Download - Cake eating problem
![Page 1: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/1.jpg)
Using Pontryagin’s Maximum Principle
![Page 2: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/2.jpg)
Cake-eating Problem Andy Lim receives his favorite chocolate cake on his
birthday celebration. He wants to maximize his consumption on the chocolate cake before his siblings arrive within 40 minutes.
His problem is represented by:
Subject to x’ = y; x(0) = 20; x(40) = 0where t is time;
x is the size of the cake (state variable); andy is his consumption (control variable).
dtty40
0 2max
2
![Page 3: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/3.jpg)
The Hamiltonian:
where λ is the co-state variable
We must satisfy the following conditions:
1. Maximize H with respect to y
![Page 4: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/4.jpg)
2. Costate equation
3. State equation
4. End Point conditions
![Page 5: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/5.jpg)
Solving:
This gives us:
![Page 6: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/6.jpg)
The costate and state equations:
Note that y = λ
![Page 7: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/7.jpg)
Integrating and x’ yields to and
respectively.
Substituting:
At t = 0, x = 20
![Page 8: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/8.jpg)
At t = 40, x = 0
So our optimal solutions are: ,
and
![Page 9: Cake eating problem](https://reader038.vdocuments.net/reader038/viewer/2022100501/559f82241a28ab45318b45bc/html5/thumbnails/9.jpg)
Interpretation If Andy maximizes his consumption on the cake, the
size of the cake changes by .
His consumption decreases at a constant rate of and his marginal utility is also .