12. * Fix a, b E R and consider a sequence ( ) =(, 11, 13,...) of variables in R generated by the following recursive relation: 1+1 =a + bus (k > 0) This recursion is called a discrete-time linear system with initial condition to and control inputs tio, u1, U2,.... We want to choose control inputs which take the system from the initial condition to to a future condition in such that the average input energy E is minimized: E(U,U1,...,U-1) = (a) Let a = 1 and b = 2. Calculate the optimal values of o, , u, taking the system from initial condition Xo = 3 to 2 = 9. (b) Give the optimal values of up and u (in terms of a, b, and the initial condition to) for taking the system to 2 = 0. (c) Solve the linear system: give x in terms of a, b, 2o, k and the control inputs uo,...,.-1. (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 to, u1, 2, ..., in terms of a, b, and the initial condition for taking the system to Xn+1 = 0. 12. * Fix a, b E R and consider a sequence ( ) =(, 11, 13,...) of variables in R generated by the following recursive relation: 1+1 =a + bus (k > 0) This recursion is called a discrete-time linear system with initial condition to and control inputs tio, u1, U2,.... We want to choose control inputs which take the system from the initial condition to to a future condition in such that the average input energy E is minimized: E(U,U1,...,U-1) = (a) Let a = 1 and b = 2. Calculate the optimal values of o, , u, taking the system from initial condition Xo = 3 to 2 = 9. (b) Give the optimal values of up and u (in terms of a, b, and the initial condition to) for taking the system to 2 = 0. (c) Solve the linear system: give x in terms of a, b, 2o, k and the control inputs uo,...,.-1. (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 to, u1, 2, ..., in terms of a, b, and the initial condition for taking the system to Xn+1 = 0