Symmetry of Solutions to the Generalized 1-D Optimal Sojourn Time Control Problem

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1 Symmetry of Solutions to the Generlized -D Optiml Sojourn Time Control Problem Wei Zhng nd Jinghi Hu School of Electricl nd Computer Engineering Purdue University West Lfyette IN706 zhng70 Abstrct The optiml sojourn time control (OSTC) problem tries to find the feedbck control lw with resonble cost tht cn keep the system stte in certin subset of the stte spce clled the sfe set for the longest time under rndom perturbtions The rdil symmetry property of its solutions hs been proved previously when the stte dimension is higher thn one However the proof does not pply in the one dimensionl cse This pper generlizes the OSTC problem by introducing rbitrry weighting functions for both the cost function nd the performnce mesure function nd shows tht the solution to the -D generlized OSTC problem is rdilly symmetric Our proof cn lso be etended to the multi-dimensionl cse Numericl simultion is performed to verify the proof I INTRODUCTION Sfety is criticl issue in mny engineering problems such s intelligent trnsporttion systems 67] control of Unmnned Aeril Vehicles (UAV) ] chemicl processes etc In these pplictions the system stte is required to sty in certin subset of the stte spce clled the sfe set; nd whenever the system stte gets into the unsfe set the system crshes or t lest some costly procedure must be invoked to bring the system bck to order Sfety of deterministic systems is reltively esy to mintin by properly designed feedbck control strtegies However in prctice systems re often perturbed by rndom noises which my steer the stte outside the sfe set despite the control efforts Therefore for such systems meningful problem is to find feedbck control lw with resonble cost tht cn keep the rndomly perturbed system sttes in the sfe set for s long s possible This type of control problem is clled the optiml sojourn time control (OSTC) problem nd will be the focus of this pper While the generl stochstic optiml control problem hs been widely studied (83]) the OSTC problem is reltively new It is first proposed in ] nd is lter etended to the generl setting of stochstic hybrid systems in ] By ssuming the symmetry of the solutions the OSTC problem with one-dimensionl stte spce is simplified nd n nlyticl solution is obtined in ] It turns out tht the symmetry property becomes even more importnt in finding the optiml solution for the multi-dimensionl cse Hence before further studying the solutions of the problem it is necessry to formlly prove this property By using the symmetriztion method 3] the symmetry property is proved when the stte dimension in ] However when certin multi-dimensionl concepts such s surfce nd volume nd some key steps in the proof such s the ppliction of the isoperimetric inequlity re not pplicble Thus n lterntive proof for the cse is needed In ddition the OSTC problem ] ssumes tht the control cost is the integrl of the norm of the control nd the performnce mesure to be mimized is just simple verge of the system sojourn time over the sfe set However for some pplictions the control cost my not be eqully weighted nd the sojourn time in some regions in the sfe set my be more importnt thn the others Thus it is desirble to generlize the OSTC problem to llow some weighting functions for nd The contributions of this pper re threefold Firstly the generlized OSTC problem is formulted where the control cost nd performnce mesure re etended to incorporte some weighting functions Secondly we prove tht with rbitrry symmetric performnce weighting nd constnt control weighting functions the solution to the one-dimensionl generlized OSTC problem is symmetric Finlly since the wy we del with the newly-introduced performnce weighting function does not depend on the stte dimension our proof cn be etended to the higher-dimensionl cse Thus together with our previous contribution ] the results of this pper estblish completely the rdil symmetry property of the solutions to the generlized