>/ Ò�4/Å8 ã Ù Ð 1 / = Ð =>/ Ó�4/ = Ð =>/ Ó�4C�4D1 Ð ; Ñ 4Ü�4C�4D/ 79ÛO;>4 Ð 8:4 Ð 79?#=>C�4DC�/ Ñ C�4C�4

    à97 Ð 15Ó Ð 1 Ù�4�E'Ú3=>C�4DÛOE�7�=>;>E�1×ØXC Ð =0Ò Ð Ô'4�8�Ü544�8>8/ Ó�4¦Ò Ð 7 Ð'Ñ 4;>/ 2�Ù�=>/ E�7

    E'Ú Ð . Ð ;?�8C�4 Ð . Ð ;?¿/Å8

    v+1/R.ØXC�Ù98�=>C�46ÛOE4OþBÛO/ 4C�4 Ð . Ð ;?32 8

    Ü98:A�ÛC�/ÅÛDÓ Ð 1 Ù�4L/ 7Z=>C�4L.�E�;>Ô�4/ Ó'4DÚ(Ù�79ÛG=>/ E�7|/Å80à�Õ�4�? Ð =¦Ù�7�/ =A�×5övE .�4C Ð =*ÛOE�4GþBÛO/ 4; Ð 4OÖ5E�;:=X/Å8

    ρÝ Ð 79?¿=>C�4DÛOE�8=XE'Ú�4OÕ�4/ 7 Ñ

    =>C Ð =D4OÖ5E�;:=6/Å8 Ñ E$Ó�47�4�?*2Ad

    -�8:Ò Ð 1 1dÛOE�;>;>4�8:Ü5E'79?�8v=>E@1 Ð ; Ñ 4ÛOE�8=GFG×3öv4/ÅÛOÙ�1 Ð ;�Ý,ÒÙ�1 =>/ Ü�1 A/ 7 Ñv, ρ,

    Ð 79?c2A Ð|Ñ / Ó�4C Ð =�8 Ð Ò�4#ÛOE�7�8= Ð 7�=/Å8L4 ã Ù�/ Ó Ð 1 4Ex?�/ Ó/Å?�/ 7 Ñ =>C�4@ÛOE4OþBÛO/ 44/ E�72A=>C Ð =08 Ð Ò�4�ÛOE�798= Ð 7�=�×�(¦E�/ 7 Ñ =>C�/Å8X8:C�E .v8l=>C Ð =X.0C�4C�4¦;>4/ E�7 Ð 1�4OÖ54�ÛG=8Ð ;>4xÙ�7�/ Ò�Ü�E�;:= Ð 7�=û-(7�E'=Z2�4�Û Ð Ù98:4

    R/Å8#1 Ð ; Ñ 4x2�Ù�=Z; Ð =>C�4C Ð 7BE�7�4$FGÝ'=>C�4�8:E�1 Ù�=>/ E�7¿2�4�ÛOE�Ò�4�8tÓ'4A�8:/ Ò�Ü�1 4�× � 8�8:EE�7 Ð 8R/Å8�ÒÙ9Û>C Ð 2�E Ó'4v÷Ù�; Ð 1,4OÕ=>4AZÛOE�ÒáÜ�E�7�4Ô5Ý�Ò Ð Ô/ 7 Ñ =>C�4DÜ�8:A�Û>C�/ÅÛ�Ó Ð 1 Ù�4�2�4 Ð ?�/ Ö5444Oá8:Ù�Ò Ð 2�1 AÁ8=>/ 1 1�?�4�ÛO;>4 Ð 8:/ 7 Ñ F0Ú(Ù�79ÛG=>/ E�7ÉE'Ú�=>C�4B8= Ð =>4¿Ó Ð 1 Ù�4

    R,4O=Û'×3æÁ4¿44

    C�446E�7 Ð 7Z4OÕ=>4C Ð =v/Å8XÒ�E�;>4D8:Ù�2�=>1 4�×| Ù98= Ð 8 Ð . Ð ;?�8�Û Ð 7LC Ð Ó�4�25E'=>C Ð Ò�E�7�4O= Ð ;>A Ð 79? Ð ;>4/ E�7 Ð 1ÛOE�Ò�Ü�E�7�4C�46Ü�E .�4/ Ñ Ò Ð =>/ ÷ Ð =>/ E�7*Û Ð 7Z254?�/ 1 Ù�=>4�?Z/ Úl=>C�48 Ð 79ÛG=>/ E�7Z/Å8 Ñ / Ó�4E�EÚ(;>4 ã Ù�41 A�×� Ð =>C Ð 7�/ 47�4�2 8 ' Û Ð ;>1 4O=�34O=:=>4E Ð ?�4 Ð =>C

