# hw4 - problem by adding the dynamical constraint with a...

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ME254/ECE217C, HW 4 Winter 2009 Due Monday Feb 9 at 1pm in Prof. Bamieh’s mailbox 1. You are given a control system described by ¨ x ( t ) = u ( t ) , x (0) = - 2 2 , ˙ x (0) = 5 2 , which will be steered to the target set Γ = { ( x, ˙ x ) R 2 : x 2 + ˙ x 2 = 1 } (a circle) in unit time, while minimizing the control energy J = Z 1 0 u 2 ( t ) dt . Derive the necessary conditions for optimality and ﬁnd the optimal control by solving the diﬀerential equations in the resulting TPBVP. There are two ways to do this problem: (a) Ignore what we did with optimal control, convert directly to a calculus of variations
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Unformatted text preview: problem by adding the dynamical constraint with a Lagrange multiplier function, and then solve the resulting Euler-Lagrange equations. (b) Add u as a state using the equation ˙ u ( t ) = v ( t ), and now v becomes the input to be optimized. This is now in the standard form for optimal control problems. In fact, it is a minimum energy state transfer problem, but with only part of the states speciﬁed at the end points Use both methods above to solve this problem and compare the two. 1...
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