A generalized model of flocking with steering
AA generalized model of flocking with steering
Guy A. Djokam and Muruhan Rathinam Department of Mathematics and Statistics, University of MarylandBaltimore County Department of Mathematics and Statistics, University of MarylandBaltimore County
Abstract
We introduce and analyze a model for the dynamics of flocking and steering of afinite number of agents. In this model, each agent’s acceleration consists of flockingand steering components. The flocking component is a generalization of many of theexisting models and allows for the incorporation of many real world features suchas acceleration bounds, partial masking effects and orientation bias. The steeringcomponent is also integral to capture real world phenomena. We provide rigoroussufficient conditions under which the agents flock and steer together. We also providea formal singular perturbation study of the situation where flocking happens muchfaster than steering. We end our work by providing some numerical simulations toillustrate our theoretical results.
The emergence of phenomena such as flocking of birds, schooling of fish and swarming ofbacteria have attracted considerable attention by mathematicians, scientists and engineersin the recent years. See [2, 5, 6, 7, 13, 14], and references therein. Studying these phenomenanot only help us understand the natural world, but also help us better engineer systems suchas unmanned aerial vehicles. In [18], Viscek and his team introduced a novel discrete timedynamics to investigate the emergence of self ordered motion. In Viscek’s model, all agenthave the same absolute velocity and at each step, they adjust their orientation based on theirneighbors orientation. Inspired by this model, Cucker and Smale proposed the celebratedcontinuous time model [6], which led to many other subsequent studies. The Cucker-Smale(CS) model is: dx i dt = v i ,dv i dt = αN N (cid:88) j =1 a ij ( v j − v i ) , (1.1)1 a r X i v : . [ m a t h . D S ] F e b here N is the number of agents, x i and v i are the position and velocity of agent i , and the influence a ij of agent j on agent i is assumed to be symmetric ( a ij = a ji ) and is a function ofthe Euclidean distance (cid:107) x i − x j (cid:107) between i and j , so that a ij = φ ( (cid:107) x i − x j (cid:107) ). The function φ was chosen to be φ ( r ) = K ( a + r ) β , so that it was positive and non increasing.Cucker and Smale defined flocking by the condition thatsup t ≥ (cid:107) x i ( t ) − x j ( t ) (cid:107) < ∞ and that lim t →∞ (cid:107) v i ( t ) − v j ( t ) (cid:107) = 0for every pair ( i, j ) of agents. The analysis of the CS model is based on the parameter β andit is shown [6] that if β < / β ≥ / dx i dt = v i dv i dt = α ( v i − v i ) (1.2)where v i = (cid:80) Nj =1 a ij v j is a convex combination of the influences of all agents j on agent i sothat (cid:80) j a ij = 1 and a ij ≥
0. In this model, α > a ij are taken to besome function of the pairwise distances of the following form: a ij ( x ) = φ ( (cid:107) x i − x j (cid:107) ) (cid:80) k φ ( (cid:107) x i − x k (cid:107) )where φ is a nonnegative function of distance. This form of a ij lead to lack of symmetry( a ij (cid:54) = a ji ) and necessitated Motsch and Tadmor to introduce some new ideas into theanalysis of flocking; in particular the concept of maximal action by a skew-symmetric matrixand the notion of an active set .Our study is based on a finite number of agents where each agent follows a similar rulethough parameters appearing in these rules may vary from agent to agent. The notion ofthe presence of leader agents is an important concept and has been investigated in [13, 16].It is important to mention the development of continuum models which arise as limitingmodels when the number of agents approaches infinity. These models are based on partialdifferential equations that describe the evolution of the density of the agents that formed2he system. See [4, 11, 13] and reference therein. It must be noted that flocking modelsusually are concerned with a number of agents moving in the physical space and Newton’slaws dictate that such systems have a second order dynamics so that it is the accelerationsof agents that are usually controlled. Models of first order self-organized systems commonlyarise in other applications such as opinion dynamics models or flocking situations whereone may reasonably assume that agents can directly control their velocities. See[12, 17] forinstance.In this manuscript, we further generalize the MT model in ways that are inspired by theability to account for acceleration bounds, masking effects as well as orientation bias. Weendeavor to keep the model as general and flexible as possible while ensuring flocking be-havior. Moreover, despite these generalizations, we