# Bangbang and Singular Contr ols Suzanne Lenhar Univ ersity of ennessee Kno xville Depar tments of Mathematics Suppor ted NIH ant Lecture PDF document - DocSlides

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1 brPage 2br BangBang Contr ol solution to prob lem that is linear in the control frequently in olv es discontin uities in the optimal control Lecture4 2 brPage 3br Contd rest where is witching function Maximiz r t at if if ID: 22465

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## Presentations text content in Bangbang and Singular Contr ols Suzanne Lenhar Univ ersity of ennessee Kno xville Depar tments of Mathematics Suppor ted NIH ant Lecture

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Bang-bang and Singular Contr ols Suzanne Lenhar Univ ersity of ennessee Kno xville Depar tments of Mathematics Suppor ted NIH ant Lecture4 .1/ ??

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Bang-Bang Contr ol solution to prob lem that is linear in the control frequently in olv es discontin uities in the optimal control Lecture4 .2/ ??

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Contd. %$ rest where is witching function. Maximiz .r .t. at (*) if if if Lecture4 .3/ ??

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If 1 2 1 2 is not sustained er an inter al of time then the control is bang-bang Bang-bang alw ys at the xtreme alues of the control set. u* Lecture4 .4/ ??

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If : ; : ; er an inter al of time the alue of is singular The choice of ust be obtained from other inf or mation than max .r .t. . The times when the OC witches from to or vice-v ersa or witches to singular control are called witch times (Sometimes difﬁcult to ﬁnd). Lecture4 .5/ ??

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Example KL O%P max sales KU stoc can be rein ested to xpand capacity or sold or re en ue fr action of stoc to be rein ested KL Lecture4 .6/ ??

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Y%Z Xba when Y d when Y d is decreasing and On Lecture4 .7/ ??

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Switch time If r t If y{z Lecture4 .8/ ??

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Example ~€ †‡ ‹%Œ Š“’ †– †— †‡ ‹%Œ no tr ansv ersality condition †‡ Lecture4 .9/ ??

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ž % is witching function and it will witch from to at most once at on If on DE That solution does not satisfy Lecture4 .10/ ??

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There ust be one witch. On ¢ DE ust be contin uous at ռ ռ Switching function at ¢ Lecture4 .11/ ??

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Lecture4 .12/ ??

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Basic esour ce Model (Clar 95) Fisher y control (eff or t), pr ice and catchability max discounted proﬁt re en ue cost Lecture4 .13/ ??

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Singular control occurs when the coefﬁcient of control is ero er time inter al (s witching function is ero). when singular case Lecture4 .14/ ??

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Suppose on time inter al. Solv or + Lecture4 .15/ ??

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rom necessar conditions 365 798 <>= @BA <>= after substituting in Set xpressions or equal cancel @BA Lecture4 .16/ ??

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er in olving control cancel NPO NPQ optimal state should satisfy this equation when in the singular control case (ﬁnd and use state DE to ﬁnd ). Lecture4 .17/ ??

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Simple case []\ [a` ignor ing cost of ﬁshing Singular case []\ []\ [a` dur ing singular case [a` Lecture4 .18/ ??

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Dur ing singular case If singular case Lecture4 .19/ ??

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Back to Bior eactor Model state (bacter ia) |a} v€ |†… |†… |†… control (n utr ient input) Lecture4 .20/ ??

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Ž Ž If on time inter al, singular control ma satisfy †™ Ž using DE †™ ] Ž †™ Lecture4 .21/ ??

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† is str ictly decreasing function. Thus cannot be maintained on an inter al. No singular case here One witch occurs when † can sho if and sufﬁciently large there is witch. Lecture4 .22/ ??

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If or large Where is deca ate of contaminant and is ro wth ate of bacter ia and sufﬁciently large there is witch u* Otherwise i.e no utr ient eeding. Lecture4 .23/ ??