NECESSARY AND SUFFICIENT CONDITIONS FOR NEAR- OPTIMALITY HARVESTING CONTROL PROBLEM OF STOCHASTIC AGE-DEPENDENT SYSTEM WITH POISSON JUMPS
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1 IJRRS 4 M wwweom/vome/vo4ie/ijrrs_4 NCSSRY N SUFFICIN CONIIONS FOR NR- OPIMLIY RVSING CONROL PROBLM OF SOCSIC G-PNN SYSM WI POISSON JUMPS Xii Li * Qimi Z & Jiwei Si Soo o Memi Come Siee NiXi Uiveiy YiC 75 P R Ci Fy o Iomi Come Siee No Uiveiy o Nioiie YiC 75 P R Ci *Coeoi o -mi: imiz89@iom BSRC I i e we oie e-oim vei oo oem o oi e-eee yem wi Poio jm e mi im o e e i o ei eey we iie oiio o eoimiy iie y e vei oo oem e oo o e mi e i e o e' viio iie ome eime o e e e joi oee e i I ô ' om Boe- vi-gy' ieiy e eey iie oiio o i yem e eie eme i ive o iio Keywo: Soi e-eee oio yem; Poio jm; Ne-oimiy; Neey oiio; iie oiio MR: 65B5;65;9B7;6 INROUCION N MOL Soi ieei eio e vey imo i moe o iooi emi yi eoomi yem Soi ieei eio wi jm ve ee o my iio i e eoomi ie eve e o iee eieei e y o oi e-eee yem eeive e e o eio I i e we e oi o ioe e-oim vei oo oem o oi e-eee yem wi Poio jm oow: w N m P Q wee Q : ; i e ie eey i e eiiy iev e oe mee me oow: : e eiy o e oio o e ime ; oio o e ime ; : e vee eiiy io o e oio ime ; m eme i e oio o e ime ; : e iii e iiio o e e oio; P o oio ime ; ; : e vee moiy io o e : e io o : e : e vei eo io wi i e oo vie i e moe iie: : L Q : e i Q wee i L Q i ; N : e oi eio eei o ee eviome o e oio yem mi ee emiio im o e eei oje o o Fo e yem LS W XJ W ie e oveee o e emi-imii e meo Li P W iveie e mei yi Fo e yem wio Poio jm we 66
2 IJRRS 4 M Sáez-Bee & O Some me o Feee Z M iveie e oveee o mei yi e oveee o mei oio e oveee o e emi-imii e meo ee 45 Z Li Nie 6 ie e eiee iee eoei iiy Z 7 ie e eiee iee o o oio Fo eemiii yem we ee ve ee my wo o e oim oo oem Fo eme Lo 8 ie oim vei oo oem o e-eee omei yem o eie Zo 9 Ce e o oim oo eeivey S i iveie oim vei o oie e-eee oio ymi we ow ee ve ee ew e o e oi o e-oim oo oem o oi ee oweve o e e o o owee ee i ie wo o oo o oi e-eee yem wi Poio jm So e im o i e i o y e-oimiy vei oo oem o oi e-eee yem wi Poio jm Moe eiey we ei eey we iie oiio o e-oimiy ee oiio e eie i em o joi oe oeoi o e oi i ieei eio omoe ey mimm oiio o e mioi O mi e i e o e e' viio iie i e i eeio o 4- ee e emie o i e wi oe oow: I eio we ive e mio oio ome i eiiio emm I eio eio 4 we ei eey we iie oiio o e oimiy I eio 5 we ovie eme o ie o e PRLIMINRIS OF PROBLM I i e e V L L wee e eeize i eivive V i Sooev e L V V V i e e o V We eoe y e om i V V eeivey; y e iy o ewee V V y e o i K i e ee ie e Fo oeo B K ei e e o oe ie oeo om K io we eoe y B e ie-smi om ie B BWB oo i e e P e omee oiiy e wi iio iyi e oiio ie i i iei i oio wie oi P - e Le e Wiee oe eie o omee oiiy e P i i ve i e ee ie e K wi ieme ovie oeo Le e e -e eee y e i mie eive o o we me e Poio oe N i ieee o e Bowi moio ee wio o o eeiy e eee o o e ime iev i e ve io i eie oow: V Sie e ojeive o i e i o y e-oim e oim oo o e yem we ive e eie eiiio o e-oimiy ive i Zo 67