believe that many real world phenomenamay not be captured by a model that only incorporates flocking mechanisms without whatwe call steering . By steering, we mean additional acceleration by each agent which accountsfor their individual responses to other external influences such as the need to compensatefriction and gravity, pursuit of targets and evasion of danger.The paper is organized as follows. In section 2 we motivate our generalized flockingmodel via the need for acceleration bounds, the presence of masking effects and orientationbias. We introduce the open loop and closed loop aspects of the flocking model. Once theflocking part of the model is described, we show that in the presence of friction the velocitiesof all agents asymptotically approach zero. This and other considerations motivate us tothe introduction of the steering forces. We also briefly discuss existence and uniquenessof solutions. In section 3 we provide an analysis of our model and prove some sufficientconditions on flocking. Section 4 investigate the leading order behavior of the flocking andsteering model via a formal singular perturbation approach when flocking is much fasterthan steering. Numerical simulations are provided in Section 5 that illustrate our analysis. We first discuss the generalization of the flocking model and then include steering. Weobserve that the Motsch-Tadmor model has two aspects. First is the velocity alignmentaspect which is given by: ˙ v i = α ( v i − v i ) where α > v i = (cid:80) Nj =1 a ij v j ,is a (time dependent) convex combination of v , . . . , v N . Regardless of the nature of thiscombination, in the velocity space, the acceleration of agent i is always pointed towards apoint in the convex hull of all the velocities. The second aspect of the model involves how a ij depend on the positions x , . . . , x N . We note that throughout this paper (cid:107) z (cid:107) stands forthe Euclidean norm of a vector z ∈ R d . We start with the reasonable assumption that the magnitude of the acceleration (cid:107) ˙ v i (cid:107) of anyagent i may not exceed a certain predetermined value, say A >
0. It is readily observedthat in the Motsch-Tadmor model of (1.2), the acceleration of agent i is always given by α ( v i − v i ) and since α > t and i , this does not readily allow for thecondition α (cid:107) v i − v i (cid:107) ≤ A to be satisfied. Simply relaxing the model to allow for α to depend3n i and t , readily provides for the condition on acceleration bound to be α i ( t ) ≤ A (cid:107) v i ( t ) − v i ( t ) (cid:107) , which can always be satisfied since agent i choose a time varying value for α i ( t ). Thus, onemay regard α i ( t ) as a scalar control input from agent i . The only condition on each agent i is that the agent accelerates in a direction parallel to v i − v i and pointing in the same senseso that α i ( t ) >
0. A simple feedback law that each agent i can implement may take the form α i ( t ) = ξ i ( v i ( t ) − v i ( t )) (2.1)where ξ i : R Nd → [0 , ∞ ). Then the condition on acceleration bound becomes ξ i ( u ) ≤ A/ (cid:107) u (cid:107) .Motivated by this discussion, we state the following assumption. Assumption 1
For i = 1 , . . . , N , the functions ξ i : R Nd → (0 , ∞ ) are C (continuouslydifferentiable), strictly positive and there exists A > such that ξ i ( u ) ≤ A/ (cid:107) u (cid:107) , for u (cid:54) = 0 , i = 1 , . . . , N. (2.2)We note that the C assumption helps ensure existence uniqueness of solutions. A simpleexample of ξ i is given by ξ i ( u ) = A (cid:112) a + (cid:107) u (cid:107) i = 1 , . . . , N, (2.3)where a > In the CS model, the influence of agent j on i is given by the form a ij = φ ( (cid:107) x i − x j (cid:107) ) whereasin the MT model it is given by a ij = φ ( (cid:107) x i − x j (cid:107) ) / (cid:88) k φ ( (cid:107) x i − x k (cid:107) )where φ : [0 , ∞ ) → [0 , ∞ ). This form assumes that the influence of j on i is a function of allthe pairwise distances. This specific form is not general enough to model masking effects. Inorder to explain this, we refer to Figure 1. In the position space, if a third agent l is presentin the line segment joining agents i and j , then the influence of j on i may be lesser thanif there were no agents present. This motivates a very general form of position dependencefor a ij . Additionally, the effect of agent j on agent i will depend on the orientation of thefield of view of agent i . It is natural to consider the orientation of agent i as the unit vector v i / (cid:107) v i (cid:107) . However, this is undefined when v i = 0. To avoid singularities, we consider agent i ’s orientation u i to be a C function of v i , so that u i = σ i ( v i ) where σ i : R d → ¯ B d where ¯ B d is the closed unit ball in R d . An example of σ i is given by σ i ( u ) = u (cid:112) (cid:107) u (cid:107) + b i . b i is a nonzero real number. These two observations suggest the following form for a ij : a ij = φ ij ( x ; σ i ( v i )) (2.4)where x = ( x , . . . , x N ) ∈ R Nd , σ i : R d → ¯ B d and φ ij : R Nd × R d → [0 , ∞ ). We note that ¯ B d is the unit ball in R d . Thus the influence of agent j on agent i can be a nuanced function ofthe positions of all the agents as well as the velocity of agent i . We state our assumptionson φ ij . Assumption 2
