Controllability and optimal control of the transport equation with a localized vector field
CControllability and optimal control of the transport equation with alocalized vector field*
Michel Duprez and Morgan Morancey and Francesco Rossi Abstract — We study controllability of a Partial DifferentialEquation of transport type, that arises in crowd models. Weare interested in controlling such system with a control beinga Lipschitz vector field on a fixed control set ω .We prove that, for each initial and final configuration, onecan steer one to another with such class of controls only if theuncontrolled dynamics allows to cross the control set ω .We also prove a minimal time result for such systems. Weshow that the minimal time to steer one initial configuration toanother is related to the condition of having enough mass in ω to feed the desired final configuration. I. INTRODUCTIONIn recent years, the study of systems describing a crowdof interacting autonomous agents has draw a great interestfrom the control community (see e.g. the Cucker-Smalemodel [4]). A better understanding of such interaction phe-nomena can have a strong impact in several key applications,such as road traffic and egress problems for pedestrians.Beside the description of interaction, it is now relevant tostudy problems of control of crowds , i.e. of controlling suchsystems by acting on few agents, or on the crowd localizedin a small subset of the configuration space.Two main classes are widely used to model crowds ofinteracting agents. In microscopic models , the positionof each agent is clearly identified; the crowd dynamicsis described by a large dimensional ordinary differentialequation, in which couplings of terms represent interactions.In macroscopic models , instead, the idea is to represent thecrowd by the spatial density of agents; in this setting, theevolution of the density solves a partial differential equationof transport type. This is an example of a distributedparameter system . Some nonlocal terms can model theinteractions between the agents. In this article, we focus onthis second approach.To our knowledge, there exist few studies of control of thiskind of equations. In [7], the authors provide approximatealignment of a crowd described by the Cucker-Smale model[4]. The control is the acceleration, and it is localizedin a control region ω which moves in time. In a similar *This work was supported by Archim`ede Labex (ANR-11-LABX-0033)and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the“Investissements d’Avenir” French Government programme managed by theFrench National Research Agency (ANR). The second and third authorsacknowledge the support of the ANR project CroCo ANR-16-CE33-0008. Aix Marseille Universit´e, CNRS, Centrale Marseille, I2M, LSIS, Mar-seille, France. [email protected] Aix Marseille Universit´e, CNRS, Centrale Marseille, I2M, Marseille,France. [email protected] Aix Marseille Universit´e, CNRS, ENSAM, Universit´e de Toulon, LSIS,Marseille, France. [email protected] situation, a stabilization strategy has been established in [2],by generalizing the Jurdjevic-Quinn method to distributedparameter systems.In this article, we study a partial differential equation oftransport type, that is widely used for modeling of crowds.Let ω be a nonempty open connected subset of R d ( d ≥ ),being the portion of the space on which the control isallowed to act. Let v : R d → R d be a vector field assumedLipschitz and uniformly bounded. Consider the followinglinear transport equation (cid:40) ∂ t µ + ∇ · (( v + ω u ) µ ) = 0 in R d × R + ,µ ( · ,