3 IJRRS 4 M Sáez-Bee & O Some me o Feee 68 eiiio -oim Fo ive i e -oim i V o o iiey m wee i io o iyi e eime i e eo o I C o ome ieee o e o C e i e e-oim wi oe oo i e we wi wie L o o o L eie : F o we me e oowi: Q Q L o wee i eee y zeo o e oio i oe oioy ieeie wi ee o ee ei o K K m Q m m we o 4 e o 5 L 6 ee e o e mioi io i ive y : F Ue e mio ee i ie oio L wi ove wee L eoe e ie e o -e oee Fo y wi i oeoi e jeoy We ioe e joi eio o o
4 IJRRS 4 M Sáez-Bee & O Some me o Feee oem e joi eio e wie : N Q I i we ow e mio Moeove ie e oe y ee ei o Noe e oe i e joi oe oeoi o e oi e-eee yem e joi eio mi oe oy oe - e oio F C ieee o e oio o e joi eio iy e oowi eime: Q Le e e' viio iie wi wi e e i e ee Q C Lemm e Le V e omee mei e F : V R e owe emioio io oe om eow I o e ee ei V e o y ee ei V F i F V i F F ii iii F F o V Fo we eie P : Wee i e o mee o e Leee mee wi e oiiy mee I i we ow i omee mei e ee4 NCSSRY CONIIONS OF NR-OPIMLIY i i e mi e o i e I i eio we eive eey oiio o oo o e e-oim Lemm Fo y ee i o o wi e oeoi jeoie i o Poo Fi we me yi I ô ' om o o y we oi 69
5 IJRRS 4 M Sáez-Bee & O Some me o Feee 64 N N N wee : Coiei e i em i e i o we ve m e y 5 o e ieiy we i m Le eime e i em i e i o Now e o e eo em o y Boe-vi-Gy' ieiy we ve
6 IJRRS 4 M Sáez-Bee & O Some me o Feee K K K o ome oiive o K K yi N N o e ie o we ve N N N N O O e y Boe-vi-Gy' ieiy ee ei oiive o K 4 K we ve K K N N Wi 6 o e ove ive i K Bi 5 we ive e oowi emm I ive e oiiy o e oio o e joi eio wi ee o e oo vie I y ey oe i ovi e eey oiio Lemm Fo y iyi ee i o o y o wi e oeoi jeoie e oio o e oeoi joi eio i o 7 Poo Now e ove 7 Noi iie e oowi w oi ieei eio:
7 IJRRS 4 M Sáez-Bee & O Some me o Feee 64 Q N Le e e oio o e oowi ie oi ieei eio S: N Wee i e iiive io o y Noe e eiee iee o oio o e ove eio e veiie y mio e e we ve 8 Wee Noe e i ie em o 8 i oe e o O e oe yi I ô ' om o i eeio we oi 9 m We oee o eime e eo em i e i ie o 9 Fi
8 IJRRS 4 M Sáez-Bee & O Some me o Feee 64 Fom Lemm i e o y i eiy e Ui imi me eveoe ove we eiy ove 4 I oow om 5 Ne we oee o eime e i em i e i ie o 9 By 5 i e i oe we oi We oee o eime e i em i e i ie o 9 Fom Lemm mio 5 8 m I oow om 4 9 Now we ve omee e oo o Lemm eoem me 6 o ee i o o y -oim e eey oiio
