Mean-Field Sparse Jurdjevic–Quinn Control Marco Caponigro∗, Benedetto Piccoli† Francesco Rossi‡ Emmanuel Tr´elat§ January 6, 2017 7 1 Abstract 0 Weconsidernonlineartransportequationswithnon-localvelocity,describingthetime-evolutionofa 2 measure,whichinpracticemayrepresentthedensityofacrowd. Suchequationsoftenappearbytaking n themean-fieldlimitoffinite-dimensionalsystemsmodellingcollectivedynamics. Wefirstgiveasenseto a dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending J on the measure. 5 Then, we address the problem of controlling such equations by means of a time-varying bounded controlactionlocalizedonatime-varyingcontrolsubsetwithboundedLebesguemeasure(sparsityspace ] C constraint). Finite-dimensional versions are given by control-affine systems, which can be stabilized by the well known Jurdjevic–Quinn procedure. O Inthispaper,assumingthattheuncontrolleddynamicsaredissipative,wedevelopanapproachinthe h. spiritoftheclassicalJurdjevic–Quinntheorem,showinghowtosteerthesystemtoaninvariantsublevel t oftheLyapunovfunction. ThecontrolfunctionandthecontroldomainaredesignedintermsoftheLie a derivativesoftheLyapunovfunction,andenjoysparsitypropertiesinthesensethatthecontrolsupport m is small. [ Finally, we show that our result applies to a large class of kinetic equations modelling multi-agent dynamics. 1 v 6 1 Introduction and main result 1 3 1 1.1 The context 0 . In recent years, the study of collective behavior of a crowd of autonomous agents has drawn a great interest 1 0 fromscientificcommunities,e.g.,incivilengineering(forevacuationproblems[16,26]),robotics(coordination 7 of robots [9, 24, 29, 33]), computer science and sociology (social networks [25]), and biology (animals groups 1 [5, 14, 20]). In particular, it is well known that some simple rules of interaction between agents can promote v: formation of special patterns, like lines in ants formations and migrating lobsters, or V-shaped formation in i migrating birds. This phenomenon is often referred to as self-organization. X Beside the problem of analyzing the collective behavior of a “closed” system [15], it is interesting to r understand what changes of behavior can be induced by an external agent (e.g., a policy maker) to the a crowd. For example, one can try to enforce creation of patterns when they are not formed naturally, or break the formation of such patterns [11, 12, 19, 32, 27]. This is the problem of control of crowds, that we address in this article in a specific case. From the mathematical point of view, problems related to models of crowds are of great interest. From the analysis point of view, one needs to pass from a big set of simple rules for each individual to a model capable of capturing the dynamics of the whole crowd. This can be solved via the so-called mean-field ∗Conservatoire National des Arts et M´etiers, E´quipe M2N, 292 rue Saint-Martin, 75003, Paris, France ([email protected]). †DepartmentofMathematicalSciencesandCenterforComputationalandIntegrativeBiology,RutgersUniversity,Camden, NJ08102,USA([email protected]). ‡AixMarseilleUniversit´e,CNRS,ENSAM,Universit´edeToulon,LSIS,Marseille,France([email protected].) §Sorbonne Universit´es, UPMC Univ. Paris 6, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005, Paris,France([email protected]). 