0) = µ in R d , (1)where µ ( t ) is the time-evolving measure representing thecrowd density and µ is the initial data. The control is thefunction ω u : R d × R + → R d . The function v + ω u represents the velocity field acting on µ . System (1) is a firstapproximation for crowd modeling, since the uncontrolledvector field v is given, and it does not describe interactionsbetween agents. Nevertheless, it is necessary to understandcontrollability properties for such simple equation. Indeed,the results contained in this article will be instrumental to aforthcoming paper, where we will study more complex crowdmodels, with a non-local term v [ µ ] .We now recall the precise notion of approximate control-lability for System (1). We say that System (1) is approxi-mately controllable from µ to µ on the time interval (0 , T ) if for each ε > there exists ω u such that the correspondingsolutions to System (1) satisfies W p ( µ ( T ) , µ ) (cid:54) ε . Thedefinition of the Wasserstein distance W p is recalled inSection II.To control System (1), from a geometrical point of view,the uncontrolled vector field v needs to send the support of µ to ω forward in time and the support of µ to ω backwardin time. This idea is formulated in the following Condition: Condition 1 (Geometrical condition)
Let µ , µ be twoprobability measures on R d satisfying:(i) For all x ∈ supp( µ ) , there exists t > such that Φ vt ( x ) ∈ ω, where Φ vt is the flow associated to v , i.e. the solution to the Cauchy problem (cid:40) ˙ x ( t ) = v ( x ( t )) for a.e. t > ,x (0) = x . (ii) For all x ∈ supp( µ ) , there exists t > such that Φ v − t ( x ) ∈ ω . Remark a r X i v : . [ m a t h . A P ] N ov tem of Condition 1 is not satisfied, there exists a whole sub-population of the measure µ that never intersects the controlregion, thus, we cannot act on it.We denote by U the set of admissible controls, thatare functions ω u : R d × R + → R d Lipschitz in space,measurable in time and uniformly bounded. If we imposethe classical Carath´eodory condition of ω u being in U ,then the flow Φ v + ω ut is an homeomorphism (see [1, Th.2.1.1]). As a result, one cannot expect exact controllability,since for general measures there exists no homeomorphismsending one to another. We then have the following result ofapproximate controllability. Theorem 1
Let µ , µ be two probability measures on R d compactly supported absolutely continuous with respect tothe Lebesgue measure and satisfying Condition 1. Thenthere exists T > such that System (1) is approximatelycontrollable at time T from µ to µ with a control ω u in U . The proof of this result will be given in Section III. Afterhaving proven approximate controllability for System (1), weaim to study the minimal time problem, i.e. the minimal timeto send µ to µ . We have the following result. Theorem 2
Let µ , µ be two probability measures, withcompact support, absolutely continuous with respect to theLebesgue measure and satisfying Condition 1.We say that T ∗ is an admissible time if it satisfies (a) For each x ∈ supp( µ ) T ∗ (cid:62) inf { t ∈ R + : Φ vt ( x ) ∈ ω } . (b) For each x ∈ supp( µ ) T ∗ (cid:62) inf { t ∈ R + : Φ v − t ( x ) ∈ ω } . (c) There exists a sequence ( u k ) k of C ∞ -functions equalto in ω c such that lim k →∞ [Φ v + u k t µ ]( ω ) (cid:62) − lim k →∞ [Φ v + u k t − T ∗ µ ]( ω ) . (2) Let T be the infimum of such T ∗ . Then, for all T > T ,System (1) is approximately controllable from µ to µ attime T . The proof of this Theorem is given in Section IV.
