An Introduction to Optimal Control Applied to Disease Models - PowerPoint PPT Presentation

An Introduction to Optimal Control Applied to Disease Models Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics Lecture1 – p.1/37

☎ ✞ ✞ ☎ ☎ � � ✍ ✆ ✠ ☎ ☞ � ✟ ✂ � ✝ � ✝ ✆ ☎ ✎ ✆ ☎ ✂ ✝ ☎ ✂ ✁ Example ✁✄✂ Number of cancer cells at time (exponential growth) State ✁✄✂ Drug concentration Control ✁✄✂ ✁✄✂ ✁☛✡ known initial data �✌☞ minimize ✁✄✂ where the first term represents number of cancer cells and the second term represents harmful effects of drug on body. Lecture1 – p.2/37

Optimal Control Adjust controls in a system to achieve a goal System: Ordinary differential equations Partial differential equations Discrete equations Stochastic differential equations Integro-difference equations Lecture1 – p.3/37

✒ ✕ ✓ ✑ ✔ ✘ ✒ ✑ ✗ ✖ ✓ ✓ ✔ ✏ ✓ ✏ ✓ ✓ ✘ ✔ ✔ ✓ ✓ ✏ ✘ Deterministic Optimal Control Control of Ordinary Differential Equations (DE) ✑✄✒ control ✑✄✒ state State function satisfies DE Control affects DE ✑✄✒ ✑✄✒ ✑✄✒ ✑✄✒ Goal (objective functional) Lecture1 – p.4/37

Basic Idea System of ODEs modeling situation Decide on format and bounds on the controls Design an appropriate objective functional Derive necessary conditions for the optimal control Compute the optimal control numerically Lecture1 – p.5/37

Design an appropriate objective functional –balancing opposing factors in functional –include (or not) terms at the final time Lecture1 – p.6/37

Big Idea In optimal control theory, after formulating a problem appropriate to the scenario, there are several basic problems : (a) to prove the existence of an optimal control, (b) to characterize the optimal control, (c) to prove the uniqueness of the control, (d) to compute the optimal control numerically, (e) to investigate how the optimal control depends on various parameters in the model. Lecture1 – p.7/37

✢ ✜ ✜ ✜ ✩ ✛ ✫ ✜ ✪ ✢ ✫ ✙ ✬ ✜ ★ ✢ ✚ ✛ ✜ ★ ✢ ★ ✜ ✜ ✙ ✯ ✜ ✮✯ ✜ ✢ ✚ ✛ ✚ ✜ ✣ ✤ ✥ ✦ ✧ ✢ ★ ✢ ✧ ✙ Deterministic Optimal Control- ODEs Find piecewise continuous control ✚✄✛ and associated state variable to maximize ✚✄✛ ✚✄✛ ✚✄✛ subject to ✚✄✛ ✚✄✛ ✚✄✛ ✚☛✭ and Lecture1 – p.8/37

✴ ✴ ✴ ✰ ✱ ✵ ✱ ✴ ✱ ✱ ✰ ✵ ✱ ✴ ✴ ✱ ✰ ✵ Contd. Optimal Control ✲✄✳ achieves the maximum Put ✲✄✳ into state DE and obtain ✲✄✳ ✲✄✳ corresponding optimal state ✲✄✳ ✲✄✳ , optimal pair Lecture1 – p.9/37

✺ ✹ ✻ ✷ ✺ ✼ ✶ ✷ ✺ ✺ ✸ ✷ ✷ ✻ ✻ ✺ ✷ ✷ ✶ ✺ ✶ Necessary, Sufficient Conditions Necessary Conditions ✸✄✹ If , are optimal, then the following conditions hold . . . Sufficient Conditions If ✸✄✹ , ✸✄✹ and (adjoint) satisfy the conditions . . . ✸✄✹ ✸✄✹ then , are optimal. Lecture1 – p.10/37

Adjoint like Lagrange multipliers to attach DE to objective func- tional. Lecture1 – p.11/37