1 process, that permits to consider the limit of a set of ordinary differential equations (one for each agent) to a partial differential equation (PDE in the following) for the whole crowd [32]. The resulting equation is a transport equation with non-local velocity, of the form ∂ µ+∇·(f[µ]µ)=0, (1) t where µ is the measure representing the density of agents, ∇· is the divergence operator and f[µ] is a vector fielddependingonthemeasure,takingintoaccountinteractionsbetweenagents. Manykineticequationsare of this form. We recall fundamental properties of such classes of equations in Section 2. We just highlight here that we assume in the following that f[µ] is a bounded Lipschitz vector field for any µ, Lipschitz with respect to the Wasserstein distance W , p ∈ [1,+∞), as a function of µ. This ensures existence and p uniqueness of the solution of the associated Cauchy problem [3, 31]. Since such an equation generates a semigroup, we will use the notation etfµ to denote the unique solution of (1) at time t with initial data µ . 0 0 We recall existence results for (1) in Section 2. 1.2 Control of transport equations with non-local velocity To control systems described by (1), we assume to act only on a small part of the crowd. Since agents are indistinguishablewhenoneonlyknowsµ(t)attimet,controlscanonlybestate-dependentandcannotfocus on specific agents. For this reason, we model the control action by means of a vector field g, and a control gain u(t,x) localized in a small control set ω(t) (itself depending on time), modeling our choice of the gain on the vector field. We choose the control gain u and the control set ω, both of them varying in time while the vector field g is fixed, and it depends on the density µ. The resulting control system is given by ∂ µ+∇·((f[µ]+χ ug[µ])µ)=0. (2) t ω Here, the function χ is the indicator function of ω, defined almost verywhere by χ (x) = 1 if x ∈ ω and ω ω χ (x)=0 otherwise. ω We focus on the following continuous sparse space constraint: we assume to act only on a small portion of the configuration space, with finite strength. Accordingly, we assume the following constraints. Here, given a measurable subset ω of Rd, we denote by |ω| its Lebesgue measure. Control Constraints (U) Fix c>0. For each time t≥0, we have: Sparsity space constraint: |ω(t)|≤c, (3) Finite strength: (cid:107)u(t,·)(cid:107) ≤1. (4) L∞ Control sparsity constraints have been first introduced in [11, 12], for a population with a finite number of agents. The sparsity space constraint was considered in [32]. In the mean-field approach, this is actually the most natural sparsity constraint when one wants to use space-dependent vector fields and to act on “small sets”. A sparsity population constraint was also considered in [32]. For the sparsity space constraint, one can easily deal both with measures that are absolutely continuous with respect to the Lebesgue measure, and with measures containing singular (Dirac) parts. We denote by P (Rd) the space of probability measures on Rd with compact support and by Pac(Rd) the subspace of c c probability measures on Rd that are absolutely continuous with respect to the Lebesgue measure. In the following, we will define a control strategy that satisfies the following property: if the initial data, at time 0, is absolutely continuous with respect to the Lebesgue measure, then it remains absolutely continuous for any positive time. This does not prevent µ(t) of converging to some Dirac mass as t→+∞, as this is the case for consensus problems for crowd models. In this paper, our objective is to generalize the Jurdjevic–Quinn stabilization method [28] to mean-field controlled equations, under the sparsity constraint (U) described above. Following the Jurdjevic–Quinn approach, we assume to have available a Lyapunov function V for which: 2 • the uncontrolled dynamics f[µ] gives no increase of V; • the control ug[µ] allows one to increase-decrease V, except for some specific configurations of the population µ in the subset Z of the set of measures with compact support P (Rd) defined as the set c on which the Lie derivatives of V vanish (see the precise definition