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assn5 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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16.323 Handout #5 Prof. J. P. How April 10, 2008 Due: April 24, 2008 16.323 Homework Assignment #5 1. Suppose that the stock of a species of fish increases at a constant rate when they are not harvested. Fishing regulations set a maximum permitted rate for the harvesting of fish, which is proportional to the size of the stock. The system model is then that: x ˙ = x ux 0 u h where x ( t ) is the normalized size of the stock of fish, ux is the harvesting rate, and h < 1 is the maximum proportional rate. Suppose that initially x = 1 and that the terminal time T is fixed. The goal is to determine how to maximize the total number of fish caught, so the cost functional is given by T min J uxdt = 0 and obviously x ( T ) is free. (a) What is the optimal choice of u , and how does that depend on the costate?
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