❁ ❀ ❀ ❀ ❈ ✿ ❊ ❀ ❉ ❁ ❊ ✽ ❋ ❀ ❇ ❁ ✾ ✿ ❀ ❇ ❁ ❇ ❀ ❀ ✽ ■ ❀ ❍■ ❀ ❁ ✾ ✿ ✾ ❀ ❂ ❃ ❄ ❅ ❆ ❁ ❇ ❁ ❆ ✽ Deterministic Optimal Control- ODEs Find piecewise continuous control ✾✄✿ and associated state variable to maximize ✾✄✿ ✾✄✿ ✾✄✿ subject to ✾✄✿ ✾✄✿ ✾✄✿ ✾☛● and Lecture1 – p.12/37

❑ ❯ ❖ ❚ ◗ P ◗ ❏ ❚ ◗ ▼ ❙ ❙ ❙ ◆ ❖ ❚ ◗ P ❯ ❖ ❱ ❲ ◆ ◆ P ❚ ◗ ❏ ❑ P ▲ ❑ P ▼ P P P ❘ P ▼ ◗ ❑ ❏ ❏ ❑ ◆ P ◗ ▼ ❚ ❖ Quick Derivation of Necessary Condition Suppose is an optimal control and ◆✄❖ corresponding state. variation function, . u*(t)+ ah(t) u*(t) t ◆✄❖ ◆✄❖ another control. state corresponding to , ◆✄❖ ◆✄❖ Lecture1 – p.13/37

❞ ❳ ❩ ❪ ❴ ❝ ❜ ❳ ❭ ❫ ❴ ❨ ❡ ❛ ❭ ❪ ❫ ❬ ❭ ❳ ❪ ❫ ❴ ❪ ❝ ❜ ❭ ❳ ❴ ❫ ❨ ❴ ❭ ❫ ❳ ❨ ❩ ❩ ❬ ❭ ❩ ❪ ❫ ❴ ❨ ❨ ❫ ❞ ❴ ❳ ❣ ❬ ❭ ❳ ❪ ❩ ❴ ❨ ❵ ❜ ❭ ❳ ❢ Contd. At , ❵✌❛ y(t,a) x * (t) x 0 t all trajectories start at same position when control Maximum of w.r.t. occurs at . Lecture1 – p.14/37

✇ ❥ ❦ ❥ ③ ♠ ① ❥ ② ① ♠ ❤ ❤ t ❤ ⑨ ♠ ❦ ④ ❦ ♠ ❥ ④ ④ ⑥ ❥ ② ⑦ ♠ ❦ ✈ ❤ ❥ ③ ♠ ✈ ❥ ② ⑩ ① ✈ ♠ ⑥ ✈ ② ❥ ① ❤ ❤ t ✇ ✉ ① t s ❦ ❦ ❤ ♠ ♠ ✐ ❥ ❥ ✈ ⑥ ❥ ② ✉ ❦ ④ ⑥ ❥ ✈ ❥ ❥ ② ✉ ① ❤ ♠ ♠ ❦ ✐ ❥ Lecture1 – p.15/37 ✈❷❶ ♠❧③ ♠❧③ ♠⑧⑦ ♠❧③ gives ①⑤④ ①⑤④ ♠❧③ to our Contd. ♥♦♥♦♥♣♥rq ❥❧❦ Adding

➄ ❻ ❹ ➑ ➋ ➇ ❻ ➉ ❺ ➎ ➅ ➃ ❺ ➄ ➃ ❸ ➃ ❺ ➀ ➏ ❿ ❾ ❼ ❻ ❻ ➄ ➃ ➇ ➋ ➌ ❹ ➋ ➇ ➁ ➊ ❻➓ ➉ ❺ ➆ ➃ ➅ ➃ ➄ ❹ ➎ ❻ ➋ ➇ ❻ ❺ ❺ ➎ ➍ ➃ ❹ ➁ ➊ ➊ ➇ ❻ ➉ ❺ ➊ ➅ ❻ ❼ ❺ ➃ ➄ ➄ → ➀ ❿ ❾ ➊ ➁ ➋ ❻ ❹ ➄ ➄ ❻ ➎ ❹ ➋ ❻ ❺ ➃ ➌ ❹ ❻ ➄ ➌ ❹ ➋ ➇ ➁ ➊ ❻ ❻ ❺ ➒ ❿ Lecture1 – p.16/37 ❹➂➁⑤➃ ❹➂➁ ❹➂➁ ❻❧➔ . ❹➂➁ here we used product rule and ➆➈➇ ➆➐➇ ❻⑧➍ ❹➂➁ ❹➂➁⑤➃ ❹➂➁⑤➃ ❻❧➒ ❹➂➁⑤➃ ❹➂➁ ❹➂➁ Contd. ❻❽❼ ❹❧❺