9 IJRRS 4 M Sáez-Bee & O Some me o Feee m o wee i e oio o joi eio i e mioi io Poo Le e wo e o omee i Se : Fom mio i i ey o ee J i oio o eowe wi e mei eie y yi e' viio iie wi ee i miie oo 7 o y wee i me i oim o e yem wi e ew o io Le e ie Fo y eie e ie viio o oewie e 8 Fom 8 Lemm yo' eio we eive 9 Le we oie mio Lemm Le we e ie
10 IJRRS 4 M Sáez-Bee & O Some me o Feee 645 Se : Neey oiio o Le ee e y i e we eime e oowi ieee e we ve So e eio 7 i ove 4 SUFFICIN CONIIONS OF NR-OPIMLIY I i eio we wi ove e e-mimm oiio o e mioi io i iie oiio o e-oimiy e iio mio eoem 4 Le e e e-oim oio o e e eio i e oio o e joi eio oeoi o me i ove o e i o ome 4 e 4 i Wee i o wi i ieee om Poo Le i eie ew mei o oow 4 y Wee 44 y eie io I o y: I ime omio ow y I I wi imie I i oio o wi ee o By 4 e' emm ee ei
11 IJRRS 4 M Sáez-Bee & O Some me o Feee m wee 46 y By me i e ove e ie mimm oiio 46 imie oiwie mimm oiio mey o e 47 m Ui mio we ove 48 y y By e oviy o we ve 49 By iei o ie oi 48 e we oi 4 O e oe y yi I ô ' om eeivey o y i eeio we oi 4 Comii 4 4 i e e e we oi 4 Sie i iy o e eie e oow 5 N XMPL Coie e oowi oi e-eee yem:
12 IJRRS 4 M Sáez-Bee & O Some me o Feee e N eewio o o eeiy o eee o o e e-oim oo oem i I e yem i e Bowi moio We e i oem i o omio y i L W V ooev e wi eeme iyi e oy oiio ove m Cey e oeo iy e mio i o yem o ove i oem we wie ow e mioi io: So e joi eio i 5 N Sovi eio 5 5 o y miie oo i we ve Lemm Lemm e 6 i vi wi i e eey iie oiio o e-oimiy i i oo oem e oeoi mioi e I i -oim e eey oiio o 5 i 5 wee i o Fo eme i we e
13 IJRRS 4 M Sáez-Bee & O Some me o Feee wee i iiey m mee e i ie -oim oo Moeove ee o e oio o iyi e mio 6 we oe So we i 5 i o e iie oiio o 5 6 CKNOWLGMNS e o wo ie o e eeee o ei vey e eio wi ey imove i e e o wo ie o e eeee o ei vey e omme wi ey imove i e e ee w oe y e Nio N Siee Foio No 64; 64 Ci RFRNCS LS W XJ W Coveee o e emi-imii e meo o oi e-eee oio eio wi Poio jm M Moe R Li WK P Q W Nmei yi o oi e-eee oio eio wi Poio jm J M QM Z C Coveee o mei oio o oi e-e oio yem ie Memi Comio QM Z Coveee o mei oio o o oi e-eee i yem wi Movi wii oomi Moei WJ M QM Z Coveee o e Semi-imii e Meo o Soi e-eee Poio io wi Movi Swii Lee Noe i Come Siee QM Z W Li ZK Nie iee iee eoei iiy o oi e-eee oio M Com QM Z CZ iee Uiee o So Soio o Soi e-eee Poio Lee Noe i Come Siee ZX Lo Oim vei oo oem o e-eee omei yem o eie M Com C Zo MS W P Zo Oim Coo o vei o e-eee Peo-Pey Syem M Com Moe JJ Ce ZR e Oim oo o oie e-iie oio yem M Com S i Oim vei o oie e-eee oio ymi M XY Zo Soi e-oim oo: Neey iie oiio o e-oimiy SIM J Coo Oim I e No ove miimiio oem B me M So NS B Mezei Neey oiio o oimiy o iio wi omoo i Soi K Bi N Ke B Mezei Neey iie oiio o e-oimiy i Soi oo o FBS Syem Coo Lee
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