in (8)). We will then define a sparse control strategy, steering the population exactly to the set Z, in complete analogy with the standard finite-dimensional Jurdjevic–Quinn method. The vector fields f and ug defined on the space Rd play the role of derivatives for the function V[µ], in the following sense: the vector field f induces an infinitesimal change in V that can be estimated as the derivative lim V[etfµ]−V[µ]. This limit needs to be well-defined, and this is why we will assume from now t→0 t on the following regularity assumptions on f: for any µ ∈ P (Rd), the function t (cid:55)→ V[etfµ] is of class C2. c Then, in analogy with the definition of a Lie derivative in finite dimension, we give the following definition. Definition 1.1. We define the Lie derivative of V along f as the limit V[etfµ]−V[µ] L V[µ]= lim . (5) f t→0 t Requiring the non-increase of V along the flow of f is equivalent to require dissipativity for the system. Definition1.2. Wesaythatthesystem (2)isdissipativeifthereexistsaLyapunovfunctionV :P (Rd)→R c such that t(cid:55)→V[etfµ] is of class C2 and L V[µ]≤0 ∀µ∈P (Rd). (6) f c Regularity conditions need to be satisfied as well for the controlled vector field ug. For this reason, we define the space U of admissible control functions u, and from now on, we assume that U = Lip(Rd,R). In what follows, we impose the following regularity assumption: for all µ ∈ P (Rd) and u ∈ U, the function c t(cid:55)→V[etugµ] is of class C2. As a consequence, the limit V[etugµ]−V[µ] L V[µ]= lim (7) ug t→0 t is well defined. Remark 1. Accordingly, the notion of Lie derivative can be extended to piecewise constant functions t(cid:55)→u(t,·)∈U. Definition 1.3. Assume that V : P (Rd) → R is such that t (cid:55)→ V[etfµ] and t (cid:55)→ V[etugµ] are of class C2 c for all µ∈P (Rd) and u∈U. We define c Z =(cid:8)µ∈P (Rd) | L V[µ]=L V[µ]=0 ∀u∈Lip(Rd,R)(cid:9). (8) c f ug Note that the definition of Lie derivative implies the multiplicative property L V[µ]=kL V[µ] and L V[µ]=kL V[µ]. (9) kf f kug ug ThiscontinuityconditionalsoimpliesadditivityofLiederivatives. Indeed,onecaneasilyseethatet(u+u(cid:48))g = etug+o(t)etu(cid:48)g, which in turn implies that L V[µ]=L V[µ]+L V[µ]. (u+u(cid:48))g ug u(cid:48)g While conditions (5)-(7) are equivalent to differentiability of V along f and ug, we also need a kind of differentiability for V along directions of the dynamics. By the additivity property, we can state it as follows: there exists K >0 such that, for all µ∈P (Rd) and u∈U, we have c |L V[µ]|≤K(cid:107)u(cid:107) . (10) ug L1(µ) 3 This yields a metric for the space of controls u similar to the zero-order metric in a more general sub- Riemannian structure for metrics on the space of diffeomorphisms on a manifold [4] (see also [1]). The main difference here is that we choose the L1 norm weighted with respect to the measure µ(t), and not with respect to the Lebesgue measure. While conditions (5)-(7)-(10) hold for a fixed µ ∈ P (Rd), we also require the continuity of the Lie c derivatives of first and second order. We require that lim L V[µi]=L V[µ], lim L2V[µi]=L2V[µ], f f f f i→+∞ i→+∞ lim LugV[µi]=LugV[µ], lim L2ugV[µi]=L2ugV[µ], (11) i→+∞ i→+∞ lim L L V[µi]=L L V[µ], f ug f ug i→+∞ for all µi and µ ∈ P (Rd), with µi (cid:42) µ (weak convergence of measures), i.e., lim (cid:82) φdµi = (cid:82) φdµ c i→+∞ for every φ ∈ C0(Rd,R). The conditions (11) imply in particular that L V[µ] exists and satisfies c f+ug L V[µ]=L V[µ]+L V[µ], i.e., that additivity holds also for the vector field f +ug. f+ug f ug Remark 2. Clearly, the choice of the set of admissible controls U has an impact on the set of admissible functionals V for which (7)-(10)-(11) are satisfied. We choose here the