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A collisionless singular Cucker-Smale model with decentralized formation control

In this paper we address the design of decentralized feedback control laws inducing consensus and prescribed spatial patterns over a singular Cucker-Smale type model. The control design consists of a feedback term regulating the distance between each agent and pre-assigned neighbours. Such a design represents a multidimensional extension of existing control laws for 1d platoon formation control. For the proposed controller, we study consensus formation, collision-avoidance and formation control features in terms of energy estimates for the closed-loop system. Numerical experiments in 1, 2 and 3 dimensions assess the different features of the proposed design.

represented by a subset of states which are updated via "physical" interaction rules [5](attraction, repulsion, alignment, etc.), or by means of a control/game framework [6], [7].
In this work, we are concerned with the design of dynamic interactions and external control laws for nonlinear MAS representing the physical motion of a swarm of unmanned aerial vehicles (UAVs). Our mathematical modelling of the swarm is inspired by animal collective dynamics, where large populations of birds and fish normally exhibit self-organization behaviour such as flocking, swarming, milling, and alignment. In particular, we are interested in prescribing nonlinear dynamics for the swarm which can lead to self-organized flocking together with collision-avoidance features -which are fundamental in the case of UAVs-. The emergence of collisionless flocking behaviour, understood as a configuration in which agents travel with the same constant velocity reducing the swarm motion to a planar/spatial translation, that is already a complex dynamic equilibrium of interest on its own right. However, besides applications in modelling of animal behaviour, it remains of limited practical interest in the context of UAVs, as it does not provide information about the swarm spatial configuration to be translated. Hence, it is desirable to endow the MAS dynamics with additional forcing terms which can also induce the formation of a given spatial configuration. In this paper, we propose a dynamical MAS model including collisionless flocking, together with a control law inducing spatial formation. The control action we propose is inspired by the literature concerning string stability for autonomous vehicles, and consists of a decentralized feedback law which requires a reduced amount of communication between agents.
Starting from a Cucker-Smale type model, we study collision avoidance in the framework of [29], [30], [31], [32], where singular interaction kernels have been proposed. Such interaction kernels blow-up whenever two agents are located at the same position, providing an adequate framework for the study of collision-less control laws. Finally, in order to induce spatial configurations over the swarm, we propose the use of a decentralized controller [33], [34], similar to those who have been derived in [35] in the context of 1D vehicle platoons avoiding string instability [36].
We extend some of these ideas to 2D and 3D formation control.
In applications, decentralized control schemes have a clear economical benefit, but they are not necessarily sacrificing performance, when compared to more complex solutions [4]. Additionally, in higher dimensions, self organizing agents and their control take relevance if the extra states are interpreted as a feature of the agent that is prone to be measured and/or used for synchronization [37].
The control of MAS, and in particular drone and/or robot swarms is an intensive research topic (see the recent survey [38]), with applications ranging from spacecraft formation [39], [21], robot self-organization [40], surveillance [41] and localization [42], to digital media arts [43]. Along this line, in Section VII we present different numerical simulation for planar and spatial formations, mimicking the display of UAV swarms forming the Olympic Rings symbol, as in the Intel project shown at the opening ceremony of the XXIII Olympic Winter Games, 2018, in PyeongChang, South Korea [44]. Full simulation videos can be accessed https://youtu.be/C7UDGRudsyA. The rest of the paper is structured as follows. In Section II we present our main system, which is a Cucker-Smale model with singular interactions and a decentralized feedback control.
In Section III we introduce a total energy functional E(x, v) for our system and show that the total energy functional is not increasing in time. Section IV presents a result concerning the collision-avoidance behaviour of the controlled system, and Section V is devoted to provide a flocking estimate showing the velocity alignment between individuals as time goes to infinity and the relative positions are uniformly bounded in time. In Section VI we show a result regarding the formation control, to conclude with different numerical experiments in Section VII.

