Let be the value function for the dynamic programming problem (example 2 32) Assume that cent f is bounded on X Atilde X cent G(x) is nonempty for every x X Show that the plan x (x0, x 1, x 2, ) Icirc (x0) is optimal if and only if it satisfies Bellman 's equation v(x 1) f(x t , x t 1) Icirc sup2 v...

Assume x is optimal so that x x for every x 0 This implies using 239 0 1 x 0 1 x where x 1 2 ...

Let be the value function for the dynamic programming problem (example 2.32). Assume that ¢ f is

Question:

Let

be the value function for the dynamic programming problem (example 2.32). Assume that
€¢ f is bounded on X Ã— X
€¢ G(x) is nonempty for every x ˆŠ X
Show that the plan x* = (x0, x*1, x*2, ...) ˆŠ Î“ (x0) is optimal if and only if it satisfies Bellman's equation
v(x*1) = f(x*t , x*t+1) + Î²v(x*t+1), t = 0, 1, 2,... (13)