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hw5_sol - Question 1-1 Part i Show that ut =-B e-A t Wc(0...

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Question 1 Part i: Show that u t = - B e - A t W - 1 c (0 , T x i transfers the state x 0 = ¯ x i to x T = 0. Note that the solution to the ODE, ˙ x t = Ax t + Bu t , is, as was shown repeatedly and in various contexts, x ( t ) = e At x 0 + e At t 0 e - Bu ( τ ) . To prove the result, it suffices to substitute u ( τ ) for the given policy and to evaluate x ( t ) for t = T . First, substitute u ( · ). We have x ( t ) = e At ¯ x i + e At t 0 e - B ( - B ) e - A τ W - 1 c (0 , T x i dτ. Note that W c (0 , T ) is a fixed matrix and not a function of τ . Hence, x ( t ) = e At ¯ x i - e At t 0 e - BB e - A τ dτ W - 1 c (0 , T x i . Now, evaluating this expression for t = T yields, x ( T ) = e AT ¯ x i - e AT T 0 e - BB e - A τ dτ W - 1 c (0 , T x i , = e AT ¯ x i - e AT W c (0 , T ) W - 1 c (0 , T x i , = 0 , which completes the proof. Part ii: We again proceed via substitution. Fix x 0 = 0 and show x T = ¯ x f . x ( t ) = e At x 0 + e At t 0 e - Bu ( τ ) dτ, = e At t 0 e - BB e - A τ W - 1 c (0 , T ) e - AT ¯ x f dτ, = e At t 0 e - BB e - A τ dτ W - 1 c (0 , T ) e - AT ¯ x f , x ( T ) = e AT T 0 e - BB e - A τ dτ W - 1 c (0 , T ) e - AT ¯ x f , = e AT W c (0 , T ) W - 1 c (0 , T ) e - AT ¯ x f , = ¯ x
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