II. PROBLEM STATEMENT
Let x i (t) and v i (t) be the position and velocity of i-th individual at time t ∈ R + , respectively. Then our main system reads as subject to the initial data Here r ij denotes the Euclidean distance between i and j-th individuals at time t, i.e., r ij (t) := |x j (t) − x i (t)|, and K and M are nonnegative constants. The first term on the right hand side of (1) 2 represents a nonlocal velocity alignment forces which is weighted by the communication function ψ in such a way that closer individuals have stronger influence than further ones. The second term on the right hand side of (1) 2 serves as a control u := (u 1 , · · · , u n ), which is given by In the original Cucker-Smale model [8], the regular weight function ψ(r) = 1/(1 + r 2 ) β/2 is considered without the control, i.e., the system (1) with M = 0. Depending on the exponent β, which determines the short/long-range interaction regimes, the conditional/unconditional flocking estimates are obtained in [8]. After that work, the conditional flocking estimates are refined in [26], [27].
In our current work, inspired by the recent work [30], we consider the following singular influence function ψ : (0, +∞) → (0, +∞) which is given by ψ(r) = 1 r α , α > 0. In [30], a critical value of the exponent α leading to global regularity of solutions or finite-time collision between individuals is obtained. More precisely, if α ≥ 1 and the individuals are placed in different positions initially, then there is no collision between them governed by the system (1) with M = 0, and this immediately provides the global existence and uniqueness of solutions to the system (1) with M = 0. On the other hand, as shown in [45], for α ∈ (0, 1) the particles may collide and even stick together in a finite time. Since there is no reason to have the singular interactions through the control, we take into account the following regular one for the weight function in the control: For notational simplicity, we set for w = (w 1 , · · · , w n ) ∈ R dn

III. PRELIMINARIES: A PRIORI ESTIMATES
In this section, we present a priori estimates of the average quantities and total energy, which will be significantly used for the flocking estimate.
We first set total energy functional and its dissipation rate be a classical solution to the system (1) in the time interval [0, T ]. Then we have the explicit from of the averages: and the non-increasing total energy estimate: Proof: By the definition of the average quantities together with the fact that n i=1 u i = 0, it is obvious to get We next estimate the kinetic energy. A straightforward computation yields where ·, · denotes the standard inner product in R d . Here, by substituting indices i and j, and using the symmetry of the weight function ψ, I 1 can be easily estimated as For the estimate of I 2 , we obtain Combining the estimates (3), (4), and (5), we conclude the desired result.
Remark 3.1: Since the velocity average is conserved in time, i.e., v c (t) = 0, the time derivative of the kinetic energy can be rewritten as This yields that we can rewrite the total energy estimate appeared in Lemma 3.1 as Then we obtain the following uniform-in-time estimate: In particular, we obtain ≡ 0 for all t ≥ 0, and this subsequently gives Without loss of generality, throughout this work, we may assume that v c (0) = 0 and x c (0) = 0, if necessary, we may consider the translation frame ( for all t ≥ 0.

