Optimal Control of PDEs

Optimal Control of PDEs Suzanne Lenhart University of Tennessee, Knoville Department of Mathematics Lecture1 p.1/36

Outline 1. Idea of diffusion PDE 2. Motivating Eample 3. Big picture of optimal control of PDEs 4. Details of motivating eample 5. Beaver Eample (joint work with Bhat) Lecture1 p.2/36

Diffusion description of movement arising from many short movements in random directions Suppose an organism moves along a line: moving a distance to left with prob. 1/2 moving a distance to right with prob. 1/2 p(, t) = probability that the organism is at location at time t Lecture1 p.3/36

movement Lecture1 p.4/36 from both sides subtract and divide by

"! % # #! ( # % ) # ( ' # & Lecture1 p.5/36 scale Impose the scaling $ % $ * % $ % $ % % $

, + 9, - 6 =2 < 9, 3 ; 2-6. +87 5 / / Lecture1 p.6/36 limit, go to 0 41goes to As 2 41 1 3. / 0 23 1./0 - +87 5 1 41 1 3 4 / 0. / - 41 1 3./0 2 3 41 1 3 4 / 0 ;. / : - to goes > 2 4 / Diffusion Equation 5 / /? 5 1

L K @ @ A J L K L K P L K N Q N N J J J L K @ J More space dimensions @CE E @CD D @CB B A @ H I8J @GF where includes growth, decay or control features Diffusion depend on space and time @ D D H I8J @OB D H I J @OB B H M I J A @GF H I8J Lecture1 p.7/36

Y R Z R R R U V T T T Y ^ [ R V c [ R V V Y R R T Z U Motivating Eample Consider the problem of harvesting: RGS WX on on R`_ W[ \ à b in (side boundary) W[ \ Y ] Note logistic growth diffusion coefficient harvesting control. and constant. The function is the WX Lecture1 p.8/36

w g rs d q z y f u p v i d } ƒ f Objective functional { e t f h g e f on jkml which is a discounted "revenue less cost" stream. We seek to g e f ~ over controls,. (set p =1) Lecture1 p.9/36

Background There is no complete generalization of Pontryagin s Maimum Principle in the optimal control of PDEs. We will see how to handle this. Lecture1 p.10/36

Š ˆ ˆ Set-up Start with a PDE with solution and control ; with operator as your favorite partial differential operator: 8ˆ in ˆ along with Œ Lecture1 p.11/36

Ž Ž Ž Ÿ Ÿ Ž Ž œ Objective The objective functional represents the goal of the problem: seek to find the optimal control an appropriate set such that in with objective functional ž š Lecture1 p.12/36

ª «ª ª Steps Eistence of an Optimal Control To derive the necessary conditions, we need to differentiate the objective functional with respect to the control, i.e., differentiate the map Note that usually contributes to so we also must differentiate the map, Lecture1 p.13/36

¹ ±² ³ º» µ Steps continued The map is differentiable in the directional derivative sense: The function is called the sensitivity" of the state with respect to the control. We need the sensitivity PDE to find the adjoint PDE. Lecture1 p.14/36

À Á Â Å Å continued Formally, the sensitivity is in the direction of The sensitivity solves a PDE, which is linearized version of the state PDE: ¼¾½ ¼¾. È É Ç Æ À Å Ã Ä Lecture1 p.15/36

Ê Í Ë ÊË Ì Î Ï Ê Ð Ñ Ì Ì Ò Ò How to find adjoint becomes from integration by parts Final time condition at Ê Î nonhomogeneous term integrand of J state Lecture1 p.16/36

Ó Õ Õ Ö á Ó Ó à â ÙÚ Ø Û ã à ß Þ Ý Ü ä Õ å Steps continued Differentiate the objective functional respect to at, for min problem Ö Ô Õ with Ö Ô Õ Ô Õ Use the adjoint problem and to simplify and obtain the eplicit characterization Ö ä ç Ô æ Lecture1 p.17/36

Optimality System State PDE Problem Adjoint PDE Problem coupled with the optimal control characterization NUMERICAL METHOD iterative method with forward and backward sweeps Lecture1 p.18/36

ï è ð è è è ë ì ê ê ê ï ô ñ è ì ø ñ è ì ì ï è è ê ð ë Motivating Eample, again Consider the problem of harvesting: ègé íî on on è`õ íñ ò ö` in (side boundry) íñ ò ï ó Note logistic growth diffusion coefficient harvesting control. and constant. The function is the íî Lecture1 p.19/36