For ≤ i, j ≤ N , φ ij : R Nd × R d → (0 , ∞ ) are C and strictly positive.Moreover, φ ij are shift invariant in position: φ ij ( x + y, x + y, . . . , x N + y ; u ) = φ ij ( x , x , . . . , x N ; u ) (2.5) ∀ x ∈ R Nd , ∀ y ∈ R d , ∀ u ∈ ¯ B d . Additionally, σ i : R d → ¯ B d are C . We note that the shift invariance assumption is reasonable since the influence of agent j onagent i must only depend on the relative positions of all the agents, but not on their absolutepositions. As before, the C assumption helps ensure existence uniqueness results. The strictpositivity assumptions on φ ij are utilized in our flocking results and are a statement of lackof complete masking. That is, each agent has a nontrivial influence on every other agentregardless of the relative configuration. i jkl u i Figure 1:
Masking effect and orientation bias. The agents j and k are equidistant from agent i . Nevertheless, agent l contributes to masking effect which diminishes agent j ’s influenceon agent i . On the other hand, agent i is moving to the right and in agent i ’s field of viewagent j is in a more prominent position than agent k , which diminishes agent k ’s influenceon agent i . .3 The open loop and closed loop models It is instructive to consider our general model as forming two layers. The first layer, is the“open loop” model given by ˙ x i = v i , ˙ v i = α i ( v i − v i ) ,v i = N (cid:88) j =1 a ij v j ,a ij ≥ , N (cid:88) j =1 a ij = 1 , α i ≥ α i and a ij are considered to be given functions of t , which can be regarded as controlinputs from agent i . The second layer of our model specifies how α i and a ij are chosen asfunctions of positions and velocities, thus “closing the loop”. The closed loop model thuscontains the equations ˙ x i = v i ˙ v i = α i ( v i − v i ) v i = N (cid:88) j =1 φ ij ( x ; u i ) v j u i = σ i ( v i ) α i = ξ i ( v i − v i ) (2.7)for i = 1 , . . . , N , where ξ i and φ ij satisfy Assumptions 1 and 2 respectively. In the real physical world, forces such as aerodynamic friction are present. We will considera form of friction which is proportional to some power of the velocity. We have the following(open loop) system: ˙ x i = v i ˙ v i = α i ( t )( v i − v i ) − c i (cid:107) v i (cid:107) r v i , (2.8)where r ≥
0. The following lemma shows very trivial asymptotic behavior.
Lemma 1
Suppose { x i ( t ) , v i ( t ) } Ni =1 is a C solution of the system (2.8) . Then for each i lim t →∞ v i ( t ) = 0 . Remark 1
We note that the Lemmas 5 and 6 given in the appendix will be frequently usedin the proofs of the results in this paper. roof We define an energy of the system by E = max ≤ j ≤ N E j where E j = (cid:107) v j (cid:107) . Thenby Lemmas 5 and 6 E ( t ) is absolutely continuous and dE/dt ( t ) = dE i /dt ( t ) for almost all t where i = i ( t ) is an index of the maximum. Thus, for almost all t , dEdt = (cid:104) v i , ˙ v i (cid:105) = (cid:104) v i , α i (¯ v i − v i ) − c i (cid:107) v i (cid:107) r v i (cid:105) = − c i (cid:107) v i (cid:107) r +2 + α i (cid:104) ¯ v i , v j (cid:105) − α i (cid:107) v i (cid:107) = − c i (cid:107) v i (cid:107) r +2 + α i (cid:88) j a ij (cid:104) v i , v j (cid:105) − α i (cid:107) v i (cid:107) ≤ − c i (cid:107) v i (cid:107) r +2 − α i (cid:107) v i (cid:107) + α i (cid:107) v i (cid:107) (cid:88) j a ij (cid:107) v j (cid:107) ≤ − c i (cid:107) v i (cid:107) r +2 where we have used the Cauchy-Schwartz inequality and the fact that (cid:107) v i (cid:107) ≥ (cid:107) v j (cid:107) for all j .We also note that the index i in general varies with t . Letting c = min i c i , we have dE ( t ) dt ≤ − c (cid:107) v i (cid:107) r +2 ≤ − r +1 c ( E ( t )) r +1 . (2.9)Multiplying both side by ( E ( t )) − r − , we have( E ( t )) − r − dE ( t ) dt ≤ − r +1 c − r dE ( t ) − r dt ≤ − r +1 c integrating the last inequality from 0 to t after some algebra manipulation, we have( E ( t )) − r − ( E (0)) − r ≥ r r ct which implies that E ( t ) ≤ E (0)) − r + 2 r r ct ) r . Thus E ( t ) → t → ∞ .Thus the addition of the friction shows that the asymptotic velocities go to zero. Thisobservation and other real world considerations show that in addition to pure flocking, thereshould be a “steering” component to each agent’s acceleration. In reality a group of agents may want to follow a desired trajectory in addition to stayingtogether as a flock. This necessitates an extra “steering” term. Thus, each agent i may havean extra acceleration β i ( t ) which contributes to steering. This steering term can also act tocancel other external forces such as friction and gravity. We interpret β i ( t ) in the followingas the steering component in excess of friction and gravity.7his leads to the system ˙ x i = v i , ˙ v i = α i ( v i − v i ) + β i ,v i = N (cid:88) j =1 a ij v j ,a ij ≥ N (cid:88) j =1 a ij = 1 , α i ≥ x i = v i ˙ v i = α i ( v i − v i ) + β i v i = N (cid:88) j =1 φ ij ( x ; u i ) v j u i = σ i ( v i ) α i = ξ i ( v i − v i ) (2.11)for the closed loop with steering. Assumption 3
