Connections in control strategy

P oceedings o he Es onian Academy o Sciences, 2014, 63, 1, 26–32 doi: 10.3176/p oc.2014.1.05 A ailable online a www.eap.ee/p oceedings Connec ions in con ol s a egy Maido Rahulaa∗and Pe Vaˇ s´ ıkb aIns i u e o Ma hema ics, Uni e si y o Ta u, J. Lii i 2, 50409 Ta u, Es onia bIns i u e o Ma hema ics, B no Uni e si y o Technology, Facul y o Mechanical Enginee ing, Technick´ a 2, 616 69 B no, Czech Republic; [email p o ec ed].cz Recei ed 15 Augus 2013, e ised 20 Sep embe 2013, accep ed 14 Oc obe 2013, a ailable online 14 Ma ch 2014 Abs ac . We p esen an in ini esimal in e p e a ion o he con ol heo y, pa icula ly o he pa conce ning dynamic sys ems. We use he o iginal concep o a bundle connec ion, which lies in he idea o ib e anspo a ion along a pa h on he base mani old. The con ol o a p ocess leads also o he anspo a ion o ib es, and he con ol s a egy, i.e. he choice o a sui able sys em con ol in o de o op imize he p ocess co esponds o he choice o a pa h on he base mani old. The iple o c ucial e ms o con ol, aim–con ol–s a egy, ansla es in he e ms o connec ions as ib e–connec ion–cu e. Such a scheme is qui e con incing, bu i also wo ks well in dynamic sys ems analysis. Key wo ds: con ol heo y, connec ion. 1. INTRODUCTION When con olling a sys em, we no only apply one con ol model bu also y o ind a mo e sui able con ol model among he possible ones, i.e. we sea ch he con ol s a egy. We dis inguish he ollowing s ages: con olled p ocess – con ol co ec ion – s a egy choice. Le us desc ibe he ma hema ical se ing. Le X,Y, and Zbe h ee ec o ields and le us deno e by a =exp X,b σ =exp σ Y,c τ =exp τ Z he app op ia e lows. I we unde s and he low as a mo ion, he ec o ield can be seen as s opping he mo ion a he p ecise momen (s op-scene). Sho ly, a ec o ield is an in ini esimal ep esen a ion o he low. The low c τ o he ec o ield Z ep esen s he con olled p ocess (i is also possible o eplace i by a anspo o an a bi a y enso ield along he low c τ ). Fu he mo e, he low b σ ac s on he ec o ield Z low c τ as ollows; see [1,3]: c τ Ãb σ c τ b−1 σ ,ZÃZ σ . This co esponds o he change o con ol. I in addi ion he low a ac s on b σ , we ha e a con ol s a egy b σ Ãa b σ a−1 ,YÃY . ∗Co esponding au ho , [email p o ec ed] M. Rahula and P. Vaˇs´ ık: Connec ions in con ol s a egy 27 Conce ning he s a egy {XÃ{YÃZ}}, i is ob ained as a composi ion o he ac ion o b σ on c τ and he ac ion o a on b σ , c τ Ãb σ c τ b−1 σ Ã(a b σ a−1 )c τ (a b σ a−1 )−1 . The ec o ield Yplays he ole o he one con olled by he ec o ield Xand he ole o he con olling ield o e Z. The main goal o he pape is o desc ibe he ans o ma ion o he pa ame e s when he p ocess Zis changed acco ding o he s a egy app op ia e o he ec o ield Xunde he ac ion o he ec o ield Y. 2. VECTOR FIELDS Le Mbe a smoo h mani old. The de i a i es o a unc ion :M→Ralong he ec o ields X,Y, and Z a e de ined by X . = ( ◦a )0 =0,Y . = ( ◦b