prob_set4

# prob_set4 - Problem Set 4 ARE 261 Question 1 Consider the...

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Problem Set 4 ARE 261 September 30, 2002 Question 1 Consider the following optimization problem: max S ( y,u )= n 1 X i =0 F ( x i ,y i ,u i ) x (1) where y 0 is given and where y i +1 y i = u i xi =0 ,...,n 1 (2) Note that x can be interpreted as the number of units of time and F ( x i i i ) is a f ow (pro F ts per unit of time). The optimization problem (equations 1 and 2) can be written as a La- grangian, L = n 1 X i =0 μ F ( x i i i ) x λ i ( y i +1 y i u i x ) (3) Write the derivative of the Lagrangian with respect to the control, state and co-state variables. Question 2 De F ne the Hamiltonian as: H i ( x i i i i F ( x i i i )+ λ i u i (4) Write the Lagrangian in terms of the Hamiltonian Rewrite the F rst order conditions in terms of the Hamiltonian. What is the termial value of λ ,( λ n 1 )? 1

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Question 3 Suppose we included a “scrap function,” f ( y n ) , to the original problem, so that the problem becomes max S ( y,u )+ f ( y n ) (5) What is the terminal value of λ ? Question 4 Take the limit as
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## This note was uploaded on 08/01/2008 for the course ARE 263 taught by Professor Karp during the Fall '06 term at Berkeley.

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prob_set4 - Problem Set 4 ARE 261 Question 1 Consider the...

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