Using Pontryagin’s Maximum Principle

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

The Hamiltonian:

where λ is the co-state variable

We must satisfy the following conditions:

1. Maximize H with respect to y

2. Costate equation

3. State equation

4. End Point conditions

Solving:

This gives us:

The costate and state equations:

Note that y = λ

Integrating and x’ yields to and

respectively.

Substituting:

At t = 0, x = 20

At t = 40, x = 0

So our optimal solutions are: ,

and

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 .