hw4_sol

hw4_sol - ME254/ECE271C Winter ’09 Solutions to HW 4 1....

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Unformatted text preview: ME254/ECE271C Winter ’09 Solutions to HW 4 1. Note: These solutions were written several years back, so the terminology may not exactly match that being used in class this quarter, but you should be able to figure things out from context. The technique also uses a short cut that’s somewhat special to this problem. Letting ˙ x 1 = x 2 , ˙ x 2 = u , we have the calculus of variations problem: min x 1 ,x 2 Z 1= t f ( ˙ x 2 ) 2 dt 3 ˙ x 1 = x 2 x 1 ( t f ) 2 + x 2 ( t f ) 2 = 1 x 1 (0) =- 2 √ 2 ,x 2 (0) = 5 √ 2 Let the Lagrange multiplier function associated with the differential constraint be λ . Then, we seek an extremal for: L := ( ˙ x 2 ) 2 + λ ( t )[ ˙ x 1- x 2 ] . The E-L equations are: 2¨ x 2 =- λ ˙ λ = 0 → λ ( t ) = c 1 Hence, 2¨ x 2 =- c 1 ⇒ ˙ x 2 =- 1 2 c 1 t + c 2 x 2 ( t ) =- 1 4 c 1 t 2 + c 2 t + 5 √ 2 . (1) Since ˙ x 1 = x 2 , we have x 1 ( t ) =- 1 12 c 1 t 3 + 1 2 c 2 t 2 + 5 √ 2 t- 2 √ 2 . (2) From the target set we have x 1 (1)...
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This note was uploaded on 03/23/2009 for the course ME 254 taught by Professor Bamieh during the Winter '09 term at UCSB.

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hw4_sol - ME254/ECE271C Winter ’09 Solutions to HW 4 1....

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