Optimal Control of Distributed Systems with Conjugation by Ivan V. Sergienko, Vasyl S. Deineka, Naum Z. Shor
By Ivan V. Sergienko, Vasyl S. Deineka, Naum Z. Shor
This paintings develops the technique based on which sessions of discontinuous services are utilized in order to enquire a correctness of boundary-value and preliminary boundary-value difficulties for the instances with elliptic, parabolic, pseudoparabolic, hyperbolic, and pseudohyperbolic equations and with elasticity conception equation platforms that experience nonsmooth recommendations, together with discontinuous solutions.With the root of this technique, the monograph exhibits a continual dependence of states, particularly, of suggestions to the enumerated boundary-value and preliminary boundary-value difficulties (including discontinuous states) and a dependence of answer strains on allotted controls and controls at sectors of n-dimensional area limitations and at n-1-dimensional function-state discontinuity surfaces (i.e., at suggest surfaces of skinny inclusions in heterogeneous media). Such a side offers the life of optimum controls for the pointed out structures with J.L. Lions' quadratic rate functionals.
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Extra resources for Optimal Control of Distributed Systems with Conjugation Conditions
9i' . i! le--07 le--08 . 4 0 45 Figure 41" Variance of frequency 1 estimates, p = 3 case, ct - 2 FREQUENCY ESTIMATION AND THE QD METHOD 0 . 0001 '".. ';" . ,. ,9" ~ " / \9 ,. ,,. ~ ' /, ", . ,. , . ~, ~ ,,9" (D O , , ~' "". ~" , ..... -:: t] . ~_=<. "M" ~ / ~ q ~ , , + ~ _ . . ~ " : " ~ / ',',J.. J ~.. -. " .. 4 :. i \ '.. ,--g , B :. ~: ~Ct ::> ~9. ',. x~ . 001 ,--4 --*--- -m-- --- .... 4 / ! / / ! ~. 05 _,, _ , i. 01 %,/ ~ 0 . 001 c~ ~ . s . I dB ' x '). -- .... ~,. . . i '~ "~ , 0 i ~0 i , le-05 /" ~ t i .
D. p=3 ESTIMATION AND THE QD METHOD 47 case Figs. 37-48 show the results of the algorithms with the double sinusoid input in the case where p = 2. We show both the variance and normalized variance for each of the two frequency estimates, and we show results for both the unmodified algorithm and the modified algorithm with both c~ = 2 and o~ = 6. 0 . 001 . . '-~---~--"k . ',' le--05 ", , '~-.. I ---~ -- - ".. - ':,,. , 0 ....... ~i , l | "~, " ~.. i ] .... ~. -'" \',? 4 0 45 Figure 37" Variance of frequency 1 estimates, p - 3 case.
7-24 present graphs of the variance and normalized variance versus Cramer-Rao bound (CRB). We can see that increased values of c~ tend to alleviate the problem around f - 1/4, moving it further and further to the left of the frequency axis as c~ increases. 0 . 0001 ; --~-- ~ ..... le--05 > le-06 9,, " ~ ",'""k . . x . m . -~. "x .... /" t/" ~.. le--07 9. ~ . - ~,-- . ~. 05 1 0 . 2 i z i. e d ~ , " ~ . _ . ~ . 45 0 Figure 7: Variance of frequency estimates, p = l case, ct - 2 0 . ~\. , . ~ ,- , , 1"~.