Remark u k are used to store the mass in ω . Thus, condition(2) means that at each time t there is more mass that hasentered ω that mass that has exited. This is the minimalcondition that we can expect in this setting, since controlcan only move masses, without creating them.This paper is organized as follows. In Section II, werecall some properties of the continuity equation and theWasserstein distance. Sections III and IV are devoted toprove Theorems 1 and 2, respectively. We conclude withsome numerical examples in Section V. II. T HE CONTINUITY EQUATION AND THE W ASSERSTEINDISTANCE
In this section, we recall some properties of the continuityequation (1) and of the Wasserstein distance, which will beused all along this paper.We denote by P c ( R d ) the space of probability measuresin R d with compact support, and by P acc ( R d ) the subsetof P c ( R d ) of measures which are absolutely continuouswith respect to the Lebesgue measure. First of all, we givethe definition of the push-forward of a measure and of theWasserstein distance. Definition 1
Denote by Γ the set of the Borel maps γ : R d → R d . For a γ ∈ Γ , we define the push-forward γ µ ofa measure µ of R d as follows: ( γ µ )( E ) := µ ( γ − ( E )) , for every subset E such that γ − ( E ) is µ -measurable. Definition 2
Let p ∈ [1 , ∞ ) and µ, ν ∈ P acc ( R d ) . Define W p ( µ, ν ) = inf γ ∈ Γ (cid:40)(cid:18)(cid:90) R d | γ ( x ) − x | p dµ (cid:19) /p : γ µ = ν (cid:41) . (3) Proposition 1 W p is a distance on P acc ( R d ) , called the Wasserstein distance . The Wasserstein distance can be extended to all pairs ofmeasures µ, ν compactly supported with the same mass µ ( R d ) = ν ( R d ) (cid:54) = 0 , by the formula W p ( µ, ν ) = | µ | /p W p (cid:18) µ | µ | , ν | ν | (cid:19) . For more details about the Wasserstein distance, in particularfor its definition on the whole space of measures P c ( R d ) , werefer to [8, Chap. 7].We now recall a standard result for the continuity equation: Theorem 3 (see [8])
Let T ∈ R , µ ∈ P acc ( R d ) and w be a vector field uniformly bounded, Lipschitz in space andmeasurable in time. Then the system (cid:40) ∂ t µ + ∇ · ( wµ ) = 0 in R d × R ,µ ( · ,
0) = µ in R d (4) admits a unique solution µ in C ([0 , T ]; P acc ( R d )) . More-over, it holds µ ( · , t ) = Φ wt µ for all t ∈ R , where the flow Φ wt ( x ) is the unique solution at time t to (cid:40) ˙ x ( t ) = w ( x ( t ) , t ) for a.e. t (cid:62) ,x (0) = x . (5)In the rest of the paper, the following properties of theWasserstein distance will be helpful. Property 1 (see [6])
Let µ, ν ∈ P acc ( R d ) . Let w : R d × R → R d be a vector field uniformly bounded, Lipschitz in spaceand measurable in time. For each t ∈ R , it holds W pp (Φ wt µ, Φ wt ν ) (cid:54) e ( p +1) L | t | W pp ( µ, ν ) , (6) Here, P acc ( R d ) is equipped with the weak topology, that coincides withthe topology induced by the Wasserstein distance W p , see [8, Thm 7.12]. here L is the Lipschitz constant of w . Property 2
Let µ, ν, ρ, η some positive measures satisfying µ ( R d ) = ν ( R d ) and ρ ( R d ) = η ( R d ) . It then holds W pp ( µ + ρ, ν + η ) (cid:54) W pp ( µ, ν ) + W pp ( ρ, η ) . (7)Using the properties of Wasserstein distance given in Section1 of [6], we can replace W p by W in the definition of theapproximate controllability.III. P ROOF OF T HEOREM ω contains the support of both µ , µ . Proposition 2
