3. Another simple stochastic optimal control problem. Consider a system with state space X= {1,2}, input space U={1,2}, and disturbance space W={1,2}, transition function ft(x,u,w)=u Page 1 of 3 MECH 6326 Optimal Control and Dynamic Programming Homework \#2 (so that xt+1=ut ), and cost functions gt(x,u,w)=10(t+1)20x=u=1,w=1,x=1,u=2,otherwisegT(x)={0x=2,otherwise. Suppose x0=1,T=4, and wt are independent and identically distributed random variables with P(wt=1)=0.3,P(wt=2)=0.7. (a) Based on your results from Problem 1, how many open-loop, closed-loop state feedback, and closed-loop disturbance state feedback policies are there? This gives are rough idea of how many computations would need to be performed to compute the optimal policy by brute force. (b) Compute the optimal policy and value functions using the dynamic programming algorithm for (i) state feedback and (ii) state and disturbance feedback