OSTC problem with n rbitrry stte spce dimension The rest of the pper is orgnized s follows In section II the generlized OSTC problem is formulted To better present the proof the problem is trnsformed into nother form in Section III Section IV reviews some bsic definitions nd properties of the symmetriztion procedures Then in Section V we prove the symmetry property of the -dimensionl generlized OSTC problem A numericl emple is given in Section VI to verify the results II GENERALIZED OSTC PROBLEM Let be the stte of dynmicl system nd let be the sfe set of the system which is bounded open connected domin in The derivtive of stte is subject to the perturbtions of the white noises %$&(')+* ; nd one wnts to find feedbck control lw tht keeps in for the longest time Mthemticlly the bove system cn

2 d d d be described by the following stochstic differentil eqution + %$ () s the first eit time of Define * % ' from The epected vlue of is denoted by where denotes the epecttion tken under the initil condition & Thus is the epected sojourn time of in strting from By the stndrd results of stochstic clculus ] is the solution to the PDE $ &%(' &)* + () with the boundry condition -/0 In prticulr is bounded nd second order differentible with piecewise continuous second order derivtives As discussed erlier one wishes to mimize the epected sojourn time of the stte in the sfe set Without ny constrint on the problem will obviously result in the trivil solution of n infinite Thus well-posed problem is to find control with t most certin mount of cost tht cn mimize the weighted verge of over the sfe set Compred with ] the generlized OSTC problem dopts more generl control cost function defined s: nd more generl performnce function of defined s: ( 7& : while in ] only the specil cse is studied Here the weighting functions nd re positive nd ; =% is required to be monotoniclly incresing on )BÄ with ; C% D For emple ; FEGH I for some positive integer J Given control the sojourn time is lwys positive nd is the unique solution to the PDE () Therefore is essentilly the weighted verge sojourn time of over the sfe set under the control which is resonble performnce mesure Given the bove definitions two formultions of the generlized OSTC problem re given below Problem (Generlized OSTC Problem): Find the piecewise continuous tht chieves the lrgest with n cost LK 7 for some positive constnt M Problem (Dul Generlized OSTC Problem): Find the piecewise continuous with the lest cost subject to N for some positive constnt O C% Remrk : Under zero control P0 it still tkes the stte some time to eit ie C%RQ So O C% is required in Problem to void trivil solution The bove two problems re dul to ech other nd solving one is equivlent to solving the other ] In the rest of this pper for convenience we my switch freely between these two formultions III PROOF SETUP We now study Problem with ie the onedimensionl cse Then becomes n intervl SUTVWT7 nd the sojourn time is the solution to the ODE ) ) (3) =SUT Eqution (3) gives one-to-one correspondence between nd In prticulr cn be recovered from by Z with boundry conditions CT () Thus to show the symmetry of the solution to Problem we need only to show the symmetry of the corresponding To this purpose we write in terms of s: Z \ 78 () This substitution is vlidted by the following lemm proved in ] Lemm : Suppose tht F SUTVWT7_^ is the epected sojourn time corresponding to some optiml control tht solves the -D version of Problem (or ) Then V stisfies ) nd =SUT CT ; ) is second order differentible with piecewise continuous second order derivtives; I 3) KI e nd whenever ; ) ` ) cb dfehg g e g 7jikA for n rbitrry positive integrble function SUTVWT7^ with e g g e g d t those where Prt 3) of Lemm gurntees tht the substitution () is vlid for every optiml nmely the one corresponding to n optiml control Moreover if we define l&h SUTVWT7 s the set of ll SUTVWT7m^n> stisfying the bove properties then Lemm implies tht we need only to focus on the functions in l&h SUTVWT7 to find the optiml ones Hence n equivlent