set of Lipschitz functions because existence is then ensured for (2) (see [3, 31]). Note that reducing the space of admissible controls to some proper subset of Lip(Rd,R) (such as C∞(Rd,R)) may enlarge the set of functionals V for which the regularity conditions (5)-(7)-(10)-(11) are c satisfied. In Section 3, we will enforce the decrease of the functional V by a steepest descent method on the space U, by (approximately) solving an optimization problem in the space of Lipschitz functions. FollowingtheclassicalLyapunovtheoryforfinite-dimensionalsystems,weneedtoimposesomeconditions ensuring compactness of trajectories. In finite dimension, this is often imposed by requiring V to be proper, i.e., lim V(x) = +∞, hence the fact that dV(x(t)) ≤ 0 implies compactness. In the present mean- |x|→+∞ dt field setting, instead, such a condition cannot be imposed by a simple evaluation of the function V: in the case where V is the variance of the measure, one can have measures µ with arbitrarily small variance and arbitrarily large support. For this reason, we impose compactness of trajectories by assuming that the dynamics of the system, i.e., the vector fields f and g, have a compact support. More precisely, we assume the existence of a ball B(0,R) such that for all µ∈P (B(0,R)), f[µ](x)=g[µ](x)=0 ∀x(cid:54)∈B(0,R). (12) c Note that this condition implies that µ(t) ∈ P (B(0,R)) for any t ≥ 0. Then, all assumptions described c above will be assumed to hold for measures in P (B(0,R)) only. c Remark 3. Theresultsofthispapercanbestatedfortransportequationswithnon-localterms(2)defined on bounded manifolds without boundary, or on bounded manifold with no-flux boundary condition. Summing up, we make the following assumptions on the system (2) and on the Lyapunov functional V: 4 Assumptions (H) The vector fields f,g :P (Rd)→Lip(Rd,Rd) and the Lyapunov function V :P (Rd)→R satisfy Assump- c c tions (H) if: • there exists R>0 such that f,g satisfy the compact support property (12); • there exist L>0, Q>0 and p≥1 such that, for all µ,ν ∈P (B(0,R)) and for all x,y ∈Rd, c |f[µ](x)−f[µ](y)|≤L|x−y|, |g[µ](x)−g[µ](y)|≤L|x−y|, (13) |f[µ](x)−f[ν](x)|≤QW (µ,ν), |g[µ](x)−g[ν](x)|≤QW (µ,ν); p p • the functions t (cid:55)→ V[etfµ] and t (cid:55)→ V[etugµ] are of class C1 for all µ ∈ P (B(0,R)) and u ∈ U; in c particular, the Lie derivatives (5) and (7) exist; • the uncontrolled system is dissipative, i.e., L V[µ]≤0 for any µ∈P (B(0,R)); f c • the Lie derivative (7) satisfies the Lipschitz condition (10) and the continuity condition (11) for any µ∈P (B(0,R)). c Remark 4. TheuniformLipschitzpropertyoff andg in(13)andtheuniformcompactnessoftheirsupport in(12)implythatthereexistsM >0suchthat(cid:107)f[µ](cid:107) ≤M and(cid:107)g[µ](cid:107) ≤M foranyµ∈P (B(0,R)). L∞ L∞ c These facts imply existence and uniqueness of the solution of the Cauchy problem for (2) (see, e.g., [3, 31]). We recall existence results in Section 2. 1.3 The main result The main idea of our control strategy is to choose the controller to make V decrease along trajectories. We will do this choice with a steepest descent method, similarly to the finite-dimensional approach described in [11, 12, 13]. Since the space of admissible controls χ ug[µ] is infinite dimensional, we restrict ourselves to a finite- ω dimensional set by imposing the following structure. Consider the class of Lipschitz mollified indicator functions χη :R→R, defined by [a,b]  1 for x∈[a,b], 0 for x(cid:54)∈[a−η,b+η], χη (x)= [a,b] x−a+η for x∈[a−η,a]; −xη+b+η for x∈[b,b+η], η and then, consider the d-dimensional version of such functions. Given a = (a1,...,ad),b = (b1,...,bd) and x=(x1,...,xd) in Rd, we define χη (x)= min χη (xi). (14) [a,b] [ai,bi] i=1,...,d Now, for any choice of the three parameters (a,b,η), we take ω = ω(a,b,η) as the multi-interval [a1 − η,b1 +η]×[a2 −η,b2 +η]×...