Control Techniques for Complex Networks in .NET Creator barcode 3/9 in .NET Control Techniques for Complex Networks

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Control Techniques for Complex Networks using .net framework todraw code 39 full ascii in web,windows application bar code Draft copy April 22, 2007. de ned in (9.53). Following the proof of Proposition 9.

4.3, for the (n + 1)-period problem we express the minimum over all admissible input sequences as the minimum rst over admissible sequences {U (1), U (2), . .

. } for a given U (0) U (x), and then minimize over U (0). This gives,.

min Ex V0 (X(n + 1)) + c(X(t)). = c(x) + min Ex V0 (X(n + 1)) + t=1. c(X(t)). = c(x) + min Ex U (0). U (1),U (2),...

. E V0 (X(n + 1)) + c(X(t)) . F1 . . We have by the structure of barcode 3/9 for .NET the MDP model,. n U (1),U (2),...

n 1 c(X( t)) . F1 = min E X(1) V0 (X(n))+ E V0 (X(n+1))+ c(X(t)). and the right hand side is precisely Vn (X(1)) by the induction hypothesis. Moreover, by the induction hypothesis the minimum on the right hand side is achieved using U (1) = (X(1)), U (2) = (X(2)), . .

. . n 1 n 2 We thus obtain the desired representation, n.

min Ex V0 (X(n + 1)) + t=0. c(X(t)) = c(x) + min Pu .net framework 39 barcode Vn (x) = Vn+1 (x). u U (x).

Moreover, the policy achievi ng the minimum is Markov as claimed. We obtain a suggestive corollary on applying the bound, 1 inf lim sup Ex n n. c(X(t)). 1 min Ex lim sup n n c(X(t)). (9.54). where the in mum and the min VS .NET Code 3/9 imum are over all policies. Exercise 9.

7 contains a generalization of Corollary 9.5.2 to allow non-zero initialization in the VIA.

Corollary 9.5.2.

If V0 0 then for every x, 1 lim sup Vn (x) x . n n Proof. We obtain the lower bound (9.

54) since the policy can depend upon n in the minimum on the right hand side, while this is not true for the in mum on the left hand side. The corollary follows since the minimum is achieved to give Vn (x), and the . in mum is by de nition x .

Control Techniques for Complex Networks Draft copy April 22, 2007. This gives hope that { } w ill converge to an average-cost optimal policy. n In-spite of the positive message conveyed in Corollary 9.5.

2, the initialization V0 should be chosen with care. Proposition 9.5.

3. c-myopic policy. If V0 0, then the policy obtained from value iteration is the 0 .

Consequently, when V0 0 it visual .net barcode 3 of 9 is possible that the controlled chain is transient under for some n, such as n = 0; We have seen in several examples, such as n Example 4.4.

2, that the c-myopic policy may not be stabilizing in network models. There is one special case under which the policy n de ned in Proposition 9.5.

1 is stationary. Proposition 9.5.

4. n 1, If (h , ) solves the ACOE (9.4a) and V0 = h , then for each.

Vn = h + n . Hence, there exists a single barcode 3/9 for .NET stationary policy minimizing the right hand side of (9.52) for each n.

Proof. The proof is by induction: this holds by de nition when n = 1, and then for arbitrary n since is constant. Proposition 9.

5.4 suggests that an initialization approximating the solution to the ACOE might result in quickened convergence. We assume in Theorem 9.

5.5 below that at least one regular policy 1 exists, and that the function V0 solves a relaxation of the ACOE, min Pu V0 (x) V0 (x) c(x) + ,. u U (x). where is a constant. Equiv .NET 39 barcode alently, the Poisson inequality holds for some policy 1 , P 1 V0 V0 c + .

(9.55) The existence of a pair (V0 , 1 ) satisfying (9.55) is a natural stabilizability assumption on the model, and we nd below that this initialization ensures that the VIA generates stabilizing policies, in the sense that Poisson s inequality holds for each of the feedback laws { } obtained from the algorithm.

n To simplify notation, for each n we denote Pn = P , and we let En denote n the expectation operator induced by the stationary policy with feedback law . Let n bn := Vn+1 Vn denote the incremental value, and n = supx bn (x). Theorem 9.

5.5. (Performance bounds for VIA) Suppose the initialization V0 satis es (9.

55). Then, the upper bounds { n } are nite and non-increasing: For each n, the average cost under the policy satis es, n. x n n ,.
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