The steering functions β i : [0 , ∞ ) → R d for i = 1 , . . . , N are continuous. We briefly discuss existence and uniqueness of solutions of the open loop and closed loopmodels (2.10) and (2.11). The open loop model is linear and non-autonomous and hence itis adequate to assume that α i ( t ), a ij ( t ) and β i ( t ) are all continuous in time. The closed loopmodel is of the form ˙ z = F ( z ) + β ( t )where z = ( x , . . . , x N , v , . . . , v N ) ∈ R Nd and F is C by our assumptions on φ ij and ξ i .Again if we assume β i ( t ) to be continuous in t then for any given initial condition for z (0),we are assured of a unique solution in an open maximal interval of time containing 0.In order to discuss flocking behavior, it is important to ensure that the forward maximalinterval of existence is [0 , ∞ ). When the steering is open-loop with Assumption 3 it is shownin Lemma 7 that the forward maximal interval is infinite. When steering is considered tobe closed-loop, that is some function of position and velocity, then a different analysis isneeded. First we define some relevant concepts and state some useful lemmas. Given the positions x i ( t ) and velocities v i ( t ) (where i = 1 , . . . , N ) of agents, we denote by d X ( t ) and d V ( t ) the8iameters in position and velocity spaces R Nd : d X ( t ) = max i,j (cid:107) x j ( t ) − x i ( t ) (cid:107) d V ( t ) = max i,j (cid:107) v j ( t ) − v i ( t ) (cid:107) . (3.1)The system { x i ( t ) , v i ( t ) } i = 1 , ..., N is said to converge to a flock if the following twoconditions hold: sup t ≥ d X ( t ) < ∞ , lim t →∞ d V ( t ) = 0 . (3.2)We define d β ( t ), the diameter in the “steering space” by d β ( t ) = max i,j (cid:107) β j ( t ) − β i ( t ) (cid:107) . (3.3)The flocking analysis in this paper uses the notion of active sets developed in [13]. Recallthat a ij ( t ) denotes the influence of agent j on agent i at time t and that a ij ( t ) ≥ (cid:80) Nj =1 a ij ( t ) = 1. Given θ >
0, it is instructive to consider the set of all agents who influencea given agent 1 ≤ p ≤ N by an amount greater than or equal to θ . This is known as the active set Λ p ( θ ) for agent p : Λ p ( θ ) = { j | a pj ≥ θ } . (3.4)For a pair of agents p and q , the common active set Λ pq ( θ ) is simply the intersection Λ p ( θ ) ∩ Λ q ( θ ). The global active set Λ( θ ) is the intersection of all the active sets:Λ( θ ) = (cid:92) p Λ p ( θ ) . (3.5)The following lemma from [13] is critical. Lemma 2 [13] Let S be and antisymmetric matrix, S ij = − S ji with | S ij | ≤ M . Let u, w ∈ R n be two given vectors with positive entries, u i , w i ≥ and let U , W denoted their respectivesums, U = (cid:80) i u i and W = (cid:80) j w j . Fix θ > and let λ ( θ ) denoted the number of ”activeentries” of u and w at the level θ , in sense that, λ ( θ ) = | Λ( θ ) | Λ( θ ) = { j | u j ≥ θU and w j ≥ θW } . Then for every θ > , we have |(cid:104) Su, w (cid:105)| ≤
M U W (1 − λ ( θ ) θ )For our analysis, in addition to Lemma 2, we need the following simple lemma about theconvex hull of a finite set of points in R d . Lemma 3
Let { v i } Ni =1 be a set of vectors in R d and let Ω be their convex hull. If v p and v q delimit the diameter of the convex hull, (that is v p and v q are furthest apart), then for each v ∈ Ω (cid:104) v p − v q , v − v q (cid:105) ≥ . p v q v j . .. H pq Figure 2:
Illustration of the lemma
Proof
Let the diameter of Ω equal (cid:107) v p − v q (cid:107) . We first show that (cid:104) v p − v q , v i − v q (cid:105) ≥ ∀ i. Let H pq be the hyper plane passing through v q and is perpendicular to v p − v q . (See Figure2). Suppose there is some i such that (cid:104) v p − v q , v i − v q (cid:105) <
0. This shows that v i and v p will be on opposite sides of the hyper plane H pq , implying that (cid:107) v i − v p (cid:107) > (cid:107) v q − v p (cid:107) , acontradiction. Given any v ∈ Ω, there exist a i ≥ i = 1 , . . . , n such that (cid:80) Ni =1 a i = 1and v = (cid:80) Nj =1 a i v i . Hence (cid:104) v p − v q , v − v q (cid:105) = (cid:104) v p − v q , N (cid:88) i =1 a i ( v i − v q ) (cid:105) ≥ . We shall suppose that Assumption 3 holds.
