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Approximation of the Value Function for a Class of Differential Games with Target

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HAL Id: inria-00073757

https://hal.inria.fr/inria-00073757

Submitted on 24 May 2006

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Approximation of the Value Function for a Class of Differential Games with Target

Odile Pourtallier, Mabel M. Tidball

To cite this version:

Odile Pourtallier, Mabel M. Tidball. Approximation of the Value Function for a Class of Differential

Games with Target. RR-2942, INRIA. 1996. �inria-00073757�

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a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

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Odile POURTALLIER and Mabel TIDBALL

N˚ 2942

July 1996

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¡ Z„^"{k^

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³¹irn t

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t

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t

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hm^w] ¡ Z\dfx0Z<dfqS9ki#hf]_lSqkj ;

j›Z\^²o„{kl

¢

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leg„n

t

l#9jkZ\^®qk^"xlSn

t

js^w{k] l#°j›Z\^®{kdmŽeZSjZ„iSn

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qsd

t

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º^j

!

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!

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!

]l _ Kk#d

dm

l j S

jkZ„^"n

!

]l _ ` k#d

pq = !

]_:`

S m

(18)

¸…lS{2iTqkjk{kdpxjshpuošleqsdpjsdp©r^q›xirhpiS{O³„¡9^

t

^)<…n„^X

]lR_O`

Sr !

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iSn t jkZ\^{k^Žeg\hpiS{sdp§"irjsdplSn

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]m_ ³

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_#]l _d (lLoo

=Έ\‹D

¡ Z„^"{k^ 9

;ž]_^"iSn„q*jkZ\^xlSne©rlShpg„jsdplSnoš{sl t g„xj³„j›Z„i#j2dfq

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ac]l0U\_]\_\(m

˜ ^i#hfqkl t ^ <…n\^j›Z\^¯g„n„xjsdplSn

iSq9jkZ\^xlSnejsdfnvg\lSg„q2o„dm^wx^¡2dfqk^i6F®n\^ž¯g„n„xjkdmlenl#



q›g„x0ZjkZ„irj4

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m

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]hm_$o

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" _ v

v l

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g#_

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l oJpq r

³

-

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¡ Z„dpx0ZŽrdp©r^wq¶irn_g„o„o…^"{

¢

lSg„n

t

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t

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Y2Z„^lS{k^"]Tq B\¤p‹irn t B\¤èˆ\³\ŽSdm©S^irng„o„o…^"{ ¢ lSg„n t «le{*jkZ\^qk^"xlSn t js^w{k] l#jkZ\^ t ^wxle]_o…lSqkdpjsdplSn =çˆ-MD6¤

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l

dfn



³…iSn t jkZ\^"lS{k^"] B\¤èˆTiro„o\hpdp^"q*«le{

l

dpn#

³š¡ Z\^"{k^r³„j›Z\^ qk^j

 dpq

j›Z\^xle]To\hm^w]_^"nejkiS{su²dfnjkZ\^ t dfq›x{k^js^q›o„irx^l#j›Z\^^"n„hpiS{sŽS^ tt dpq›x{k^jk^žj›ir{kŽr^"j"#

³„j›Z„i#j2dfq

#

` lŽ 1]lfd

# _

d

irn t

 ` pqr

#

m

(19)

ÀvÄ “ À kyl

#]hm_ +

]hm_

4,+$%+ Q-!45 up x (+

luo

 x n$ &

-

]lR_

]l _"-

1 ¼

=Έ 6 D

n &)0

4G'%- 45 -#

x

+ )(+ #

l x

&"n$ nOE -R100' 4

·*^«lS{k^žqkjkiS{sjkdpn\ŽTj›Z\^o„{klelrlrjkZ„^jkZ\^"lS{k^"] ¡9^¡9iSnej9jkl²qsj›i#js^i_js^wx0Z„n\dfxirh¹hm^w]T]_iD

#%À ª” kyl --

l oJpq r

n$ &s&$) ‰nO2L! )ƒ 0!&(+4

-

]lR_

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¼ d

=Έ'`D

-

]lR_ K]

_]lR_q-

1 ¼ m

=ΈBD

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-v

]lR_

v l 2_

v

]l _

v l

d%ac]lfdhecdhg1_Nj - v

v l l ¼ ¼

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+

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t

dp{k^"xj2xlSnšqs^w±eg„^"n„x^žlrS=Έ'`D°iSn

t

l#oš{sleoš^w{sj´u

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¤

?@?

