6-hw116

6-hw116 - the quantity Z 1 u t 2 dt Find the function u t...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #6 Last modiﬁed: October 26, 2007 Due Thursday, Nov. 1 at class if the World Series ends in 4 or 5 games. Due Saturday, Nov. 3 at 5PM if the Series continues to Wednesday night. 1. Luenberger, Section 3.13, problem 16. 2. Luenberger, Section 3.13, problem 13. Hint: diagonalize the Gram matrix. 3. Contemplating a run for public oﬃce after 1 decade, you establish a trust fund that, at time t = 0, has a balance x (0) = 1 million dollars. It is managed by a skillful group of undergraduate investors who bring in income proportional to the balance, so after a decade it will grow on its own to a balance of e . However, you feel you will need x (1) = e 2 million dollars to win the election, and you decide to add in your own personal funds at a rate u ( t ) So the diﬀerential equation for the trust fund balance is ˙ x ( t ) = x ( t ) + u ( t ) . Suppose that the government penalizes candidates who contribute to their own campaign funds by subtracting from their available free broadcast time

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Unformatted text preview: the quantity Z 1 u ( t ) 2 dt. Find the function u ( t ) that achieves your goal while minimizing the reduc-tion in broadcast time. 4. Luenberger, Section 3.13, problem 20. 1 5. Luenberger, Section 3.13, problem 23. 6. As an application of the preceding problem, suppose that you have a con-tract to produce hydrogen gas by electrolysis over a period of 1 year. You do this by using a direct current u ( t ). To meet the delivery requirement, Z 1 u ( t ) dt ≥ c 1 . An additional requirement, to encourage production late in the year, is Z 1 tu ( t ) dt ≥ c 2 . You want to meet both requirements while minimizing Z 1 u ( t ) 2 dt, which is proportional to the electric energy dissipated in your system. Solve this problem, and identify for what sets of values of c 1 and c 2 you might overfulﬁll either the ﬁrst or second requirement. 2...
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6-hw116 - the quantity Z 1 u t 2 dt Find the function u t...

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