A water agency faced with quantity-quality management of a groundwater resource

« A Water Agency faced with Quantity-quality Management of a Groundwater Resource »

Katrin ERDLENBRUCH Mabel TIDBALL Georges ZACCOUR

DR n°2012-09

❆ ❲❛#❡% ❆❣❡♥❝② ❢❛❝❡❞ ✇✐#❤ /✉❛♥#✐#②✲/✉❛❧✐#② ♠❛♥❛❣❡♠❡♥# ♦❢ ❛

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❏❊▲ ❝❧❛""✐✜❝❛#✐♦♥✿ ❍✷✸✱ ◗✶✺✱ ◗✷✺✳

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❝✐❡%

✶ ■♥#$♦❞✉❝#✐♦♥

❚❤❡ ♠❛♥❛❣❡♠❡♥' ♦❢ ❣*♦✉♥❞✇❛'❡* ✐/ ❛ '②♣✐❝❛❧ ❝♦♠♠♦♥✲♣♦♦❧ *❡♥❡✇❛❜❧❡ *❡/♦✉*❝❡ ♣*♦❜❧❡♠ ✇❤❡*❡

/❡✈❡*❛❧ ✉/❡*/ ❤❛✈❡ '♦ /❤❛*❡ ❛ /❛♠❡ *❡/♦✉*❝❡ /'♦❝❦✱ ✇✐'❤✱ ❤♦✇❡✈❡*✱ ♦♥❡ ❛❞❞✐'✐♦♥❛❧ ✐♠♣♦*'❛♥' ❢❡❛✲

'✉*❡✱ ♥❛♠❡❧②✱ '❤❡ 9✉❛❧✐'② ♦❢ '❤❡ /'♦❝❦✳ ❚❤❡*❡❢♦*❡✱ ❛♥② ❛''❡♠♣' '♦ *❡❣✉❧❛'❡ '❤❡ ✉/❡ ♦❢ ✇❛'❡*✱

❤❛/ '♦ '❛❝❦❧❡ '❤❡ ❡①'❡*♥❛❧✐'✐❡/ *❡❧❛'❡❞ '♦ ❜♦'❤ 9✉❛♥'✐'② ❛♥❞ 9✉❛❧✐'②✳ ■♥ '❤✐/ ♣❛♣❡*✱ ✇❡ ❝♦♥/✐❞❡*

❛♥ ❡♥❞♦❣❡♥♦✉/ ♣♦❧❧✉'✐♦♥ ❡①'❡*♥❛❧✐'② ❢*♦♠ ❛❣*✐❝✉❧'✉*❛❧ ♣*♦❞✉❝'✐♦♥ ❛♥❞ ❞✐/❝✉// ♦♣'✐♠❛❧ 9✉❛♥'✐'②✲

9✉❛❧✐'② *❡❣✉❧❛'✐♦♥ ❜② ❛ ✇❛'❡* ❛❣❡♥❝② ✇✐'❤ *❡/'*✐❝'❡❞ *❡❣✉❧❛'♦*② ♣♦✇❡*✳

❆ /✐❣♥✐✜❝❛♥' ❧✐'❡*❛'✉*❡ ❤❛/ /♦ ❢❛* ❛♥❛❧②③❡❞ '❤❡ ♥❡❡❞ ❢♦* ♣✉❜❧✐❝ ✐♥'❡*✈❡♥'✐♦♥ '♦ *❡❣✉❧❛'❡ ♣*✐✲

✈❛'❡ ❡①♣❧♦✐'❛'✐♦♥ ♦❢ ❣*♦✉♥❞✇❛'❡*✳ ❯/✐♥❣ ❛ /✐♠♣❧❡ 9✉❛♥'✐'② ♠♦❞❡❧ ✇✐'❤ /'♦❝❦ ❛♥❞ ♣✉♠♣✐♥❣ ❝♦/'

❡①'❡*♥❛❧✐'✐❡/✱^✶ ●✐//❡* ❛♥❞ ❙❛♥❝❤❡③ ❬✺❪ ❛*❣✉❡❞ '❤❛' '❤❡ ❞✐✛❡*❡♥❝❡ ❜❡'✇❡❡♥ '❤❡ ❝♦♠♣❡'✐'✐✈❡ ❛♥❞ '❤❡

♦♣'✐♠❛❧ ♦✉'❝♦♠❡ ✐/ '♦♦ /♠❛❧❧ '♦ ❥✉/'✐❢② ♣♦❧✐❝② ✐♥'❡*✈❡♥'✐♦♥ ✭/❡❡ ❑♦✉♥❞♦✉*✐ ❬✻❪ ❢♦* ❛ /✉*✈❡②✮✳ ❍♦✇✲

❡✈❡*✱ '❤❡ ❝♦♥/✐❞❡*❛'✐♦♥ ♦❢ ♠♦*❡ ❝♦♠♣❧✐❝❛'❡❞ *❡/♦✉*❝❡ ♣*♦❜❧❡♠/ ❛♥❞ ♦'❤❡* ❡①'❡*♥❛❧✐'✐❡/ ❤❛/ /❤♦✇♥

'❤❛' ♣✉❜❧✐❝ ✐♥'❡*✈❡♥'✐♦♥ ❝❛♥ ❜❡ ♥❡❝❡//❛*②✱ ❡✳❣✳✱ ✇❤❡♥ /❡✈❡*❛❧ *❡/♦✉*❝❡/ ❛*❡ ❧✐♥❦❡❞ '♦ ❡❛❝❤ ♦'❤❡*

✭❩❡✐'♦✉♥✐ ❛♥❞ ❉✐♥❛* ❬✶✼❪✮✱ ✇❤❡♥ ❣*♦✉♥❞✇❛'❡* ❤❛/ ❛ ❜✉✛❡* ✈❛❧✉❡ ❛❣❛✐♥/' /✉*❢❛❝❡ ✇❛'❡* /❝❛*❝✐'②

∗❈♦""❡$♣♦♥❞✐♥❣ ❛✉,❤♦"✿ ❦❛,"✐♥✳❡"❞❧❡♥❜"✉❝❤❅✐"$,❡❛✳❢"

†❑✳ ❊"❞❧❡♥❜"✉❝❤ ❛♥❞ ▼✳ ❚✐❞❜❛❧❧ ❛❝❦♥♦✇❧❡❞❣❡ ✜♥❛♥❝✐❛❧ $✉♣♣♦", ❢"♦♠ ,❤❡ ❋"❡♥❝❤ ◆❛,✐♦♥❛❧ ❘❡$❡❛"❝❤ ❆❣❡♥❝② ,❤"♦✉❣❤ ❣"❛♥, ❆◆❘✲✵✽✲❏❈❏❈✲✵✵✼✹✲✵✶✳

✶❚❤❡ $,♦❝❦ ❡①,❡"♥❛❧✐,② ❛"✐$❡$ ❜❡❝❛✉$❡ ,❤❡ ❡①,"❛❝,✐♦♥ ♦❢ ❡❛❝❤ "❡$♦✉"❝❡ ✉$❡" ✐$ ❝♦♥$,"❛✐♥,❡❞ ❜② ,❤❡ ,♦,❛❧ ❣"♦✉♥❞✲

✇❛,❡" $,♦❝❦❀ ,❤❡ ♣✉♠♣✐♥❣ ❝♦$, ❡①,❡"♥❛❧✐,② ❛"✐$❡$ ❜❡❝❛✉$❡ ,❤❡ ❝♦$, ♦❢ ♣✉♠♣✐♥❣ ❣"♦✉♥❞✇❛,❡" ❞❡♣❡♥❞$ ♦♥ ,❤❡ ❧❡✈❡❧

♦❢ ,❤❡ ❣"♦✉♥❞✇❛,❡" ,❛❜❧❡✱ $❡❡✱ ❡✳❣✳✱ M"♦✈❡♥❝❤❡" ❛♥❞ ❇✉", ❬✶✵❪✳

✶

✭!"♦✈❡♥❝❤❡" ❛♥❞ ❇✉"- ❬✶✵❪^✷✮✱ ♦" ✇❤❡♥ 5✉❛❧✐-② ✐9 -❛❦❡♥ ✐♥-♦ ❛❝❝♦✉♥- ✭❘♦9❡-❛✲!❛❧♠❛ ❬✶✷❪✮✳

❈♦♥❝❡"♥✐♥❣ ✇❛-❡" 5✉❛❧✐-②✱ ❛ ❢♦❝❛❧ ♣♦✐♥- ✇❛9 -❤❡ ✐99✉❡ ♦❢ 9❛❧-✇❛-❡" ✐♥-"✉9✐♦♥ ✐♥ ❝♦❛9-❛❧ ❛5✉✐❢❡"9

✭9❡❡✱ ❡✳❣✳✱ ❈✉♠♠✐♥❣9 ❬✷❪✱ ❩❡✐-♦✉♥✐ ❛♥❞ ❉✐♥❛" ❬✶✼❪✱ ❉✐♥❛" ❛♥❞ ❳❡♣❛♣❛❞❡❛9 ❬✸❪✱ ❚9✉" ❛♥❞ ❩❡♠❡❧

❬✶✸❪✱ ▼♦"❡❛✉① ❛♥❞ ❘❡②♥❛✉❞ ❬✾❪✮✳ ❲✐-❤ -❤❡ ✐♥-❡♥9✐✜❝❛-✐♦♥ ♦❢ ❛❣"✐❝✉❧-✉"❛❧ ♣"♦❞✉❝-✐♦♥✱ ✐♥❧❛♥❞

"❡9♦✉"❝❡9 ❛"❡ ✐♥❝"❡❛9✐♥❣❧② -❤"❡❛-❡♥❡❞ ❜② 5✉❛❧✐-② ❞❡❣"❛❞❛-✐♦♥✱ ✈✐❛ ♥✐-"❛-❡ ✐♥✜❧-"❛-✐♦♥✳ ❇❡❝❛✉9❡

❣"♦✉♥❞✇❛-❡" "❡9♦✉"❝❡9 ❛"❡ ♦❢-❡♥ ✉9❡❞ ❢♦" ❞"✐♥❦✐♥❣ ✇❛-❡"✱ -❤❡ ✐99✉❡ ✐9 ♦❢ ✐♠♣♦"-❛♥❝❡ ❛❧9♦ ♦✉-9✐❞❡

-❤❡ ❛❣"✐❝✉❧-✉"❛❧ 9❡❝-♦"✳ ❋♦" ✐♥9-❛♥❝❡✱ 5✉❛❧✐-② ✐9 ❛❞❞"❡99❡❞ ❜② 9❡✈❡"❛❧ ❊✉"♦♣❡❛♥ ♣♦❧✐❝✐❡9✱ 9✉❝❤

❛9 -❤❡ ❲❛-❡" ❋"❛♠❡✇♦"❦ ❉✐"❡❝-✐✈❡ ✭❉✐"❡❝-✐✈❡ ✷✵✵✵✴✻✵✴❊❈✮✱ ✇❤✐❝❤ ✜①❡9 -❤❡ ♦❜❥❡❝-✐✈❡ ♦❢ ✧❣♦♦❞

✇❛-❡" 5✉❛❧✐-②✧ ✐♥ ✷✵✶✺✱ -❤❡ ❉✐"❡❝-✐✈❡ ♦♥ -❤❡ ♣"♦-❡❝-✐♦♥ ♦❢ ❣"♦✉♥❞✇❛-❡" ❛❣❛✐♥9- ♣♦❧❧✉-✐♦♥ ❛♥❞

❞❡-❡"✐♦"❛-✐♦♥ ✭❉✐"❡❝-✐✈❡ ✷✵✵✻✴✶✶✽✴❊❈✮ ♦" -❤❡ ◆✐-"❛-❡9 ❉✐"❡❝-✐✈❡ ✭❉✐"❡❝-✐✈❡ ✾✶✴✻✼✻✴❊❊❈✮✱ ✇❤✐❝❤

9♣❡❝✐✜❝❛❧❧② -❛❝❦❧❡9 ♣♦❧❧✉-✐♦♥ ❢"♦♠ ❛❣"✐❝✉❧-✉"❛❧ ♣"♦❞✉❝-✐♦♥✳

❆ ❧❛"❣❡ ❧✐-❡"❛-✉"❡ ❡①✐9-9 ♦♥ -❤❡ ✐99✉❡9 ♦❢ ♥✐-"❛-❡ ♣♦❧❧✉-✐♦♥ ❛♥❞ ♥♦♥✲♣♦✐♥- 9♦✉"❝❡ ♣♦❧❧✉-✐♦♥ "❡✲

9✉❧-✐♥❣ ❢"♦♠ ❛❣"✐❝✉❧-✉"❛❧ ❛❝-✐✈✐-②✱ ✐♥❝❧✉❞✐♥❣ ❞②♥❛♠✐❝ ♠♦❞❡❧9 ✭❡✳❣✳✱ ❨❛❞❛✈ ❬✶✻❪✱ ❳❡♣❛♣❛❞❡❛9 ❬✶✹❪✮✳

❨❡-✱ ❛9 ❑♦✉♥❞♦✉"✐ ❬✻❪ 9-❛-❡9✱ -❤❡9❡ ♠♦❞❡❧9 ✧❣❡♥❡"❛❧❧② ❛✈♦✐❞ -❤❡ "❡❧❛-✐♦♥9❤✐♣ ❜❡-✇❡❡♥ ❝♦♥-❛♠✐✲

