party 10. This problem is optional for bonus +2 points. (Points scored above 10 on this homework can make up for points missed on previous homeworks.) Fix a, b e R and consider a sequence (2.) = (20, 11, 12,...) of variables in R generated by the following recursive relation: Tk+1 = azt + buk (k > 0) This recursion is called a discrete-time linear system with initial condition To and control inputs to, t1, 12,.... We want to choose control inputs which take the system from the initial condition do to a future conditions, such that the average input energy E is minimized: Eu, ... NI (a) Let a = 1 and 5-2. Calculate the optimal values of t, 1t, taking the system from initial condition to 3 to Is = 9. (b) Give the optimal values of u and t, in terms of a, b, and the initial condition 1o) for taking the system to 2-0. (e) Solve the linear system: give x in terms of a, b, c, k and the control inputs o.-. (The value of with all control inputs is called the natural response, a value with non-zero control inputs is called a forced response) (d) Give the optimal values of us in terms of a, b, and the initial condition for taking the system to 7--1 - 0 party 10. This problem is optional for bonus +2 points. (Points scored above 10 on this homework can make up for points missed on previous homeworks.) Fix a, b e R and consider a sequence (2.) = (20, 11, 12,...) of variables in R generated by the following recursive relation: Tk+1 = azt + buk (k > 0) This recursion is called a discrete-time linear system with initial condition To and control inputs to, t1, 12,.... We want to choose control inputs which take the system from the initial condition do to a future conditions, such that the average input energy E is minimized: Eu, ... NI (a) Let a = 1 and 5-2. Calculate the optimal values of t, 1t, taking the system from initial condition to 3 to Is = 9. (b) Give the optimal values of u and t, in terms of a, b, and the initial condition 1o) for taking the system to 2-0. (e) Solve the linear system: give x in terms of a, b, c, k and the control inputs o.-. (The value of with all control inputs is called the natural response, a value with non-zero control inputs is called a forced response) (d) Give the optimal values of us in terms of a, b, and the initial condition for taking the system to 7--1 - 0