σ )0 σ =0,Z . = ( ◦c τ )0 τ =0. The ec o ield Yis anspo ed along he low o X, which can be unde s ood as an in ini esimal in e p e a ion o such anspo a ion (s op-scene) – he b acke o ec o ields, i.e. Lie de i a i e LXY= [X,Y]. Rema k 1. One can ob ain he b acke o wo ec o ields [X,Y] = XY −YX by double di e en ia ion o a unc ion along he ec o ield low a b σ a−1 w. . . σ and hen w. . . : ◦(a b σ a−1 )−1(.)0 σ =0 −→ −¡Y( ◦a )¢◦a−1 (.)0 =0 −→ (XY −Y X) . Nex , he anspo o an a bi a y smoo h enso ield along a ec o ield low is de ined by he Lie– Maclau in se ies. Fo example, he anspo o a ec o ield Zalong he low b σ is de ined by ZÃZ σ =Z+Z0 σ +Z00 σ 2 2+... = ∞ ∑ k=0 Z(k) σ (k) k!, whe e he coe icien s Z(k)=LYZ(k−1) ,k=1,2,..., a e Lie de i a i es o Zwi h espec o Y. In ou si ua ion, he ec o ield Xplays h ee oles: 1. The ec o ield Xi sel causes he p ocess as a mo ion in i s low a . 2. The ope a o LX ans o ms he con ol YÃ[X,Y]. 3. The ope a o LLXde ines he con ol s a egy – con ol o con ol LYÃ[LX,LY] = L[X,Y]. No e ha he abo e equali y can be ob ained om he Jacobi iden i y: £[X,Y],Z¤+£[Y,Z],X¤+£[Z,X],Y¤=0 o equi alen ly £[X,Y],Z¤=£X,[Y,Z]¤−£Y,[X,Z]¤. No e ha he p ocess can be in luenced only by some ou e p ocess, no by i i sel . Indeed, i we admi ha he ope a o LXac s on he con ol by he p ocess X, we ob ain: LXX= [X,X] = 0. We assign he ollowing ope a o s o he ec o ields X,Y, and Z: 1. he ope a o Zimplemen ing he p ocess (mo ion in he low c τ ); 2. he ope a o LYimplemen ing he con ol o he p ocess ZÃ[Y,Z](mo ion o he low c τ in he low b σ ); 3. he ope a o LLXde ining he con ol s a egy LY(mo ion in he low a o a mo ion o he low c τ in he low b σ ). 28 P oceedings o he Es onian Academy o Sciences, 2014, 63, 1, 26–32 3. CONTROL AND CONNECTION We will ollow he no a ions app op ia e o he heo y o connec ions on ib ed mani olds; see, e.g., [1,4]. Le us conside a ec o bundle π :M1→Mwi h n-dimensional base mani old Mand -dimensional ib es. The s anda d ib e is isomo phic o R . On a neighbou hood U⊂M1we ha e local coo dina es (ui ,u α ), whe e (ui)deno es he base coo dina es and (u α ) he ib e coo dina es. P ecisely, ui=¯ui◦ π , whe e ¯uideno es he local coo dina es on he neighbou hood π (U)⊂M. The coo dina es (u α )a e he coo dina es o R .La in indices i,j, . . . ange om 1 o n, G eek indices α , β , . . . ange om n+1 o n+ . We de ine wo ec o ields: Y=y α∂α and Z=z α∂α . He e z α a e he unc ions depending on he ib e coo dina es u α only, while y α a e he unc ions o all coo dina es (ui ,u α ). The low b σ =exp σ Yis de ined on he neighbou hood Uby a sys em o ODEs du α d σ =y α (ui ,u β ).