sol3 - EE292 Spring 2006 Analysis& Control of Markov...

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Unformatted text preview: EE292 Spring 2006 Analysis & Control of Markov Chains April 29, 2006 Prof. Ben Van Roy Homework Assignment 3 : Solutions 1 See attached code. Note that we use the fact that argmax p ≥ exp(-γp )( p + C ) = 1 /γ-C for C > . Below is a plot of optimal price versus x and k . 200 400 600 800 1000 5 10 15 20 500 1000 1500 2000 k Optimal Price (x max = 20, N = 1000, p = 0.1, γ = 400) x p * Figure 1: Optimal Price From the plot it is clear that maximal price sensitivity is at x = 0 , t = 0 . 4.7 a) Letting z k = x k + γe k-1 , z k + 1 = x k +1 + γe k = x k + u k-w k + γe k = x k + γe k-1 + u k-(1-γ ) e k = z k + u k-(1-γ ) e k Since e k-1 = ∑ k-1 i =0 γ i w k-1-i , e k-1 is known to the controller at time k . Thus, defining z k as the new state variable does not result in the loss of state information (i.e. we still have a system with 200 400 600 800 1000 5 10 15 20 2000 4000 6000 8000 10000 12000 14000 k x J Figure 2: Optimal Expected Revenue perfect state information). The new cost function at stage perfect state information)....
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This note was uploaded on 02/15/2011 for the course EECS 6.231 taught by Professor Bertsekas during the Spring '10 term at MIT.

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sol3 - EE292 Spring 2006 Analysis& Control of Markov...

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