Control of the Continuity Equation with a Non Local Flow

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Niveau: Supérieur, Doctorat, Bac+8
Control of the Continuity Equation with a Non Local Flow Rinaldo M. Colombo? Michael Herty† Magali Mercier‡. September 29, 2009 Abstract This paper focuses on the analytical properties of the solutions to the continuity equa- tion with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posed- ness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with re- spect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows. 2000 Mathematics Subject Classification: 35L65, 49K20, 93C20 Keywords: Optimal Control of the Continuity Equation; Non-Local Flows. 1 Introduction We consider the scalar continuity equation in N space dimensions { ∂t?+ div ( ? V (?) ) = 0 (t, x)?R+ ? RN ?(0, x) = ?0(x) x?RN (1.1) with a non local speed function V . This kind of equations appears in numerous examples, a first one being the supply chain model introduced in [3, 4], where V (?) = v ( ∫ 1 0 ?(x) dx ) , see Section 3.

  • differentiability properties

  • local operator

  • r0 ?

  • weak entropy

  • well blow

  • differentiability

  • nonlinear local

  • ?0 ?


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49bta8At This paper focuses on the analytical properties of the solutions to the continuity equa-tion with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posed-ness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with re-spect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows. 2000MathematicsSubjectClassi cation:35L65, 49K20, 93C20 Keywords: Equation; Non-Local Flows. ntinuityOptimal Control of the C o
Introduction
We consider the scalar continuity equation inNspace dimensions ( ητt(0ηπ+R=)vidη (η(R()η)) (= 0Lπ R)R2RN+×RN(1.1) with anon localspeed function kind of equatio. This appears in numerous examples, s n a first one being the supply chain model introduced i [ 3 here(η) =NR(()η(R) dR, n , 4], w see Section 3. Another example comes from pedestrian trac, in which a reasonable model can be based on (1.1) with the functional(η) =N(η)N~(R4nT.tcoieeeS)s,t,oughouhr our assumptions are modeled on these examples. Other analytically similar situations are found in [ 5], where a kinetic model for pedestrians is presented, and in [ 23], which concerns the Keller{Segel model. In the following we postulate assumptions on the functionwhich are satisfied in the cases of the supply chain model and of the pedestrian model, but not for general functions. In particular, we essentially require below thatis anon localfunction, see (2.1). Therstquestionweaddressisthatofthewellposednessof(1.1).Indeed,weshow in Theorem 2.2 that (1.1) admits a unique local in time weak entropy solution on a time NeLiC;LfiW.edSvibIL5LiJ,ifi3,S0bzdLdS05-0.2LpfdS8TcLESafbLfiNfVSaSeNie,5d aSH,dSWpMLdU22bS.,,24L26SbNVSd,AWFBH4LNVSbIbivSdeifibiaL z,3,.FG6NF2IESD4Rb,icC,GNiDbcSWSRfbLdWS6S9Nccb-,SDifiedSvibI,bciDSRSifdevSbiI Cbefifgf6LaiWWSJcdRLb;10MWvR.Rg--NcvSaMdS-9-3,;´6SSRS6bbMLbNdL,;Sh.9dgSWWiJ.
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intervalε. For allLinε, we call>tthe nonlinear local semigroup that associates to the initial conditionη(the solution>tη(of (1.1) at timeL. As in the standard case,>tturns out to beL0ti.zpsch{Li Then, we look for the Ga^teaux differentiability of the mapη(7!>tη(, in any direc-tionJ(and for allL2ε. Aweak Ga^teaux differentiability of the semigroup generated by (1.1) is proved along any solutioη2C0(ε; (W0;0\W0;×)(RN;R)and in any direc-n tionJ(2L0(RN;Rtionsumpreovs.More,libiiantowllfotytsrednussaregnordillreeFu). the Ga^teaux derivative of>tatη(in the directionJ(is uniquely characterized as weak entropysolutiontothefollowinglinearnon{localCauchyproblem,thatcanbeformally obtainedbylinearizing(1.1): ( τtJ+ divJ∂(η(J) J(0π R) = J((R)+)ηD(η)) (= 0Lπ R)R2RεN×RN(1.2) whereη(Lπ R) = (>tη()(Rlbmeplorolacnenoofthnessosedellp).2(1oslawehtT.),suh needed to be proved, see Proposition 2.8. Remark that, in both (1.1) and ( 1.2), solutions are constructed