(1) Indeed, now we can see he connec ion be ween dynamic sys ems, see [2], and he con olling pa ame e s (ui). As men ioned abo e, hese pa ame e s a e li ed om he base π (U)⊂M o he neighbou hood U⊂M1, i.e. ui=¯ui◦ π . On e e y ib e, he ec o ield Yinduces a amily o ajec o ies – phase po ai . When he ib e is changed, he ec o ield Ychanges oo and so does he phase po ai , i.e. he con ol {YÃZ}. A ques ion a ises: how do he pa ame e s (ui)a ec he con olling p ocess? Le us conside he coo dina e map Φ:(ui ,u α )Ã(ui ,s,I κ ),k=n+2,...,n+ , whe e sis a canonical pa ame e , i.e. LYs=0, and I κ is a sys em o −1 independen in a ian s o he ec o ield Y. The coo dina es (ui ,I κ ) o m a comple e sys em o local in a ian s o Yon he mani old M1. Now we can de ine he subme sion o he mani old M1on o he ib e R , ϕ :M1→R :(ui ,u α )Ã(s,I κ ). A ib e o he subme sion ϕ has he dimension nand o ms he amily o he in eg al su aces which de ine a ho izon al dis ibu ion 4h.Thus on he ib a ion π , a ze o o sion connec ion s uc u e 4h⊕4 is de ined. Le us conside he adap ed basis (XiX α ) = µ ∂ ∂ uj ∂ ∂ u β ¶·Ã δ j i0 Γ β i δβ α !,µ ω i ωα ¶=µ δ i j0 −Γ α j δα β ¶.µduj du β ¶, whe e he ec o ields Xi= ∂ ∂ ui+Γ α i ∂ ∂ u α o m a base o he dis ibu ion ∆hand he o ms ωα =du α −Γ α idui anish on he dis ibu ion ∆h. The numbe o pa ame e s Γ α iequals n and hey de ine he dis ibu ion ∆huniquely. On he o he hand, he pa ame e s Γ α ia e de e mined by se ing he unc ions ϕα equal o a cons an on he ib es o he subme sion ϕ , mo e p ecisely by hei di e en ials: d ϕα = ϕα idui+ ϕα β du β = ϕα β (du β +¯ ϕβ γϕγ idui) = ϕα βωβ =⇒Γ α i=¯ ϕα γϕγ i, whe e he coe icien s o d ϕα a e he pa ial de i a i es o ϕα .The ma ix ( ϕα β )is he in eg a ing ma ix wi h espec o he o ms ωα and i s in e se is (¯ ϕβ α ). M. Rahula and P. Vaˇs´ ık: Connec ions in con ol s a egy 29 Theo em 1. The ec o ield Y is p ojec ed by he subme sion ϕ :M1→R on o he ec o ield T ϕ Y on he s anda d ib e R . In he coo dina es (s,I),whe e s deno es he canonical pa ame e and I is a sys em o he base in a ian s, he ec o ield T ϕ Y ep esen s he ope a o ∂ s . = ∂ ∂ s. The ec o ield Z is exp essed uniquely in he basis ( ∂ s, ∂ I)and he p ocess con olled by Z is,in he coo dina e sys em (s,I),desc ibed by he unc ions s◦ ϕ and I ◦ ϕ . These unc ions depend on he pa ame e s u α and he con olling pa ame e s ui. P oo . A amily o he ib es co esponding o he subme sion ϕ is de ined by he solu ion o he sys em o di e en ial equa ions (u α ) σ = ϕα ( σ ,ui ,u β ); see sys em (1). Fu he mo e, an a bi a y sec ion o he ib a ion π can be ex ended in o he sys em o imp imi i i y app op ia e o he low bs, i.e. he amily o he ib es co esponding o he subme sion ϕ . The ec o ield Yis ϕ -p ojec ed on he ib e R . An in eg able dis ibu ion ∆h=Ke T ϕ in he ib a ion π de ines a ze o cu a u e connec ion and hus on he neighbou hood U he basis and he co-basis o he dis ibu ion ∆his de ined as ollows: Xi= ∂ i+Γ α i ∂α , ωα =du α −Γ α idui . Le us ecall ha an a bi a y ec o ield ¯ Xon he base mani old Mcan be li ed om M o he ho izon al dis ibu ion ∆h: ¯ X=¯xi¯ ∂ iÃX=xiXi,whe e xi=¯xi◦ ϕ . In ou no a ions, he basis Xi ep esen s he ope a o s ¯ ∂ i om he neighbou hood π (U)li ed o he dis ibu ion ∆h. I