♥❛-✐♦♥ ❛♥❞ ✇❛-❡"✲✉9❡ ❞❡❝✐9✐♦♥9✳ ❚❤❡ ❛99❡99♠❡♥- ♦❢ ❤♦✇ ♠✉❝❤ ❣"♦✉♥❞✇❛-❡" 9❤♦✉❧❞ ❜❡ ♣✉♠♣❡❞

✐9 ❛❜9❡♥- ❢"♦♠ -❤❡9❡ ♠♦❞❡❧9✧✳ ❚❤❡ ✜"9- ✇♦"❦ -❤❛- ❜"✐♥❣9 -♦❣❡-❤❡" -❤❡9❡ ❛9♣❡❝-9 ✐♥ ❛ ❣❡♥❡"❛❧

❞②♥❛♠✐❝ 9❡--✐♥❣ ✐9 ❘♦9❡-❛✲!❛❧♠❛ ✭❬✶✶❪ ❛♥❞ ❬✶✷❪✮✳ ❙❤❡ ❝♦♥9✐❞❡"9 -❤❡ ✐♠♣❛❝- ♦❢ ❝♦♥-❛♠✐♥❛♥- ❞✐9✲

❝❤❛"❣❡9 ♦♥ ❣"♦✉♥❞✇❛-❡" 5✉❛❧✐-② ❛♥❞ ✐♥ ♣❛"-✐❝✉❧❛" -✇♦ 9♣❡❝✐❛❧ ❡✛❡❝-9✿ -❤❡ 9-♦❝❦ ❞✐❧✉-✐♦♥ ❡✛❡❝-

✇❤✐❝❤ ❞❡9❝"✐❜❡9 -❤❡ ❜❡♥❡✜❝✐❛❧ ✐♠♣❛❝- ♦❢ ✇❛-❡" ✈♦❧✉♠❡ ♦♥ ✇❛-❡" 5✉❛❧✐-②✱ ❛♥❞ -❤❡ ❝♦♥-❛♠✐♥❛-✐♥❣

✈❡❝-♦" ❡✛❡❝- ✐♥ ✇❤✐❝❤ ❝♦♥-❛♠✐♥❛♥-9 ✐♥✜❧-"❛-❡ ♠♦"❡ ❡❛9✐❧② ✐♥-♦ -❤❡ 9♦✐❧ ✇❤❡♥ ❝❛""✐❡❞ ✇✐-❤ ✐""✐❣❛✲

-✐♦♥ ✇❛-❡"✳ ❘♦9❡-❛✲!❛❧♠❛ 9❤♦✇9 -❤❛- ♣✉❜❧✐❝ "❡❣✉❧❛-✐♦♥ 9❤♦✉❧❞ ❛❞❞"❡99 ❜♦-❤ 5✉❛♥-✐-② ❛♥❞ 5✉❛❧✐-② -♦ ❜❡ ♦♣-✐♠❛❧✳ ❙❤❡ ❛❧9♦ ♥✉♠❡"✐❝❛❧❧② ❝♦♥✜"♠9 -❤❛- ♣♦❧✐❝② ✐♥-❡"✈❡♥-✐♦♥ ✐9 ❥✉9-✐✜❡❞ ❡✈❡♥ ✐❢ ❣❛✐♥9

❢"♦♠ 5✉❛♥-✐-② "❡❣✉❧❛-✐♦♥ ❛"❡ 9♠❛❧❧✱ ❛9 ✐♥ ●✐99❡" ❛♥❞ ❙❛♥❝❤❡③ ❬✺❪✱ ❜❡❝❛✉9❡ ♦❢ -❤❡ ✐♠♣♦"-❛♥❝❡ -♦

♠❡❡- 5✉❛❧✐-② 9-❛♥❞❛"❞9✳

❍♦✇❡✈❡"✱ ❘♦9❡-❛✲!❛❧♠❛ ✭✷✵✵✸✮ ❛♥❞ ♠♦9- ♦-❤❡" ❛"-✐❝❧❡9 ❝♦♥9✐❞❡" ❞②♥❛♠✐❝ -❛①❛-✐♦♥ ❛9 -❤❡ ♦♥❧② -♦♦❧ ❢♦" ♣♦❧✐❝② ✐♥-❡"✈❡♥-✐♦♥✳^✸ ❆❧-❤♦✉❣❤ ❛ ❞②♥❛♠✐❝ -❛① ❤❛9 ❛ ❝♦♥❝❡♣-✉❛❧ ❛♣♣❡❛❧✱ ✐- ✐9 5✉✐-❡ ✉♥"❡✲

❛❧✐9-✐❝ ✐♥ ♠❛♥② "❡❛❧✲❧✐❢❡ ❝♦♥-❡①-9✳ ■♥❞❡❡❞✱ ✐- "❡5✉✐"❡9 -❤❛- -❤❡ "❡❣✉❧❛-♦" ❝❤♦♦9❡9 ❛♥ ♦♣-✐♠❛❧ ♣♦❧✐❝② -❤❛- ❝❤❛♥❣❡9 ❝♦♥-✐♥✉♦✉9❧②✱ ❞❡♣❡♥❞✐♥❣ ♦♥ -❤❡ ✐♥❞✐✈✐❞✉❛❧ ❛❝-✐♦♥9 -❛❦❡♥✳ ❘♦9❡-❛✲!❛❧♠❛ ♣♦✐♥-9 ❛- 9♦♠❡ ✐♠♣❧❡♠❡♥-❛-✐♦♥ ♣"♦❜❧❡♠9 ❜✉- ❢♦❝✉9❡9 ♦♥ -❤♦9❡ ❧✐♥❦❡❞ -♦ ✐♥❢♦"♠❛-✐♦♥❛❧ ❝♦♥9-"❛✐♥-9 ♦♥ ✐♥✲

❞✐✈✐❞✉❛❧ ♣"♦❞✉❝-✐♦♥ ❛♥❞ ♣♦❧❧✉-✐♦♥ ❢✉♥❝-✐♦♥9✳ ■♥ -❤✐9 ♣❛♣❡"✱ ✇❡ 9-✉❞② -❤❡ ❝❛9❡ ✇❤❡"❡ -❤❡ ✇❛-❡"

"❡❣✉❧❛-♦" ✐♠♣♦9❡9 ❝♦♥9-❛♥- ♣♦❧✐❝✐❡9 ♦✈❡" ❛ "❡❛9♦♥❛❜❧❡ -✐♠❡ ♣❡"✐♦❞✱ ❢♦" ❡①❛♠♣❧❡ ❛ ②❡❛"✱ ✇❤✐❝❤

❝♦""❡9♣♦♥❞9 -♦ -❤❡ ❧❡♥❣-❤ ♦❢ ❛ ✜9❝❛❧ ❡①❡"❝✐9❡✳

❲❡ ❝♦♥9✐❞❡" ❛ ❣"♦✉♣ ♦❢ ✐""✐❣❛-✐♥❣ ❢❛"♠❡"9 ✉9✐♥❣ -❤❡ 9❛♠❡ ❣"♦✉♥❞✇❛-❡" "❡9♦✉"❝❡✳ ❋❡"-✐❧✐③❡"

✉9❡❞ ❜② -❤❡ ❢❛"♠❡"9 ❧❡❛❝❤❡9 ✐♥-♦ -❤❡ ❣"♦✉♥❞✇❛-❡" ❛♥❞ ❝❛✉9❡9 ♥✐-"❛-❡ ♣♦❧❧✉-✐♦♥✱ ♠✐-✐❣❛-❡❞ ❜② -❤❡ 9-♦❝❦ ❞✐❧✉-✐♦♥ ❡✛❡❝- ❛♥❞ -❤❡ ♥❛-✉"❛❧ ❞❡❝❛② "❛-❡ ♦❢ -❤❡ ❝♦♥-❛♠✐♥❛♥-✳ ❲❡ ❛99✉♠❡ -❤❛- -❤❡

❢❛"♠❡"9 ❛"❡ ❛-♦♠✐9-✐❝ ♣❧❛②❡"9 ✇❤♦ ♦♣-✐♠✐③❡ -❤❡✐" ✐♥❞✐✈✐❞✉❛❧ ♣❛②♦✛9 ✇✐-❤♦✉- -❛❦✐♥❣ ✐♥-♦ ❛❝❝♦✉♥- -❤❡ ✐♠♣❛❝- ♦❢ -❤❡✐" ❞❡❝✐9✐♦♥9✱ ✐✳❡✳✱ ✇❛-❡" ✇✐-❤❞"❛✇❛❧ ❛♥❞ ✉9❡ ♦❢ ❢❡"-✐❧✐③❡"9✱ ♦♥ -❤❡ 9-♦❝❦ ♦❢ ✇❛-❡"

❛♥❞ ✐-9 5✉❛❧✐-②✳ ■♥ ♦"❞❡" -♦ ✐♥9✉"❡ ❛ 9✉9-❛✐♥❛❜❧❡ ✉9❡ ♦❢ -❤❡ "❡9♦✉"❝❡✱ ❛ ✇❛-❡" ❛❣❡♥❝② ✐9 ✐♥ ❝❤❛"❣❡

♦❢ "❡❣✉❧❛-✐♥❣ -❤❡ 5✉❛♥-✐-② ❛♥❞ 5✉❛❧✐-② ♦❢ -❤❡ ❣"♦✉♥❞✇❛-❡"✳ ❘❡❣✉❧❛-✐♦♥ -❛❦❡9 -❤❡ ❢♦"♠ ♦❢ -❛①

✭♦" 9✉❜9✐❞②✮ ♦♥ ✇❛-❡" ✇✐-❤❞"❛✇❛❧ ❛♥❞ ♣♦❧❧✉-✐♦♥ ✭✐✳❡✳✱ ✉9❡ ♦❢ ❢❡"-✐❧✐③❡"9✮✳ ❲❡ 9❤❛❧❧ ❝♦♥9✐❞❡" ❛♥❞

❝♦♥-"❛9- -❤❡ "❡9✉❧-9 ♦❢ -❤❡ ❢♦❧❧♦✇✐♥❣ -❤"❡❡ 9❝❡♥❛"✐♦9✿

▲❛✐##❡③✲❢❛✐(❡ #❝❡♥❛(✐♦✿ ❆9 -❤❡ ♥❛♠❡ 9✉❣❣❡9-9✱ ✐♥ -❤✐9 ❝❛9❡ -❤❡ ✉9❡ ♦❢ ✇❛-❡" ✐9 ♥♦- "❡❣✉❧❛-❡❞✳

❚❤✐9 9❝❡♥❛"✐♦ ✐9 9❡❡♥ ❛9 ❛ ❜❡♥❝❤♠❛"❦✳

❘❡❣✉❧❛1✐♦♥ ✇✐1❤ ❜✉❞❣❡1 ❝♦♥#1(❛✐♥1✿ ■♥ -❤✐9 9❝❡♥❛"✐♦✱ -❤❡ ✇❛-❡" ❛❣❡♥❝② ✐9 ❡♥❞♦✇❡❞ ✇✐-❤

✷❘❡❞✉❝✐♥❣ ❣(♦✉♥❞✇❛,❡( -,♦❝❦- ,❤❡♥ ❣❡♥❡(❛,❡- ,❤❡ -♦✲❝❛❧❧❡❞ (✐-❦✲❡①,❡(♥❛❧✐,②✱ -❡❡ 5(♦✈❡♥❝❤❡( ❛♥❞ ❇✉(, ✶✾✾✸ ❬✶✵❪✳

✸❆- ❛(❣✉❡❞ ❜② 5(♦✈❡♥❝❤❡( ❛♥❞ ❇✉(, ❬✶✵❪✱ ♣❡(♠✐, ❛❧❧♦❝❛,✐♦♥ ❞♦❡- -♦❧✈❡ ♥❡✐,❤❡( ,❤❡ (✐-❦ ❡①,❡(♥❛❧✐,② ♥♦( ,❤❡ ❝♦-,

❡①,❡(♥❛❧✐,②✳

✷

❛ ❜✉❞❣❡& ❛& &❤❡ (&❛)& ♦❢ &❤❡ ♣❧❛♥♥✐♥❣ ❤♦)✐③♦♥ ❛♥❞ ♠✉(& ❜❛❧❛♥❝❡ ✐&( ❜♦♦❦ ❛& &❤❡ ❡♥❞ ♦❢ &❤❡ ✜(❝❛❧

❡①❡)❝✐(❡✳

❘❡❣✉❧❛&✐♦♥ ✇✐&❤ ♥♦ ❜✉❞❣❡& ❝♦♥/&0❛✐♥&✿ ■♥ &❤✐( (❝❡♥❛)✐♦✱ &❤❡ ✇❛&❡) ❛❣❡♥❝② ❞♦❡( ♥♦&

❞✐(♣♦(❡ ♦❢ ❛ ❜✉❞❣❡& ❛♥❞ ✐&( ♣)♦❜❧❡♠ ✐( (✐♠♣❧② &♦ ✜♥❞ &❤❡ ♦♣&✐♠❛❧ &❛① ♦) (✉❜(✐❞② ♦❢ ✇❛&❡) ✇✐&❤✲

