Frontiers in Adaptive Control Part 10

Tham khảo tài liệu 'frontiers in adaptive control part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 216 Frontiers in Adaptive Control n r- and 7 r r . lie IX. 21 In addition from Assumption a c and applying the inequality 12 a straightforward calculation shows that for some constant C . 22 Thus from Assumption T and Tn maps . n into itself. We fix anarbitrarynumber I I- I and define the function H .r II I .I 1 for where li l ị - I . Let d be the space of measurable functions II X IB with norm I . .11 H dl __ Observe that the norms Illi- and II- In are equivalent because In- ll dln 1 -d I I n 23 A consequence of Lemma 2 in Van Nunen Wessels 1978 is that the inequality 12 implies respectively that the operators Tn and T II LX are contractions with modulus I with respect to the norm II- In . for all II. I II7 7 ln lh lln 24 7 - In l - ln a-s- 25 Hence from 21 for each II IX we have III V - v i w TV - Tn 1V w Tn iV - Tn iKIliv . Id 26 Now let 11 and be arbitrary and define I limsup V - Cullũ and í limsupn_ oo IIV - K illw- Observe that l X. and l X see Proposition 22 and 23 . Then from 26 I 1Í11I Slip . I 7 1 -7 . - 1 n 27 J- n oo and liinsnp 7 l - 71 . r 28 1 TI J-OQ Cost estimation when d has a density In this part we suppose the existence of a density of 7 as stated below. We will then start step by step the proof of Theorem . Estimation and Control of Stochastic Systems under Discounted Criterion 217 Assumption a s i A. b The distribution 0 Ị is absolutely continuous with respect to the Lebesgue measure on lì Á and has a density function . That is - I r HKs. R . 29 B Under this context from 4 we have - I a. . ElK 30 Let 1 I be independent realizations observed up to time t of . s with the unknown density and r s s Iz s E be an arbitrary estimator of J such that - l as II - -X . 31 Defining for each II _ L 0 I Hi BIT1 . 32 B the relation 18 becomes C I . 1. .s . c IK. 33 Now let us define the approximate discrepancy function for each lì c L as see 9 l . I 1 I r _ I i l C .r

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