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hw52 - μ ε for value V s x Then one divides the sum over...

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MAE 288A Assignment 2 Due Thursday, 22 Oct., 2009 1. (10) Consider the discrete-time stochastic control problem with dynam- ics ξ t +1 = t + Bu t + w t , ξ t 0 = x and cost criterion J ( t 0 , x ; u · ) = E T - 1 summationdisplay t = t 0 bracketleftbigg 1 2 ξ T t t + 1 2 u T t Du t bracketrightbigg + 1 2 ξ T T T . Let u · be a minimizing control, and let the value function be denoted by V ( t, x ) for all t ∈ { t 0 , t 0 + 1 , . . . , T } and x IR n . Let the control take values in k , and let the noise process consist of independent, identically distributed, normal random variables (taking values in IR n ) with mean 0 and covariance matrix Q . Consider the special case A = bracketleftbigg 0 2 3 0 bracketrightbigg , B = bracketleftbigg 0 3 bracketrightbigg , C = bracketleftbigg 1 0 0 1 bracketrightbigg , D = 2 E = bracketleftbigg 4 0 0 4 bracketrightbigg , Q = bracketleftbigg 2 0 0 1 bracketrightbigg . Let T = 4 and t 0 = 1. Find V ( t, x ) and the optimal feedback control for t ∈ { 1 , 2 , 3 } . 2. (10) Complete the second half of the proof of the theorem (finite time-

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Unformatted text preview: μ ε for value V ( s, x ). Then, one divides the sum over [ s, T − 1] into segments over [ s, t − 1] and [ t, T − 1]. The cost over [ t, T ] with this controller is certainly higher than the value given initial time t and the state at that time. 3. (5) Let f 1 , f 2 be convex functions mapping IR n into IR . Let g . = f 1 ⊕ f 2 . That is, let g ( x ) = f 1 ( x ) ⊕ f 2 ( x ) = max { f 1 ( x ) , f 2 ( x ) } for all x ∈ IR n . Prove that g is convex. 1 4. (5) Suppose f : IR → IR is convex. Let ¯ x ∈ argmin { f } . Is f monoton-ically increasing on [¯ x, ∞ )? If so, prove it. If not, provide a counterex-ample. 2...
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