inC0(ε;L0(RN;R)forehereostl,wemT.ontilusos,otreferyvokzurK see [ 21, Definition 1]. Indeed, this definition of solutions is more demanding than that of weak solutions. Besides, it allows us to apply the results in [ 15], used in the subsequent part concerning the differentiability of solutions with respect to the initial datum. However, we note that in the case of the standard transport equation (5.2) and with the regularity conditions assumed below, the two notions of solution coincide, see Lemma 5 1. . We recall here the well known standard (i.e. local) situation: the semigroup generated by a conservation law is in generalnotdifferentiable inL0, not even in the scalar 1D case, see [ 9, Section 1]. To cope with these issues, an entirely new differential structure was introduced in [ 9], and further developed in [ 6, 10], also addressing optimal control problems, see [ 11, 14]. However, the mere definition of the shift differential in the scalar 1Dcasetakesaloneaboutapage,see[14,p.89{90].Wereferto[7,8,18,24,25]for further results and discussions about the scalar one{dimensional case. Then, we introduce a cost functionJ:C0(επL0(RN;R)!Rand, using the differen-tiability property given above, we find anecessary conditionon the initial dataη(in order to minimizeJalong the solutions to (1.1) associated toη( stress that the present nec-. We essary conditions are obtained within the functional setting typical of scalar conservation laws, i.e. withinL0andL× reflexivity property is ever used.. No The paper is organized as follows. In Section 2, we state the main results of this paper. The differentiability is proved in Theorem 2.10 and applied to a control in supply chain management in Theorem 3.2. The sections 3 and 4 provide examples of models based on (1.1), and in Section 5 we give the detailed proofs of our results.
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NotationandMainResults
DenoteR+= [0π+[,R+= ]0π+[ and byε, respectivelyεxeeith,av[ltnre0π T[, respec-tively[0π T[, forT π Txe>0. The open ball inRNcentered at 0 with radiusis denoted ex byψ(0π onehtecudortniewe,orrmhertFu).rms: kNk1= ess supN(R)πkNk1+krNk1π W1;1=kNkρL L L ρ2RN Nk1;1+r+πkNk1;1=kN k=kNk W+krρNkL1ρNkL1+rkρNkL1ξ 1 W L 1 L
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2.1ExistenceofaWeakEntropySolutionto(1.1) Let:L0(RN;R)!C1(RN;RN) be a functional, not necessarily linear. straightforward A extensionof[21,Denition1]yieldsthefollowingdenitionofweaksolutionsfor(1.1).
DC Vti).(WVFixη(2L×(RN;R). A weak entropy solution to (1.1) onεis a bounded measurable mapη2C0(ε;L0(RN;R)ontolutiovsouˇzkrKasihcihw clo (  )0wL))(R)ξ ητt(0πη+Rdvi=)ηη((QR()Lπ R) =hereQ(Lπ R) =( η( Here, differently from [ 21, Definition 1], we require the full continuity in time. Introduce the spaces 0 X= (L0\L×\57)(RN;R) andX= (L\57)R; [0π ])for >0 N both equipped with theL0distance. Obviously,X L×(RN;R) for all >0. We pose the following assumptio s on, all of which are satisfied in the examples on n supply chain and pedestrian flow as shown in Section 3 and Section 4, respectively. 0 (sts a functioC2C(R+;R+) such that for allη2L0(RNπR), 7() nThere exi (η)2L×(RN;RNπ ) rρ(η)1N NNC(kηkL1(RN;R))π L(R;R) rρ(η)1N NNC(kηkL1(RN;R))π L(R;R) r+(η)C(kηk1N)ξ ρL(R;R) 1N NNN L(R;R) There exists a functionC2C0(R+;R+) such that for allη)π η+2L0(RNπR) (η)) (η+)1N NC(kη)kL1(N)kη) η+kL1(π(2.1) N R;R)R;R) L(R;R) rρ(η)) rρ(η+)1N NNC(kη)k1N)kη) η+k1Nξ L(R;R)L(R;R) L(R;R) (7))There exists a functioC2C0(R+ +su that for allη2L0(R n ;R) chNπR) , r+∂ C(kηk1N)ξ ρ(η)L(R;R) 1N NNN L(R;R) (7,):L0(RN;R)!C2(RN;RN) and there exists a functionC2C0(R;R) such that + + for allη2L0(RNπR) , r0ρ(η)C(kηk1N)ξ L(R;R) 1N NNNN L(R;R)
Condition (2.1) essentially requires thatbe anon localoperator. Note that(7,)im-plies(7))(at least locally in time) can be proved under of a solution to (1.1) . Existence only assumption(7(), see Theorem 2.2. n The stronger boundsensure additional o regularity of the solution which is required later to derive the differentiability properties, see Proposition 2 5 . .
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