ü ù û ÿ þ ù û Objective functional ú û ý ü ú û which is a discounted "revenue less cost" stream. We seek to ü ú û over controls,. Lecture1 p.20/36

Start the steps Differentiate the maps: control control state, objective functional Lecture1 p.21/36

Sensitivity Given a control, consider another control, where is a variation function and. The corresponding states and PDEs are! "! # $! % "! # $! % & ' (! % "! % #*) '! % $ ' # $! %! & ' (! "! #) '! $ '! Lecture1 p.22/36

, + + -. : 0 92 7 - -, +, < - Lecture1 p.23/36 Form difference quotients and find the corresponding PDE, : 3 8 0 0 2 3, 0 5 0 243, / 0 5 6 0 243, /10-5 0 2 3, /10 ;

> = @?? A @?? B = H D GF? A @? L K @ @ Lecture1 p.24/36, and D F I MD Assuming, as HJI CED

Y T P O O O Y Z W P W W [ P P \ \ \ \ T T P O O O Y Z \ W P _ [ ^ \ W P P ` Lecture1 p.25/36 V*W X Sensitivity and Adjoint PDE The resulting PDE for (sensitivity) on U R QSR O N VW X on on We calculate the adjoint equation: \ ] QSR O \ N O V*W X on on

pm j o e a g e Lecture1 p.26/36 Differentiate J k l prq mn kel j bc i h dfe

s { ˆ ˆ ˆ z y Ž Œ { Š ˆ z y Ž Œ { Š ˆ z y Ž Œ { Š ˆ z y Ž Œ { Š ˆ ˆ z y Ž Œ { Š ˆ ˆ z y Differentiate J continued ƒ ƒ }~ tuwv }~ Ž Ž }~ Ž }~ Ž }~ Ž }~ Lecture1 p.27/36

ž ž Lecture1 p.28/36 we, at Differentiate J By differentiating the map obtain * Ÿ š wœ š

³ ² ± µ ² ² B =0 case Linear in control bang-bang and singular if bang-bang if if singular ªƒ«ªƒ«ªƒ«¹ joint work with Joshi and Neubert Lecture1 p.29/36

Harvest Simulation Control wrt Space and time Control 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 Space 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 Time 4 Lecture1 p.30/36

Ä Ä Â Ã º º º ¾ ½ ½ ½ Ä Ä Å Å Ë È È º Æ Æ Ä Ì Ä Ë Å È º Æ Ì ¾  Á Beaver Model The population density of species is given by this model: º¼» ÀƒÇ Æ ÀƒÇ Æ ÀƒÁ ÀƒÇ Æ ºÊÉ on in À*Å Æ in À*Å Æ where is boundary of our spatial domain is the constant diffusion coefficient and are spatially dependence growth parameters Lecture1 p.31/36

Î Í Ô Ó Ò Î Ñ Ô Ó Ò Í Ñ more on model, the proportion of population to be trapped per unit time. state variable control variable. the initial density distribution. The zero boundary conditions imply the unsuitability of the surrounding habitat. ÏƒÐ Ñ Ï Ð Ñ ÎÖÕ Lecture1 p.32/36

æ á å Ý Ú Ù à ó Ü ø ø á ð ä ã Ù öõ ï ô è ú Ù á Ù Ù Beaver continued Given the control set ß á âá Ý*ß ÜÞÝ Û ØƒÙ ä à ã à we seek to minimize the cost functional: áø Ü ö Ù ò Ýñ ç Ý å ù îí éêìë that minimizes the We find the optimal control objective functional: ç Ý á ú ç Ý üwýû þ ÿ Lecture1 p.33/36

Cost functional represents the density dependent damage term represents the cost of trapping unit cost amount trapped. The term is included for discounted value of the future costs accruing over the given finite planning horizon.. Lecture1 p.34/36

, + * & % ) ' $ " # + 1 2 0 # #.. + 1 * 3 + 2 1 0 #.. + #.. ) Adjoint and OC Characterization Given an optimal control and corresponding state, the adjoint problem is given by: in (! on on * -/. * -. and ; < % 1 * 9: 4 8 576 4 Lecture1 p.35/36

background We obtained 10 years of beaver data from New York Department of Environmental Conservation Estimated parameters Ran simulation and found optimal control Will see those results in the lab later Lecture1 p.36/36