×[ad −η,bd +η]. Then we reduce the choice of the sparse control in an infinite-dimensional space of controls to the choice of three parameters (a,b,η). In what follows, we set U(a,b,η)=χ χη . (15) [a1−η,b1+η]×[a2−η,b2+η]×···×[ad−η,bd+η] [a,b] We then define the “slope function” by s (a,b,η)=|L V[µ(t)]|, t U(a,b,η)g[µ(t)] 5 which describes the instantaneous variation of V in µ(t) as a consequence of the action of the control U(a,b,η). Note that (10) and the fact that the function t(cid:55)→V[etugµ] is of class C1 imply the continuity of the slope function with respect to its arguments (t,a,b,η). Wethenapplyasteepestdescentmethodbychoosingthecontrolcorrespondingtooneofthemaximizers1 (a∗,b∗,η∗) of s in the space t Ω =(cid:8)(a,b,η) | |ω(a,b,η)|≤c and η ≥t−1(cid:9). (16) t Thecondition|ω(a,b,η)|≤cin(16)ensuresthatthespaceconstraint(3)issatisfied. WewillseeinLemma4 that the condition η ≥ t−1 implies that the control function is uniformly Lipschitz for any bounded time interval [0,θ], thus ensuring that µ(t) remains absolutely continuous with respect to the Lebesgue measure. At the same time, when t → +∞, this constraint allows to consider controls with an arbitrarily large Lipschitz constant, since Lip(χη ) = 1. This Lipschitz constraint is somehow unavoidable if one wants [a,b] η to ensure regularity of the measure µ(t) within finite time; otherwise, the steepest descent method might eithergenerateanon-Lipschitzvectorfield(forwhichexistencefor(2)holdsforsmalltimesonly)oratime- varying Lipschitz vector field converging to a non-Lipschitz vector field within finite time (see an example for a problem of crowd dynamics in Section 3). Choosing the control as the instantaneous maximizer of s may cause chattering (in time) phenomena, t as it has been already noticed in finite dimension (see [13]). For this reason, we regularize the control by means of an hysteresis: we introduce a parameter h∈(0,1) and, given the control U(a∗,b∗,η∗), maximizer ofs attimet ,wekeepitconstantoveraninterval[t ,t +δ]alongwhich(s (a∗,b∗,η∗)≥(1−h)s (a,b,η). t n n n t t Summing up, the combination of a steepest descent method with an hysteresis provides a control making V decrease and steering the density µ(t) to Z. Our main result is the following. Theorem 1 (Main Theorem). Let f,g : P (Rd) → Lip(Rd,Rd) and V : P (Rd) → R satisfy Assumptions c c (H) for some R>0. Consider the controlled transport equation with non-local velocity (cid:0) (cid:1) ∂ µ+∇· (f[µ]+χ u(t,y)g[µ])µ =0, µ(0)=µ , (17) t ω(t,y) 0 where µ ∈ Pac(Rd) is such that supp(µ ) ⊂ B(0,R). Fix the hysteresis parameter h ∈ (0,1). Fix the 0 c 0 following initial parameters n=0 and t =0. Define the following algorithm step. 0 1The method to select a maximizer plays no role in the convergence of the method. One may consider for instance the lexicographicorderinR2n+1,andchoosethesmallestmaximizer. 6 Step n At time t , choose one of the maximizers (a∗,b∗,η∗) of s (a,b,η) in the set Ω defined in (16). n tn tn Then, we have two cases: • If either s (a∗,b∗,η∗)<t−1 or Ω is empty, then choose the zero control tn n tn χ u(t,x)≡0, (18) ω(t) (thus, ω(t) needs not be defined) and let the measure µ(t), starting at µ(t ), evolve according to (17) n over the time interval [t ,t ], where t is the smallest time greater than t for which there exists n n+1 n+1 n (a¯,¯b,η¯)∈Ω(cid:48) such that s (a¯,¯b,η¯)≥2t−1 , where t tn+1 n+1 Ω(cid:48) =(cid:8)(a,b,η)∈Ω | η ≥2t−1(cid:9). (19) t t • If s (a∗,b∗,η∗)≥t−1, then choose the control defined by tn n ω(t)=[a∗1−η∗,b∗1+η∗]×[a∗2−η∗,b∗2+η∗]×···×[a∗d−η∗,b∗d+η∗], (20) u(t,·)=−χη∗ sign(L V[µ(t)]), [a∗,b∗] U(a,b,η)g[µ(t)] where