formultion of the Problem in terms of for the -D cse is Problem 3: Find ol&h SUTVWT7 tht mimize (7 subject to Z 7NK 7M Under this setting the gol of the rest of this pper is to show the following proposition Theorem : If is constnt ie no control weighting then for ny symmetric function the solution to Problem 3 is lso symmetric IV SMMETRIZATION In the clculus of vritions the symmetriztion method is powerful tool to prove tht the minimizers of functionls re symmetric functions 3] Although symmetriztion hs been widely used in mthemtics nd physics it is reltively new to the control society So before using it in the proof

3 we shll first review some bsic concepts nd properties of the clssicl symmetriztion method m f ( m ) f ( ) f ( ) Fig An emple of Schwrz symmetriztion Roughly speking symmetriztion is wy of mpping nonsymmetric function to symmetric one while preserving certin properties Fig illustrtes the geometric ide of symmetriztion through one-dimensionl emple Here TV nd &I )U For ech ) denote by / : & the level set of The symmetriztion is performed by symmetrizing the level sets of More precisely for ech ) contins two points nd tht my be nonsymmetric S M ' nd let the with respect to the origin Define function tke the vlue t the two symmetric points where F )Uh^ is some nonnegtive function Typiclly is chosen so tht the resulting stisfies certin properties such s preserving certin integrls If & the bove procedure becomes the -dimensionl Schwrz symmetriztion 0] s is the cse in Fig In this cse we denote the resulting symmetric function by insted of More generlly we cn define the symmetriztion of functions on multi-dimensionl spces s follows Definition : Let ^ be nonnegtive mesur- / + ble function on Define nd denote by hs the sme Lebesgue mesure (volume) s Lebesgue mesure of the set is finite for every the symmetriztion of is defined s Here N+ ^ where the bll centered t the origin tht If the then ( m is nonnegtive function Remrk : In the literture the definition of symmetriztion usully does not involve ie Here we introduce generl function to be consistent with the proof in Section V The conventionl definition cn be thought of s specil cse of our definition If the symmetriztion is clled the Schwrz symmetriztion nd hs the following properties 0] Lemm : Let nd stisfy the conditions specified in Definition Denote by the superscript the Schwrz symmetriztion opertor of set or function Then ) is rdilly symmetric ie W & ; M ) If =% is continuous function then 8 7 3) If nd re continuous then 7OK 8 7& 7h Note tht Lemm cn not be pplied directly in the proof of this pper s the integrnd in () involves the first nd second order derivtives of Insted some creful construction of is necessry V PROOF OF SMMETR The bsic ide of the proof cn be described s follows For ny cndidte solution cl&h SUTVWT7 we wish to find symmetric ol&h SUTVWT7 such tht nd K If such lwys eists for rbitrry l&h SUTVWT7 then the solution to Problem 3 must be symmetric More precisely we shll prove the following theorem which is equivlent to Theorem Theorem : For ny function ol&h SUTVWT7 there eists symmetric function ol&h SUTVWT7 such tht nd ] 7 (7oK ] $ 78 (6) 78 (7) for rbitrry positive symmetric function The key of the proof of Theorem is the construction of the symmetric function tht stisfies the two requirements According to the discussion in Section IV this cn be done by properly choosing the function =% Following this ide we now go over the symmetriztion process to derive the necessry conditions for the desired =% A( ) c A( ) Fig Symmetriztion of V f ( ) r V ( r) f ( ) According to Lemm K S i So is concve on SUTVWT7 Since CT =SUT j chieves its mimum vlue t unique point &% in SUTVWT7 s is shown in Fig Thus % Define nd + B for SUTVH%< B for %WT7