Let µ , µ ∈ P acc ( R d ) be such that supp( µ ) ⊂ ω and supp( µ ) ⊂ ω . Then, for all T > ,System (1) is approx. contr. at time T with ω u in U .Proof: We assume that d := 2 , T := 1 and ω := (0 , ,but the reader will see that the proof can be clearly adapted toany space dimension. Fix n ∈ N ∗ . Define a := 0 , b := 0 and the points a i , b i for all i ∈ { , ..., n } by induction asfollows: suppose that for i ∈ { , ..., n − } the points a i and b i are given, then a i +1 and b i +1 are the smallest valuessatisfying (cid:82) ( a i ,a i +1 ) × R dµ = n and (cid:82) ( b i ,b i +1 ) × R dµ = n . Again, for all i ∈ { , ..., n − } , we define a i, := 0 , b i, := 0 and supposing that for a j ∈ { , ..., n − } the points a i,j and b i,j are already defined, a i,j +1 and b i,j +1 are the smallestvalues such that (cid:82) A ij dµ = n and (cid:82) B ij dµ = n , where A ij := ( a i , a i +1 ) × ( a ij , a i ( j +1) ) and B ij :=( b i , b i +1 ) × ( b ij , b i ( j +1) ) . Since µ and µ have a massequal to and are supported in (0 , , then a n , b n (cid:54) and a i,n , b i,n (cid:54) for all i ∈ { , ..., n − } . We give inFigure 1 an example of such decomposition. x x a a a a ... ... a n − a n − a n a a a ... n · · ·· · · a i a i ... a ij a i ( j +1) ... /n a i ( n − a i +1 · · ·· · · a n − ... a n − ... a n Fig. 1. Example of a decomposition of µ . If one aims to define a vector field sending each A ij to B ij , then some shear stress is naturally introduced to theinterfaces of the cells. To overcome this problem, we firstdefine sets (cid:101) A ij ⊂⊂ A ij and (cid:101) B ij ⊂⊂ B ij for all i, j ∈{ , ..., n − } . We then send the mass of µ from each (cid:101) A ij to each (cid:101) B ij , while we do not control the mass contained in A ij \ (cid:101) A ij . More precisely, for all i, j ∈ { , ..., n − } , wedefine, a − i , a + i , a − ij , a + ij the smallest values such that (cid:82) ( a i ,a − i ) × ( a ij ,a i ( j +1) ) dµ = (cid:82) ( a + i ,a i +1 ) × ( a ij ,a i ( j +1) ) dµ = n and (cid:82) ( a − i ,a + i ) × ( a ij ,a − ij ) dµ = (cid:82) ( a − i ,a + i ) × ( a + ij ,a i ( j +1) ) dµ = n × (cid:0) n − n (cid:1) . We similarly define b + i , b − i , b + ij , b − ij . We finally define (cid:101) A ij := [ a − i , a + i ) × [ a − ij , a + ij ) and (cid:101) B ij := [ b − i , b + i ) × [ b − ij , b + ij ) . The goal is to build a solution to System (1) such that thecorresponding flow Φ ut satisfies Φ uT ( (cid:101) A ij ) = (cid:101) B ij , (8)for all i, j ∈ { , ..., n − } . We observe that we donot take into account the displacement of the mass con-tained in A ij \ (cid:101) A ij . We will show that the correspondingterm W ( (cid:80) ij Φ v + uT µ | A ij \ (cid:101) A ij , (cid:80) ij µ | B ij \ (cid:101) B ij ) tends to zerowhen n goes to the infinity. The rest of the proof is dividedinto two steps. In a first step, we build a flow and a velocityfield such that its flow satisfies (8). In a second step, wecompute the Wasserstein distance between µ and µ ( T ) showing that it converges to zero when n goes to infinity. Step 1:
We first build a flow satisfying (8). For all i ∈{ , ..., n − } , we denote by c − i and c + i the linear functionsequal to a − i and a + i at time t = 0 and equal to b − i and b + i at time t = T = 1 , respectively i.e. c − i ( t ) = ( b − i − a − i ) t + a − i and c + i ( t ) = ( b + i − a + i ) t + a + i . Similarly, for all i, j ∈ { , ..., n − } , we denote by c − ij and c + ij the linear functions equal to a − ij and a + ij at time t = 0 and equal to b − ij and b + ij at time t = T = 1 , respectively, i.e. c − ij ( t ) = ( b − ij − a − ij ) t + a − ij and c + ij ( t ) = ( b + ij − a + ij ) t + a + ij . Consider the application being the following linear com-bination of c − i , c + i and c − ij , c + ij in (cid:101) A ij , i.e. x ( x , t ) := a + i − x a + i − a − i c − i ( t ) + x − a − i a + i − a − i c + i ( t ) a + ij − x a + ij − a − ij c − ij ( t ) + x − a − ij a + ij − a − ij c + ij ( t ) , (9)when x ∈ (cid:101) A ij . Let us prove that an extension of theapplication ( x , t ) (cid:55)→ Φ ut ( x ) := x ( x , t ) is a flow associatedto a velocity field u . We remark that t (cid:55)→ x ( x , t ) is C andis solution to (cid:40) dx ( x ,t ) dt = α i ( t ) x ( x , t ) + β i ( t ) ∀ t ∈ [0 , T ] , dx ( x ,t ) dt = α ij ( t ) x ( x , t ) + β ij ( t ) ∀ t ∈ [0 , T ] , here for all t ∈ [0 , α i ( t ) = b + i − b − i + a − i − a + i c + i ( t ) − c − i ( t ) , β i ( t ) = a + i b i − a − i b + i c + i ( t ) − c − i ( t ) ,α ij ( t ) = b + ij − b − ij + a − ij − a + ij c + ij ( t ) − c − ij ( t ) , β ij ( t ) = a + ij b − ij − a − ij b + ij c + ij ( t ) − c − ij ( t ) . For all t ∈ [0 , , consider the set C ij ( t ) := [ c − i ( t ) , c + i ( t )) × [ c − ij ( t ) , c + ij ( t )) . We remark that C ij (0) = (cid:101) A ij and C ij ( T ) = (cid:101) B ij . On C ij := { ( x, t ) : t ∈ [0 , T ] , x ∈ C ij ( t ) } , we thendefine the velocity field u by u ( x, t ) = α i ( t ) x + β i ( t ) and u ( x, t ) = α ij ( t ) x + β ij ( t ) , for all ( x, t ) ∈ C ij ( x = ( x , x ) ). We extend u by a C ∞ and uniformly bounded function outside ∪ ij C ij , then having u ∈ U . Then, System (1) admits a unique solution and theflow on C ij is given by the expression (9). Step 2:
We now prove that the refinement of the gridprovides convergence to the target µ , i.e. W ( µ , µ ( T )) −→ n →∞ . (10)We remark that (cid:82) (cid:101) B ij dµ ( T ) = (cid:82) (cid:101) B ij dµ = ( n − n . Hence, by defining R := (0 , \ (cid:83) ij (cid:101) B ij , we also have (cid:82) R dµ ( T ) = (cid:82) R dµ = 1 − ( n − n . It comes that W ( µ , µ ( T )) (cid:54) n (cid:80) i,j =1 W ( µ × (cid:101) B ij , µ ( T ) × (cid:101) B ij )+ W ( µ × R , µ ( T ) × R ) . (11)We estimate each term in the right-hand side. Since we dealwith absolutely continuous measures, using Proposition 2,there exist measurable maps γ ij : R → R , for all i, j ∈{ , ..., n − } , and γ : R → R such that γ ij µ × (cid:101) B ij ) = µ ( T ) × (cid:101) B ij and γ µ × R ) = µ ( T ) × R . In the first term, for each i, j ∈ { , ..., n − } , observe that γ ij moves masses inside B ij only. Thus W ( µ × (cid:101) B ij , µ ( T ) × (cid:101) B ij ) = (cid:82) (cid:101) B ij | x − γ ij ( x ) | dµ ( x ) (cid:54) ( b + i − b − i + b + ij − b − ij ) ( n − n . (12)Concerning the second term in (11), observe that ¯ γ moves asmall mass in the bounded set ω . Thus it holds W ( µ × R , µ ( T ) × R ) (cid:54) (cid:82) R | x − γ ( x ) | dµ ( x ) (cid:54) √ (cid:16) − ( n − n (cid:17) = 4 √ n − n . (13)We thus have (10) by combining (11), (12) and (13).In the rest of the section, we remove the constraints supp( µ ) ⊂ ω and supp( µ ) ⊂ ω , now imposing Condition1. First of all, we give a consequence of Condition 1. Lemma 1
If Condition 1 is satisfied for µ , µ ∈ P c ( R d ) ,then the following Condition 2 is satisfied too: Condition 2
There exist two real numbers T ∗ , T ∗ > anda non-empty open set ω ⊂⊂ ω such that(i) For all x ∈ supp( µ ) , there exists t ∈ [0 , T ∗ ] suchthat Φ vt ( x ) ∈ ω . (ii) For all x ∈ supp( µ ) , there exists t ∈ [0 , T ∗ ] suchthat Φ v − t ( x ) ∈ ω . Proof:
We use an compactness argument. Let µ ∈P c ( R d ) and assume that Condition 1 holds. Let x ∈ supp( µ ) . Using Condition 1, there exists t ( x ) > such that Φ vt ( x ) ( x ) ∈ ω. Choose r ( x ) > such that B r ( x ) (Φ vt ( x ) ( x )) ⊂⊂ ω , that exists since ω is open. Bycontinuity of the application x (cid:55)→ Φ vt ( x ) ( x ) (see [1, Th.2.1.1]), there exists ˆ r ( x ) such that x ∈ B ˆ r ( x ) ( x ) ⇒ Φ vt ( x ) ( x ) ∈ B r ( x ) (Φ vt ( x ) ( x )) . Since µ is compactly supported, we can find a set { x , ..., x N } ⊂ supp( µ ) such that supp( µ ) ⊂ N (cid:91) i =1 B ˆ r ( x i ) ( x i ) . Thus the first item of Lemma 1 is satisfied for T ∗ := max { t ( x i ) }} and ω := N (cid:91) i =1 B r ( x i ) (Φ vt ( x i ) ( x i )) . The proof of the existence of T ∗ is similar.We now prove that we can store nearly the whole mass of µ in ω , under Condition 2. Proposition 3
Let µ ∈ P c ( R d ) satisfying the first item ofCondition 2. Then there exists ω u ∈ U such that supp( µ ( T ∗ )) ⊂ ω. (14) Proof:
Let k ∈ N ∗ . We denote by α := d ( ω, ω ) , ω := { x ∈ R d : d ( x , ω ) < α/ } and S k := { x ∈ R d : d ( x , ω ) < α/ k } . We define θ k a cutoff function on ω ofclass C ∞ satisfying (cid:54) θ k (cid:54) ,θ k = 1 in S ck ,θ k = 0 in ω . (15)Define u k := ( θ k − v. (16)We remark that the support of u k is included in ω . Let x ∈ supp( µ ) . Define t ∗ ( x ) := inf { t ∈ R + : Φ vt ( x ) ∈ ω } (cid:54) T ∗ . Consider the flow y := Φ vt ( x ) associated to x withoutcontrol, i.e. the solution to (cid:40) ˙ y ( t ) = v ( y ( t )) ,y (0) = x and the flow z k := Φ u k + vt ( x ) associated to x with thecontrol u k given in (16), i.e. the solution to (cid:40) ˙ z k ( t ) = ( v + u k )( z k ( t )) = θ k ( z k ( t )) × v ( z k ( t )) ,z k (0) = x . (17)e now prove that the range of z k for t ≥ is included inthe range of y for t ≥ . Consider the solution γ k to thefollowing system (cid:40) ˙ γ k ( t ) = θ k ( y ( γ k ( t ))) , t (cid:62) ,γ (0) = 0 . (18)Since θ k and y are Lipschitz, then System (18) admits asolution defined for all times. We remark that ξ k := y ◦ γ k is solution to System (17). Indeed for all t (cid:62) (cid:40) ˙ ξ k ( t ) = ˙ γ k ( t ) × ˙ y ( γ k ( t )) = θ k ( ξ k ( t )) × v ( ξ k ( t )) ,ξ k (0) = y ( γ k (0)) = y (0) . By uniqueness of the solution to System (17), we obtain y ( γ k ( t )) = z k ( t ) for all t (cid:62) . Using the fact that (cid:54) θ (cid:54) and the definition of γ k , wehave γ k increasing ,γ k ( t ) (cid:54) t ∀ t ∈ [0 , t ∗ ( x )] ,γ k ( t ) (cid:54) t ∗ ( x ) ∀ t (cid:62) t ∗ ( x ) . We deduce that, for all x ∈ supp( µ ) , { z k ( t ) : t (cid:62) } ⊂ { y ( s ) : s ∈ [0 , t ∗ ( x )] } . We now prove that for all k large enough, there exists t ∈ (0 , t ∗ ( x )) such that for all s > t , then Φ u k + vs ( x ) ∈ ω . Consider B := B α/ (Φ vt ∗ ( x )) ⊂ ω . By continu-ity, there exists β > such that Φ vt ( x ) ∈ B for all t ∈ ( t ∗ − β, t ∗ ) . For all s ∈ [0 , t ∗ − β ] , we can find r ( s ) > such that B r ( s ) ( φ vs ( x )) ⊂ ω c . By compactnessof { φ vs ( x ) : s ∈ [0 , t ∗ − β ] } , there exists a finite subcover { B r ( s i ) ( φ vs i ( x )) } (cid:54) i (cid:54) n of { φ vs ( x ) : s ∈ [0 , t ∗ − β ] } . Wedenote by R := min { r ( s i ) } . Let k be such that α/ k < R .Thus (cid:26) Φ v + u k s ( x ) = Φ vs ( x ) , for all s (cid:54) t ∗ − β, Φ v + u k s ( x ) ∈ B ⊂ ω , for all s > t ∗ − β. There exists a ball B r ( x ) , such that Φ v + u k s ( x ) ∈ ω forall x ∈ B r ( x ) and s > t ∗ − β . Thus, by compactnessof supp( µ ) , for k large enough, Φ v + u k T ∗ ( x ) ∈ ω for all x ∈ supp( µ ) .The third step of the proof is to restrict a measurecontained in ω to a measure contained in a hypercube S ⊂ ω . Proposition 4
Let µ ∈ P c ( R d ) satisfying supp( µ ) ⊂ ω. Define S an open hypercube strictly included in ω andchoose δ > . Then there exists ω u ∈ U such that thecorresponding solution to System (1) satisfies supp( µ ( δ )) ⊂ S. Proof:
