Introductory Lecture 2 Suzanne Lenhart University of Tennessee, - PowerPoint PPT Presentation

Introductory Lecture 2 Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics Supported by NIH grant Lecture2 – p.1/25

✛ ☎ ✘ ✜ ✗ ✜ ☎ ✞ ✟ ✟ ✟ ✘ ✎ ✞ ✙ ✖ ☎ ✄ ✠ ✌ ✒ ✟ � ✘ ✠ ✚ ✑ ✎ ☎ ✟ ✘ ☎ ✠ ✟ ✖ ✎ ✞ ☎ ✘ ✜ ✙ ✜ ✟ ✞ ✦ ✔ ✥ ✚ ✌ ✒ ✤ ✣ ✣ ✣ ✢ ☛ ✡ ✔ ✖ ✟ ✟ ✓ ✎ ✕ ✔ ✓ ✆ ✌ ✒ ✟ ✟ ✞ ✑ ✎ ✟ ✆ ✎ ✖ ☞ ☛ ✡ ✠ ✟ ✟ ✞ ☎ ✟ ✄ ✂ ✁ ☎ ✆ ✙ ✞ ✎ ✜ ☞ ☛ ✡ ✔ ✟ ✙ ✗ ✜ ✟ ✟ ☎ ✎ ✆ ☎ ✄ ✘ ☛ ✆ ✔ ✟ ✖ ☎ ✆ ✟ ✑ ✄ Lecture2 – p.2/25 is final payoff. What change results? same as before ☎✍✌ ☎✝✆ ☎✍✌ ☎✍✌✏✎ Salvage Term ☎✍✌✏✎ ☎✝✆ ☎✍✌ ☎✍✌ ☎✝✘ where . . .

✹ ✪ ✧ ✺ ✩ ✻ ✷ ✼✽ ✩ ✲ ✹ ✬ ✴ ✫ ✪ ✼✽ ✬ ✧ ★ ✬ ✬ ✪ ✩ ★ ✩ ✪ ✫ ✩ ✮ ★ ✩ ★ ✶ ✯ ✰ ✱ ✮ ✴ ★ ✩ ✵ ✩ Lecture2 – p.3/25 ★✸✷ ★✭✬ ✲✳✬ ★✭✬ Only change Example

✾ ❄ ✿ ❄ ❁ ❈ ✿ ✾ ❁ ❏ ❂ ▼ ❉ ✾ ❅ ❀ ❋ ❃ ✾ ❃ ❂ ❁ ◆ ❂ ❁ ❃ ❀ ❀ ❁ ❏ ✾ Example ✿✍❀ Number of cancer cells at time (exponential growth) State ✿✍❀ Drug concentration Control ✿✍❀ ❁❇❆ ✿✍❀ known initial data ✾❊❉ ❁▲❑ ✿✍❀ ●■❍ where the first term is number of cancer cells at final time and the second term is the harmful effects of drug on body. Lecture2 – p.4/25

❖ ❯ ❘ ❖ P ❵ ❛ ❜ ❝ ❘ ❙ ❳ ❖ ❞ ❙ ❙ ❳ ❚ ❯ ◗ P ❫ ❣ ❳ ❙ ❞ ❖ ❖ ❳ ❯ ❙ ❪ ❤ ❘ ❱ ❳ ❖ ◗ ❘ ❖ ❳ P ❳ ❨ ❨ P ❖ ❩ P ❱ ❘ ❬ ❖ ❘ ❯ ❨ ❨ ❱ ❖ ❪ ❩ ❯ ❘ ❖ ❭ P ❭ P ✐ Lecture2 – p.5/25 ❯❢❥ transversality condition ❘❴❫ here at ❯❲❱ ❙✸❤ ❙✭❚ ❯❢❡

⑥ ♦ ♣ ♦ ⑥ ❧ ④ ⑩ ③ r q ♣ ⑨ q ⑧ ② ✇ ♣ r ① q ♣ ♦ ❦ ❶ r ③ ♠ ♦ ⑥ r q ♣ ♦ t ✇ q ♦ ③ ♦ r q ♦ ❧ ✉ ✉ ❷ ③ ⑧ ✇ q ❧ ⑤ r ✈ ♣ ♦ ❧ ❦ ✇ q ♦ ❧ ♠ ❦ ❧ ① ✉ t ❦ s r q ♣ ♦ ⑤ ❧ q r ⑥ ♦ ④ ③ ⑧ ② ✇ ♣ r ① q ♣ ⑥ ♣ ③ ⑤ ❧ ⑦ ❹ ⑤ ❧ ④ ③ ② ✇ q Lecture2 – p.6/25 ③❲⑥ ❦♥♠ t✸❸ Contd.