4 % Then nd re monotone functions with inverse ] SUTVH%= nd ++ ] %/WT7 ) respectively Define / SUTVWT7= for ech \ )& consists of two points nd & ++ whose distnce is denoted by : S ++ &S RQI) According to the concvity property RQ) nd Therefore cn be written s S H Thus to symmetrize we shll let nd define Then Li) (8) Ri) () ' (0) B () where is n unknown function to be decided in order for to meet the required conditions in Theorem Since the symmetrized function is defined in terms of it is convenient to use chnge of vribles to epress the integrls in Theorem in terms of Recll the derivtive formul of n inverse function we hve 7 Furthermore we cn derive tht S ')7 7 S Therefore the integrl over SUTVWT7 on the left-hnd side of eqution (6) cn be splitted on two subintervls to derive where \ 7 S V S In deriving the bove two equtions (8) is used to determine the sign of ech term As for the right-hnd side of eqution (6) by the symmetry property of we hve \ Z From (0) nd () we cn further obtin $ V () Hence () reduces to S % g d < V (3) By equting the integrnd in (3) with we set up the following ODE tht the desired needs to stisfy: % () Here we hve used () to determine the sign of ech term Besides the ODE () in order for the ssocited to belong to l&h SUTVWT7 =% must stisfy some etr conditions First of ll since in () corresponds to T nd CT is required to be zero we must hve C% ) () In ddition the requirement tht KI implies tht K) (6) Moreover we ssume tht is incresing in ie RQI) (7) Under the bove three conditions the ODE () reduces to $ where &% g g +) C%) (8) d b d b The so- nd lution of eqution (8) yields which in turn determines s ' ])(+* - / 0 / + ' The function cn be obtined by integrting (+ 3- / 0 / ) () : (0) Since stisfies the conditions () (6) nd (7) the constructed in () belongs to l&h SUTVWT7 nd stisfies the condition (6) Thus the only thing left now is to show tht lso stisfies condition (7) To this purpose we introduce the following lemm Lemm 3: The 6 defined s follows 6 () hs the following properties: ) 6 ; ) 6 ; 3) db 6 d b &S 6 Proof: The first two re obvious thus we only prove the third one By Cuchy-Schwrz inequlity the following is true for nonnegtive numbers 7 s nd 8 s

5 db b S 6 Hence we hve 6 S S Integrting both sides over B ) we hve d d 6 d S b b d S b d S d &S S b b S &S S 6 &S 6 M () where in deriving the first inequlity we use (8) nd the fct tht S RK) 7 Therefore the desired result cn be obtined by dividing both sides of () by RS nd letting ^ Lemm 3 cn be used to prove n importnt property of the function defined in (0) Lemm : The solution to the ODE () stisfies for ll )& Proof: For ny )& by prt ) of Lemm 3 we hve 6 6 B Tking squre root multiplying both sides by 6 db nd pplying prt 3) of Lemm 3 we hve 6 which implies Hence ' S)6 + ' S)6 S)6+ ' 6 (+* - / 0 / S &S ' &S 6 ' (+ 3- / 0 / +&S 6+ + ' ( 3- / 0 / (+* - / 0 / ( - * / 0 / ( 3- / 0 / 6 ' B ( * - / 0 / where we hve used prt ) of Lemm 3 Then by the epression of in () we hve ' ])(+* - / 0 / + ' ' ] ( - * / 0 / % 6 ' (+ 3- / 0 / ( * - / 0 / 7 which is the desired result With Lemm we re redy to prove Theorem Proof: Proof of Theorem ] Given n rbitrry l&h SUTVWT7 let where nd re relted by (0) nd is defined by (0) The wy we compute gurntees tht Moreover C% implies tht \ l&h SUTVWT7 nd nd by Lemm \ Therefore 7 ' + ' + = ' + \ 7 stisfies (6) which (7h (3) where we hve used the symmetric property of nd the nondecresing property of ; The lst step of the bove derivtion follows directly from the definition of the Schwrz symmetriztion which preserves the function vlue during the symmetriztion process Applying prt 3) of Lemm to (3) the desired result cn be obtined: ol&h SUTVWT7 nd 7 7 Thus stisfies both (6) nd (7) This completes the proof of the theorem As result Theorem is lso proved In summry the proof of Theorem consists of two prts The first prt proves tht the whose corresponding stisfies (6) hs the property The second prt uses this result nd the property of the Schwrz symmetriztion (Lemm ) to derive the inequlity (7) The properties in Lemm hold for n rbitrry Eucliden spce thus only the first prt of the proof depends on the stte dimension Since ] proves tht the corresponding to the symmetriztion of multi-dimensionl stte spce ( ) stisfies the following theorem follows immeditely Theorem 3: Suppose tht the sfe set is rdilly symmetric If is constnt over nd is positive rdilly symmetric function defined on the solution to the generl OSTC problem (Problem or ) is rdilly symmetric for n rbitrry stte dimension VI NUMERICAL VERIFICATION In this section numericl emple is given to verify our proof in Section V To this end we will numericlly solve -D OSTC problem without ssuming the symmetry property to see whether the solution is symmetric In prticulr we focus on Problem The gol is to find control tht minimizes subject to the -D PDE (3) nd ` 7 N To del B