Theorem 1
Let ( x ( t ) , v ( t )) ∈ R Nd × R Nd be a C solution of the open loop (2.10) . At time t , let d V ( t ) = (cid:107) v p ( t ) − v q ( t ) (cid:107) . Fix an arbitrary θ > and let λ pq ( θ ) be the number of agentsin the common active set Λ pq ( θ ) associated with the influence matrix a ij ( t ) of the system.Let α ( t ) = min i α i ( t ) . Then for almost all t , the diameters of the system, d X ( t ) , d V ( t ) and d β ( t ) satisfy : ddt d X ( t ) ≤ d V ( t ) ddt d V ( t ) ≤ − α λ pq ( θ ) θ d V ( t ) + d β ( t ) . (3.6)10 emark 2 In Theorem (1) , we note that p and q are functions of t , and so is λ pq ( θ ) . Thistheorem is a generalization of Theorem 3.4 of [13] where α i ( t ) was independent of i and t .One needs Lemma 3 to handle the extra terms that appear in our analysis. Proof
By Lemma 5 d X ( t ) is absolutely continuous. We choose i = i ( t ) and j = j ( t )such that d X ( t ) = (cid:107) x i ( t ) − x j ( t ) (cid:107) for all t . Using Lemma 6 we obtain (cid:12)(cid:12)(cid:12)(cid:12) ddt ( d X ( t )) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:107) x i − x j (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) = 2 |(cid:104) x i − x j , v i − v j (cid:105)| . Hence (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x i − x j (cid:107) ddt (cid:107) x i − x j (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) = 2 |(cid:104) x i − x j , v i − v j (cid:105)| ≤ (cid:107) x i − x j (cid:107)(cid:107) v i − v j (cid:107) . This yields that for almost all t ddt d X ( t ) ≤ (cid:107) v i − v j (cid:107) ≤ d V ( t ) . Note that if for some t > d X ( t ) = 0 and d X ( t ) is differentiable, then ddt d X ( t ) = 0.For the second inequality, we again proceed by using Lemmas 5 and 6. Let p = p ( t ) and q = q ( t ) be such that d V ( t ) = (cid:107) v p − v q (cid:107) for all t . Then (for almost all t ) ddt ( d V ( t )) = ddt ( (cid:107) v p − v q (cid:107) ) = 2 (cid:104) v p − v q , ˙ v p − ˙ v q (cid:105) = 2 (cid:104) v p − v q , α p ( v p − v p ) − α q ( v q − v q ) (cid:105) + 2 (cid:104) v p − v q , β p − β q (cid:105) = 2 α p (cid:104) v p − v q , v p − v p (cid:105) − α q (cid:104) v p − v q , v q − v q (cid:105) + 2 (cid:104) v p − v q , β p − β q (cid:105) . We proceed by assuming WLOG that α p ≤ α q and write ddt ( (cid:107) v p − v q (cid:107) ) = 2 α p (cid:104) v p − v q , v p − v q (cid:105) − α p (cid:107) v p − v q (cid:107) − α q − α p ) (cid:104) v p − v q , v q − v q (cid:105) + 2 (cid:104) v p − v q , β p − β q (cid:105) . Using Lemma (3), Cauchy-Schwartz inequality and the fact that (cid:107) β p − β q (cid:107) ≤ d β , we have: ddt ( (cid:107) v p − v q (cid:107) ) ≤ α p (cid:104) v p − v q , v p − v q (cid:105) − α p (cid:107) v p − v q (cid:107) + 2 (cid:107) v p − v q (cid:107) d β . Moreover v p − v q = N (cid:88) j =1 a pj v j − v q = N (cid:88) j =1 a pj ( v j − v q )= N (cid:88) j =1 a pj ( v j − N (cid:88) i =1 a qi v i ) = N (cid:88) i,j a pj a qi ( v j − v i ) . Hence ddt ( (cid:107) v p − v q (cid:107) ) ≤ α p N (cid:88) i,j a pj a qi (cid:104) v p − v q , v j − v i (cid:105) − α p (cid:107) v p − v q (cid:107) + 2 d β (cid:107) v p − v q (cid:107) . u i = a pi , w i = a qi , and the anti symmetric matrix S ij = (cid:104) v p − v q , v i − v j (cid:105) . Since | S ij | ≤ d V , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) i,j a pj a qi (cid:104) v p − v q , v j − v i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d V (1 − λ pq ( θ ) θ )Therefore we have ddt ( (cid:107) v p − v q (cid:107) ) ≤ α p d V (1 − λ pq ( θ ) θ ) − α p (cid:107) v p − v q (cid:107) + 2 d β (cid:107) v p − v q (cid:107) Noting that v p and v q are such that (cid:107) v p ( t ) − v q ( t ) (cid:107) = d V ( t ) and that α p ( t ) ≥ α ( t ) bydefinition, we have ddt ( d V ( t ) ) ≤ − α d V λ pq ( θ ) θ + 2 d β d V . An argument similar to the one used in deriving the first inequality proves (1). Thefollowing corollary is immediate.
