Optimal Control of Discrete Models Suzanne Lenhart University of - PowerPoint PPT Presentation

Optimal Control of Discrete Models Suzanne Lenhart University of Tennessee, Knoxville Department of Mathematics SL5 – p.1/15

Introduction For many populations, births and growth occur in regular times each year (or each cycle). Discrete time models are well suited to describe the life histories of organisms with discrete reproduction and/or growth. SL5 – p.2/15

✆ ✞ ✠ � ✄ ✁ ✟ � ✂ ✄ ☎ ✝ � ✝ � ✆ Another example For example, the Beverton-Holt stock-recruitment model for a population at time is SL5 – p.3/15

✏ ✔ ✎ ✑ ✎ ✓ ✎ ✏ ✡ ✒ ✡ ✑ ✎ ✒ ✑ ✍ ✎ ✒ ✎ ✔ ✡ ✡ ✍ ✏ ✡ ✎ ✎ ✑ ✑ Age structured models A population may be divided into separate discrete age classes. At each time step, a certain proportion of each class may survive and enter the next age class. Individuals in the first age class originate through reproduction from other classes. ☛✌☞ ☛✌☞ ☛✌☞ ☛✌☞ ☛✌☞ ☛✌☞ ☛✌☞ ☛✌☞ SL5 – p.4/15

✤ ✮ ✮ ✰ ✱ ✕ ✱ ✥ ✱ ✧ ✣ ✲ ✱ ✳ ✣ ✧ ✣ ✱ ✧ ✭ ✱ ✧ ✚ ✯ ✛ ✩ ✛ ✚ ✙ ✚ ✩ ✣ ✤ ✥ ✱ ✚ ✩ ✩ ✚ ✛ ✪ ✭ ✧ ✩ ✭ ✮ ✧ ✛ ✱ Simple Control Problem ✦★✧ ✖✘✗ ✛✢✜ subject to for ✛✬✫ state ✧✵✴ control , one less component SL5 – p.5/15

❁ ❉ ✶ ❃ ✺ ❇ ✺ ✼ ❇ ✷ ❈ ✺ ✺ ❃ ❊ ✺ ✽ ✽ ✽ ✺ ❋ ❉ ❂ ❂ ❂ ✺ ❂ ✸ ❀ ✶ ✷ ✸ ✺ ❁ ✺ ✽ ✽ ✽ ✺ ✾ ✼ ✸ ❁ ❂ ✺ ❂ ✹ ✽ ✽ ✽ ✼ ✷ ❆ ✷ Control Problem Given a control and initial ✶✌✹✻✺ ✶✌✼ ✶✿✾ state , the state equation is given by the difference equation ❂❄❃✬❅ for . Note that the state has one more component than the control ✹✻✺ SL5 – p.6/15

❘ ❑ ❑ ▲ ❲ ◆ ❑ ❯ ◗ ● ❚ ❍ ▼ ■ SL5 – p.7/15 ❘❱❯ ❘✢❙ ◆P❖ ❍❏▼ ❍❏■ Goal

❭ ❤ ❢ ❢ ❳ ❣ ❡ ❳ ❪ ❵ ❳ ❨ ❳ ❡ ❤ ❬ ❳ ❤ ❨ ❦ ❩ ❬ ❧ ✐ ❴ ❭❢ ❭ ❳ ❝ ❨ ❢ ❧ ❪ ❳ ❭ ❫ ❴ ❵ ❪ ❜ ❩ ❞ ❬ ❳ ❭ ❪ ❞ ❫ ❴ ❭ ❨ ❫ ❨ ❤ Hamiltonian ❩❏❬ for ❳❱❭ ❳✬❛ ❳❱❭ Notice the indexing on the adjoint. Necessary conditions ❵❥✐ at SL5 – p.8/15

① ① ✉ ⑨ ⑧ ⑥ ⑦ ⑨ r ② r ♠ ⑥ ⑤ ① t ✈ ③ r q ② r q ⑥ ✈ ✉ t ⑩ q ♣ ❶ ⑨ Simple Control Problem ✇★① ♥✘♦ r✢s subject to for r✬④ The control is the input (growth or decay). What optimal control is expected? (min state and size of control) SL5 – p.9/15

