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sol1 - EE292E Analysis Control of Markov Chains Prof Ben...

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EE292E Spring 2008 Analysis & Control of Markov Chains April 10, 2008 Prof. Ben Van Roy Homework Assignment 1 : Solutions 1.16 (a) Each state is a set S k ⊂ { 2 , ...N } . The allowable states at stage k are those sets S k of cardinality k . The allowable controls are u k ∈ { 2 , ..., N } - S k . This control represents the multiplication of the term ending in M u k - 1 by the one starting in M u k . The system equation is S k +1 = S k u k . The terminal state is S N = { 2 , ..., N } , with cost 0 . The cost at stage k is given by the number of multiplications: g k ( S k , u k ) = n a n u k n b where a = max { i ∈ { 1 , . . . , N + 1 }| i / S k , i < u k } b = min { i ∈ { 1 , . . . , N + 1 }| i / S k , i > u k } For example, let N = 3 and M 1 be 1 × 10 , M 2 be 10 × 1 , M 3 be 1 × 10 . The order ( M 1 M 2 ) M 3 corresponds to controls u 1 = 2 and u 2 = 3 , giving cost: ( u 1 = 2) : n 1 n 2 n 3 = 10( a = 1 , b = 3) ( u 2 = 3) : n 1 n 3 n 4 = 10( a = 1 , b = 4) with a total cost of 20 . whereas M 1 ( M 2 M 3 ) gives: ( u 1 = 3) : n 2 n 3 n 4 = 100( a = 2 , b = 4) ( u 2 = 2) : n 1 n 2 n 4 = 100( a = 1 , b = 4) with a total cost of 200 !
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