U is given in (15), and let the measure µ(t), starting at µ(t ), evolve according to (17) over n the time interval [t ,t ], where t is the smallest time greater than t satisfying at least one of n n+1 n+1 n the following conditions: – either s (a∗,b∗,η∗)≤ t−n+11; tn+1 2 – or there exists (a¯,¯b,η¯)∈Ω(cid:48) such that tn+1 s (a∗,b∗,η∗)≤(1−h)s (a¯,¯b,η¯). (21) tn+1 tn+1 If t is finite, then go to Step (n+1). n+1 If t =+∞, then keep the control (18) or (20) over the time interval [t ,+∞). n+1 n For this control strategy, the control χ u satisfies the control constraint (U), the unique solution µ(t) of ω (17) is such that µ(t)∈Pac(Rd) for any t∈[0,+∞), and µ(t) converges to Z ∩P (B(0,R)), i.e. c c • lim inf W (µ(t),ν)=0, t→+∞ ν∈Z∩Pc(B(0,R)) p • or equivalently, there exists a choice ν(t)∈Z ∩P (B(0,R)) for each t≥0 such that µ(t)(cid:42)ν(t), i.e. c for all φ∈C∞(Rd) it holds lim (cid:82) φd(µ(t)−ν(t))=0. c t→∞ Remark 5. The three threshold time-dependent functions used in the definition of control algorithm in Theorem 1 satisfy t−1/2 < t−1 < 2t−1. One can easily see that they can be replaced with three positive functions satisfying φ (t)<φ (t)<φ (t) converging to 0 as t→+∞. In particular, the functions can take 1 2 3 finite values for t=0, by maybe allowing one control to be active on the starting interval [0,t ]. 1 The rest of the paper is structured as follows. In Section 2, we recall the main definitions and results for transportPDEswithnon-localvelocitiesas(1)and(2). InSection3,wediscusssomeexamplesofdynamics oftheform(2),andweshowexplainsomedifferenceswithrespecttothefinitedimensionalsetting. Theorem 1isprovedinSection4. InSection5,westudyageneralizationoftheTheorem1toasystemoftheform(2) with several control potentials. Finally, in Section 6, we present an application of Theorem 1 to the control of kinetic multi-agent systems. 7 2 Transport equations with non-local velocities In this section, we recall existence and uniqueness results for (1) and (2). In (1), the variable µ∈P (Rd) is c a probability measure on Rd. The term f[µ] is called the velocity field and it is a non-local term. Since the value of a measure at a single point is not well defined, it is important to observe that f[µ] is not a function depending on the value of µ in a given point, as it is often the case in the setting of hyperbolic equations in whichf[µ](x)=f(µ(x)). Instead,onehastoconsiderf asanoperatortakinganasinputthewholemeasure µ and giving as an output a global vector field f[µ] on the whole space Rd. These operators are often called “non-local”, as they consider the density not only at a given point, but in a whole neighbourhood. We first recall two useful definitions to deal with measures and solutions of (1), namely the Wasserstein distance and the push-forward of measures (for more details see, e.g., [35]). Definition 2.1. Given two probability measures µ and ν on Rd and p∈[1,+∞), the p-Wasserstein distance between µ and ν is (cid:26)(cid:90) (cid:27)1/p W (µ,ν)=inf |x−y|pdπ(x,y)|π ∈Π(µ,ν) , p R2d where Π(µ,ν) is the set of transference plans from µ to ν, i.e., of the probability measures π on Rd×Rd such that Proj #π =µ and Proj #π =ν with Proj :(x,y)(cid:55)→x and Proj :(x,y)(cid:55)→y. x y x y ThetopologyinducedbyW onthespaceofprobabilitymeasuresP(X)onacompactspaceX coincides p with the weak-∗ topology of measures (see [35, Theorem 7.12]). As a consequence of condition (12), each trajectory µ(t) of the controlled system (2) is contained in the compact space P (B(0,R)) (compact if c endowedwiththeWassersteintopology). Thus, from now on, we will state equivalently convergence with respect to the weak-∗ topology of measures and with respect to the Wasserstein distance. We