Corollary 1 If λ ( θ ) is the number of elements in the global active set Λ( θ ) and if α denotesthe infimum of α i ( t ) over i and t ≥ then ddt d X ( t ) ≤ d V ( t ) . (3.7a) ddt d V ( t ) ≤ − αλ ( θ ) θ d V ( t ) + d β ( t ) . (3.7b) We shall suppose that Assumptions 1, 2 and 3 hold. As observed in Section 2.6 theseAssumptions guarantee existence and uniqueness of a solution to the closed-loop equationson the time interval [0 , ∞ ). Moreover, this solution is C in time t . We note that, if allsteering terms β i are equal for all t , then d β ( t ) = 0 and the system of inequalities given by(3.7a) and (3.7b) show that the diameter d V is decreasing in time. Even in this case, in orderto show flocking, one needs stronger inequalities. To that end, we shall modify the ideasfrom Ha et al [10] and also from Motsch and Tadmore [13] in order to prove the flockingresults. We define the function ψ : [0 , ∞ ) → (0 , ∞ ) by ψ ( r ) = min ≤ i,j ≤ N min { φ ij ( x ; u ) | (cid:107) x l − x k (cid:107) ≤ r, u ∈ ¯ B d and 1 ≤ l, k ≤ N } . (3.8)In order to see that the minimum exists, we observe that by shift invariance (Assumption2), { φ ij ( x ; u ) | (cid:107) x l − x k (cid:107) ≤ r, u ∈ ¯ B d } = { φ ij ( x ; u ) | x = 0 , (cid:107) x l − x k (cid:107) ≤ r, u ∈ ¯ B d } { ( x, u ) ∈ R Nd × ¯ B d | x = 0 , (cid:107) x l − x k (cid:107) ≤ r, u ∈ ¯ B d } is a compact set and that φ ij are continuous. Since φ ij are strictly positive by Assumption2, it follows that ψ is strictly positive. Moreover, it is also clear that ψ is a decreasing (non-increasing) function. Since ψ is decreasing, it is also positive and measurable, and hence (cid:82) r ψ ( r ) dr < ∞ and (cid:82) ∞ ψ ( r ) dr ≤ ∞ are well-defined. Lemma 4
Let α be the infimum of α i ( t ) over i and t ≥ . Suppose that (cid:90) ∞ d β ( t ) dt < ∞ . Then α > . Proof
Let M be defined by M = d V (0) + (cid:90) ∞ d β ( t ) dt. Then from (3.7b) it follows that d V ( t ) ≤ M for all t ≥
0. Hence, for all t ≥ ≤ i ≤ N , (cid:107) v i ( t ) − v i ( t ) (cid:107) ≤ d V ( t ) ≤ M , where we have used the fact that v i is in the convexhull of all velocities v j . Now α = inf { ξ i ( v i ( t ) − v i ( t )) | t ≥ , ≤ i ≤ N } , ≥ min { ξ i ( u ) | ≤ (cid:107) u (cid:107) ≤ M, ≤ i ≤ N } > , where we have used the fact that ξ i are continuous by Assumption 1. Theorem 2
Consider the closed loop system (2.11) . Suppose ψ is defined by (3.8) and that (cid:90) ∞ d β ( t ) dt < ∞ and lim t →∞ d β ( t ) = 0 . Further suppose that the initial diameters satisfy d V (0) + (cid:90) ∞ d β ( t ) dt < αN (cid:90) ∞ d X (0) ψ ( s ) ds. (3.9) Then the solution ( x ( t ) , v ( t )) flocks. In particular, if (cid:82) ∞ ψ ( s ) ds = ∞ , then the condition oninitial diameters is always satisfied. Proof
At any given time, by choosing θ ( t ) = (cid:112) ψ ( d X ( t )), one readily obtains that thenumber of elements in the global active set is N , and hence the inequality ddt d V ( t ) ≤ − αN ψ ( d X ( t )) d V ( t ) + d β ( t ) . We define the energy functional E : R Nd × R Nd → R E ( d X ( t ) , d V ( t )) = d V ( t ) + αN (cid:90) d X ( t )0 ψ ( s ) ds, (3.10)13he time derivative of the energy functional satisfies˙ E = ˙ d V + αN d V ψ ( d X ) ≤ d β . Hence E ( d V ( t ) , d X ( t )) − E ( d V (0) , d X (0)) ≤ (cid:90) t d β ( s ) ds, Which implies that d V ( t ) − d V (0) ≤ − αN (cid:90) d X ( t )0 ψ ( s ) ds + αN (cid:90) d X (0)0 ψ ( s ) ds + (cid:90) t d β ( s ) ds. We deduce that d V ( t ) − d V (0) ≤ αN (cid:90) d X (0) d X ( t ) ψ ( s ) ds + (cid:90) t d β ( s ) ds. (3.11)By the assumption (3.9), there exists d ∗ (independent of t ) such that (cid:90) ∞ d β ( t ) dt + d V (0) ≤ αN (cid:90) d ∗ d X (0) ψ ( s ) ds. (3.12)Replacing this inequality in (3.11), we obtain that d V ( t ) ≤ αN (cid:90) d X (0) d X ( t ) ψ ( s ) ds + αN (cid:90) d ∗ d X (0) ψ ( s ) ds ≤ αN (cid:90) d ∗ d X ( t ) ψ ( s ) ds. Since d V ( t ) ≥