d “ Ä¹ÄžÄ ’

ÀeÄ “ À — Y,lo„{sl'©S^jkZ\dfq j›Z\^le{s^w] ¡9^ <…{›qkjGdfnSjk^"{›o„{k^j

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€

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¼ d _ ` k#d dm

l o

 d

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#

m

=Έ#Œ3D

Æ ^"{k^r³

f]lfd

¼ d _O`T]

_]l _

]l _Z1

]l _m

‚žqsdfn\Ž²^w±egši#jsdplSn*=ΈB3D°l#¶hm^w]T]_iJB\¤p‹r³„j›Z\^¯iSxj j›Z„i#j

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dfqGnvg\hph,lSn

 ¢ ^wxiSg„qs^lr =s‹,bD6³

dpj xle]_^"q

- ]l d ¼ d _q-

1 ¼

dl o



m

}nj›Z\^qkiS]^©S^dfn½³

Zširq*jsl ¢ ^¡ {sdpjsjk^"nWirq

f]lfd

¼ d _ `2

]lR_ V ]lR_ 1‚]lR_ V ]_Z1] _d

(20)

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luo

#

³š¡ Z„^"{k^

Io

#

dfq qkgšx0Zj›Z„i#j

l0

…¤

˜

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¯{klS]-^w±vg„i#jkdmlen =k‹,b3D

iSn

t

g„qsdfn\ŽTjkZ„^¯irxj2j›Z„i#j

-]lR_ ] _c-

…

…³\¡9^l

¢

j›i#dfn

-

c]l d ¼ d _c-

d

¡ Z„^"{k^i#ŽSirdpn¹³

dpqži²xlen„qkjkirnej j›Z„i#jG¡°^ t ln\lrjG¡9iSnSjGjslo„{k^"xdpqk^r¤ ž^©S^hplSo\dfn\Ž²jkZ„^lSo…^"{›i#jklS{

iSn t g„qsdfn\Ž t ^ <…n\dpjsdplSn„q =Jb3D°lr

iSn t =?‰3D°lr

³

]lR_

x"irn ¢ ^¡ {kdmjkjs^wn«le{2i#hph

l

dpn



]lR_ `a‡†‰ˆ

E

&„'

Š €

C

]lfdhecdhg1_Z1

]l defdg#_

]l d\-efdg#_}

]\_

d

¡ Z„^"{k^

C

]lfdhecdhg1_ ` ]l dhecdhg1_]

S

]l d ¼ d __m

°le] ¢ dfn\dfn\ŽjkZ\dfq¶hfirqkj¶^"±vg„irjsdplSn¡2dmj›ZT^"±vg„i#jkdmlen„q=s‹w‰3Dirn t =k‹QMD6³SiSn t {s^w]dfn t dfn\Žj›Z„i#j7«le{7¯g„n„xjsdplSn„q

! ]hmd m_

irn t -]mdQm_

³„jkZ„^"{k^^y\dpqkjkq

e iSn t

g

qkgšx0Zj›Z„i#j

† ˆ

E

&„'

Š !

]efdg#_ V‡†‰ˆ

E

ƒ&„'

Š ]efdg#_

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¡9^xiSn¡ {kdmjk^ž«lS{

l

dfn



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C

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1

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]hm_

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=ΈPD

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]lR_

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]md efdC g#_C C

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1

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]l d\ - eC d g1_C

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=Έ b3D

iSn t jkZ\^wqs^j´¡9lThpiSqsj*^"±vg„irjsdplSn„q9jklrŽr^"jkZ\^w{*hm^wi t jsl²j›Z\^dpn„^"±vg„i#hpdpj´u

]hm_

]hm_

S

S

C

1 ¼

¼ C

m

=Έ#‰3D

(21)

˜

^<…n„irhmhpug„qk^jkZ\^¯iSxj2j›Z„i#j

C

ƒ

6

‰

¼

1 ¼ ] 1 ¼

_d

jklTl ¢ j›i#dfnj›Z„i#j*«le{

l

dpn

 ³

]m_

]hm_

I] 1 ¼

_m

Y2Z„dpq*^wn t q2j›Z\^o„{klelrlrjkZ„^lS{k^"] B\¤p‹r¤

?@?