  • ¤G² ¸�¬�õXG '�º�¬ ¶5¬�¸eE�? Ð A�×��7@Ò�E�;>4�ÛOE�7�=>4

  • �/ Ð 7 ' ÛO/ 4/ E�7-�å�æjålFXÙ�79?�4; Ð ÛG=�vE9×�/����7 � � á 4 ú1² ×

    Ì �����!���� ¯Ì u6� �����v���vu6� �9�������(������! B3øå�;>E�Ò 8:A�8=>4

  • ¤Gà ¸�¬�õXG '�º�¬ ¶5¬�¸e�F Ð ;>4¦=>C�4D;>EE'=G8�E'Ú,=>C�4 ã Ù Ð ?�; Ð =>/ÅÛ�4 ã Ù Ð =>/ E�7

    E(µ) := µ2 − rµ−[(

    du2

    R2− δ)2

    + r

    (du2

    R2+ δ

    )]= 0,

    .0C�/ÅÛCûC Ð 8D;>4 Ð 1l;>E�E'=86E'ÚX7�4 Ñ�Ð =>/ Ó�4�Ü�;>E�?�Ù�ÛG=�×,æÁ4�=>C�44OÚ(E�;>4BÛ Ð 7ÉÛOE�79ÛO1 Ù9?�4�=>C Ð ==>C�4L8=>4 Ð ?�A#8= Ð =>4

    (R̂, û)/Å8 Ð 8 Ð ?�?�1 46Ü�E�/ 7�=�×

    �34O=µ1

    2�4#=>C�4@Ü�E�8:/ =>/ Ó�4#4C�4 |�Ð ÛOE�2�/ Ð 7#Ò Ð =>;>/ ÕÁ-@��>�FtÚ(E�;X=>C�4L8=>4 Ð ?�A#8= Ð =>4

    (R̂, û).�4DC Ð Ó�4

    8:/ Ñ 7(v2) =

    8:/ Ñ 7 (− a12a11 − µ2

    )=

    8:/ Ñ 7

    2du30R30

    du20R̂2

    + (δ + r)− µ2

    > 0,

    8:Ev2

    /Å80Ü�E�8:/ =>/ Ó�4�×�ØXC�4D8:/ Ñ 7@E'Ú�=>C�4DÜ�;>E�?�Ù9ÛG=v1v2

    /Å8 Ñ / Ó�4C@Ü5E�8:/ =>/ Ó'4�× *× ú × (×

    o u6� �����v���vu6� �9�������(������! B9�34O=

    (R0, u0)2�4 Ð Ü�E�/ 7�=l/ 7 Ð 7�4C�E�E�?DE'Ú

    (R̂, û)Ð 8e?�4Oà97�4�?/ 7�/t;>E�Ü5E�8:/ =>/ E�7

    ����Ýe8:/ =>Ù Ð =>4�?çE�7�=>C�4Z8= Ð 2�1 4@Ò Ð 7�/ Ú(E�1Å?É=>E(R̂, û)

    8:Ù9ÛC�=>C Ð =R0 < R̂

    Ý Ð 7�?ç1 4O=ωL = (RL, uL)

    2�4*=>C�4|=>; Ð â4�ÛG=>E�;>AçÜ Ð 8>8:/ 7 Ñ =>C�;>E�Ù Ñ C(R0, u0)

    Ð 79? Ñ E�/ 7 Ñ =>E(R̂, û).

    ü¦8:/ 7 ÑωL

    .�4D.0/ 1 1,?�4Oà97�461 Ð =>4C�46Ú(Ù�79ÛG=>/ E�7us

    E�7(0, R̂).æû4L?�44C�4L8:E�1 Ù�=>/ E�7

    ωLE'Ú¦=>C�4*8:A�8=>4C�4�=>; Ð âÄ4�ÛG=>E�;>A

    (RL, uL)E�7

    (a−, b−);>4

  • �C�4�=>; Ð â4�ÛG=>E�;>A

    ωL/Å8,2�E�Ù�79?�4�?�×$æÁ4

    Ð 7 Ð 1 A÷C�4L=.�E#ÛOE�Ò�Ü5E�7�4;>/ÅÛG=>1 AD/ 79ÛO;>4 Ð 8:/ 7 Ñ E�7

    J× öv48:Ù�Ò�4�=C Ð =

    uL(J)/Å80Ù�7�25E�Ù�79?�4�?�×9�34O=

    ū > 2û2�4 Ð

    ;>4 Ð 1�7�Ù�Ò2�4C@=>C Ð =

    du2

    R̂2+ c̄u− (δ + r) > 0 Ú(E�;04 Ð Û>C u > ū.