❞)❛✇❛❧ ❛♥❞ ❢❡)②✐❧✐③❡) ✉(❡✳ ❚❤✐( ❛❧❧♦✇( ✉( &♦ ❜❡&&❡) ✉♥❞❡)(&❛♥❞ &❤❡ )❡(✉❧&( ✐♥❝❧✉❞✐♥❣ &❤❡ ❜✉❞❣❡&

❝♦♥(&)❛✐♥&✳

■♥ ❛❧❧ (❝❡♥❛)✐♦(✱ ✇❡ )❡&❛✐♥ ❛ ♠♦❞❡ ♦❢ ♣❧❛② = ❧❛ ❙&❛❝❦❡❧❜❡)❣ ✇❤❡)❡ &❤❡ )♦❧❡ ♦❢ ❧❡❛❞❡) ✐( ❛((✉♠❡❞

❜② &❤❡ ✇❛&❡) ❛❣❡♥❝② ❛♥❞ &❤❡ ❢❛)♠❡)( ❛)❡ &❤❡ ❢♦❧❧♦✇❡)(✳ ❆( &❤❡ ❝♦♥(✐❞❡)❡❞ ♣❧❛♥♥✐♥❣ ❤♦)✐③♦♥ ✐(

(❤♦)& ✭❛ ✜(❝❛❧ ②❡❛)✮✱ ✇❡ (✉♣♣♦(❡ &❤❛& &❤❡ ✇❛&❡) ❛❣❡♥❝② (❡❡❦( ❝♦♥(&❛♥& &❛① ♦) (✉❜(✐❞② ♣♦❧✐❝✐❡(✳

❲❡ ❜❡❧✐❡✈❡ &❤❛& ❝♦♥(&❛♥& ♣♦❧✐❝✐❡( ❛)❡ ♠♦)❡ )❡❛❧✐(&✐❝✱ ❢)♦♠ ❛♥ ✐♠♣❧❡♠❡♥&❛❜✐❧✐&② ♣♦✐♥& ♦❢ ✈✐❡✇✱ &❤❛♥

&✐♠❡✲✈❛)②✐♥❣ ♦♥❡(✳ ■♥❞❡❡❞✱ ✐& ✇♦✉❧❞ ❜❡ ♣❡❝✉❧✐❛) &♦ )❡D✉✐)❡ &❤❛& &❤❡ ✇❛&❡) ❛❣❡♥❝② ♣)♦❞✉❝❡( ❛

♥❡✇ &❛① )❛&❡ ❛& ❡❛❝❤ ✐♥(&❛♥& ♦❢ &✐♠❡ &❤)♦✉❣❤♦✉& &❤❡ ❞✉)❛&✐♦♥ ♦❢ ❛ ✜(❝❛❧ ❡①❡)❝✐(❡✳

❚❤❡)❡❢♦)❡✱ ✇❡ ❝♦♥(&)✉❝& ❛♥ ♦♣❡♥✲❧♦♦♣ ❞②♥❛♠✐❝ ❙&❛❝❦❡❧❜❡)❣ ❣❛♠❡ &♦ ♠♦❞❡❧ ❢❛)♠❡)(✬ ♦♣&✐♠❛❧

❞❡❝✐(✐♦♥( ✐♥ &❤❡ ❢❛❝❡ ♦❢ &❤❡(❡ ❝♦♥(&❛♥& ✐♥❝❡♥&✐✈❡ ♣♦❧✐❝✐❡(✳^✹

❲❡ ✜♥❞ &❤❛&✱ ✉♥❞❡) ❣✐✈❡♥ ❝♦♥❞✐&✐♦♥(✱ &❤❡)❡ ✐( ✐♥❞❡❡❞ ❛ (❡& ♦❢ ❝♦♥(&❛♥& ♦♣&✐♠❛❧ ♣♦❧✐❝✐❡( ✇❤✐❝❤

❢✉❧✜❧❧( ❛❧❧ &❤❡ ❝♦♥(&)❛✐♥&( &❤❡ ❲❛&❡) ❆❣❡♥❝② ❤❛( &♦ )❡(♣❡❝&✳ ■♥ ♦✉) (✐♠♣❧❡ ❡①❛♠♣❧❡✱ &❤❡ ♦♣&✐♠❛❧

♣♦❧✐❝②✲♠✐① ❝♦♥(✐(&( ✐♥ ❛♥ ✐♥♣✉&✲&❛① ♦♥ ✇❛&❡) ✇✐&❤❞)❛✇❛❧( ❛♥❞ ❛♥❞ ✐♥♣✉& (✉❜(✐❞② ♦♥ ❢❡)&✐❧✐③❡) ✉(❡✳

❚❤❡ ♣❛♣❡) ✐( (&)✉❝&✉)❡❞ ❛( ❢♦❧❧♦✇(✳ ■♥ (❡❝&✐♦♥ ✷ ✇❡ ♣)❡(❡♥& &❤❡ ♣)♦❜❧❡♠✱ ❛ (✐♠♣❧✐✜❡❞ ❛❣)♦✲

❡❝♦♥♦♠✐❝ ♠♦❞❡❧ ✐♥❝❧✉❞✐♥❣ ❛ ❣)♦✉♥❞✇❛&❡) )❡(♦✉)❝❡✳ ■♥ (❡❝&✐♦♥ ✸✱ ✇❡ ♣)❡(❡♥& &❤❡ ❙&❛❝❦❡❧❜❡)❣

❣❛♠❡ ❛♥❞ ❝❤❛)❛❝&❡)✐③❡ ✐&( (♦❧✉&✐♦♥✳ ■♥ (❡❝&✐♦♥ ✹ ✇❡ ❝♦♥(✐❞❡) &✇♦ ❡①❛♠♣❧❡( ❛♥❞ ❝♦♠♣✉&❡ &❤❡

♦♣&✐♠❛❧ &❛①❛&✐♦♥ ♣♦❧✐❝② ✐♥ &❤✐( ❝♦♥&❡①&✳ ❋✐♥❛❧❧②✱ ✐♥ &❤❡ ❧❛(& (❡❝&✐♦♥✱ ✇❡ ❝♦♥❝❧✉❞❡ ❛♥❞ ❣✐✈❡ (♦♠❡

♣❡)(♣❡❝&✐✈❡( ❢♦) ❢✉&✉)❡ )❡(❡❛)❝❤✳

✷ ❚❤❡ ▼♦❞❡❧

✷✳✶ ❚❤❡ ❋❛(♠❡(*

❈♦♥(✐❞❡) ❛ ❣)♦✉♣ ♦❢ N ❢❛)♠❡)( ❣)♦✇✐♥❣ ❛ (✐♥❣❧❡ ❛❣)✐❝✉❧&✉)❛❧ ♣)♦❞✉❝& ❛♥❞ ❧♦❝❛&❡❞ ❛❜♦✈❡ ❛ (❛♠❡

❣)♦✉♥❞✇❛&❡) )❡(♦✉)❝❡✳ ❚✐♠❡t✐( ❝♦♥&✐♥✉♦✉( ❛♥❞ &❤❡ ♣❧❛♥♥✐♥❣ ♣❡)✐♦❞ ❣✐✈❡♥ ❜② &❤❡ ✐♥&❡)✈❛❧[0, T]✳

❚❤❡ ❛❣)✐❝✉❧&✉)❛❧ ♣)♦❞✉❝&✐♦♥ yi(t), ♦❢ ❢❛)♠❡) i = 1, ...N, ❛& &✐♠❡ t ∈ [0, T], ❞❡♣❡♥❞( ♦♥ &✇♦

✐♥♣✉&(✱ ♥❛♠❡❧②✱ &❤❡ D✉❛♥&✐&② ♦❢ ❢❡)&✐❧✐③❡) (♣)❡❛❞ ♦♥ ❝✉❧&✐✈❛&❡❞ ❧❛♥❞✱ f_i(t)✱ ❛♥❞ &❤❡ ✈♦❧✉♠❡ ♦❢

✐))✐❣❛&✐♦♥ ✇❛&❡)✱ w_i(t)✱ &❤❛& ❡❛❝❤ ❢❛)♠❡) ♣✉♠♣( ✐♥ &❤❡ ❣)♦✉♥❞✇❛&❡) )❡(♦✉)❝❡✳ ❲❡ ❛((✉♠❡ &❤❛&

&❤❡ ♣)♦❞✉❝&✐♦♥ ❢✉♥❝&✐♦♥ yi(wi(t), fi(t))✐( ✐♥❝)❡❛(✐♥❣ ✐♥ ❜♦&❤ ✐♥♣✉&(✱ ❜✉& ❛& ❞❡❝)❡❛(✐♥❣ )❡&✉)♥( &♦

(❝❛❧❡✱ &❤❛& ✐(✱

∂y_i

∂w_i ≥0,∂y_i

∂f_i ≥0,∂²y_i

∂w²_i ≤0,∂²y_i

∂f_i² ≤0, ∂²y_i

∂w_i∂f_i ≥0. ✭✶✮

❚❤❡ ♣♦(✐&✐✈❡ (✐❣♥ ♦❢ &❤❡ ❧❛(& ❞❡)✐✈❛&✐✈❡ ♠❡❛♥( &❤❛& ✐))✐❣❛&✐♦♥ ✇❛&❡) ❛♥❞ ❢❡)&✐❧✐③❡)( ❛)❡ ❝♦♠♣❧❡✲

♠❡♥&❛)② ✐♥♣✉&(✳

❙♦✐❧ ❢❡)&✐❧✐③❛&✐♦♥ ❛♥❞ ✇❛&❡) ♣✉♠♣✐♥❣ ❛)❡ ❝♦(&❧②✳ ❚❤❡ ❢❡)&✐❧✐③❛&✐♦♥ ❝♦(&c_f(·)✱ ✇❤✐❝❤ ✐♥❝❧✉❞❡(

&❤❡ ♣✉)❝❤❛(✐♥❣ ❝♦(& ♦❢ ❢❡)&✐❧✐③❡)( ❛♥❞ &❤❡✐) ❧❛♥❞ ❛♣♣❧✐❝❛&✐♦♥✱ ❞❡♣❡♥❞( ♦♥ &❤❡ D✉❛♥&✐&② ♦❢ ❢❡)&✐❧✐③❡)

✉(❡❞✳ ❲❡ ❛((✉♠❡ &❤❛& &❤✐( ❝♦(& ✐( ❝♦♥✈❡① ❛♥❞ ✐♥❝)❡❛(✐♥❣✱ ✐✳❡✳✱

∂c_f

∂fi

≥0, ∂²c_f

∂f_i² ≥0. ✭✷✮

❚❤❡ ❝♦(& ♦❢ ♣✉♠♣✐♥❣ ❛♥❞ ❞✐(&)✐❜✉&✐♥❣ ✇❛&❡)c_w(·) ❞❡♣❡♥❞( ♦♥ &❤❡ ✈♦❧✉♠❡ ✇✐&❤❞)❛✇♥✱ ❛♥❞ ♦♥

&❤❡ ❞❡♣&❤ ♦❢ &❤❡ ❛D✉✐❢❡)✱ ❞❡♥♦&❡❞D(t)✳ ❲❡ ❛((✉♠❡ &❤❛&c_w(·)✐( ❥♦✐♥&❧② ❝♦♥✈❡① ✐♥ ✐&( ❛)❣✉♠❡♥&(

✹❋♦" ❛ ❣❡♥❡"❛❧ ❢❡❡❞❜❛❝❦ ❙.❛❝❦❡❧❜❡"❣ ♠♦❞❡❧ 0❡❡ ❢♦" ❡①❛♠♣❧❡ ❳❡♣❛♣❛❞❡❛0 ✶✾✾✺ ❬✶✺❪ ✮✳

✸

❛♥❞ #❛$✐#✜❡# $❤❡ ❢♦❧❧♦✇✐♥❣ ❛##✉♠♣$✐♦♥#✿

∂cw

∂w_i ≥0, ∂²cw

∂w_i² ≥0, ∂cw

∂D ≥0, ∂²cw

∂D² ≥0, ∂²cw

∂w_i∂D ≥0. ✭✸✮

❚❤❡ ✜6#$ ❞❡6✐✈❛$✐✈❡ #$❛$❡# $❤❛$ $❤❡ ❝♦#$ ♦❢ ♣✉♠♣✐♥❣ ✇❛$❡6 ✐# ✐♥❝6❡❛#✐♥❣ ✐♥ $❤❡ ✐♥♣✉$ w_i✳ ❚❤❡

#❡❝♦♥❞ ❞❡6✐✈❛$✐✈❡ ✐♠♣❧✐❡# $❤❛$ $❤❡ ♠❛6❣✐♥❛❧ ❝♦#$ ♦❢ ♣✉♠♣✐♥❣ ✐# ✐♥❝6❡❛#✐♥❣✳ ❚❤✐# ❝❛♥ ❜❡ ❥✉#$✐✜❡❞

❜② $❤❡ ❢❛❝$ $❤❛$ $❤❡ ❝♦♥#✉♠♣$✐♦♥ ♦❢ ❡♥❡6❣② ✐♥❝6❡❛#❡# ♥♦♥✲❧✐♥❡❛6❧② ✐♥ $❤❡ ✈♦❧✉♠❡ ♦❢ ♣✉♠♣❡❞ ✇❛$❡6✳