❽ ❾ ❾ ❺ ❽ ❽ ❾ ➀ ❿ Well Stirred Bioreactor Contaminant and bacteria present in spatially uniform time varying concentrations ❻✸❼ concentration of contaminant ❻✸❼ concentration of bacteria bioreactor rich in all nutrients except one ❻✸❼ concentration of input nutrient bacteria degrades contaminant via co-metabolism. Lecture2 – p.7/25

➁ ➃ ➃ ➣ → ➔ ➅ ➒ ➐ ➁ ➓ ➅ ➒ ➃ ➂ ➅ ➄ ➑ ➅ ➉ ➔ ➅ ➔ ➜ ➛ ➙ ↕ ➔ ➊ ↕ ➉ ➄ ↔ ➅ ➅ ➉ ➃ ➁ ➁ ➉ ➈ ➋ ➃ ➁ ➅ ➅ ➌ ➈ ➅ ➈ ➅ ➉ ➐ ➅ ➁ ➅ ➐ ➊ ➆ ➑ ➂ ➃✍➄ ➅➇➆ ➃✝➉ ➃✍➄ ➅❇➊ ➃✍➄ ➃✝➉ ➅➇➆ where ➍➏➎ ➃✍➄ ➃✍➄ ➃✍➄ where is control and are known. Objective functional: ➃✝➉ ➅➇➆ ➃✍➄ ➃✍➄ Find to maximize ➃✝➉ ➅➇➆ ➃✝➉ maximize bacteria and minimize input nutrient cost. Lecture2 – p.8/25

➝ ➠ ➠ ➯ ➡ ➨ ➝ ➞ ➸ ➞ ➠ ➝ ➝ ➫ ➞ ➟ ➠ ➭ ➭ ➫ ➢ ➠ ➡ ➥ ➦ ➧ ➢ ➩ ➨ ➫ ➭ ➠ ➯ ➩ ➞✸➟ ➞✭➭ ➝➤➢ ➲➵➳ ➝✸➢ ➞✭➺ penalizes large values of at final time . Can eliminate variable and work with only. Lecture2 – p.9/25

➻ ➮ ➪ ➻ ➘ ➚ ➼ ➻ ➷ ➾ ➬ ➪ ➷ ➚ ➼ ➬ ➱ ➾ ➪ ➻ ➾ ➾ ➴ ➻ ➷ ➚ ➼ ➬ ➮ ➱ ➪ ➽ ➬ ➷ ➼ ➼ ➼ ➾ ➽ ➼ ➪ ➶ ➶ ➾ ➻ ➽ ➹ ➚ ➶ ➚ ➶ ➾ ➾ ➼ ➾ ➻ ➘ ➾ ➾ ➴ ➽ ➹ ➚ ➾ Lecture2 – p.10/25 at ➼❲➽

✃ Ý ❮ à ß Þ Û Ï Ò Ð ✃ Ù Ò á Ð Ø Õ Ð × Ö ❮ ã Ï à Ï Ï Þ ❮ ✃ æ ❒ Û Ü Ù Ð à Ð × Ï Õ ❮ à ß Þ Û Ï Ò Ð ✃ Ù ß ❮ Û × Ï ❒ Û Ú Ù ✃ Ø Õ Ð Ö ❮ ❮ Ó Õ Ï ✃ ❮ ❒ Ï ❰ ❮ ✃ Ü ✃ ❮ Ò Ð ✃ Ù â Õ Ï Ò Ï á à Ò Ý ✃ Ð Ù Ï Û Þ ß Lecture2 – p.11/25 ÒÔÓ numerically. known ❰ÑÐ Ð❊å Ðèç Solve for ❐❢❒ ❐❢❒ ❐ä❒

õ ï í î ô õ ì í î ð ô ñ ò ó ö ô õ ï é ë é ñ ê ë ì í î ï ð ò ò ó ø ì í î ï ð ñ ô Problems What if: Need additional constraint ì✸÷ Lecture2 – p.12/25

ÿ � þ ✡ ✌ ý � ù ù ✌ þ ✍ ☞ ✎ � ✏ ✁ ✡ ✆ ✝ ✞ ✟ ✑ ✡ ☞ ✡ ✌ ù ☞ ✍ ✍ ☞ ü ù ý þ ù ÿ þ � ù ù ÿ ☞ ✑ � ✌ ÿ ✡ ✑ ✄ ☎ ✆ ✝ ✞ ✟ ✑ Fishery Model ú❢û ý ✂✁ population level of fish ý ✂✁ harvesting control Maximizing net profit: ✠☛✡ where is discount factor, terms represent profit ☞✒✑ from sale of fish, diminishing returns when there is a large amount of fish to sell and cost of fishing. are positive constants. Lecture2 – p.13/25