➺ ➺ ➛ ➙ ➫ ➽ ➦ ↕ ➣ ↔ ➣ ➙ ➽ ➝ ➫ ➫ ➩ ➦ ➳ ➺ ➳ ➹ ➺ ➺ ➡ ➺ ↕ ↕ ➫ ➣ ➧ ➨ ➤ ➜ ➼ ➨ ➽ ➦ ➫ ➸ ➦ ➭ ➵ ➲ ➣ ➩ ➫ ➥ ➳ ➭ ➤ ➦ ➤ ➫ ➭ ➳ ➞ ➤ ➭ ➯ ➣ ➤ ➡ ➣ ↕ ➫ ➶ ➜ ➝ ➼ ↕ ↕ ➭ ➤ ↕ ➠ ➧ ➫ ➨ ➯ ➭ ➤ ↕ ➠ ➫ ➩ ➤ ➫ ➫ ↔ ➣ ↕ ➙ ➛ ➜ ➫ ➨ ➠ ↕ ➤ ➲ ➠ ➦ ➺ ➳ ➫ ↕ ↕ ➫ ➦ ➡ ↕ ➠ ➠ ➫ ➸ ➦ ➭ ➵ ➲ ➣ ➠ ➞ ➥ ➠ ➡ ➠ ↕ ➤ ➳ ➠ ➫ ➦ ➧ ➨ ➤ ↕ ➩ ➠ Lecture1 – p.17/37 ➸➻➺ . ➦➂➲ ➝➟➥ ➠➢➡ . ➦➂➲ ➾♦➾♦➾♦➾➪➚ ➦➂➲ ➦➂➲ ➦➂➲ ➦➂➲ terms are terms are ➸➻➺ ➦❧➧ ➠➢➡ ➝➟➞ ➝➟➥ ➠➢➡ ➝➟➞ Arguments of Arguments of ➦➂➲ Contd.

➮ Ö Ó ß ❒ Þ Ø Ô Ò à ✃ Ý Ü ❮ Û Õ ❒ ❐ ❮ Ó Ö ❒ Ô Õ Ò Ô Ò ß ❐ ✃ Ø á Þ ❒ Ò Û ❒ ❒ ❰ Ò Ô Ó Û â Ï ❐ Ô â ➱ ➮ Ú ➬ ã ➘ Õ Û Ú Ò ✃ ➮ Ô ❒Ù Ô Õ Ô ➮ Ó Ò ❒ Ñ ❮ ❐ ✃ ➮ Lecture1 – p.18/37 adjoint equation for all transversality condition ✃➂❐ ❒❧Ø ✃➂❐ ❒×Ö ✃➂❐ ✃➂❐ arbitrary variation Optimality condition. s.t. ✃➂❐⑤Ò ➴✄➷ Ð➟Ñ Ð➟Þ Contd. Choose Ð➟Þ ❒❽❮ ✃➂❐ ✃➂❐

ï é è ä è î ï ó è ë ð ñ ì ô ñ ë ð ë é ò ë ó ï ï ç ô ê ò å ð ë æ ç æ è æ é ê ë ä æ é ç æ è ê ê ä æ ç æ è ê é Using Hamiltonian Generate these Necessary conditions from Hamiltonian ä✄å ä✄å éíì integrand (adjoint) (RHS of DE) maximize w.r.t. at optimality eq. adjoint eq. éíì transversality condition Lecture1 – p.19/37

÷ ú õ ù ÷ ú ü ÷ û ÷ ö û ù ö ÷ ö õ ü ÷ ö Converted problem of finding control to maximize objective functional subject to DE, IC to using Hamiltonian pointwise. For maximization at as a function of õø÷ For minimization at as a function of õø÷ Lecture1 – p.20/37

þ ÿ ✄ ý ý ý ý þ � ÿ þ þ ✂ þ ý � ÿ � þ þ ÿ � þ ✁ Two unknowns and introduce adjoint (like a Lagrange multiplier) Three unknowns , and nonlinear w.r.t. Eliminate by setting and solve for in terms of and Two unknowns and with 2 ODEs (2 point BVP) + 2 boundary conditions. Lecture1 – p.21/37