IV. NON-COLLISIONAL BEHAVIOR: GLOBAL REGULARITY
In this section, we provide the non-collisional behavior of the system (1), inspired by the recent work [30], when α ≥ 1. This fact together with Cauchy-Lipschitz theory implies the global existence and uniqueness of smooth solutions to the system (1)-(2).    Proof: [Proof of Theorem 4.1] We first fix a time T > 0, and we will show that there is no collision between particles until that time. Since there are no particles colliding at the initial time, there exists a t 0 ∈ (0, T ] such that the smooth solution uniquely exists until that time. Let us assume that t 0 < T , i.e., t 0 is the first time of collision of any particles. We then set [l] to be the set of all indices j ∈ {1, · · · , n} where the j-th particle collides with l-th particle, i.e., and some positive constant δ > 0. Due to our assumption, |[l]| > 1. We set Our goal is to show that this cannot happen, We Estimate of I 1 : By exchanging the indices i and k, we find In a similar fashion as the above, we get This together with the fact that Estimate of I 2 : A straightforward computation gives where L δ is the Lipschitz constant of ψ in the interval (δ, ∞). On the other hand, it follows from Remark 3.1 that Thus we can estimate I 2 as where C 2 > 0 is given by Estimate of I 3 : We first notice that for any i ∈ {1, . . . , n} On the other hand, we get Thus we find and furthermore we obtain Hence we have Combining all of the above estimates, we find Since if we set i.e., We now apply Gronwall's inequality on the time interval (s, due to (7). We now estimate Note that if J is bounded from above by some constant J * > 0, then we have Then this leads to a contradiction since the right hand side of the above inequality is bounded, Then it follows from (8) that Note that b(t, s) has the following properties: By using these properties, we estimate J i (t, s), i = 1, 2 as  In this section, we provide a rigorous flocking estimate for the system (1). The proof follows a similar idea to the one used for a regular communication weight ψ. In the regular case we conclude that v 2 → 0 from the fact that v 2 is integrable on [0, ∞). However we may do it only because we know that v 2 is sufficiently regular (which is ensured by the regularity of ψ). In the case of singular ψ we need to put some additional effort into proving sufficient, uniform-in-time, regularity of v 2 . We do it by showing that the derivative of v 2 is a sum of an integrable function and a bounded function, which implies that v 2 is a sum of an absolutely continuous function and of a Lipschitz continuous function. Therefore v 2 is a uniformly continuous function and its integrability ensures that v 2 → 0 as t → ∞. Then there exists a unique smooth solution to the system (1)- (2). Furthermore, we assume that one of the two following hypotheses holds: Then we have On the other hand, under our main assumptions, we can find some constant This together with (10) yields thus, we get This further implies Subsequently, we obtain Time-asymptotic velocity alignment behavior: It follows from Lemma 3.1 and Remark 3.1 due to E ≥ 0. Furthermore, we obtain thanks to (12) and the zero momentum condition (6). Thus we get This together with the estimate in the proof of Lemma 3.1 and (14) yields On the other hand, I 2 can be estimated as for some C > 0, due to (11) and the energy estimate. Thus I 2 is bounded on [0, ∞). Now we come back to (16) to see that the derivative of v 2 is a sum of an integrable function (−D) due to (13) and of a bounded function M I 2 . Hence we have where f 1 is absolutely continuous and f 2 is Lipschitz continuous. Both absolutely continuous and Lipschitz continuous functions are uniformly continuous and thus v 2 is uniformly continuous.
After recalling from (15) that v 2 is also integrable, we conclude that v 2 → 0 with t → ∞.