✦✮ ✙ ✢ ✣ ✥ ✩ ✦ ✛ ✧ ✣ ✤ ✢ ✜ ✤ ✭ ✓ ✣ ✢ ✛ ✙ ✥ ✗ ✖ ✕ ✔ ✓ ✜ ✫ ✫ ✦ ✯ ✜ ✰ ✢ ✥ ✤ ✙ ✗ ✖ ✕ ✔ ✭ ✦ ✛ ✧ ✣ ✢ ✜ ✛ ✙ ✥ ✗ ✖ ✕ ✔ ✰ ✢ ✩ ✣ ✓ ✰ ✓ ✣ ✦ ✣ ✣ ✦ ✢ ✣ ✥ ✢ ✥ ✩ ★ ✜ ✛ ✧ ✤ ✦ ✦ ✜✢ ✥ ✤ ✙ ✣ ✜✢ ✛ ✤ ✗ ✖ ✕ ✔ ✜✢ ✩ ✢ ✛ ✭ ✣ ✥ ✩ ✦ ✢ ✤ ✜ ✤ ✙ ✭ ✣ ✜ ✙ ✪ ✥ ✗ ✖ ✕ ✔ ✬ ✣ ✓ ✢ ✫ ✫ ✣ ✓ ✰ Lecture2 – p.14/25 ✘✚✙ Contd.

✺ ✱ ✲ ✳ ✴ ✲ ✳ ✵ ✶ ✱ Contd. Solve for numerically. Need control bounds ✷✹✸ ✻✽✼ Ref: B D Craven book Control and Optimization Lecture2 – p.15/25

■ ❑ ❋ ❑ ❉ ✾ ▼ ▼ ❉ ❆ ❉ ■ ❈ ❇ ■ ❋ ◆ ❇ ■ ❇ ❈ ❙ ■ ❉ ❆ ❘ ❚ ◗ ✾ P ❖ ❋ ■ ■ ❆ ❋ ❆ ✿ ❀ ❁ ❙ ❋ ■ ❈ ❉ ❈ ❊ ❋ ● ❇ ❍ ❇ ❉ ▲ ❋ ❊ ❈ ❉ ❈ ❆ ❑ ❈ ❏ ❉ ❈ ❋ ■ ❇ ■ Interpretation of Adjoint ❂❄❃ ❆✹❇ ❉✽■ ❂❄❅ ( Definition of value function ) ❆✹❇ ❆✹❇ ❉❯■ Units: money/unit item in profit problems. Lecture2 – p.16/25

❱ ❫ ❳ ❨ ❳ ❬ ❱ ❜ ❬ ❳ ❛ ❬ ❳ ❵ ❲ ❲ ❭ ❬ ❨ ❳ ❪ ❲✹❳❩❨ = marginal variation in the optimal objective functional value of the state value at . “Shadow price” additional money associated with addi- tional increment of the state variable ❲✹❳ for all ❳❞❝ ❪❴❫ “If one fish is added to the stock, how much is the value of the fishery affected ?" Lecture2 – p.17/25

❡ ❦ ❣ ❥ ❤ ❦ ❣ ❢ ❤ ♥ ✐ ❥ ❤ ♣ ♣ ❣ ❢ ❤ ✐ ❥ ❤ ❦ ♠ ❣ ❥ ❤ ❦ ♠ ♥ ❦ ❤ ❦ ❧ ♠ ❣ ❥ ❤ ❦ ✐ ❣ ❢ ♥ ❥ ✐ ❥ ❤ ❦ ♦ ❣ ❣ ✐ ❥ ❤ ❤ Lecture2 – p.18/25 Original value + adjoint ❢✹❤ ❢✽❤ ❡❴❢ New value Approximate

✉ q ③ ④ ⑤ ⑩ ⑨ ⑥ ⑩ t s t s ❶ ① ❷ ✉ t ❸ ❹ ❺ ⑩ ✉ s t s ② ✇ ❶ ① q r s t r ❶ ✉ ✉ ① ⑩ r ✉ ✉ t ❹ ❶ q r s t r ✉ ① ✉ ✉ q Principle of Optimality If is an optimal pair on and , ✉❩✈ ✉❩✇ ✉❩✈ ✉❩✇ then is also optimal for the problem on : ⑥⑧⑦ x (t, x*(t )) x 0 t t0 t t1 Lecture2 – p.19/25

➀ ➇ ➅ ➁ ❾ ➆ ❿ ❻ ➁ ➃ ❾ ➈ ❿ ➂ ➈ ➅ ❿ ❻ ➅ ❼ ➅ ➊ ❿ ❻ ❾ ❻ ❼ ❽ ❾ ❼ ❽ ➂ ❻ ❾ ➀ ➁ ➂ ❽ ❽ ❾ ❽ ➁ Existence of Optimal Controls “Sufficient conditions to guarantee existence of OC" Suppose satisfy ➃✹➄ ➃✹➄❩➆ ➃✹➄❩➉ is maximized w.r.t. at plus set of controls compact jointly concave in and bounded state functions Lecture2 – p.20/25

↔ ➓ ➔ ➐ ➍ ➎ ➏ ➏ ➣ → ➔ ➎ ➛ ➒ ➑ ➐ ➏ ➐ ➌ ➋ ➎ ➏ → For details about existence of OC see Macki and Strauss book Fleming and Rishel book Back to exercise example ➍❩➎ ➍➙↕ To guarantee the maximum value of would be finite, need a priori bound on state , control . Lecture2 – p.21/25