    ' / 79ÛO4uL(J)

    /Å8vÙ�7�25E�Ù�79?�4�?�Ý�=>C�44D/Å8a ∈ (t−,+∞)

    8:Ù9Û>CZ=>C Ð =uL(a) > ū.

    æÁ4?�4Oà97�4

    b+ = sup {b > a |uL(t) > ūÚ(E�;04 Ð Û>C

    t ∈ [a, b)}.��A¿=>C�4LÛOE�7�=>/ 7Ù�/ =A#E'ÚuL

    Ýb+ > a.

    å�E�;X4 Ð ÛCt ∈ [a, b+)

    .�4DC Ð Ó�4

    u̇L(t) = uL(t)(d(uL(t))2

    (RL(t))2+ c̄uL(t)− (δ + r)

    )>

    > ū(d(uL(t))2

    R̂2+ c̄uL(t)− (δ + r)

    )> 0,

    =>C�44OÚ(E�;>4uL

    /Å8#8=>;>/ÅÛG=>1 Aý/ 79ÛO;>4 Ð 8:/ 7 Ñ E�7[a, b+).

    ØXC�44

    b+ = +∞.å�E�;

    bn → b+Ú(;>E�Ò -@��7�F�.�4DE�2�= Ð / 7 Ð ÛOE�7�=>; Ð ?�/ÅÛG=>/ E�73Ý

    0 < 2û < ū < uL(a) ≤ û,8:E¦.�4�à97 Ð 1 1 ALÛOE�79ÛO1 Ù9?�4t=>C Ð =

    uL(J)Û Ð 7�7�E'=e2�4�Ù�72�E�Ù�79?�4�?�×��7 Ð ?�?�/ =>/ E�7L.�4�C Ð Ó'4

    ωL(J) ⊂ Ω0Ý�.0C�44

    Ω0 := (0, R̂)× (0, ū+ 1).�¦E$. .�4DÜ�;>E$Ó�4�=>C Ð =limt↘t−

    ωL(t) = (0, 0).

    ØXC�/Å80Ò�4 Ð 798X=>C Ð =v=>C�4ÛOE�7�=>;>E�1 Ñ E�4�8�=>E¿÷C Ð =/ = Ñ E�4�8�=>E*/ 7�à97�/ =AÁ/× 4�×

    R → 0. æÁ4#8:Ù�Ü�Ü�E�8:4=>C Ð =X=>C�446/Å8 Ð 8:4 ã Ù�4C Ð =

    ωL(tn)?�E4�8�7�E'=vÛOE�7Ó�4E(0, 0).

    ØXC�4L8:4O=Ω0

    /Å8vÛOE�Ò�Ü Ð ÛG= Ð 79?ωL(tn) ∈ Ω0

    Ú(E�;04 Ð ÛCn ∈ � � =>C�44OÚ(E�;>4=>C�44�/Å8 Ð 8:Ù�298:4 ã Ù�4C�E�Ù�=1 E�8>8¦E'Ú Ñ 4

  • G�õ¦¬9:G�v¬ ® ¬�¹XO

    E�Ü�44�?@Ü�1 Ð 7�4L8:4O=�Ý9ÛOE�7�= Ð / 7�/ 7 Ñ 7�EB8=>4 Ð ?�A@8= Ð =>4�8X4 Ð ÛCZ8:E�1ÅÙ=/ E�7E�7j/ =8@Ò Ð Õ�/ Ò Ð 1�/ 7�=>4Ó Ð 1�E'Ú4OÕ�/Å8=>4C�/Å8ÛOE�;>E�1 1 Ð ;>AZ=>E

    Ωm =[ 1m, R̂− 1

    m]× [ 1

    m, ū+ m−1

    m]Ý�Ú(E�;04 Ð Û>CZÜ�E�8:/ =>/ Ó�46/ 7�=>4 Ñ 4;>/ÅÛG=>1 AB/Å7�ÛO;>4 Ð 8:/ 7 Ñ E�7

    (t−,+∞)=>C�44OÚ(E�;>4

    limn→+∞R(tn) = R1 < R̂;>4�8:Ü�4�ÛG=>/ Ó�4C t ∈ (t−,+∞) =>C�44OÚ(E�;>4 limn→+∞ u(tn) < û+1.�Ä=v;>4EB8:C�E$.j=>C Ð =

    (R1, u1) = (0, 0)Ú(E�;X=>C�46Ú(E�1 1 E .0/ 7 Ñ =.�E¿Û Ð 8:4�8

    ¦ (R1, u1) ∈ [0, R̂]× {0}¦ (R1, u1) ∈ {0} × [0, ū+ 1].�7x=C�4�à9;8=DÛ Ð 8:4�Ý�8:Ù�Ü�Ü�E�8:4=>C Ð =