Remark 5.2:
It is clear from Theorem 5.1 that Furthermore, we find

VI. PATTERN FORMATION
In this section we prove that if the particles do not collide asymptotically then they form a pattern induced by the control. We discuss briefly the problem with asymptotic collisions in Remark 6.1.
We first provide an enhancement of Young's inequality that will be significantly used later for the spatial pattern formation estimate.
Lemma 6.1: Let a 1 , . . . , a n−1 be a set of vectors in R d . Then where δ ∈ (0, 1) is a sufficiently small number.
Proof: Let δ > 0 be a small number to be specified later. We take 1 = 1 − δ and use Young's inequality with 1 to obtain Then it is easy to prove by induction that (1 + δ i )/2 < i < 1 − δ i provided that δ is sufficiently small (for example δ = 1/4). Thus, by the recursive definition of i , for all i = 1, ..., n − 1, we find 0 < i + 1/(4 i−1 ) ≤ 1 − δ n . Hence, by Young's inequality, we have This provides the desired result.
We are now in a position to state the asymptotic spatial pattern formation result.
for all i, j ∈ {1, . . . , n}. Then there exists a limit lim t→∞ x(t) =: x ∞ satisfying August 13, 2018 DRAFT Proof: For notational simplicity, in the rest of the proof, we denote Observe that by assumption (17), there exists a minimal distance between particles ρ > 0 on the time interval [t 0 , ∞) for sufficiently large t 0 . Thus we have for all t ∈ [t 0 , ∞). Consequently, this implies Let us first show that for all i = 1, . . . , n. It follows from (1) 2 that d dt We then estimate each summand on the right-hand side separately. Clearly, we get Furthermore, by using Young's inequality with > 0, we get for any > 0. We finally estimate the term including the control u as By Lemma 6.1, there exists a positive constant δ > 0 such that Combining the above estimates with suitably chosen > 0, we end up with d dt By (15), the kinetic energy E 1 is integrable and by the following inequality for some C > 0 independent of t, the left-hand side of (20) is a derivative of a bounded function.
, and thus we conclude (19) is proved. This together with the fact that we fixed 0 as the center of mass (recall (6)) and the help of basic linear algebra implies that x i (t) is convergent with t → ∞, and its limit x ∞ i satisfies (18) for all i = 1, · · · , n. This completes the proof.  (17). Indeed, if we consider the simplest case of two particles on a line, with z 1 = −1 then the resulting pattern has to be of the form x ∞ 2 = x ∞ 1 + 1 > x ∞ 1 . However if initially x 1 (0) > x 2 (0), then at some point the particles change order, and thus, collide, which is impossible due to Theorem 4.1. It is also clear intuitively: the particles are forbidden from colliding and if the control would result in a collision, the singularity of the communication weight ψ prevails and the pattern cannot be formed. We numerically investigate this issue, see Figure 4 below. The one-dimensional case is special in the sense that the collisions are unavoidable if the order of the particles has to be changed. It is however a much more complex question in d ≥ 2.

VII. NUMERICAL EXPERIMENTS
In this section we present simulations for planar and spatial configurations with our model.
we define the angle θ of travel with respect to the x axis, that is, θ = arctan(v y /v x ). If we chose the desired inter-particle spacings, z i , to be , sin θ − π 9 i > n 2 our model and control should achieve a bird-like pattern in steady state. Figure 1 left shows the trajectories followed by the agents on the x − y plane. Figure 1 right shows the plot of min i,j |x i (t) − x j (t)|. A few particles are initially very close to each other but the plot and its zoomed in view reveal that no collisions occur. In Figure 2 we show the energy decomposition of the system and its dissipation. It can be noted that the second part of Lemma 3.1 is satisfied. b) A single circle: Now we consider a set of inter-particle spacings z i in the control term that achieves a circle formation pattern with a single agent at the center. Figure 3 left shows the trajectories for n = 50 agents on the plane, with the model parameters as in the previous example. The initial conditions are such that some agents are close to each other and about to collide. Figure 3 right shows the plot of min i,j |x i (t) − x j (t)| over time, illustrating that the singular influence term causes the collision-avoidance. However, there is at least one pair of agents involved in near misses in two opportunities.  for i = 1, 2, 3, x 4 (0) = −1, v i (0) = (−1) i i/4 for i = 1, 2, 3 and v 4 (0) = 1. The desired formation is given by the selection z i = −2 for all i, which should put the agent with position x 4 (t) over the rest, considering the initial conditions. It can be noted that the only case where the agents do not collide is for the singular weight with α = 1.5. In that case, the particles are collapsing together but never really colliding, since Theorem 5.1 prohibits collisions in finite time.
We also provide animations of these cases at https://tinyurl.com/yapwy924 d) The Olympic Rings at PyeongChang 2018: In figure 5 we present time snapshots of the trajectories followed by a system of 50 agents in the 3 dimensional space. The parameters are chosen as before but with z i selected to obtain a final pattern that describes the Olympic Rings.
At each snapshot we plot the position of every agent as points and their instantaneous velocities as vectors. Initially, the agents are located at random positions and satisfying v c (0) = 0 ∈ R 3 .
Moreover, we plot the desired final pattern, the Olympic Rings formation, as empty circles on the plane with the third coordinate equal to 0.
It is possible to observe for some agents that at t = 0.5[s] the magnitudes of their velocities are greater than their initial ones. This is not, however, inconsistent with dv c (t)/dt = 0.