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 v is a bounded functional on X (i e , v B(X)) that satisfies the equation for every x X The previous exercise showed that the value function satisfies Bellman 's equation The next exercise shows that every optimal plan must satisfy Bellman 's equation at each stage u(x) sup U(x) yeG(x)

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

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 v is a bounded functional on X (i.e., v ˆŠ B(X)) that satisfies the equation

for every x ˆŠ X.
The previous exercise showed that the value function satisfies Bellman's equation. The next exercise shows that every optimal plan must satisfy Bellman's equation at each stage.

u(x)= sup U(x) yeG(x)

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