❚❤❡ #✐❣♥ ♦❢ $❤❡ $❤✐6❞ ❞❡6✐✈❛$✐✈❡ ❝❛♣$✉6❡# $❤❡ ✐❞❡❛ $❤❛$ $❤❡ ❧❛6❣❡6D✭♠❡❛♥✐♥❣ $❤❛$ ✇❛$❡6 ♠✉#$ ❜❡

❧✐❢$❡❞ ❛ ❧♦♥❣❡6 ❞✐#$❛♥❝❡ $♦ #✉6❢❛❝❡✮✱ $❤❡ ❤✐❣❤❡6 $❤❡ ❝♦#$ ♦❢ ♣✉♠♣✐♥❣ ✇❛$❡6✳ ■♥❝6❡❛#✐♥❣ ♠❛6❣✐♥❛❧

6❡$✉6♥# ❛6❡ ❛##✉♠❡❞ $❤6♦✉❣❤ ^∂_∂D^2cwi2 ≥0.❋✐♥❛❧❧②✱ $❤❡ ♥♦♥ ♥❡❣❛$✐✈❡♥❡## ♦❢ $❤❡ ❧❛#$ ❞❡6✐✈❛$✐✈❡ ♠❡❛♥#

$❤❛$ $❤❡ ♠❛6❣✐♥❛❧ ❝♦#$ ♦❢ ♣✉♠♣✐♥❣ ✇❛$❡6 ♠✐❣❤$ ❜❡ ✐♥❝6❡❛#✐♥❣ ✐♥D✳

❚❤❡ ❢❛6♠❡6# ❛6❡ ♣6✐❝❡✲$❛❦❡6# ❛♥❞ $❤❡ ♣6✐❝❡p_i♦❢ $❤❡ ❛❣6✐❝✉❧$✉6❛❧ ♣6♦❞✉❝$ ✐# ❝♦♥#$❛♥$ $❤6♦✉❣❤✲

♦✉$ $❤❡ #❤♦6$ ❞✉6❛$✐♦♥ ♦❢ $❤❡ ♣❧❛♥♥✐♥❣ ❤♦6✐③♦♥✳ ■♥❞❡❡❞✱ $❤❡ ♣❡6✐♦❞ ❝♦♥#✐❞❡6❡❞ ✐# ❞❡✜♥❡❞ ❛# ❛

#❡❛#♦♥ ♦6 ❛ ✜#❝❛❧ ②❡❛6✳ ❋✉6$❤❡6✱ $❤❡ ❢❛6♠❡6# ❛6❡ #✉❜❥❡❝$ $♦ ♣✉❜❧✐❝ ♣♦❧✐❝✐❡# ♦❢ $❤❡ ✇❛$❡6 ❛❣❡♥❝②✱

♥❛♠❡❧②✱ $❤❡② ♣❛② ❛ $❛① τ ♦♥ $❤❡ ✉#❡ ♦❢ ♣♦❧❧✉$✐♥❣ ❢❡6$✐❧✐③❡6✱ ❛♥❞ ❛ $❛① φ ♦♥ ✐♥❞✐✈✐❞✉❛❧ ✇❛$❡6

✇✐$❤❞6❛✇❛❧#✳ ❈♦♥#❡E✉❡♥$❧②✱ $❤❡ i✬# ❛❣❡♥$ ♣6♦✜$ 6❡❛❞# ❛# ❢♦❧❧♦✇#✿

π_i= Z T

0

(piy_i(wi(t), fi(t))−c_w(D(t), wi(t))−c_f(fi(t))−τ f_i(t)−φw_i(t))dt. ✭✹✮

❲❡ ♠❛❦❡ $❤❡ ❢♦❧❧♦✇✐♥❣ $✇♦ 6❡♠❛6❦#✿

✶✳ ❚❤❡6❡ ✐# ♥♦ ❝♦♥❝❡♣$✉❛❧ ❞✐✣❝✉❧$② ✐♥ ❡①$❡♥❞✐♥❣ $❤❡ ♠♦❞❡❧ $♦ ❛♥ ♦❧✐❣♦♣♦❧✐#$✐❝ #❡$$✐♥❣ ✇❤❡6❡

$❤❡ ❢❛6♠❡6# ❝♦♠♣❡$❡ ✇✐$❤ ❛♥ ❤♦♠♦❣❡♥♦✉# ♣6♦❞✉❝$ K ❧❛ ❈♦✉6♥♦$✳ ❆❝$✉❛❧❧②✱ ✇❡ ❝❛♥ ❛❧#♦

❝♦♥#✐❞❡6 ❛ ❞✐✛❡6❡♥$✐❛$❡❞ ♣6♦❞✉❝$ ✭❡✳❣✳✱ ♦6❣❛♥✐❝ ❛♥❞ 6❡❣✉❧❛6✮✱ ✇❤❡6❡ $❤❡ ♣6✐❝❡ ❞♦❡# ♥♦$ ♦♥❧②

❞❡♣❡♥❞ ♦♥ $❤❡ E✉❛♥$✐$② ♣✉$ ♦♥ $❤❡ ♠❛6❦❡$✱ ❜✉$ ❛❧#♦ ♦♥ $❤❡ E✉❛❧✐$② ♦❢ ✐66✐❣❛$✐♦♥ ✇❛$❡6 ❛♥❞

$❤❡ E✉❛♥$✐$② ♦❢ ❢❡6$✐❧✐③❡6 ✉#❡❞ ✐♥ ❢❛6♠✐❣✳ ❖❜✈✐♦✉#❧②✱ $❤❡ ♠♦6❡ #♦♣❤✐#$✐❝❛$❡❞ $❤❡ ♠♦❞❡❧✱ $❤❡

♠♦6❡ ❝♦♠♣❧❡① $❤❡ ❝♦♠♣✉$❛$✐♦♥ ♦❢ $❤❡ ❡E✉✐❧✐❜6✐✉♠ ♣♦❧✐❝✐❡#✳

✷✳ ●✐✈❡♥ $❤❡ #❤♦6$✲$❡6♠ ♣❧❛♥♥✐♥❣ ❤♦6✐③♦♥✱ ✇❡ ❞♦ ♥♦$ ❞✐#❝♦✉♥$ ❢❛6♠❡6 i✬# ♣6♦✜$✳ ■♥❝❧✉❞✐♥❣ ❛

❞✐#❝♦✉♥$ ❢❛❝$♦6 ❞♦❡# ♥♦$ ♣♦#❡ ❛♥② ♣❛6$✐❝✉❧❛6 ❞✐✣❝✉❧$②✳

✷✳✷ ❚❤❡ ❉②♥❛♠✐❝,

❚❤❡ ❞❡♣$❤ ♦❢ $❤❡ ❛E✉✐❢❡6 ❞❡♣❡♥❞# ♦♥ ✇✐$❤❞6❛✇❛❧# ❜② ❢❛6♠❡6# ❛♥❞ ♦♥ ♥❛$✉6❛❧ 6❡❝❤❛6❣❡✳ ❉❡♥♦$❡

❜② r(t) $❤❡ ♠❡❛♥ 6❡❝❤❛6❣❡ 6❛$❡ ♦❢ $❤❡ ❣6♦✉♥❞✇❛$❡6 #$♦❝❦✳ ❚❤❡ ❡✈♦❧✉$✐♦♥ ♦❢ D ✐# ❞❡#❝6✐❜❡❞ ❜②

$❤❡ ❞✐✛❡6❡♥$✐❛❧ ❡E✉❛$✐♦♥

D˙ (t) =g X

i

wi(t), r(t)

!

, D(0) =D0 ❣✐✈❡♥, ✭✺✮

✇❤❡6❡D0 ✐# ❛ ♠❡❛#✉6❡♠❡♥$ ♦❢ $❤❡ ✐♥✐$✐❛❧ ✇❛$❡6 ❞✐#$❛♥❝❡✱ ✇✐$❤

∂g

∂w_i >0, ∂g

∂r <0.

❚❤❡ E✉❛❧✐$② ♦❢ $❤❡ ❣6♦✉♥❞✇❛$❡6 ❞❡$❡6✐♦6❛$❡# ✇✐$❤ $❤❡ E✉❛♥$✐$② ♦❢ ❢❡6$✐❧✐③❡6 ✉#❡❞ ❜② ❡❛❝❤ ❢❛6♠❡6✳

❋✉6$❤❡6✱ $❤❡ ❧❛6❣❡6 $❤❡ ✈♦❧✉♠❡ ♦❢ $❤❡ #$♦❝❦ ♦❢ ✇❛$❡6✱ $❤❡ ❤✐❣❤❡6 $❤❡ ❞✐❧✉$✐♦♥ ✭♠✐$✐❣❛$✐♦♥✮ ❝❛♣❛❝✐$②✱

❛♥❞ $❤❡ ❜❡$$❡6 $❤❡ E✉❛❧✐$②✳ ❆# $❤❡6❡ ✐# ❛ ♠♦♥♦$♦♥❡ 6❡❧❛$✐♦♥#❤✐♣ ❜❡$✇❡❡♥ $❤❡ ✈♦❧✉♠❡ ♦❢ ✇❛$❡6

❛♥❞ $❤❡ ❞❡♣$❤ ♦❢ $❤❡ ❛E✉✐❢❡6✱ ✇❡ ❝❛♥ ♠♦❞❡❧ $❤❡ ❡✈♦❧✉$✐♦♥ ♦❢ E✉❛❧✐$② ❛# ❢✉♥❝$✐♦♥ ♦❢ D❛♥❞ ♦❢ $❤❡

✇❛$❡6 ✇✐$❤❞6❛✇❛❧#✳ ▼♦6❡ #♣❡❝✐✜❝❛❧❧②✱ $❤❡ ❡✈♦❧✉$✐♦♥ ♦❢ ✇❛$❡6 E✉❛❧✐$② ✐# ♠♦❞❡❧❡❞ ❜② $❤❡ ❢♦❧❧♦✇✐♥❣

❞✐✛❡6❡♥$✐❛❧ ❡E✉❛$✐♦♥✿

Q˙(t) =h X

i

fi(t), D(t)

!

, Q(0) =Q0 ❣✐✈❡♥, ✭✻✮

✹

✇❤❡#❡Q0 ✐% ❛ ♠❡❛%✉#❡♠❡♥* ♦❢ *❤❡ ✐♥✐*✐❛❧ ✇❛*❡# .✉❛❧✐*②✱ ✇✐*❤

∂h

∂f_i <0, ∂h

∂D <0.

❲❡ ❞♦ ♥♦* ♠❛❦❡ ❢♦# *❤❡ ♠♦♠❡♥* ❛♥② ❛❞❞✐*✐♦♥❛❧ ❛%%✉♠♣*✐♦♥ ♦♥ h✱ ❜✉* %✐♠♣❧② ♥♦*❡ *❤❛* *❤✐%

❢✉♥❝*✐♦♥ ✐% ♥♦* ♥❡❝❡%%❛#✐❧② ❧✐♥❡❛#✳

❘❡♠❛#❦✿ ❋♦# ♥♦✇✱ ♦✉# ♠♦❞❡❧✐♥❣ ♦❢ *❤❡ ✇❛*❡# .✉❛❧✐*② ❡✈♦❧✉*✐♦♥ ❞♦❡% ♥♦* ❛❝❝♦✉♥* ❢♦# ❛♥②

❛❜❛*❡♠❡♥* ❛❝*✐✈✐*② *❤❛* *❤❡ ❢❛#♠❡#% ❛♥❞✴♦# *❤❡ ✇❛*❡# ❛❣❡♥❝② ♠❛② ✉♥❞❡#*❛❦❡ *♦ ✐♠♣#♦✈❡ *❤❡

.✉❛❧✐*② ♦❢ ✇❛*❡#✳ ■♥❞❡❡❞✱ ✐* ✐% *❡❝❤♥✐❝❛❧❧② ♣♦%%✐❜❧❡ *♦ ✐♥✢✉❡♥❝❡ *❤❛* .✉❛❧✐*② ❜②✱ ❡✳❣✳✱ ❢❛✈♦✉#✐♥❣ *❤❡

✉%❡ ♦❢ ♣❧❛♥*% ❝♦♥*❛✐♥✐♥❣ ♥✐*#♦❣❡♥✲✜①✐♥❣ %②♠❜✐♦*✐❝ ❜❛❝*❡#✐❛✳ ❚❤✐% ✐% *❤❡ ❝♦♥❝❡♣* ♦❢ ❣#❡❡♥ ♠❛♥✉#❡✿

❢♦# ✐♥%*❛♥❝❡✱ ✇❤✐*❡ ♠✉%*❛#❞ ✭❙✐♥❛♣✐% ❛❧❜❛✮✱ ✈❡*❝❤❡% ✭❱✐❝✐❛✮✱ ♣❤❛❝❡❧✐❛ ♦# #❛♣❡%❡❡❞ ✭❇+❛%%✐❝❛ ♥❛♣✉%✮

❛#❡ ❛❜❧❡ *♦ ✜① ♥✐*#♦❣❡♥ ✐♥ *❤❡ ✜❡❧❞✳ ❚❤❡② ❛#❡ %❡* ✉♣ ❛❢*❡# *❤❡ ♠❛✐♥ ❤❛#✈❡%*✱ ✐♥ ❛✉*♦♠♥ ❛♥❞

