Problem 2 (20 points}: Consider the following system 3U: + 1] = Axe] + sum = are] + [ b1 52 ] [ 3:53 1 , where A E R2X2,b1,bg E R2. Suppose both (21.51] and (21,53) are controllable. {a} (5 points] Starting from z[] = 0, what is the minimum energy required to steer the System state to :c[3] = mam at time k = 3 using control it; alone [hence agile] = U for I: = 0,1,2]? Write your answer in terms of A, b13331. (b) [5 points] Starting from 9.:[3] = eds\" at time it = 3, what is the minimum energy required to steer the system to a:[] = 3:19:32 at time k = 6 using oontrol ug alone {hence uk] = U for k = 3.4, 5)? Write your answer in terms ofA1b2,:v.-,t331,;rdg. (e) [10 points) Using your answers in the previous two parts to answer the following question. Suppose :r[[i] = U at time it = 0 and the goal is to reach m[] = ed\": at time k = 6. You are allowed to onlz,r use 1&1 at times it = 0,1,2, and only use u: at time k = 3,4,5. Note that there is no requirement on $[3] at k = 3. What is the minimum total energy required over the whole time horizon? Write your answer in 111311118 Of .4, b1, b2, Ida-,2