Question

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8. Use dynamic programming to solve the following discrete-time optimal control problem: minimizeukozzou subject to Xt+1 = X+ + a ut, X% = 1, Xã = 0. Here a is a positive parameter.

Answer 1

0 Solution ; given that: ezi nimicola 0 Subjecled to *+1 = atau ao = 1, az=0 Here we have to minimize this by dynamic programming Dynamic programming meary solve the given problem by making it into Subpasts and Solving each Subpart we can get the final answer on the given question. let cy apply Same 능혼는 *H=&t + x4 a=0 &z=0 a is a positive parameleo. Scanned with CamScanner

So od to F dor do tauo a = 1444 → t=1 alt =ajtalli *2 = (1+240)+QU, →@ t=2 42 agth = a2 + 3 &z= 1+40o + xu, tabr where x3=0 So tauo tay, tau=0 --aug then count |Cuod*741°76). Y ** from eano x=1truo 21- Ho = Scanned with CamScanner

Eq" @ = dy = 1+900+au. x = atau - kg - 1 eqr4 a2=- Elu2 So ear 6 will be - Tu +4 +4 ] ** ((a?+1-24 + 46+47-2+ et] (a-² ca-b2 - daz ( 243 +24,*-*X; X2 – 2x,+) + d2 [a;+x3–4122-21) = 1 / 2 ( 1 + 27 + 27-2, 22-x] This is the actual problem after Substitueting the given values we get thes form so e torm & converted to at form. CS Scanned with CamScanner