now define the push-forward of measures. Definition 2.2. Given a Borel map γ : Rd → Rd, the pushforward of a measure µ ∈ P (Rd) is defined by c γ#µ(A)=µ(γ−1(A)) for every measurable subset A of Rd. We now recall an existence and uniqueness result for (1) (see a complete proof in [32]). Theorem 2. We assume that, for every µ ∈ P (Rd), the velocity field f[µ] is a function of (t,x) with the c regularity f[·]:P(Rd) −→ Lip(Rd)∩L∞(Rd) µ (cid:55)−→ f[µ] satisfying the following assumptions: • there exist functions L(·) and M(·) in L∞(R) such that loc (cid:107)f[µ](t,x)−f[µ](t,y)(cid:107)≤L(t)(cid:107)x−y(cid:107), (cid:107)f[µ](t,x)(cid:107)≤M(t)(1+(cid:107)x(cid:107)), for every µ∈P (Rd), every t∈R and all (x,y)∈Rd×Rd; c • for a given p∈[1,+∞), there exists a function K(·) in L∞(R) such that loc (cid:107)f[µ]−f[ν](cid:107) ≤K(t)W (µ,ν), L∞(R;C0(Rd)) p for all (µ,ν)∈(P (Rd))2. c Then, for every µ0 ∈P (Rd), the Cauchy problem c ∂ µ+∇·(f[µ]µ)=0, µ =µ , (22) t |t=0 0 has a unique solution µ(·)∈C0(R;P (Rd)), where P (Rd) is endowed with the weak-∗ topology of measures. c c Moreover, t (cid:55)→ µ(t) is Lipschitz in the sense of the Wasserstein distance W . Moreover, if µ0 ∈ Pac(Rd), p c then µ(t)∈Pac(Rd) for every t∈R. c 8 Furthermore, for every T >0, there exists C >0 such that T W (µ(t),ν(t))≤eCTtW (µ(0),ν(0)), (23) p p for all solutions µ and ν of (22) in C0([0,T];P (Rd)). c Finally, the solution µ of the Cauchy problem (22) can be made explicit as follows. Let Φ(t) be the flow of diffeomorphims of Rd generated by the time-dependent vector field f[µ], defined as the unique solution of the Cauchy problem Φ˙(t)=f[µ(t)]◦Φ(t), Φ(0)=Id , or in other words, Rd ∂ Φ(t,x)=f[µ(t)](t,Φ(t,x)), Φ(0,x)=x. t Then, we have µ(t)=Φ(t)#µ , 0 that is, µ(t) is the push-forward of µ under Φ(t). 0 Theorem 2 can be generalized to mass-varying transport PDEs, that is, in presence of sources (see [30]). We now observe that Theorem 2 can be applied to (2) as well, under Assumptions (H) and provided that the control u be a Lipschitz function of the space variable for all times. Corollary 1. Under Assumptions (H), if u is a uniformly Lipschitz function of x on the time interval [0,θ], satisfying the constraint (U), then, given any initial data µ(0) = µ ∈ P (B(0,R)), the equation (2) has a 0 c unique solution µ(·)∈C0([0,θ],P (B(0,R))). Moreover, if µ ∈Pac(B(0,R)), then µ(t)∈Pac(B(0,R)) for c 0 c c every t∈[0,θ]. Denoting by Ψ the flow of diffeomorphims of Rd generated by the time-dependent vector field f[µ]+u(t,x)g[µ], we have µ(t)=Ψ(t)#µ . 0 Proof. It suffices to check that Theorem 2 can be applied to the vector field f[µ]+u(t,x)g[µ]. As already stated,theexistenceofauniformboundM for(cid:107)f+ug(cid:107) isaconsequenceoftheuniformLipschitzproperty L∞ and of the uniform boundedness of the support of both f[µ] and g[µ], together with the bound (cid:107)u(cid:107) ≤ 1 L∞ imposed by (U) in (4). Similarly, we have a uniform bound on the Lipschitz constant Lip(f +ug). Indeed, by (3), Lip(f[µ(t)]+ug[µ(t)])≤L+Lip (u)(cid:107)g[µ(t)](cid:107) +(cid:107)u(cid:107) Lip(g[µ(t)]) x L∞ L∞ ≤2L+Lip (u)M. (24) x Finally, we have (cid:107)f[µ]+ug[µ]−f[ν]+ug[ν](cid:107) ≤(cid:107)f[µ]−f[ν](cid:107) +(cid:107)u(cid:107) (cid:107)g[µ]−g[ν](cid:107) L∞ L∞ L∞ L∞ ≤W (µ,ν)+1·QW (µ,ν) p p =2QW (µ,ν). p This proves the corollary. We end this section with an estimate of the L∞ norm of the solution µ(t) to (1), when it is absolutely continuous with respect to the Lebesgue measure. Proposition 1. Let µ(·) be the unique solution of (1) for a given Lipschitz vector field f with µ ∈Pac(Rd). 