➁ ❺ ❼ ❷ ❷ ❿ ➃ ➃ ➈ ❸ ➉ ➊ ❹ ➊ ➊ ❷ ➁ ❸ ➉ ➊ ➉ ❸ ➇ ❷ ➇ ❿ ❷ ❸ ❸ ❼ ❷ ➃ ❸ ❹ ❺ ➆ ❾ ❷ ❿ ❾ ❷ ➀ ➁ ❷ ➄ ➇ ❼ ❷ ❿ ❷ ➅ ➆ ➍ ➁ ❷ ❸ ➇ ❷ ❿ SL5 – p.10/15 Optimality Conditions for at ❷✬➂ Our necessary conditions are ❷✬➂ ❷✬➂ ❻❽❼ ➁➌➋

➓ ↕ ➟ → ➓ ➎ ➓ ➝ ➎ ↔ ➒ ➔ ➜ ➛ ➙ ➜ ↕ ↔ ➓ ➣ → ➒ ➟ → ➔ ➏ ➎ ➓ ➒ ➓ continuing for ➎➐➏✬➑ ➏✬➑ →➞➝ and ➎✵➟ ➎✵➟ ➎➠➟ solve algebraic equations SL5 – p.11/15

➯ ➯ ➦ ➯ ➦ ➳ ➫ ➦ ➫ ➡ ➫ ➵ ➦ ➦ ➡ ➨ ➧ ➦ ➸ ➡ ➳ Control Answers optimal state values , ➢➥➤ ➢➥➩ ➢➥➭ optimal control values , , . ➢➥➲ ➢➥➤ ➢➥➩ SL5 – p.12/15

➷ ➼ Õ ➺ ➬ ➼ Ñ ➮ Ô ➪ ➷ ❐ Ó ➴ ➮ ➷ ➷ ➾ ➴ ➪ ➴ ➴ Ò ➷ ➘ ➶ ➘ ❐ ➺ ➼ ➾ ➼ ➪ ➺ ➘ Ò ➷ ➪ ➴ ❐ Ó ➬ ➶ ➺ ➴ ➷ ➷ ➷ ➴ Ó ➼ ➪ ➺ ➘ ➻ ➷ ➷ ➴ ➴ ➷ ➷ ➾ ➼ ➪ ➮ ➴ ➾ ➼ ➴ ➷ ➼ ➷ ✃ ➼ ➪ ➺ ➘ ➻ ➹ ➶ ➪ ❐ ➼ ➴ Ñ Ï ➴ ➷ ➷ ➴➷ ❰ ➴ ❮ ➶ Ð ➴ ❮ ➴ ✃ ➷ ➷ ➴➷ ❰ ➴ ❮ ➴ ❒ ➶ ✃ ➷ ❐ ➴➷ SL5 – p.13/15 time steps. ➾❱➴ ➮P➱ . ➮Ù➱ states, and ➺✌➻➽➼ ➾❱➴➷ Define the objective functional as ➺P➬ ➾Ø➴ ➺✌➻➽➼ ➺✌➻➽➼ controls, ➾❱➴➷ System Case ÓPÖ ➾✢× ➾✬➚ There are ➺✌➻➽➼ for

æ Ú Û î ß ç ê Ú ß ç Ý í í í Û Ú ß ç ê Ú å à Ú ß á í Ý á ß ñ ð Û Ú ß Þ í Ú í æ î â ç Ý à á á á à î ß ß î ß à á ã ã à Ú ß ä à á á ß ß Þ à æ Ú ß â Ý à á á Ú ß æ Û å â á àá ß Þ ã à Ú ß â Ý à á á Ú ñ ß à Ü ç ì Þ ã ß ç ê Þ Ü SL5 – p.14/15 ãÙä ãPä ÚØàá Ú❱àá Úòàá Ü❏Ý✌Þ at Ý✵Þ Ú❱àá Ü❏Ý✌Þ ã✌ï Ú✬ë continued çéè

ô û ✡ ☛ û ô û õ ✁ û ô ó ☛ û ô ✞ ✝ ☞ ó ☞ ☞ ✌ ✞ ☞ ☞✌ ô þ û õ ☛ ☞ ☛ þ ✞ ý ☞ þ ✍ ✞ ý ó ☞ ☞ ✌ ✌ ✌ ☞ ✌ þ ✡ ✌ ✂ õ ✁ û ó � ÿ þ ý ú ✄ ù ø ÷ ö ✌ õ ✌ ☞ ô ✌ û ó ☞ ✠ ☛ ✞ ✡ þ û ô û ó ✁ û ☞ ó ✁ þ ✟ û ó û ó ✞ ✝ ✞ SL5 – p.15/15 pest population to be Simple pest control problem for for û✢ü good population and for . controlled with û ✆☎ û ✆☎