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# 2007final - 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, #32 May 24, 2007 16.323 Final Exam This is a closed-book exam, but you are allowed 3 pages of notes (both sides). You have 3 hours. There are five 5 questions with the relative values clearly marked.
1. (total 15pts) John Howe, who hasn’t taken 16.323 and has no idea about the Riccati equation, is taking the final exam for an optimization class. The instructor loves to challenge the students by asking interesting, though possibly unexpected, questions, gave the following LQR problem in their exam, although it hasn’t been covered in class: minimize J ( x 2 + ρu 2 ) dt (1) 0 subject to x ˙ = ax + bu, x (0) = x 0 . (2) The only hint the instructor gave is that the optimal solution should have the form: u ( t ) = kx ( t ). Avoiding panic, John managed to solve the problem using what he learned in class as follows: (a) (5pts) Express the optimal cost J in terms of k and other given parameters. (b) (5pts) Find k that minimizes the above expression of J . (c) (5pts) John realizes that the same result can be derived from the Riccati equation, while talking to one of his committee

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2007final - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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