❞❡%*#♦②❡❞ ✐♥ ✇✐♥*❡#✳ ■♥ %♦♠❡ ❊✉#♦♣❡❛♥ ❝♦✉♥*#✐❡%✱ ❢❛#♠❡#% ✇❡#❡ ❡❧✐❣✐❜❧❡ *♦ ❛ ❞❛♠❛❣❡ ♣❛②♠❡♥* ❢♦#

*❤❡ ✐♥*#♦❞✉❝*✐♦♥ ♦❢ *❤❡%❡ ♥✐*#♦❣❡♥ ✜①✐♥❣ ♣❧❛♥*%✳^✺

✷✳✸ ❚❤❡ ❲❛(❡) ❆❣❡♥❝②

❲❤❡#❡❛% *❤❡ ❞❡✜♥✐*✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝*✐✈❡ ❢✉♥❝*✐♦♥ ❢♦# ❛ ❢❛#♠❡# ✐% ❡❛%②✱ *❤❡ *❛%❦ ♦❢ ❞♦✐♥❣ %♦ ❢♦# *❤❡

✇❛*❡# ❛❣❡♥❝② ✐% ♥♦* *❤❛* %*#❛✐❣❤*❢♦#✇❛#❞✳ ■❞❡❛❧❧②✱ ♦♥❡ ✇♦✉❧❞ ❧✐❦❡ *♦ ❞❡✜♥❡ ❛ ✇❡❧❢❛#❡ ❢✉♥❝*✐♦♥ *♦

❛%%❡%% *❤❡ ✈❛❧✉❡ *♦ %♦❝✐❡*② ♦❢ ❛♥② ♣❛#*✐❝✉❧❛# ❣♦✈❡#♥♠❡♥*❛❧ ♣♦❧✐❝②✳ ❍♦✇❡✈❡#✱ ✇#✐*✐♥❣ ❞♦✇♥ %✉❝❤

❛ ❢✉♥❝*✐♦♥ ✐% ❛ ❤✐❣❤❧② ❝♦♠♣❧❡① ♣#♦❜❧❡♠ ❢#♦♠ ❛ *❤❡♦#❡*✐❝❛❧✱ ❛% ✇❡❧❧ ❛% ❢#♦♠ ❛ ♣#❛❝*✐❝❛❧ ♣♦✐♥* ♦❢

✈✐❡✇✳ ■♥ *❤✐% ♣❛♣❡#✱ ✇❡ ❛❞♦♣* ❛ ♣#❛❣♠❛*✐❝ ❛♣♣#♦❛❝❤ ❛♥❞ ❛%%✉♠❡ *❤❛* *❤❡ ✇❛*❡# ❛❣❡♥❝② ✉%❡% ✐*%

♣✉❜❧✐❝ ♣♦❧✐❝② *♦ ❛♣♣#♦❛❝❤ ❛% ❝❧♦%❡ ❛% ♣♦%%✐❜❧❡ ♣#❡✲❞❡*❡#♠✐♥❡❞ ❧❡✈❡❧% ♦❢ .✉❛❧✐*② ❛♥❞ .✉❛♥*✐*② ♦❢

✇❛*❡# ❛* *✐♠❡ T✳ ❚❤❡%❡ ❧❡✈❡❧% ❝♦##❡%♣♦♥❞ *♦ ❛ .✉❛❧✐*② ♥♦#♠ ❛♥❞ ❛ ♠✐♥✐♠✉♠ ❛♠♦✉♥* ♦❢ ✇❛*❡#✱

✇❤✐❝❤ %❤♦✉❧❞ ❜❡ ♣#❡%❡#✈❡❞ ❢♦# ❢✉*✉#❡ ♣❡#✐♦❞%✿

Q_b(T) =Q_b, ✭✼✮

D_b(T) =D_b, ✭✽✮

▼♦#❡ ♣#❡❝✐%❡❧②✱ *❤❡ ✇❛*❡# ❛❣❡♥❝② ✇✐%❤❡% *♦ ♠✐♥✐♠✐③❡ *❤❡ ❞✐%*❛♥❝❡ ❜❡*✇❡❡♥ ❝✉##❡♥* ❛♥❞ ❞❡%✐#❡❞

.✉❛❧✐*② ❛♥❞ .✉❛♥*✐*② ❧❡✈❡❧%✱ *❤❛* ✐%✱

θ=

α(Q(T)−Q_b)²+ (1−α)(D(T)−D_b)²

, ✭✾✮

✇❤❡#❡α❛♥❞(1−α)❛#❡ ♣♦%✐*✐✈❡ ✇❡✐❣❤*% *❤❛* ♠❡❛%✉#❡ *❤❡ ✐♠♣♦#*❛♥❝❡ ♦❢ *❤❡ .✉❛❧✐*② ❛♥❞ .✉❛♥*✐*②

❣♦❛❧✱ #❡%♣❡❝*✐✈❡❧②✳ ❙✉❝❤ ♦❜❥❡❝*✐✈❡ %❡❡♠% *♦ ❜❡ ✐♥ ❧✐♥❡ ✇✐*❤ *❤❡ ♣❤✐❧♦%♦♣❤② ♦❢ ♣✉❜❧✐❝✲♣♦❧✐❝② ♠❛❦❡#%

✇❤♦ ✇♦✉❧❞ ❧✐❦❡ *♦ %❡❡ ❛ ❝❧❡❛# %*❛*❡♠❡♥* ♦❢ ✇❤❛* ❛ ❣♦✈❡#♥♠❡♥* ♣#♦❣#❛♠ ✐% ❛✐♠❡❞ ❛*✳

❚♦ ❛❝❤✐❡✈❡ ✐*% ❣♦❛❧%✱ *❤❡ ✇❛*❡# ❛❣❡♥❝② ❝❛♥ ❧❡✈② ✭❝♦♥%*❛♥*✮ *❛①❡% ♦♥ ❢❡#*✐❧✐③❡# ❛♥❞ ✇❛*❡# ✉%❡✳

❚❤❡ ❛❣❡♥❝② ✐% ❡♥❞♦✇❡❞ ✇✐*❤ %♦♠❡ ✜♥❛♥❝✐❛❧ #❡%♦✉#❝❡% ❛* *❤❡ ✐♥✐*✐❛❧ ✐♥%*❛♥* ♦❢ *✐♠❡✱ ❛♥❞ ✐% #❡.✉✐#❡❞

*♦ ❜❛❧❛♥❝❡ ✐*% ❜♦♦❦% ❛* *❤❡ ❡♥❞ ♦❢ *❤❡ ♣❧❛♥♥✐♥❣ ❤♦#✐③♦♥✳ ❚❤❡ ❡.✉✐❧✐❜#✐✉♠✲❜✉❞❣❡* ❝♦♥%*#❛✐♥* ❛*

T #❡❛❞% ❛% ❢♦❧❧♦✇%✿

0 =b0+ Z T

0

[τX

i

fi(t) +φX

i

wi(t)]dt, ✭✶✵✮

✇❤❡#❡ b0 ✐% *❤❡ ❛✈❛✐❧❛❜❧❡ ❜✉❞❣❡* ❛* *✐♠❡ 0✳ ❚❤❡ ❛❜♦✈❡ ❜✉❞❣❡* ❡.✉❛*✐♦♥ ✐% ❛♥ ✐%♦♣❡#✐♠❡*#✐❝

❝♦♥%*#❛✐♥* *❤❛* ❝❛♥ ❜❡ #❡✇#✐**❡♥ ✐♥ *❤❡ ❢♦#♠ ♦❢ ❛ %*❛*❡ ❡.✉❛*✐♦♥ ❛% ❢♦❧❧♦✇%✿

Y˙ (t) = [τX

i

f_i(t) +φX

i

w_i(t)] ✇✐*❤ Y(0) =b0 ❛♥❞ Y(T) = 0, ✭✶✶✮

✇❤❡#❡Y(t)#❡♣#❡%❡♥*% *❤❡ ❢✉♥❞% ❛✈❛✐❧❛❜❧❡ ❛* *✐♠❡ t∈[0, T]✳

❲❡ ♠❛❦❡ *❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛#✐✜❝❛*✐♦♥ #❡♠❛#❦%✿

✺■♥ ❋#❛♥❝❡ ❢♦# ❡①❛♠♣❧❡✱ .❤❡ ■♥❞❡♠♥✐&' ❝♦♠♣❡♥+❛&♦✐-❡ ❞❡ ❝♦✉✈❡-&✉-❡ ❞❡+ +♦❧+ ✭❈♦❞❡ ❞❡ ❧✬❡♥✈✐#♦♥♥❡♠❡♥.

▲■■✶✳✶✳✸✳✸✮ ❛♠♦✉♥.❡❞ .♦ ✻✵ ❡✉#♦>✴❤❛ ✐♥ ✷✵✵✸✳

✺

✶✳ ❚❤❡ %❛① (❛%❡) τ ❛♥❞ φ ❞♦ ♥♦% ✈❛(② ✇✐%❤ %✐♠❡ ❛♥❞✴♦( %❤❡ )%❛%❡ ♦❢ %❤❡ )②)%❡♠ ✭5✉❛❧✐%② ♦❢

%❤❡ ❣(♦✇♥❞✇❛%❡( ❛♥❞ ❞❡♣%❤ ♦❢ %❤❡ ❛5✉✐❢❡(✮ ❞✉(✐♥❣ %❤❡ ♣❧❛♥♥✐♥❣ ✐♥%❡(✈❛❧ [0, T]✳ ❆❧%❤♦✉❣❤

)%❛%❡ ♦( %✐♠❡ ❞❡♣❡♥❞❡♥% %❛① (❛%❡) ♠❛② ❜❡ ❝♦♥❝❡♣%✉❛❧❧② ❛%%(❛❝%✐✈❡✱ %❤❡② ❛(❡ ❞✐✣❝✉❧% %♦

✐♠♣❧❡♠❡♥% ✐♥ (❡❛❧✐%②✳ ■♥❞❡❡❞✱ ✐% ✇✐❧❧ ❜❡ ✈❡(② ❞✐✣❝✉❧% ❢♦( ♣✉❜❧✐❝ ❛❣❡♥❝✐❡) %♦ ❡①♣❧❛✐♥ ❛ ♣♦❧✐❝②

%❤❛% ❝♦♥%✐♥✉♦✉)❧② ❝❤❛♥❣❡) ♦✈❡( %✐♠❡✱ ❛♥❞ ❢❛(♠❡() ✇✐❧❧ ❤❛(❞❧② ❛❝❝❡♣% )✉❝❤ ❛ ♠❡❝❤❛♥✐)♠✳ ❲❡

❜❡❧✐❡✈❡ %❤❛% ♦✉( ❛))✉♠♣%✐♦♥ )✐♠♣❧② (❡✢❡❝%) ❛❝%✉❛❧ ♣(❛❝%✐❝❡ ✇❤❡(❡ %❛① (❛%❡) ✭❛) ✇❡❧❧ ❛) ♦%❤❡(

♣✉❜❧✐❝ )❡(✈✐❝❡ ♣(✐❝❡)✮ ❛(❡ )❡% ❝♦♥)%❛♥% ❜② ❣♦✈❡(♥♠❡♥%❛❧ ❛❣❡♥❝✐❡) ❢♦( %❤❡ ✇❤♦❧❡ ❞✉(❛%✐♦♥ ♦❢

%❤❡ ✜)❝❛❧ ❡①❡(❝✐)❡✱ %②♣✐❝❛❧❧② ❛ ②❡❛(✳ ❙✉❝❤ ❝♦♥)%❛♥% %❛① ❛♥❞ )✉❜)✐❞② ♣♦❧✐❝✐❡) ✇❡(❡ ✉)❡❞ ❜②

❑(❛✇❝③②❦ ❛♥❞ ❩❛❝❝♦✉( ❬✼❪ ✐♥ ❛ ❞②♥❛♠✐❝ ❣❛♠❡ ✇❤❡(❡ ❛ ❧♦❝❛❧ ❣♦✈❡(♥♠❡♥% ❛✐♠) ❛% ❝♦♥%(♦❧❧✐♥❣

♣♦❧❧✉%✐♦♥ ❡♠✐))✐♦♥) ❜② ❞❡❝❡♥%(❛❧✐③❡❞ ❛❣❡♥%)✳

✷✳ ❲❡ ❞♦ ♥♦% ✐♠♣♦)❡ ❛♥② )✐❣♥ ♦♥ %❤❡ ✐♥)%(✉♠❡♥%) τ ❛♥❞ φ✳ ■❢ ♦♣%✐♠✐③❛%✐♦♥ ❧❡❛❞) %♦ ♥❡❣❛%✐✈❡

✈❛❧✉❡)✱ %❤❡♥ )✉❜)✐❞✐❡) )❤♦✉❧❞ ❜❡ )❡% ✉♣ (❛%❤❡( %❤❛♥ %❛①❡)✳ ❚❤❡ )✐❣♥ ♦❢τ ❛♥❞φ✇✐❧❧ ♦❢ ❝♦✉()❡

❞❡♣❡♥❞ ♦♥ %❤❡ ♦❜❥❡❝%✐✈❡ ♦❢ %❤❡ ✇❛%❡( ❛❣❡♥❝②✳

✸✳ ❲❡ ❛))✉♠❡❞ %❤❛% %❤❡ ✇❛%❡( ❛❣❡♥❝② ♠✉)% ❜❛❧❛♥❝❡ ✐%) ❜✉❞❣❡% ❛%T✳ ■❢ %❤❡ ❛❣❡♥❝② ✐) ❛❧❧♦✇❡❞

%♦ (❡❛❧✐③❡ ❛ )✉(♣❧✉)✱ %❤❡♥ %❤❡ ❜✉❞❣❡% ❝♦♥)%(❛✐♥%✱ ❡5✉❛%✐♦♥ ✭✶✵✮ ❜❡❝♦♠❡) ❛♥ ✐♥❡5✉❛❧✐%②✱ ✐✳❡✳✱

0≤b0− Z T

0

[τX

i

f_i(t) +φX

i

w_i(t)]dt.