6 ) * S TABLE I SIMULATION PARAMETERS Pr Vlue u 0 0 u u with the constrints define ` ] nd 7 Then the -D generlized OSTC problem is equivlent to the following optiml control problem: Problem : Find tht Minimize 7 ( Subject to S ) S =SUT ) CT NM nd +=SUT +CT ) nd In our simultion we choose Define the Hmiltonin =S ) S +B According to the stndrd result of the optiml control theory the following costte equtions cn be obtined: -%$ S -%& b -%$ S -%& d -%$ S -%&(' ) &S % with boundry conditions =SUT CT The optiml control is determined by ) ) 8S B Generlly speking there is no nlyticl solution to the bove problem To solve it numericlly the mjor difficulty lies in tht the boundry conditions re not given t the sme point Here the clssicl shooting method ] is employed to tckle this two-point boundry condition problem The optiml solutions corresponding to two different re shown in Fig 3 where the prmeters used in the simultion re summrized in Tble I The simultion shows tht the solution is indeed symmetric Moreover this emple lso demonstrtes the effect of the proposed weighting functions When the weighting function decreses from to + to mintin the weighted verge sojourn time t lrger % is resulted VII CONCLUSION In this pper we generlize the optiml sojourn time control (OSTC) problem by introducing weighting functions for both the control cost function nd the performnce mesure function It is shown tht without control weighting the solution to the -D generlized OSTC problem is symmetric Numericl simultions re performed to verify the results The wy we del with the performnce weighting function does not depend on the stte dimension Therefore this pper together with ] completely proves tht without V 0 V V Fig 3 µ 0 Simultion Result 8 6 µ µ control weighting the solution to n rbitrry dimensionl generlized OSTC problem is rdilly symmetric Future reserch will focus on using the the symmetry property to nlyticlly or numericlly solve the generlized OSTC problem REFERENCES ] J Hu nd S Sstry Optiml sojourn time control within n intervl Proc Americn Control Conference vol pp Denver CO 003 ] J Hu Symmetry of solutions to the optiml sojourn time control problem Proc IEEE Int Conf Decision nd Control Seville Spin 00 3] G Poly nd G Szego Isoperimetric Inequlities in Mthemticl Physics Princeton Univ Press ] R Rffrd J Hu nd C Tomlin Adjoint-bsed optiml control of the epected eit time for stochstic hybrid systems Hybrid Systems: Computtion nd Control 8th Int Workshop (HSCC 00) Zurich Switzerlnd pp 7-7 Springer-Verlg 00 ] A E Bryson nd C Ho Applied Optiml Control Tylor nd Frncis 7 6] P Vriy Smrt Crs on Smrt Rods: Problem of Control IEEE Trnsction on Automtic Control AC-38() 3 7] J Kuchr nd L C ng Survey of Conflict Detection nd Resolution Modeling Methods IEEE Trnsctions on Intelligent Trnsporttion Systems (): ] A Bgchi Optiml Control of Stochstic Systems New ork N: Prentice-Hll 3 ] B Oksendl Stochstic Differentil Equtions: An Introduction with Applictions Springer-Verlg 6th edition 003 0] C Bndel Isoperimetric Inequlities nd Applictions Monogrphs nd Studies in Mthemtics vol 7 Pitmn (Advnced Publishing Progrm) Boston Mss 80 ] R O Sber W B Dunbr nd R M Murry Coopertive Control of Multi-Vehicle Systems using Cost Grphs nd Optimiztion Proc Americn Contr Conf volume 3 pges 7- Denver CO Jun 003 ] D Stipnovic G Inlhn R Teo nd C J Tomlin Decentrlized Overlpping Control of Formtion of Unmnned Aeril Vehicles Automtic 0(8): ] W H Fleming nd R W Rishel Deterministic nd Stochstic Optiml Control New ork N: Springer-Verlg 86