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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 #3 Prof. J. P. How March 6, 2008 Due: March 18, 2008 16.323 Homework Assignment #3 1. Find the curve x ( t ) that minimizes the functional 1 1 J ( x ) = [ x ˙ 2 ( t ) + 5 x ( t ) ˙ x ( t ) + x 2 ( t ) + 5 x ( t )] dt 0 2 and passes through the points x (0) = 1 and x (1) = 3 2. One important calculus of variations problem that we did not discuss in class has the same basic form, but with constraints that are given by an integral - called isoperimetric constraints : t f min J = g [ x , x ˙ , t ] dt t 0 t f such that : e [ x , x ˙ , t ] dt = C t 0 where we will assume that t f is free but x ( t f ) is fixed. (a) Use the same approach followed in the notes to find the necessary and bound- ary conditions for this optimal control problem. In particular, augment the constraint to the cost using a constant Lagrange multiplier vector ν , and show that these conditions can be written
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