❚♦ ❦❡❡♣ ✐% )✐♠♣❧❡✱ ✇❡ )❤❛❧❧ ❝♦♥)✐❞❡( %❤❡ ❝❛)❡ ✇❤❡(❡ %❤❡ ❜✉❞❣❡% ❝♦♥)%(❛✐♥% ✐) ❜✐♥❞✐♥❣ ❛♥❞✱ ❛) )%❛%❡❞ ✐♥ %❤❡ ✐♥%(♦❞✉❝%✐♦♥✱ ✇❡ )❤❛❧❧ ❝♦♥%(❛)% %❤❡ (❡)✉❧%) ♦❢ %❤✐) )❝❡♥❛(✐♦ %♦ %❤❡ ❝❛)❡ ✇❤❡(❡

%❤❡(❡ ✐) ♥♦ ❜✉❞❣❡% ❝♦♥)%(❛✐♥%) ❛♥❞ %❤❡ ♦♥❡ ✇✐%❤ ♥♦ (❡❣✉❧❛%✐♦♥✳

✸ ❆ ❙#❛❝❦❡❧❜❡*❣ ●❛♠❡

■♥ %❤❡ ♣(❡✈✐♦✉) )❡❝%✐♦♥✱ ✇❡ ❞❡✜♥❡❞ ❛ ✜♥✐%❡✲❤♦(✐③♦♥ ❞✐✛❡(❡♥%✐❛❧ ❣❛♠❡✱ ✇✐%❤ N + 1 ♣❧❛②❡() ✭N

❢❛(♠❡() ❛♥❞ ❛ (❡❣✉❧❛%♦(✮✳ ❚❤❡ ♠♦❞❡❧ ✐♥✈♦❧✈❡) %❤(❡❡ )%❛%❡ ✈❛(✐❛❜❧❡)✱ ♥❛♠❡❧②✱ %❤❡ 5✉❛♥%✐%② D❛♥❞

5✉❛❧✐%② Q♦❢ ✇❛%❡( ❛♥❞ %❤❡ ✇❛%❡( ❛❣❡♥❝②✬) ❜✉❞❣❡%✱Y✳ ❚❤❡ ❝♦♥%(♦❧ ✈❛(✐❛❜❧❡) ♦❢ ❛ ❢❛(♠❡( ❛(❡ %❤❡

✇❛%❡( ✇✐%❤❞(❛✇❛❧w_i ❛♥❞ %❤❡ 5✉❛♥%✐%② ♦❢ ❢❡(%✐❧✐③❡( f_i✳ ❚❤❡ ✇❛%❡( ❛❣❡♥❝② ❝❤♦♦)❡) %❤❡ %❛① (❛%❡)τ

❛♥❞ φ✱ ✇❤✐❝❤ ❝❛♥ ❛))✉♠❡ ❛♥② )✐❣♥✳

❚❤❡ ❣❛♠❡ ✐) ♣❧❛②❡❞ S ❧❛ ❙%❛❝❦❡❧❜❡(❣✳ ❚❤❡ ✇❛%❡( ❛❣❡♥❝② %❛❦❡) %❤❡ ❧❡❛❞❡(✬) (♦❧❡ ❛♥❞ ❛♥♥♦✉♥❝❡)

✐%) )%(❛%❡❣② ❜❡❢♦(❡ %❤❡ ❢❛(♠❡() ♠❛❦❡ %❤❡✐( ❞❡❝✐)✐♦♥)✳ ●✐✈❡♥ %❤❡ ❧❡❛❞❡(✬) ❛♥♥♦✉♥❝❡♠❡♥% ♦❢ %❤❡ %❛①

♣♦❧✐❝②(τ, φ)✱ %❤❡ ❢❛(♠❡()✱ ❛❝%✐♥❣ ❛) ❢♦❧❧♦✇❡()✱ ♣❧❛② ❛ ◆❛)❤ ❣❛♠❡ ❛♥❞ ❝❤♦♦)❡wi❛♥❞fi✳ ❲❡ )✉♣♣♦)❡

%❤❛% %❤❡ ❢❛(♠❡() ❡♠♣❧♦② ♦♣❡♥✲❧♦♦♣ )%(❛%❡❣✐❡)✱ %❤❛% ✐)✱ ❛% %❤❡ ✐♥✐%✐❛❧ ✐♥)%❛♥% ♦❢ %✐♠❡✱ ❡❛❝❤ ♣❧❛②❡(

❞❡❝✐❞❡) ✉♣♦♥ ❛ )%(❛%❡❣② ✇❤✐❝❤ ❞❡♣❡♥❞) ♦♥❧② ♦♥ %✐♠❡✳ ■% ✐) ✇❡❧❧ ❦♥♦✇♥ %❤❛% ♦♣❡♥✲❧♦♦♣ ❙%❛❝❦❡❧❜❡(❣

❡5✉✐❧✐❜(✐❛ ❛(❡ ✐♥ ❣❡♥❡(❛❧ %✐♠❡ ✐♥❝♦♥)✐)%❡♥%✳^✻ ❚❤✐) ♠❡❛♥) %❤❛% ❣✐✈❡♥ %❤❡ ♦♣♣♦(%✉♥✐%② %♦ (❡✈✐)❡ ❤✐) )%(❛%❡❣② ❛% ❛♥ ✐♥%❡(♠❡❞✐❛%❡ ✐♥)%❛♥% ♦❢ %✐♠❡✱ %❤❡ ❧❡❛❞❡( ✇♦✉❧❞ ❧✐❦❡ %♦ ❝❤♦♦)❡ ❛♥♦%❤❡( )%(❛%❡❣②

%❤❛♥ %❤❡ ♦♥❡ ❤❡ )❡❧❡❝%❡❞ ❛% %❤❡ ✐♥✐%✐❛❧ ✐♥)%❛♥% ♦❢ %✐♠❡✳ ❚❤❡(❡❢♦(❡✱ ❛♥ ♦♣❡♥✲❧♦♦♣ ❙%❛❝❦❡❧❜❡(❣

❡5✉✐❧✐❜(✐✉♠ ♦♥❧② ♠❛❦❡) )❡♥)❡ ✐❢ %❤❡ ❧❡❛❞❡( ❝❛♥ ❝(❡❞✐❜❧② ♣(❡❝♦♠♠✐% %♦ ❤✐) )%(❛%❡❣②✳ ■♥ %❤❡ ♣(❡)❡♥%

❣❛♠❡ ✐% )❡❡♠) ♣❧❛✉)✐❜❧❡ %♦ ❛))✉♠❡ ♣(❡❝♦♠♠✐%♠❡♥% ♦♥ %❤❡ ♣❛(% ♦❢ %❤❡ ✇❛%❡( ❛❣❡♥❝②✿ ✐♥ ♣(❛❝%✐❝❡

❛ %❛① )❝❤❡♠❡ ✐) ❞❡%❡(♠✐♥❡❞ ❛♥❞ ❛♥♥♦✉♥❝❡❞ ❢(♦♠ %❤❡ ♦✉%)❡% ❛♥❞ ✇❤❡♥ %❤❡ (❡❣✉❧❛%♦(✬) ❞❡❝✐)✐♦♥ ✐)

✐((❡✈♦❝❛❜❧❡✱ %❤❡ ❛♥♥♦✉♥❝❡♠❡♥% ✇✐❧❧ ❜❡ ❝(❡❞✐❜❧❡✳

✸✳✶ ❚❤❡ ❋♦❧❧♦✇❡*+✬ ❘❡❛❝0✐♦♥ ❋✉♥❝0✐♦♥+

❚♦ )♦❧✈❡ ❢♦( ❙%❛❝❦❡❧❜❡(❣ ❡5✉✐❧✐❜(✐✉♠✱ ✇❡ ✜()% ❞❡%❡(♠✐♥❡ %❤❡ (❡❛❝%✐♦♥ ❢✉♥❝%✐♦♥) ♦❢ %❤❡ ❢♦❧❧♦✇❡()

❛♥❞ ♥❡①% )♦❧✈❡ %❤❡ ✭♦♣%✐♠❛❧✲❝♦♥%(♦❧✮ ♣(♦❜❧❡♠ ♦❢ %❤❡ ❧❡❛❞❡(✳ ❊❛❝❤ ❢❛(♠❡( ❝❤♦♦)❡) %❤❡ ❧❡✈❡❧)

♦❢ ✐♥♣✉%)✱ wi(t) ❛♥❞ fi(t)✱ %❤❛% ♠❛①✐♠✐③❡ ♣(♦✜%)✱ ❣✐✈❡♥ ❜② ❡5✉❛%✐♦♥ ✭✹✮✳ ◆♦%❡ %❤❛% %❤❡ ✇❛%❡(

✻❙❡❡✱ ❡✳❣✳✱ ▼❛'()♥✲❍❡''-♥ ❡( ❛❧✳ ✭✷✵✵✺✮ ❛♥❞ ❇✉'❛((♦ ❛♥❞ ❩❛❝❝♦✉' ✭✷✵✵✾✮ ❢♦' ❡①❛♠♣❧❡? ✇❤❡'❡ ♦♣❡♥✲❧♦♦♣ ❙(❛❝❦✲

❡❧❜❡'❣ ❡D✉✐❧✐❜'✐❛ ❛'❡ (✐♠❡ ❝♦♥?✐?(❡♥(✳

✻

✉❛❧✐%② ❞♦❡* ♥♦% ❛♣♣❡❛- ✐♥ %❤❡ ♣❛②♦✛ ❢✉♥❝%✐♦♥ ♦❢ ❛ ❢❛-♠❡-✱ ❛♥❞ ❤❡♥❝❡ ✐% ✐* ✐--❡❧❡✈❛♥% ❢♦- %❤✐*

❛❣❡♥%✳ ❋✉-%❤❡-✱ ✇❡ *✉♣♣♦*❡ %❤❛% %❤❡ ❜✉❞❣❡% ❝♦♥*%-❛✐♥% ❛♥❞ %❤❡ ❡✈♦❧✉%✐♦♥ ♦❢ %❤❡ ✇❛%❡- ❞✐*%❛♥❝❡

❛-❡ ♣-✐✈❛%❡ ✐♥❢♦-♠❛%✐♦♥ ❞❡%❛✐♥❡❞ ❜② %❤❡ ✇❛%❡- ❛❣❡♥❝②✱ ✐✳❡✳✱ %❤❡ ❢❛-♠❡-* ❞♦ ♥♦% ♦❜*❡-✈❡ %❤❡*❡ *%❛%❡

❡ ✉❛%✐♦♥*✳ ❲❡ *❤❛❧❧ ♦♠✐% ❢-♦♠ ♥♦✇ ♦♥ %❤❡ %✐♠❡ ❛-❣✉♠❡♥% ✇❤❡♥ ♥♦ ❛♠❜✐❣✉✐%② ♠❛② ❛-✐*❡✳

❆**✉♠✐♥❣ ❛♥ ✐♥%❡-✐♦- *♦❧✉%✐♦♥✱ %❤❡ ✜-*%✲♦-❞❡- ❡ ✉✐❧✐❜-✐✉♠ ❝♦♥❞✐%✐♦♥* ❛-❡✿

∂H_i

∂w_i = p_i∂y_i(w_i, f_i)

∂w_i −∂c_w(D, w_i)

∂w_i −φ= 0, ✭✶✷✮

∂H_i

∂f_i = p_i∂y_i(w_i, f_i)

∂f_i −c^′_f(f_i)−τ = 0. ✭✶✸✮

❊ ✉❛%✐♦♥* ✭✶✷✮✲✭✶✸✮ ❛-❡ %❤❡ ✉*✉❛❧ ♦♣%✐♠❛❧✐%② ❝♦♥❞✐%✐♦♥* *%❛%✐♥❣ %❤❛%✱ ❛% %❤❡ ♦♣%✐♠✉♠✱ ♠❛-❣✐♥❛❧

-❡✈❡♥✉❡ ❢-♦♠ ♣-♦❞✉❝%✐♦♥ ❡ ✉❛❧ ♠❛-❣✐♥❛❧ ❝♦*%*✳ ■♥ ❡ ✉❛%✐♦♥ ✭✶✷✮✱ ♠❛-❣✐♥❛❧ -❡✈❡♥✉❡* ❛-❡ ❞✉❡ %♦