0 c Then d (cid:107)µ(t)(cid:107) ≤(cid:107)µ(t)(cid:107) (cid:107)∇·f(cid:107) . (25) dt L∞ L∞ L∞ Proof. The proof follows [22, Proposition 3.1]. Let ρ(t) be the density of µ(t) with respect to the Lebesgue measure. For each p∈[1,+∞), by dropping the dependence with respect to time, we write d (cid:90) ρpdx=−p(cid:90) ρp−1∇·(fρ)dx=−p(cid:90) (cid:0)ρp∇·f +ρp−1(cid:104)f,∇ρ(cid:105)(cid:1) dx. dt Since ∇·(fρp)=ρp∇·f +(cid:104)f,∇(ρp)(cid:105)=ρp∇·f +pρp−1(cid:104)f,∇ρ(cid:105), we infer that d (cid:90) (cid:90) (cid:90) ρpdx = − (p−1)ρp∇·fdx− ∇·(fρp)dx. dt The last term is zero as a consequence of the divergence theorem. Then d(cid:107)ρ(cid:107)p ≤(p−1)(cid:107)ρ(cid:107)p (cid:107)∇·f(cid:107) , dt Lp Lp L∞ which in turn implies (25) as p→+∞. 9 3 Steepest descent under population constraint induces mass con- centration In this section, we discuss a remarkable phenomenon for controlled equations of the form (2): starting from a measure µ ∈Pac(Rd), i.e., a measure that is absolutely continuous with respect to the Lebesgue measure 0 c in Rd, a time-dependent choice of the control might drive the measure outside Pac(Rd) in finite time, in c particular with emergence of Dirac deltas. In fact, we will show that such a phenomenon arises when trying to minimize a Lyapunov function V, in particular when one chooses the control u(t) as the instantaneous minimizer of the Lie derivative of V as time evolves. This example also shows that some key ideas coming from control of finite-dimensional systems cannot be extended straightforwardly to infinite dimension. In this section, we discuss the interest and the drawbacks of a control constraint different than (U), namely the following: Alternative control Constraints (U’) Fix c>0. For each time t≥0 it holds: (cid:90) Sparsity population constraint: dµ(t)≤c, (26) ω(t) Finite strength: (cid:107)u(t,.)(cid:107) ≤1. (27) L∞ The population constraint represents the idea of acting on a small part of the crowd itself, and not on a small part of the configuration space, as we require in the space constraint in (U). Even though the sparse population constraint is interesting from the theoretical point of view, it has a surprising drawback on the modelingpointofview: whenacrowdisextremelyconcentrated,theconstraint(U’)impliesthatthecontrol cannot act on the crowd anymore. This is somehow unnatural, since a crowd that is already concentrated is the best configuration to steer. On the other hand, the space constraint (U) permits to act on the whole crowd, when it is concentrated in a set of size c, i.e., exactly when it is concentrated. We now show that the population constraint also induces some formal mathematical problems when using a sparse Jurdjevic–Quinn approach. Consider the following system on the real line: ∂ µ+∂ (uµ)=0. (28) t x Thisis aparticular caseof (2)with f =0and g =1. We considerthe initialdata µ =χ , i.e., auniform 0 [0,1] probability density on the interval [0,1]. We consider the Lyapunov function (cid:90) V[µ]= x2dµ(x), i.e., the second moment with respect to zero. We have L V =0, and we have L V[µ]=0 for µ=δ only, f ug 0 i.e., Z ={δ }. Then, minimizing V is equivalent to steer µ(t) to the Dirac mass δ . 0 0 We now apply a rough form of the steepest descent method to the problem of minimizing V: given the initial measure µ , we look for a control function u that maximizes the descent L V[µ], while taking into 0 ug accountthecontrolconstraints(U’).Aneasycomputationshowsthatnooptimalchoiceforuexists. Indeed, for every ε>0, consider the C∞ function  −1 for x∈[1−c+ε,1],  u (x)= 0 for x∈(−∞,1−c]∪[1+ε,+∞), ε C∞−spline with values in [−1,0] for x∈[1−c,1−c+ε]∪[1,1+ε]. Then, for a sufficiently small time t > 0, each particle x ∈ (1−c+ε,1] is displaced to x−t while each particle x ∈ [0,1−c] undergoes no displacement. The particles in the small interval [1−c,1−c+ε] are 10