is now clea ha he ec o ield Xbeha es wi h espec o he ec o ield Yas an in ini esimal symme y, i.e. [X,Y] = 0, and hus he impac on he ec o ield Y anishes. In o he wo ds, he p ocess app op ia e o he ec o ield Zis de ined on he ib e in he coo dina es (s,I), whe e he unc ions s◦ ϕ and I◦ ϕ depend on he pa ame e s u α and he con olling pa ame e s ui. The ec o ield Xa ec s he ec o ield Zindi ec ly by means o he in a ian s o he ec o ield Y. Rema k 2. The componen s y α o he ec o ield Ydepend linea ly and homogeneously on he ib e coo dina es. Thus he de ining sys em is desc ibed by he sys em o linea di e en ial equa ions du α d σ =y α β (ui)u β . 4. APPLICATION On he bundle1 π :R3→R:(u,x,y)Ã(u) wi h he ib e coo dina es (x,y)and he con olling pa ame e (o base coo dina e) (u)we ha e he ec o ield Y= ∂ ∂ x+ux ∂ ∂ y. We de ine i s low bs=expsY , he canonical pa ame e s, and he in a ian Io Yas ollows: ½˙x=1 ˙y=ux ⇒½xs=x+s ys=y+u(xs +s2 2),½s=x I=y−ux2 2. We check ha LYs=1,LYI=0. The ajec o ies on he ib es a e pa abolas depending on he pa ame e u. 1He e, o he sake o simplici y, we deno e he local coo dina es by (u,x,y)ins ead o (u1 ,u2 ,u3)bu no e ha he ib e coo dina es (x,y)a e in no way ela ed o he componen s (xi ,y α )o he ec o ields Xand Y. 30 P oceedings o he Es onian Academy o Sciences, 2014, 63, 1, 26–32 The subme sion ϕ :R3→R:(u,x,y)Ã(s,I)p ojec s he space R3on o he plane sI. The angen mapping T ϕ is de ined by he ollowing di e en ials and by he Jacobi ma ix: ½ds =dx dI =−x2 2du −uxdx +dy,µ0 1 0 −x2 2−ux 1¶. The ec o ield Ywi h he componen s (0,1,ux)is p ojec ed o he plane sI in which i o ms he ope a o T ϕ Y= ∂ s(see Fig. 1). Thus on he bundle π a ho izon al dis ibu ion 4h=Ke T ϕ is de ined. The co-basis on 4his o he o m ½ ω 2=ds =dx = (dx −Γ2 1du) ω 3=uxds +dI =dy −x2 2du = (dy −Γ3 1du), and he connec ion coe icien s a e µΓ2 1 Γ3 1¶=µ0 x2 2¶. The adap ed basis o he dis ibu ion 4his cha ac e ized by he ollowing: X1= ∂ u+x2 2 ∂ y,µ ω 2 ω 3¶=µdx dy ¶−µ0 x2 2¶·(du). The ope a o X1commu es wi h he ec o ield Y, i.e. [X1,Y] = 0,and anishes unde he p ojec ion T ϕ , i.e. T ϕ X1=0. The co-basis admi s an in eg a ing ma ix as ollows: µ ω 2 ω 3¶=µ1 0 us 1¶·µds dI ¶⇒µ1 0 −ux 1¶·µ ω 2 ω 3¶=µds dI ¶. H H H H H H H H H H H H C C C C C C C C C CO + = - u 6 ¤¤¤¤¤¤¤¤¤¤ ¤º ¢¢¢¢¢¢¢¢¢ ¢¸ ¾ T ϕ ∂ s sI Y Fig. 1. Mapping T ϕ :Y→ ∂ s. M. Rahula and P. Vaˇs´ ık: Connec ions in con ol s a egy 31 The di ec impac o he pa ame e (u)on he ope a o Yis elimina ed. Indeed, because he p ojec ion ϕ a ge s on he ib e xy, i is possible o change he coo dina es unde he condi ion u=cons om (x,y) o (s,I), ½x=s y=us2 2+I,µ1 0 us 1¶. Using he Jacobi ma ix ( he igh -hand side), we can change he basis o he new na u al one and we ob ain he ollowing ames and co- ames: ³ ∂ ∂ x ∂ ∂ y´=³ ∂ ∂ s ∂ ∂ I´·µ1 0 −us 1¶,µdx dy ¶=µ1 0 us 1¶·µds dI ¶. Le us ocus on he ib e. No e ha he ac ion o he e ical ec o ield Zcan be unde s ood as