Bangbang and Singular Contr ols Suzanne Lenhar Univ ersity of ennessee Kno xville Depar - PDF document

Bang-bangandSingularControlsSuzanneLenhartUniversityofTennessee,KnoxvilleDepartmentsofMathematicsSupportedbyNIHgrantLecture4–p.1/?? Bang-BangControlAsolutiontoaproblemthatislinearinthecontrolfrequentlyinvolvesdiscontinuitiesintheoptimalcontrol
\n \r\n \r \n \r \r\r\n \rLecture4–p.2/?? Contd.\n !"\n #%$ !$"\n !"\n #$"\n \n #$\n !\n# restwhereisswitchingfunction.Maximizew.r.t.!at!&!& '((*)+if\n&# ,-.if\n&# -/if\n&# 0-Lecture4–p.3/?? If1\n2345361\n2789isnotsustainedoveranintervaloftime,thenthecontrolisbang-bang.Bang-bangalwaysattheextremevaluesofthecontrolset.abtu*Lecture4–p.4/?? If:\n;=�?:\n;@ABoveranintervaloftime,thevalueofC�issingular.ThechoiceofC�mustbeobtainedfromotherinformationthan“maxw.r.t.C”.ThetimeswhentheOCswitchesfromDtoEorvice-versaorswitchestosingularcontrolarecalledswitchtimes.(Sometimesdifculttond).Lecture4–p.5/?? ExampleFGHIJKLMNO%PQRmaxsalesPSTNPKUOTPJPKROstockcanbereinvestedtoexpandcapacityorsoldforrevenueNKROfractionofstocktobereinvestedUNKROLTKLMNOPVNPLecture4–p.6/?? WXY%ZY[\]^Z[_W\``ZWX\]\[Xba[Y^Wcwhen[Y\nd^\]ecaXfW][_W\[when[Y\nd^\]gcaXfWc[_W\][isdecreasingand[Y^Wc.Onhidaja[g][W\dXfWcaZfWZfYid^.Lecture4–p.7/?? SwitchtimeklIfklmnlopqmrpsr\ntIfulopslopsmvvvmklrpsrtwxrpsrtwy{z|Lecture4–p.8/?? Example}~€‚ƒ„ƒ‚‚…†‡ˆ‰Š‹%Œ†Š‹ŠŽ‡Ž‘Š“’”•ŒŽ†–‹–Ž†—‹’”†‡ˆ‰Š‹%Œ˜™‡Ž‘Š’”•Œš˜ˆ››Žˆ‡˜˜˜…’”•notransversalitycondition™‡ˆ‰Š‘˜Š’”•Œ‡˜Ž†‡ˆ‰Š‘˜…Š‹Œ‡˜ŽLecture4–p.9/?? œžŸ ¢¡£¥¤¦£§%¨ž ª©žŸ ¢¡£¥¤¦£§isswitchingfunctionanditwillswitchfromto¤atmostonce.at£œ«¬žŸ ¡¤¦§«§­«¬¨®œ¤¯on°«¬±£§If¨®²¤¯on°«¬¯³©´DE©«§œ«©œ¤ž£µ·¶µ¸¹Thatsolutiondoesnotsatisfy©¯§œ¶µ.Lecture4–p.10/?? Theremustbeoneswitch.Onº»ª¼½¾¿½ÀºÁ¢ÃÄÅƼÈÇɽÊË̾ÍÎDEͺ¾ÆÌÏÐÍÌÏÐѺÀ¼ÐľÆÒÍËmustbecontinuousat»¼ÏÐÓÑÔÀ»Õ¼ÐľÖÌÄÀ»Õ¼ÐÏÐÓÑ»¼Ì¾ÀSwitchingfunctionÌÉat»ª¼ÀºÁÂÃÄÅÆ»ª¼ÌÉ¢Ã̾ÁLecture4–p.11/?? ×ØÙÚÛ×ØÙÛÜÜÜÝÞßàØÙÚßáâäãâåàØÙãâåçæßáâÚÛèLecture4–p.12/?? BasicresourceModel(Clarkp.95)“Fishery”éêëìíîïðñóòôõùúûüûöûüýòöú÷õöcontrol(effort),ôpriceandõ“catchability”þòüúÿ
maxdiscountedprotrevenue-costLecture4–p.13/??  \n\r\n \r\nSingularcontroloccurswhenthecoefcientofcontroliszerooveratimeinterval(switchingfunctioniszero).when \r\n \r\n \r\nsingularcaseLecture4–p.14/?? Suppose !#$%#&!'(onatimeinterval.Solvefor&&' #$!))*'+#,% #$!$!-)")*'#, #$!$!-"%#!"%Lecture4–p.15/?? Fromnecessaryconditions./.0123436512798:;�@BA/CDEC5F?@FG128:;�@A8:;=2H?CDEC5F2?@Faftersubstitutingin/.Set2expressionsfor./.0equalIcancel8:;J2K=2H?AH?LCDC5F2?@5F12=@BA=2H?C?@2DEC5FLecture4–p.16/?? TerminvolvingcontrolcancelMNPONPQRSQRSTUSVWXUSVQRSMO.XUSVQUSVSUOSMQVWNoptimalstateshouldsatisfythisequationwheninthesingularcontrolcase.(ndSYandusestateDEtondZY).Lecture4–p.17/?? Simplecase[]\^_\[a`b\^c_d_`ef_gignoringcostofshingSingularcaseh[]\^f[]\^\[cbf^_i`bj\g_i\k_[a`bi^lj[\k^h_gduringsingularcase\k[a`b\k^bk\k_g`b\kbk_gLecture4–p.18/?? DuringsingularcasemnopnpqropqmnstmnmnstnpqrpqmstIfsqr,singularcaseopqmutpqmutLecture4–p.19/?? BacktoBioreactorModelvwxyzv{state(bacteria)v|a}~xv€‚ƒ„|v|†…~zy|†…~‡…}y|†…~control(nutrientinput)xvzyˆ|yzv{~ˆwxz‰‰vxz|ˆ|yzŠv~ˆ|~x}Lecture4–p.20/?? ‹ŒŽ‘’“”‘Ž•IfŽ–‘’‹—onatimeinterval,singularcontrolŒ–maysatisfy—˜Œ–†™“˜Ž–“š‹Ž–‘“–usingŽ›DEŽ–““‹—Ž–““‹’‘œž‘Ÿ –]¡“¢¡—Ž–““˜Lecture4–p.21/?? £¤¥¦§§isastrictlydecreasingfunction.Thus¤¥ª©«¬­cannotbemaintainedonaninterval.Nosingularcasehere.Oneswitchoccurswhen£¤¥¦§§¬«canshowif«®andsufcientlylarge,thereisaswitch.Lecture4–p.22/?? If¯°orlargeWhereisdecayrateofcontaminantandisgrowthrateofbacteriaandsufcientlylarge,thereisaswitchtTMu*^Otherwise±²³´i.e.nonutrientfeeding.Lecture4–p.23/??