you select it? Problem 3. Infinite horizon average cost minimization (35 points) Consider the DP problem with discrete state and control spaces S = {-00,....-3,-2,-1,0,1,2,3,....co} and allowable control set independent of the state U = {0,1,2} respectively. Period costs are gk(xk,uk,wk) = 10K+cuk+p max {0,-x-uk+wk}+ h max{0, xk+uWk} for k=0,1,...,N-1, with 1,>o the indicator function that is equal to 1 when u>0, and 0 otherwise. u>0 K a fixed cost incurred whenever the indicator function 1,>0 is equal to one c is the variable cost per unit of uk p is cost per unit of negative xk+1 h is the cost per of positive x+1 Wk is a discrete random variable with probability mas function Prob( w = 0)=0.1, =1)=0.3, Prob(w = 2)=0.5, Prob(w = 3)=0.1. Prob(wk and dynamics are x+1 = xk+uk - Wk 3.1 Write the transition probabilities Prob(xnext=xnow +i|unow = j) for i=0,1,2,3 and j=0,1,2 and Prob(next=Xnow -1|unow = j) for i=0,1,2,3 and j=0,1,2 3.2 Write a linear system that finds the differential cost h(x) and the average cost per period J, for the truncated state space x = {-k,....-3,-2,-1,0,1,2,3,...k} and policy (x)=2 for'x <0, (x)=1 for -1