    R1 6= 0×3ØXC�4¿8:A�8=>48:4�8l=>C�;>E�Ù Ñ C (R1, 0).ØXC�/Å8�ÛOE�7�=>; Ð ?�/ÅÛG=8�=>C�4�Ù�7�/ ã Ù�48,E'Ú98:E�1 Ù�=>/ E�7L/ 7 U Ú(E�;,=>C�4 4 ( ú 8:A�8=>4

  • �C�4Ü�;>E�Ü�E�8:/ =>/ E�73×�ØXC�46Ü Ð ;:=���× 2#/Å808:/ Ò�/ 1 Ð ;�=>EG��× ÐÐ 79?¿=>C�46Ü Ð ;:=��/Å8 Ð 8:/ Ò�Ü�1 4DÛOE�798:4Oáã Ù�4; Ð â4�ÛG=>E�;>A�=>C Ð =l8 Ð =>/Å8à94�83=>C�4[/�E�7�=>;>A Ð'Ñ / 72 8 Î|Ð Õ�/ ÒÙ�Ò)/t;>/ 79ÛO/ Ü�1 4=>C�4DÓ Ð 1 Ù�46E'Ú

    H4 ã Ù Ð 1Å8�=>C�4DÓ Ð 1 Ù�46E'Ú H0 Ý�C�4

  • � ¸�¬�õXG '�º�¬ ¶5¬�¸eE$Õ�/ Ò Ð =>/ E�798�8:/ Ò�/ 1 Ð ;X=>E�=>C Ð =0/ 7@=>C�4DÜ�;>EE'Ú�E'Ú�=>C�4%5�;>E�7�. Ð 1 162 8�/ 7�4 ã Ù Ð 1 / =A���É� ø0º PFe

    (R, u)t;P-S.e6ZxS@FPx^Fe�fZ�jNl�e�SZ�e6UL\�&Se

    (R0, u0) ∈◦

    ΩS\!i.l#Se6U]l�s

    yxjUL\�NeDOA(R, u)

    /Å8 Ð 8:E�1 Ù�=>/ E�7 E'Ú68:A�8=4OÒ -b"× Ð áÄ2�F/ 7 ◦Ω,=>C�44OÚ(E�;>46/ =¦8 Ð =>/Å8à94�8X8:A�8=>4C�44OÚ(E�;>4

    limt→+∞ e−rt R

    u(t) = R0

    u0.

    *× ú × (×

  • �����7 � Ý!� ��� 7�×

    � × ú ×�(¦4�ÛO/× ØXC�4x4OÖ54�ÛG=8#E'ÚÛOE�7�=>/ 7 Ñ 4/ 7 Ñ 44;>E�1Å8¦E�7Á/ 7�=>;>/ 798:/ÅÛÒ�E'=>/ Ó Ð =>/ E�73× n�Z��S\9U Á�Se6U�f\!S` wP;O/ 7 Ñ 41× n§_�e6ULVGS` PÂkf\9e6Zxf`L` P¾f#P;Ã�f\!fV�U]l#^;OÒ Ð 73× n�ZxiUL\!SZ�j¨RU «PFZxPF\9e6U�S`{�¸FW9Se6U�f\C�4Dà9;>Ò:� � 7|4/ÅÛ Ð 134/ E�73× QNS\!S#�PFVGPF\9e�g3^FU�PF\!^;P Ý�����-���F�� �9� � �����>�Ý!� ��� >�×

  • Oµ ¸�¬�õXG '�º�¬ ¶5¬�¸e;>Ù9ÛG=>Ù�;>4�× �^;f\!fVGPFe6Z�U�^xS Ý � >/ Ð 10Ü�E�1 / =>/ÅÛ/ Ñ 1 / =>÷�×/t;>/ ÷CÁÛOE�Ò�Ü�4O=>/ =>/ E�73×  wPF`D`À�fW�Z�\!S`{fuyh�^xf\!fV�U�^�l Ý�#��-@�$F�� ���F������Ý!� � 7���×

    ����×��t×�/e4Ô'E9× ¨RU «PFZxPF\9e6U�S`��¸FW9Se6U�f\E Ð 79? | × ú × ' =>/ Ñ 1 / =>÷�× ú ã Ù�/ 1 / 2�;>/ Ù�Ò Ù�7�4

  • G�õ¦¬9:G�v¬ ® ¬�¹XO

    ÆÇ È�É§Ë e·�0�� L