%❤❡ ✉*❡ ♦❢ ♦♥❡ ❛❞❞✐%✐♦♥❛❧ ✉♥✐% ♦❢ ✇❛%❡-✳ ▼❛-❣✐♥❛❧ ❝♦*%* ❛-❡ ❣✐✈❡♥ ❜② ♠❛-❣✐♥❛❧ ❝♦*%* ♦❢ ♣✉♠♣✐♥❣

❛♥❞ ❞✐*%-✐❜✉%✐♥❣ ✐--✐❣❛%✐♦♥ ✇❛%❡- ❛♥❞ ❜② %❤❡ %❛①❡* ♣❛✐❞ ♣❡- ✉♥✐% ♦❢ ✇❛%❡- ♣✉♠♣❡❞ ■♥ ❡ ✉❛%✐♦♥

✭✶✸✮✱ ♠❛-❣✐♥❛❧ -❡✈❡♥✉❡* ❞✉❡ %♦ %❤❡ ✉*❡ ♦❢ ♦♥❡ ❛❞❞✐%✐♦♥❛❧ ✉♥✐% ♦❢ ❢❡-%✐❧✐③❡- ❛-❡ ❡ ✉❛❧ %♦ ♠❛-❣✐♥❛❧

❝♦*% ♦❢ ❜✉②✐♥❣ ❢❡-%✐❧✐③❡-* ❛♥❞ %❤❡ %❛① ♣❛✐❞ ♣❡- ✉♥✐% ♦❢ ❢❡-%✐❧✐③❡-✳

❯*✐♥❣ ✭✶✷✮✲✭✶✸✮✱ ✇❡ ❝❛♥ ❡①♣-❡** w_i ❛♥❞ f_i ❛* ❢✉♥❝%✐♦♥* ♦❢ %❤❡ *%❛%❡ ✈❛-✐❛❜❧❡✱ D ❛♥❞ %❤❡

✐♥*%-✉♠❡♥%* ♦❢ %❤❡ ✇❛%❡- ❛❣❡♥❝②✳ ❉❡♥♦%❡ ❜②f˜i(D, τ, φ)❛♥❞w˜i(D, τ, φ)%❤❡*❡ -❡❛❝%✐♦♥ ❢✉♥❝%✐♦♥*✳

✸✳✷ ❚❤❡ ▲❡❛❞❡)✬+ ,)♦❜❧❡♠

❚❤❡ ❧❡❛❞❡- *♦❧✈❡* ❛♥ ♦♣%✐♠❛❧✲❝♦♥%-♦❧ ♣-♦❜❧❡♠ ✇❤✐❝❤ ✐* ♥♦% *%❛♥❞❛-❞ ❜❡❝❛✉*❡ %❤❡ ✇❛%❡- ❛❣❡♥❝②

✐* ❧♦♦❦✐♥❣ ❢♦- ❛ ❝♦♥*%❛♥% %❛① ♣♦❧✐❝② %❤-♦✉❣❤♦✉% %❤❡ ♣❧❛♥♥✐♥❣ ❤♦-✐③♦♥✳ ❚❤❡ ✇❛%❡- ❛❣❡♥❝② ❝❤♦*❡*

%❤✐* ♣♦❧✐❝② *♦ ❛* %♦ ♠✐♥✐♠✐③❡ %❤❡ ❞✐*%❛♥❝❡ ❜❡%✇❡❡♥ ♦❜*❡-✈❡❞ ❛♥❞ ❞❡*✐-❡❞ ✉❛♥%✐%② ❛♥❞ ✉❛❧✐%②

❧❡✈❡❧* ❛% %❤❡ ❡♥❞ ♦❢ %❤❡ ♣❧❛♥♥✐♥❣ ❤♦-✐③♦♥✱ T✱ %❛❦✐♥❣ ✐♥%♦ ❛❝❝♦✉♥% %❤❡ ❢♦❧❧♦✇❡-* -❡❛❝%✐♦♥* ❛♥❞ %❤❡

❡✈♦❧✉%✐♦♥ ♦❢ ❛❧❧ %❤❡ *%❛%❡ ✈❛-✐❛❜❧❡* ✭*❡❡ ❡ ✉❛%✐♦♥ ✭✶✵✮✳

❙✉❜*%✐%✉%✐♥❣ ❢♦-f˜_i(D, τ, φ)❛♥❞w˜_i(D, τ, φ)✐♥ %❤❡ ✇❛%❡- ❛❣❡♥❝②✬* ❜✉❞❣❡%✱ ✉❛♥%✐%② ❛♥❞ ✉❛❧✐%②

❡ ✉❛%✐♦♥* ❧❡❛❞* %♦

Y˙ (t) = τX

i

f˜i(D, τ, φ) +φX

i

˜

wi(D, τ, φ), Y(0) =b0, Y(T) = 0, ✭✶✹✮

D˙ = g X

i

˜

w_i(D, τ, φ), r

!

, D(0) =D0, ✭✶✺✮

Q˙(t) = h X

i

f˜_i(D, τ, φ), D(t)

!

, Q_i(0) =Q0 ❣✐✈❡♥. ✭✶✻✮

❚❤❡ ❧❡❛❞❡-✬* ❍❛♠✐❧%♦♥✐❛♥ -❡❛❞* ❛* ❢♦❧❧♦✇*✿

H_L"

D(t), µ^D(t), Q(t), µ^Q(t), Y (t), µ^Y (t), τ, φ

=µ^D(t)g X

i

˜

w_i(D(t), λ_i(t), τ, φ), r

!

+µ^Q(t)h X

i

f˜_i(D, λi, τ, φ), D(t)

!

+µ^Y (t) τX

i

f˜_i(D, λi, τ, φ) +φX

i

˜

w_i(D, λi, τ, φ)

! ,

✇❤❡-❡ %❤❡µ^D(t), µ^Q(t)❛♥❞µ^Y (t)❛-❡ ❛❞❥♦✐♥% ✈❛-✐❛❜❧❡* ❛♣♣❡♥❞❡❞ %♦ %❤❡ *%❛%❡ ✈❛-✐❛❜❧❡*D(t), Q(t)

❛♥❞ Y (t).

❆**✉♠✐♥❣ ❛♥ ✐♥%❡-✐♦- *♦❧✉%✐♦♥✱ ❛❧♦♥❣ ✇✐%❤ %❤❡ ❢♦✉- *%❛%❡ ❡ ✉❛%✐♦♥* ✐♥ ✭✶✺✮✲✭✶✻✮✱ %❤❡ ✜-*%✲♦-❞❡-

✼

♦♣"✐♠❛❧✐"② ❝♦♥❞✐"✐♦♥+ ❛,❡ ❛+ ❢♦❧❧♦✇+✿^✼

˙

µ^D = −∂H_L

∂D , µ^D(T) = 2(1−α)(D(T)−D_b), ✭✶✼✮

˙

µ^Q = −∂H_L

∂Q , µ^Q(T) = 2α(Q(T)−Q_b), ✭✶✽✮

˙

µ^Y = −∂HL

∂Y , ✭✶✾✮

Z T

0

∂HL

∂τ dt = 0, ✭✷✵✮

Z T

0

∂H_L

∂φ dt = 0. ✭✷✶✮

❘❡❝❛❧❧ "❤❛" "❤❡ ♦♣"✐♠❛❧✐"② ❝♦♥❞✐"✐♦♥+ ✐♥ ✭✷✵✮ ❛♥❞ ✭✷✶✮ "❛❦❡ "❤❡ ❢♦,♠ ♦❢ ❛♥ ✐♥"❡❣,❛❧ ❜❡❝❛✉+❡

♦❢ ♦✉, ,❡+",✐❝"✐♦♥ ♦❢ "❤❡ ❧❡❛❞❡,✬+ "❛① ♣♦❧✐❝✐❡+ "♦ ❝♦♥+"❛♥" ♦♥❡+✳ ❋✉,"❤❡,✱ ❛+ "❤❡ ✈❛❧✉❡+ ♦❢ +"❛"❡

✈❛,✐❛❜❧❡ Y (t) ❛,❡ ❣✐✈❡♥ ❛"0 ❛♥❞ T✱ "❤❡ ❛❞❥♦✐♥" ✈❛,✐❛❜❧❡ µ^Y ✐+ ❢,❡❡✳ ❋✐♥❛❧❧②✱ ✇❡ ♥♦"❡ "❤❛" "❤❡

❧❡❛❞❡,✬+ ♦♣"✐♠❛❧✐"② ❝♦♥❞✐"✐♦♥+ ✐♥❝❧✉❞❡8 ❡F✉❛"✐♦♥+ ❛♥❞ +❛♠❡ ♥✉♠❜❡, ♦❢ ✉♥❦♥♦✇♥+✳

✹ ■❧❧✉$%&❛%✐♦♥

❲❡ ✐❧❧✉+",❛"❡ ✐♥ "❤✐+ +❡❝"✐♦♥ "❤❡ "②♣❡ ♦❢ ✐♥+✐❣❤" "❤❛" ❝❛♥ ❜❡ ♦❜"❛✐♥❡❞ ✉+✐♥❣ ♦✉, ♠♦❞❡❧✳ ❚♦ ❦❡❡♣

"❤✐♥❣+ ❛+ +✐♠♣❧❡ ❛+ ♣♦++✐❜❧❡✱ ✇❡ ❛++✉♠❡ "❤❛" "❤❡ n ❢❛,♠❡,+ ❛,❡ ✐❞❡♥"✐❝❛❧✳ ●✐✈❡♥ ♦✉, +❡""✐♥❣+ ♦❢

♣,✐❝❡✲"❛❦✐♥❣ ❢❛,♠❡,+ ❧♦❝❛"❡❞ ♦♥ "❤❡ +❛♠❡ ❣,♦✇♥❞✇❛"❡,✱ "❤✐+ ❛++✉♠♣"✐♦♥ ✐+ ♥♦" +❡✈❡,❡✳

✹✳✶ #$♦❞✉❝)✐♦♥ ❢✉♥❝)✐♦♥- ❛♥❞ ❞②♥❛♠✐❝-

❲❡ ❛❞♦♣" "❤❡ ❢♦❧❧♦✇✐♥❣ ♣,♦❞✉❝"✐♦♥ ❢✉♥❝"✐♦♥✿

yi=Awifi+Bwi+Efi−K1 2f_i²−1

2M w²_i +G,

✇❤❡,❡A, B, E, K, M ❛♥❞G❛,❡ ♥♦♥✲♥❡❣❛"✐✈❡ ♣❛,❛♠❡"❡,+✳ ❙♦♠❡ ,❡+",✐❝"✐♦♥+ ♦♥ "❤❡+❡ ♣❛,❛♠❡"❡,+

✇✐❧❧ ❜❡ ,❡F✉✐,❡❞ "♦ +❛"✐+❢② "❤❡ ❝♦♥❞✐"✐♦♥+ ✐♥ ✭✶✮✱ ♥❛♠❡❧②✿

∂yi

∂w_i = Af_i+B−M w_i ≥0, ∂²yi

∂w_i² =−M ≤0,

∂²y_i

∂w_i∂f_i = A >0, ∂y_i

∂f_i =Aw_i+E−Kf_i ≥0, ∂²y_i

∂f_i² =−K <0.

❚❤❡ ,❡✈❡♥✉❡ ❢✉♥❝"✐♦♥ ♦❢ ❢❛,♠❡, i ✐+ ❣✐✈❡♥ ❜② py_i✳ ❯+✐♥❣ "❤❡ ❛❜♦✈❡ ❞❡,✐✈❛"✐✈❡+✱ ✐" ✐+ ❡❛+② "♦

✈❡,✐❢② "❤❛" ❢♦, "❤❡ ,❡✈❡♥✉❡ ❢✉♥❝"✐♦♥ "♦ ❜❡ ❝♦♥❝❛✈❡✱ ✐" ✐+ ♥❡❝❡++❛,② "♦ ❤❛✈❡ "❤❡ ❞❡"❡,♠✐♥❛♥" ♦❢ "❤❡

❍❡++✐❛♥ ♠❛",✐① ♥♦♥✲♥❡❣❛"✐✈❡✱ ✐✳❡✳✱

p²A²−p²KM ≤0. ✭✷✷✮

■♥ +♦♠❡ ♦❢ "❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡+✱ ✇❡ ✇✐❧❧ ✉+❡ "❤❡ +✐♠♣❧✐❢②✐♥❣ ❛++✉♠♣"✐♦♥ A = 0✳ ■♥ "❤❛"

❝❛+❡✱ "❤❡ ❞❡"❡,♠✐♥❛♥" ♦❢ "❤❡ ❍❡++✐❛♥ ♠❛",✐① ✐+ ♥❡❣❛"✐✈❡✱ ✐✳❡✳✱

−p²KM ≤0. ✭✷✸✮

❚❤❡ ❢❡,"✐❧✐③❡, ❛F✉✐+✐"✐♦♥ ❝♦+" ✐+ ❣✐✈❡♥ ❜②

c_f(fi) =Lf_i,

✼❆! "❤❡%❡ ✐! ♥♦ ♣❛%"✐❝✉❧❛% ♥❡❡❞ ❢♦% ✐"✱ ✇❡ ❞♦ ♥♦" ✇%✐"❡ "❤❡!❡ ❝♦♥❞✐"✐♦♥! ✐♥ ❢✉❧❧✳ ❲❡ ✇✐❧❧ ❞❡"❛✐❧ "❤❡♠ ✐♥ "❤❡