From [5, Lemma 1.1, Chap. 1] and [3, Lemma2.68, Chap. 2], there exists a function η ∈ C ( ω ) satisfying κ (cid:54) |∇ η | (cid:54) κ in ω \ S, η > ω and η = 0 on ∂ω, with κ , κ > . We extend η by zero outside of ω . S ⊂⊂ ω k . We denote by u k := k ∇ η. Let x ∈ supp( µ ) . Consider the flow z k ( t ) = Φ v + u k t ( x ) associated to x , i.e. the solution to system (cid:40) ˙ z k ( t ) = v ( z ( t )) + u k ( z k ( t )) , t (cid:62) ,z k (0) = x . The properties of η imply that n ·∇ η < C < on ∂ω , where n represents that exterior normal vector to ∂ω . We deducethat, for k large enough, n · ( v + k ∇ η ) < on ∂ω . Thus z k ( t ) ∈ ω for all t (cid:62) .We now prove that there exists K ∈ N ∗ and T ∈ (0 , δ ) such that for all k > K and t ∈ [ T, δ ] , z k ( t ) ∈ S for all x ∈ supp( µ ) . By contradiction, assume that there existsthree sequences { k n } n ∈ N ∗ ⊂ N ∗ , { t n } n ∈ N ∗ ⊂ (0 , δ ) and { x n } n ∈ N ∗ ∈ supp( µ ) satisfying k n → ∞ , t n → δ and z k n ( x n , t n ) ∈ S c . (19)Consider the function f n defined for all t ∈ [0 , δ ] by f n ( t ) := η ( z k n ( t )) . Its time derivative is given by ˙ f n ( t ) = k n |∇ η ( z k n ( t )) | + v ( z k n ( t )) · ∇ η ( z k n ( t )) . Then, using (19) and the properties of η , it holds f n ( t n ) (cid:62) ( k n κ − (cid:107) v (cid:107) L ∞ ( ω ) κ ) t n , which is in contradiction for n large enough with f k n ( t n ) (cid:54) (cid:107) η (cid:107) ∞ . Thus we deduce that, for a K ∈ N ∗ and a T ∈ [0 , δ ] , Φ v + u k t ( x ) ∈ S for all x ∈ supp( µ ) , t ∈ ( T, δ ) and k > K .We now have all the tools to prove Theorem 1. The ideais the following: we first send µ inside ω with a control u , then from ω to an hypercube S with a control u . Onthe other side, we send µ inside ω backward in time with acontrol u , then from ω to S with a control u . When boththe source and the target are in S , we send one to the otherwith a control u . Proof of Theorem 1:
Consider µ , µ satisfying Condition1. Then, by Lemma 1, there exist T ∗ , T ∗ for which µ , µ satisfy Condition 2. Define T := T ∗ + T ∗ + δ with δ > .Choose ε > and denote by T := T ∗ , T := T ∗ + δ/ , T := T ∗ and T := T ∗ + δ/ . Using Propositions 3 and 4,there exists some controls u , u , u , u ∈ U and a square S ⊂ ω such that the solutions to ∂ t ρ + ∇ · (( v + ω u ) ρ ) = 0 in R d × [0 , T ] ,∂ t ρ + ∇ · (( v + ω u ) ρ ) = 0 in R d × [ T , T ] ,ρ (0) = µ in R d (20)and ∂ t ρ + ∇ · (( v + ω u ) ρ ) = 0 in R d × [ − T , ,∂ t ρ + ∇ · (( v + ω u ) ρ ) = 0 in R d × [ − T , − T ] ,ρ (0) = µ in R d , (21)satisfy supp( u i ) ⊂ ω , ρ ( T )( S ) > − ε and ρ ( − T )( S ) > − ε. We now apply Proposition 2 to approximately steer ρ ( T ) to ρ ( − T ) inside S : this gives a control u on the timeinterval [0 , δ ] . Thus, concatenating u , u , u , u , u on thetime interval [0 , T ] , we approximately steer µ to µ .V. P ROOF OF T HEOREM µ in ω and to send it out with a rateadapted to approximate µ .Let T be the infimum satisfying Condition (2), and fix s > . We now prove that System (1) is approximatelycontrollable at time T := T + s . Consider N ∈ N ∗ , τ := T /N , δ < τ , ξ := τ − δ and τ i := i × τ . Define (cid:26) A i := { x ∈ supp( µ ) : t ( x ) ∈ [0 , τ i ) } ,B i := { x ∈ supp( µ ) : T − t ( x ) ∈ [ τ i , τ i +1 ) } , where (cid:26) t ( x ) := inf { t ∈ R + : Φ vt ( x ) ∈ ω } ,t ( x ) := inf { t ∈ R + : Φ v − t ( x ) ∈ ω } . We remark that µ × A i represents the mass of µ which hasentered ω at time τ i and µ × B i the mass of µ which needto exit ω in the time interval ( τ i , τ i +1 ) . Then, by hypothesisof the Theorem, there exists K such that (Φ v + u K τ i µ × A i ))( ω ) (cid:62) − (Φ v + u K τ i − T µ × A i ))( ω ) − ε. The function u K can be then used to store