À ª”„“ kyl -2 u4 '% +4 ')0)C ¼x I%- "!#sh r w 4 ` » ¼

ÀvÄ “ À k„i

c)

#]hm_ -+

]m_

04,+$%+ Q-!45 up

x l o #

x

n$ &

- ]lR_

]l _c-

m

n &)0

4G'%- 45 -#S+ )(+ #

l

d “ Ä¹Ä½Ä ’ ÀvÄ “ À — Y2Z\^%o„{klvl#\l#vjkZ„dpqjkZ\^"lS{k^"] t d X^"{›q½¯{klS] jkZ\^°o„{slvlrvlr\jkZ\^%oš{s^"©edplSgšq½jkZ\^"lS{k^"]

len\hpuBdfnŸjkZ\^

t

^ <…n\dpjsdplSnŸlr*j›Z\^®irg\yvdphpdpiS{su

¢

lSgšn

t

ir{kuB©'irhpg\^®oš{sl

¢

hm^w]¤

˜

^®¡9iSnejjsl<^"qkjsdf]Ti#jk^®jkZ\^

^y„o„{s^wqsdplSn

‡† ˆ

E

&('

Š €

]l defdg#_1

]l dhecdhg1_

]l d\-ecdhg1_

]\_

]l _ =6 M3D

¡ Z„dpx0ZxiSn

¢

^{s^"¡ {sdpjsjk^"n

‡†‰ˆ

E

&„'

Š €

]lfdhecdhg1_Z1

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]lfdh\ -ecdhg1_

]\_B

]lR_1 ]lfdhecdhg1_

]l _ ]l defdg#_}

]l _

`j‡† ˆ

E

&('

Š €

]l defdg#_1

]lfdhecdhg1_

]l dh\ -efdg#_0]J

]\_ V

]lR_h_1

] 1]lfdhecdhg1_

S _

]lR_

=6

‹D

€q

dpqiTº,dfo„q›x0Z\dmjk§¯g„n„xjsdplSn<iSn

t

l*\

¼,³ =?qkdfn„x^«le{

l*\

¼,³

]lfdh\o--ecdhg1_O` k D6³

¡9^Z„i•©r^

-

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¼¹¤*žniSnlSjkZ\^w{2Z„irn t ³\¯{klS] =Έ#ŒD6³

-

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²«le{

l

dpn #

¤

¸O{sle] jkZši#j³šdmj*«lShmhpl'¡ q9jkZ„irj*«lS{2i#hph

luo

#

³„irn t «le{2i#hph

e

irn t g

‡†‰ˆ

E

&('

Š - ]l defdg#_1 ]l dhecdhg1_

]l d\-ecdhg1_

]\_

]l _c-

-

#]lfdhecdhg1_c-]1a- ]lfdhecdhg1_c-

… ¼ 1 m

(22)

bvlš³v«lS{

l

dfn #

³„¡°^Z„i•©S^r³

]lR_

C

]l d ¼ d _ ` k d -

]lR_c-

d =6

ˆD

¡2dpjkZ

- C

]l d ¼ d _c-

O³„lS{2irŽSirdpn½³

]l mefmg1_ ` C

]lfdhecdhg1_Z1

]l defdg#_

]lfdh\ -efdg#_}

]\_d

¡ Z„^"{k^

C

]l dhecdhg1_ `

]l defdg#_0]

S C

]l d ¼ d _h_0m

7l'¡ jkZ„^oš{slvl#^"n t q*¡2dpjkZ^y„iSxjshpu®j›Z\^qkiS]^xle]To„g\jkirjsdplSnjkZ„irj*oš{s^"©edplSgšqshpur³\g„qkdpn„Ž_j›Z\^ž¯irxj9j›Z„i#j

]lR_

q›i#jkdpqh<š^"q*j›Z\^ ¢ lSg„n t ir{ku®©'i#hfg\^o„{sl ¢ hp^"] =k‹"‰3D0¤ ?@?

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t

df{k^"xj

xlSn„qk^"±vg\^wn„x^žl#7jkZ„^lS{k^"]Tq B„¤m‹irn

t

B„¤ˆ„¤

ÀvÄ “ À k 

4N#4 '53 '%-##!#!4O%!&' u

V O

» ¼

7+ 4(0&72 . +$°9 +,$¶

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qkjkirjs^o…lrdfnSj iSn t xlSnej›{slShpq"¤

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