❡①❛♠♣❧❡

✽

✇❤❡#❡L ✐% ❛ ♣♦%✐)✐✈❡ ♣❛#❛♠❡)❡#✳ ❚❤❡ ✐##✐❣❛)✐♦♥ ❝♦%) ✐% %♣❡❝✐✜❡❞ ❛%

c_w(wi) = (Z+CD)wi,

✇❤❡#❡Z ❛♥❞C❛#❡ ♣♦%✐)✐✈❡ ♣❛#❛♠❡)❡#%✳ ❚❤❡ )❡#♠Zwi #❡♣#❡%❡♥)% )❤❡ ❝♦%) ♦❢ ❞✐%)#✐❜✉)✐♥❣ ✇❛)❡#✱

❛♥❞ CDw_i ✐% )❤❡ ✇❛)❡#✲♣✉♠♣✐♥❣ ❝♦%) )❤❛) ❞❡♣❡♥❞% ♦♥ )❤❡ ❞✐%)❛♥❝❡ ❜❡)✇❡❡♥ )❤❡ )♦♣%♦✐❧ ❛♥❞ )❤❡

✇❛)❡#)❛❜❧❡✳ ❈❧❡❛#❧②✱ )❤❡ ❛❜♦✈❡ ❝♦%) ❢✉♥❝)✐♦♥% %❛)✐%❢② )❤❡ #❡;✉✐#❡♠❡♥)% ✐♥ ✭✸✮ ❛♥❞ ✭✷✮✳

❚❤❡ ❞②♥❛♠✐❝% ♦❢ )❤❡ ❞❡♣)❤ ❛♥❞ ;✉❛❧✐)② ♦❢ )❤❡ ❣#♦✇♥❞✇❛)❡# ❛#❡ ♠♦❞❡❧❡❞ ❛% ❢♦❧❧♦✇%✿

D˙ = X

i

w_i−r, D(0) =D0, Q˙ = −δX

i

f_i

!

D, Q(0) =Q0,

✇❤❡#❡δ ✐% ❛ ♥♦♥✲♥❡❣❛)✐✈❡ ♣❛#❛♠❡)❡#✳◆♦)❡ )❤❛) ;✉❛❧✐)② ✐% ♥♦♥✲✐♥❝#❡❛%✐♥❣✱ ❢♦# ❛❧❧t✳

✹✳✷ ❆♥❛❧②(✐❝❛❧ +❡-✉❧(- ✇✐(❤♦✉( ❝♦♠♣❧❡♠❡♥(❛+✐(②

❙✉♣♣♦%❡ ✜#%) )❤❡#❡ ✐% ♥♦ ❝#♦%%✲❡✛❡❝) ❜❡)✇❡❡♥ ✐♥♣✉)%✱ ✐✳❡✳ A= 0✳

✹✳✷✳✶ ❖♣&✐♠❛❧ ✐♥♣✉& ❛♥❞ ♣♦❧✐❝② ❝❤♦✐❝❡

●✐✈❡♥ ♦✉# ❢✉♥❝)✐♦♥❛❧ %♣❡❝✐✜❝❛)✐♦♥%✱ ❢❛#♠❡#i✬% ❍❛♠✐❧)♦♥✐❛♥ ♥♦✇ #❡❛❞%✿

Hi =p

Bwi+Efi−1

2Kf_i²−1

2M w_i²+G

−(Z+CD)wi−Lfi−τ fi−φwi.

❆%%✉♠✐♥❣ ❛♥ ✐♥)❡#✐♦# %♦❧✉)✐♦♥✱ )❤❡ ✜#%)✲♦#❞❡# ❡;✉✐❧✐❜#✐✉♠ ❝♦♥❞✐)✐♦♥% ♦❢ ❢❛#♠❡# i, i = 1, . . . , N,

❛#❡ ❣✐✈❡♥ ❜②✿

∂H_i

∂w_i = 0⇔w˜_i(τ, φ) = pB−CD−Z−φ

pM , ✭✷✹✮

∂Hi

∂f_i = 0⇔f˜_i(τ, φ) = pE−L−τ

pK . ✭✷✺✮

◆♦)❡ )❤❛) )❤❡ %♦❧✉)✐♦♥ ✐% ❢✉❧❧② %②♠♠❡)#✐❝✱ ✐✳❡✳ w˜_i(τ, φ) = ˜w(τ, φ) ❛♥❞ f˜_i(τ, φ) = ˜f(τ, φ)✱ ❢♦# ❛❧❧

i= 1, . . . N✳

34♦♣♦5✐&✐♦♥ ✶ ❋❛"♠❡"% ❛✉'♦✲"❡❣✉❧❛'❡ '❤❡✐" ✇❛'❡" ✉%❡ ✇❤❡♥ '❤❡ ✇❛'❡" ❞✐%'❛♥❝❡ ✐♥❝"❡❛%❡%✳ ❋✉"✲

'❤❡"✱ ❢❛"♠❡"% ✉%❡ ❧❡%% ✇❛'❡" ✭❢❡"'✐❧✐③❡"✮ ✇❤❡♥ '❤❡ ✇❛'❡" ✭❢❡"'✐❧✐③❡"✮ ✐♥♣✉' ✐% '❛①❡❞✳

34♦♦❢✳ ❋#♦♠ ❡;✉❛)✐♦♥% ✭✷✹✮ ❛♥❞ ✭✷✺✮✱ ✐) ✐% ♦❜✈✐♦✉% )❤❛)✿

∂w˜

∂D < 0, ∂f˜

∂D = 0,

∂w˜

∂τ = 0,∂w˜

∂φ <0,∂f˜

∂τ <0,∂f˜

∂φ = 0.

❚❤❡ ❧❛#❣❡# )❤❡ ✇❛)❡# ❞✐%)❛♥❝❡✱ ✇❤✐❝❤ ✐% %②♥♦♥②♠♦✉% )♦ ❛ ❤✐❣❤❡# ♣✉♠♣✐♥❣ ❝♦%)✱ )❤❡ ❧♦✇❡# )❤❡

❢❛#♠❡#✬% ✇❛)❡# ✉%❡✳ ❚❤❡ ❤✐❣❤❡# )❤❡ ✇❛)❡# ✭❢❡#)✐❧✐③❡#✮ )❛①✱ )❤❡ ❧♦✇❡# )❤❡ ♦♣)✐♠❛❧ ✐##✐❣❛)✐♦♥ ✭❢❡#)✐❧✲

✐③❡#✮ ✉%❡✳

✾

❆❢"❡$ %✉❜%"✐"✉"✐♥❣ ❢♦$ w˜(τ, φ) ❛♥❞f˜(τ, φ) ✐♥ "❤❡ %"❛"❡ ❡/✉❛"✐♦♥% D,˙ Q˙ ❛♥❞ Y˙✱ ✇❡ ♦❜"❛✐♥ "❤❡

❢♦❧❧♦✇✐♥❣ ❍❛♠✐❧"♦♥✐❛♥ ❢♦$ "❤❡ ❧❡❛❞❡$✿

HL = µ^DN( ˜wi−r)−µ^QδNf˜iD+µ^Y h

τ Nf˜i+N φw˜i

i

= N"

µ^D+µ^Yφ

pB−CD−Z−φ pM

+N"

−µ^QδD+µ^Yτ

pE−L−τ pK

−rN µ^D,

✇❤❡$❡µ^D(t), µ^Q(t) ❛♥❞µ^Y (t)❛$❡ ❛❞❥♦✐♥" ✈❛$✐❛❜❧❡% ❛♣♣❡♥❞❡❞ "♦ "❤❡ %"❛"❡ ✈❛$✐❛❜❧❡%D(t), Q(t)

❛♥❞ Y (t)✳ ❚❤❡ ✜$%"✲♦$❞❡$ ♦♣"✐♠❛❧✐"② ❝♦♥❞✐"✐♦♥% ❛♥❞ "❤❡ %♦❧✉"✐♦♥ ♣$♦❝❡❞✉$❡ ❛$❡ ❣✐✈❡♥ ✐♥ "❤❡

❆♣♣❡♥❞✐① ✭❆✳✶✳✶✮✳ ❙♦❧✈✐♥❣✱ ✇❡ ❣❡"✿

D(t, φ) = e^−ρtD0+ Θ(φ)(1−e^−ρt), ✭✷✻✮

Q(t, φ, τ) = Q0−Λ(τ) Z t

0

D(s)ds, ✭✷✼✮

Y (t, φ, τ) = b0

N

pKM [(pE−L−τ)τ M+ (pB−Z−φ)φK]t−φρ Z t

0

D(s)ds, ✭✷✽✮

✇❤❡$❡

Θ (φ) =

N(pB−Z−φ)−rpM N C

, Λ (τ) = δN(pE−L−τ)

pK , ρ= N C pM

❛♥❞ Z t

0

D(s)ds= D0−D(t)

ρ + Θ(φ)t.

❲❡ ❛❧%♦ ❤❛✈❡ "♦ ❝♦♥%✐❞❡$ "❤❡ %♣❡❝✐❛❧ ❝♦♥❞✐"✐♦♥% ❛% ✇❡ ❝❛♥ %❡❡ ✐♥ "❤❡ ❆♣♣❡♥❞✐①✱ ❡/✉❛"✐♦♥% ✭✹✵✮

❛♥❞ ✭✹✶✮ ✳ ❲❡ "❤❡$❡❢♦$❡ ❡♥❞ ✉♣ ✇✐"❤ ❛ %②%"❡♠ ♦❢ ✸ ❡/✉❛"✐♦♥% ✭✹✼✮✲✭✹✾✮ ❢♦$ "❤❡ "❤$❡❡ ✉♥❦♥♦✇♥%

τ, φ ❛♥❞µ^Y ✇❤✐❝❤ ✇❡ ❝❛♥ %♦❧✈❡✳ ❲❡ ❝❛♥ "❤❡♥ ✐♥%❡$" "❤❡ $❡%✉❧"% ✐♥ "❤❡ %②%"❡♠ ❞②♥❛♠✐❝% ❛♥❞ "❤❡

$❡❛❝"✐♦♥ ❢✉♥❝"✐♦♥% ♦❢ "❤❡ ❢♦❧❧♦✇❡$%✳

!♦♣♦$✐&✐♦♥ ✷ ❚❤❡ ✉$❡ ♦❢ ♦♣(✐♠❛❧ ✐♥♣✉( (❛①❡$ ❧❡❛❞$ (♦ ❛ ❜❡((❡1 ✇❛(❡1 3✉❛❧✐(② ♦✈❡1 (✐♠❡✳ ❚❤❡

✉$❡ ♦❢ (❤❡ ♦♣(✐♠❛❧ ✇❛(❡1 (❛① ❞❡❝1❡❛$❡$ (❤❡ ✇❛(❡1✲(❛❜❧❡ ❞✐$(❛♥❝❡ ♦✈❡1 (✐♠❡✱ ✐✳❡✳ ❧❡❛❞$ (♦ ❛ ❣1❡❛(❡1

❣1♦✉♥❞✇❛(❡1 ✈♦❧✉♠❡✳

!♦♦❢✳ N$♦♦❢ ✐♥ "❤❡ ❆♣♣❡♥❞✐①✱ ✭❆✳✶✳✷✮✳

✹✳✷✳✷ ◆♦ ❜✉❞❣❡& ❝♦♥$&!❛✐♥& ❝❛$❡

■❢ ✇❡ ❛%%✉♠❡ ❛✇❛② "❤❡ ❜✉❞❣❡" ❝♦♥%"$❛✐♥"✱ "❤❡ ✜$%"✲♦$❞❡$ ♦♣"✐♠❛❧✐"② ❝♦♥❞✐"✐♦♥% ♦❢ "❤❡ ❧❡❛❞❡$

❜❡❝♦♠❡✿

˙

µ^D = N

pKM

KCµ^D+µ^QδM(L+τ−pE)

, µ^D(T) = 2(1−α)(D(T)−Db),✭✷✾✮

Z T

0

∂HL

∂τ dt = 0⇔ −δµ^Q Z T

0

Ddt= 0, ✭✸✵✮

Z T

0

∂H_L

∂φ dt = 0⇔ Z T

0

µ^Ddt= 0. ✭✸✶✮

✇✐"❤D˙✱Q˙ ❛♥❞µ˙^Q ❛% ❜❡❢♦$❡✱ %❡❡ ❆♣♣❡♥❞✐①✱ ❡/✉❛"✐♦♥% ✭✸✹✮✱ ✭✸✽✮ ❛♥❞ ✭✸✾✮✳

❙♦❧✈✐♥❣ "❤❡ ❞✐✛❡$❡♥"✐❛❧ ❡/✉❛"✐♦♥% ♦❢ /✉❛♥"✐"② ❛♥❞ /✉❛❧✐"② ②✐❡❧❞%✱ ❛% ❜❡❢♦$❡✱

D(t, φ) = e^−ρtD(0) + Θ(φ)(1−e^−ρt), Q(t, φ, τ) = Q0−Λ(τ)

(D0−Θ(φ))

ρ (1−e^−ρt) + Θ(φ)t

.

✶✵