the mass of µ in ω . The meaning of the previous equation is that the storedmass is sufficient to fill the required mass for µ .We now define the control achieving approximate control-lability at time T as follows: First of all, using the samestrategy as in the Proof of Theorem 1, we can send a part of φ v + u K s − ξ µ × A ) approximately to φ v + u K − T ∗ µ × B ) during the time interval ( s − ξ, s ) . More precisely, we replace T ∗ and T ∗ by s − ξ and ξ in the proof of Theorem 1. Thus,we send the mass of µ contained in A near to the massof µ contained in B . We repeat this process on each timeinterval ( τ i , τ i +1 ) for A i to B i . Thus, the mass of µ isglobally sent close to the mass of µ in time T .V. E XAMPLE OF MINIMAL TIME PROBLEM
In this section, we give explicit controls realizing theapproximate minimal time in one simple example. Theinterest of such example is to show that the minimal time canbe realized by non-Lipschitz controls, that are unfeasible.We study an example on the real line. We consider aconstant initial data µ = [0 , and a constant uncontrolledvector field v = 1 . The control set is ω = [2 , . Our firsttarget is the measure µ = [4 , . We now consider thefollowing control strategy: u ( t, x ) = t ∈ [0 ) ∩ [ , ] ,ψ (2 + ( t − ) , + 2( t − )) t ∈ [ , ) ,ψ (2 + ( t − ) , + 2( t − )) t ∈ [ , ,ψ (2 + ( t − , + 2( t − t ∈ [2 , ) , (22)where ψ ( a, b ) is defined as follows: ψ ( a, b )( x ) = (cid:40) x − ab − a x ∈ [ a, b ] , x (cid:54)∈ [ a, b ] . (23)The choice of ψ ( a, b ) given above has the following mean-ing: the vector field ψ ( a, b ) is linearly increasing on the in-terval, thus an initial measure with constant density k [ α ,β ] with a ≤ α ≤ β ≤ b will be transformed to a measure with constant density, supported in [ α ( t ) , β ( t )] , where α ( t ) is theunique solution of the ODE (cid:40) ˙ x = v + u,x (0) = α , and similarly for β ( t ) . As a consequence, we can easilydescribe the solution µ ( t ) of (1) with control (22) andinitial data µ . For simplicity, we only describe the measureevolution and the vector field on the time interval [1 , ] inFigure 2. One can observe that the linearly increasing time-varying control allows to rarefy the mass.Two remarks are crucial: • The vector field v + ψ ( a, b ) is not Lipschitz, since it isdiscontinuous. Thus, one needs to regularize such vectorfield with a Lipschitz mollificator. As a consequence,the final state does not coincide with µ , but it can bechosen arbitrarily close to it; • The strategy presented here cuts the measure in threeslices of mass , and rarefying each of them separately.Its total time is . One can apply the same strategywith a larger number n of slices, and rarefying the massin [2 , n ] by choosing the control ψ (2 + t, n +2 t ) . With this method, one can reduce the total time to n , then being approximately close to the minimaltime T = 4 given by Theorem 2. Fig. 2. Blue: density of the measure. Red: control vector field.
VI. C
ONCLUSION
In this article, we studied the control of a transportequation, where the control is a Lipschitz vector field in afixed set ω . We proved that approximate controllability canbe achieved under reasonable geometric conditions for theuncontrolled systems. We also proved a result of minimaltime control from one configuration to another. Future re-search directions include the study of more general transportequations, namely when the uncontrolled dynamics presentsinteraction terms, such as in models for crowds and opiniondynamics. EFERENCES[1] A. Bressan and B. Piccoli.
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