an ac ion on a enso ield. Conce ning he ac ion o he ope a o Yon he ec o ield Zin he o m YÃZ= µ∂ ∂ x+ ν∂ ∂ y, wi h he componen s ( µ , ν ), we can see ha in new coo dina es i educes o he ac ion o he ope a o ∂ son he ec o ield ˜ Zdepending on he pa ame e uonly: ∂ sØ Z= µ∂ s+ ν∂ I−u µ s ∂ I. No e ha Zand ˜ Za e he same ec o ield, only exp essed in he coo dina es (x,y)and (s,I), espec i ely. The ope a o s T ϕ Yand ∂ sa e he same ope a o s exp essed in di e en coo dina e sys ems. Thus we change he con ol: {YÃZ}Ã{ ∂ sØ Z}. Rema k 3. As an example, le us conside he ope a o o o a ion Z=−y ∂ ∂ x+x ∂ ∂ y. In coo dina es (s,I),i can be w i en in he o m ˜ Z=−I ∂ s+s ∂ I+u(...), i.e. in such a o m ha some new ope a o wi h coe icien uis added. Such a p ope y holds o an a bi a y linea dynamic sys em. The con ol {YÃZ}is desc ibed in he coo dina es (x,y), while he con ol { ∂ sØ Z}is exp essed in he coo dina es (s,I). The pa ame e ua ec s he con olled ield ˜ Zdi ec ly. 5. CONCLUSION The con ol o a dynamic sys em is iewed by means o di e en ial geome y as he ec o ield Yon he bundle π :M1→Mwi h he s anda d ib e R and he base mani old M=Rn .The subme sion ϕ is de ined in such a way ha he ec o ield Yis p ojec ed o he ib e R .The dis ibu ion 4h=Ke T ϕ gi es ise o he possibili y o elimina ing he dependence o he ec o ield Yon he con olling pa ame e u. The change o a iables o (s,I), whe e sis he canonical pa ame e and Iis he in a ian o he ield Y, changes he con ol (Y→Z) o he con ol { ∂ sØ Z}, whe e he ield ∂ sno longe depends on he pa ame e uwhile he con olled ield ˜ Zdoes so. 32 P oceedings o he Es onian Academy o Sciences, 2014, 63, 1, 26–32 ACKNOWLEDGEMENTS The i s au ho was suppo ed by Es onian Ta ge ed Financing P ojec SF0180039s08. The second au ho was suppo ed by he p ojec NETME CENTRE PLUS (LO1202). The esul s o he p ojec NETME CENTRE PLUS (LO1202) we e co- unded by he Minis y o Educa ion, You h and Spo s wi hin he suppo p og amme “Na ional Sus ainabili y P og amme I”. REFERENCES 1. A anasiu, Gh., Balan, V., B ˆ ınzei, N., and Rahula, M. 2009. Di e en ial-Geome ic S uc u es. Tangen Bundles, Connec ions in Fibe Bundles, Exponen ial Law and Je Spaces. Lib okom, Moscow (in Russian). 2. Pe ko, L. Di e en ial Equa ions and Dynamic Sys ems. Sp inge , 1991. 3. Rahula, M. New P oblems in Di e en ial Geome y. Wo ld Scien i ic Publishing Co. P e. L d., 1993. 4. Rahula, M. and Vaˇ s´ ık, P. A no e on je and geome ic app oach o highe o de connec ions. Sp inge P oc. Ma h. S a is ics ( o appea ). Seos used juh imise eoo ias Maido Rahula ja Pe Vaˇ s´ ık Juh imisel on kolm aspek i: juhi a p o sess, juh i p o sess ja juh imise alik/s a eegia. Ma emaa iliseks mudeliks on kih kond, kus seos us m˜ oju ab oimu a kihil, ja baasipa amee id, milles s˜ ol ub seos us. Need m¨ a¨ a a ad juh imise s a eegia. Osu ub, e baasipa amee eid ˜ oib seos use abil o sekohe kihile suuna a, s juh i as p o sessis juhi a asse.