0, we have that the diameter in the position space is uniformly bounded. Thatis, d X ( t ) ≤ d ∗ for all t ≥
0. Defining ψ ∗ = ψ ( d ∗ ), we note that ψ ( s ) ≥ ψ ∗ for s ∈ [0 , d ∗ ].Using the inequality ddt d V ( t ) ≤ − αN ψ ( d X ( t )) d V + d β , we have that ddt d V ( t ) ≤ − αN ψ ∗ d V + d β . Which implies that: d V ( t ) ≤ e − αN ψ ∗ t d V (0) + (cid:90) t e − αN ψ ∗ ( t − s ) d β ( s ) ds Now let us show that the velocity diameter goes to zero asymptotically. The first term abovegoes to zero asymptotically in time. The second term can be written as (cid:82) t e αN ψ ∗ s d β ( s ) dse αN ψ ∗ t . There are two cases. If lim t →∞ (cid:90) t e αN ψ ∗ s d β ( s ) ds < ∞ t →∞ d β ( t ) = 0 showsthat lim t →∞ (cid:82) t e αN ψ ∗ s d β ( s ) dse αN ψ ∗ t = lim t →∞ e αN ψ ∗ t d β ( t ) e αN ψ ∗ t = lim t →∞ d β ( t ) = 0 . We consider the model given by (2.11) and investigate the scenario where flocking is muchfaster than steering. In the singular perturbation approach, we capture this by the introduc-tion of a small parameter (cid:15) . For ease of analysis, we ignore the orientation bias and assumethat a ij = φ ij ( x ). This leads us to the family of equations˙ x i = v i ˙ v i = α i (cid:15) ( v i − v i ) + β i v i = N (cid:88) j =1 φ ij ( x ) v j α i = ξ i ( v i − v i ) ∀ i = 1 , . . . , N. (4.1)Here, 0 < (cid:15) (cid:28) x i ( t, (cid:15) ) and v i ( t, (cid:15) ) for all i = 1 , . . . , N be the solution of our new model (4.1). Weexpand these solutions and some related variables of the model in a power series in (cid:15) : x i ( t, (cid:15) ) = x i, ( t ) + (cid:15)x i, ( t ) + . . .v i ( t, (cid:15) ) = v i, ( t ) + (cid:15)v i, ( t ) + . . .α i ( t, (cid:15) ) = α i, ( t ) + (cid:15)α i, ( t ) + . . .v i ( t, (cid:15) ) = v i, ( t ) + (cid:15)v i, ( t ) + . . .β i ( t, (cid:15) ) = β i, ( t ) + (cid:15)β i, ( t ) + . . . (4.2) We shall use x ( t ) to denote ( x , ( t ) , . . . , x N, ( t )) , and likewise v ( t ) and β ( t ). We are interested in characterizing the leading order terms x ( t ) and v ( t ). In what follows, we frequently omit showing the dependence on time forbrevity. Substituting the expansions (4.2) into (4.1) we obtain˙ x i, + (cid:15) ˙ x i, + . . . = v i, + (cid:15)v i, + . . . ˙ v i, + (cid:15) ˙ v i, + . . . = 1 (cid:15) ( α i, + (cid:15)α i, + . . . )(( v i, − v i, )+ (cid:15) ( v i, − v i, ) + . . . ) + β i, ( t ) + (cid:15)β i, ( t ) + . . . (4.3)15urthermore we obtain v i, = N (cid:88) j =1 φ ij ( x ) v j, v i, = N (cid:88) j =1 φ ij ( x ) v j, + N (cid:88) j =1 (cid:40) N (cid:88) l =1 d (cid:88) k =1 ∂φ ij ∂x kl ( x ) x kl, (cid:41) v j, . (4.4)We note that x ki and v ki are the k th components of the i th agent’s position and velocity.Also x ki, and x ki, denote the leading order and the next order terms of x ki and likewise for v ki, and v ki, . Balancing the terms of order (cid:15) − in (4.3), we obtain that α i, ( t )( v i, ( t ) − v i, ( t )) = 0 . (4.5)This means that α i, = 0 or v i, − v i, = 0. since α = min α i >
0, we have that v i, = v i, .Therefore N (cid:88) j =1 φ ij ( x ) v j, = v i, and hence N (cid:88) j =1 φ ij ( x ) v kj, = v ki, where we use the superscript to denote the k th component of the velocity. Fixing a compo-nent 1 ≤ k ≤ d and writing the previous equation for all agents, we obtain P ( t ) v k ( t ) = v k ( t ) , (4.6)where the matrix P is given by: P = φ ( x ) · · · φ N ( x )... . . . ... φ N ( x ) · · · φ NN ( x ) , (4.7)and v k = ( v k , , . . . , v kN, ) ∀ k = 1 , . . . , d . Since P ij = φ ij > N (cid:88) j =1 P ij = 1 , the matrix P is a stochastic matrix. Since P ij > i, j , P has eigenvector e = (1 , . . . , t corresponding to the eigenvalue 1 of multiplicity one. Thus for each k = 1 , . . . , d , (4.6) hasa unique solution for v k which is a multiple of e = (1 , . . . , t . This shows that v i, ( t ) are allequal for i = 1 , . . . , N , indicating flocking. We shall denote this flocking velocity by v f ( t ).Balancing the terms of order (cid:15) in (4.3) gives the system˙ x i, ( t ) = v i, ( t ) , ˙ v i, ( t ) = α i, ( v i, ( t ) − v i, ( t )) + α i, ( v i, ( t ) − v i, ( t )) + β i, ( t ) . (4.8)16ince v i, = v f for all i , it follows that v i, = v f for all i , and hence, from (4.4) we obtainthat v i, = N (cid:88) j =1 φ ij ( x ) v j, + N (cid:88) j =1 (cid:40) N (cid:88) l =1 d (cid:88) k =1 ∂φ ij ∂x kl ( x ) x kl, (cid:41) v f . We change the order of the summation in the second term and use the condition (cid:80) Nj =1 φ i,j ( x ) =1 to obtain that N (cid:88) j =1 (cid:40) N (cid:88) l =1 d (cid:88) k =1 ∂∂x kl φ ij ( x ) x kl, (cid:41) v f = N (cid:88) l =1 (cid:40) N (cid:88) k =1 x kl, ∂∂x kl (cid:32) d (cid:88) j =1 φ i,j ( x ) (cid:33)(cid:41) v f = 0 . Thus v i, = N (cid:88) j =1 φ ij ( x ) v j, . Substituting these results in equation (4.8), we have that for each i ˙ v f = α i, ( v i, − v i, ) + β i, . (4.9)From the first equation of (4.8) we have that˙ x i, = v i, = v f . This implies that for each i x i, ( t ) = x i, (0) + (cid:90) t v f ( s ) ds. (4.10)Hence for all i and j x i, ( t ) − x j, ( t ) = x i, (0) − x j, (0) . (4.11)It follows from (4.11) that the leading order relative positions of agents do not changewith time. Hence by the shift invariance assumption on φ ij , it follows that φ ij ( x ( t )) isindependent of t . We denote by a ij : a ij = φ ij ( x ) ∀ i, j. Since v i, = v i, , it follows that α i, = ξ i ( v i, − v i, ) = ξ i (0) >
0. Hence, for each i ,˙ v f = ξ i (0) (cid:32) N (cid:88) j =1 a ij v j, − v i, (cid:33) + β i, . (4.12)Taking the k th component in equation (4.12) we have that˙ v f,k = ξ i (0) (cid:32) N (cid:88) j =1 a ij v kj, − v ki, (cid:33) + β ki, , (4.13)17or k = 1 , . . . , d . We define for 1 ≤ i, j ≤ Nq ij = ξ i (0) a ij ∀ i (cid:54) = j,q ii = ξ i (0) a ii − ξ i (0) . The matrix Q = [ q ij ] is a transition rate matrix of a continuous time Markov chain. Moreover,since q ij = ξ (0) a ij > i (cid:54) = j , the matrix Q corresponds to an ergodic Markov chainin continuous time. Thus there exists a unique vector ( π i ) Ni =1 such that (cid:80) Ni =1 π i = 1 and N (cid:88) i =1 π i q ij = 0 . With the introduction of matrix Q , (4.13) may be written as˙ v f,k = N (cid:88) j =1 q ij v kj, + β ki, . Multiplying by π i and summing over i = 1 . . . N , and using properties of q ij and π i we obtainthat ˙ v f,k = N (cid:88) i =1 π i β ki, . (4.14)Hence the flocking velocity v f ( t ) evolves according to the equation˙ v f = N (cid:88) i =1 π i β i, . (4.15)In general, one may expect the steering terms β i to depend on x i , v i and possible t , sothat β i ( t ) = η i ( x i ( t ) , v i ( t ) , t ) (4.16)where we suppose η i : R d × R d × [0 , ∞ ) → R d is C in its arguments. Then, it follows thatthe evolution equation for v f is given by˙ v f ( t ) = N (cid:88) j =1 π i ( x ( t )) η i ( x ,i ( t ) , v f ( t ) , t ) , (4.17)where x i, ( t ) are given by x ( t ) = x (0) + (cid:90) t v f ( s ) ds. (4.18)Here x (0) = ( x (0) , . . . , x N (0)) is the initial position of the agents and we observe that π i ( x ( t )) is constant in time since φ ij ( x ( t )) is constant in time. We may summarize theleading order time evolution by the system of ODEs˙ x ( t ) = v f ( t ) , ˙ v f ( t ) = N (cid:88) j =1 π i ( x ( t )) η i ( x ,i ( t ) , v f ( t ) , t ) . (4.19)18his is a ( N + 1) d dimensional system and the leading order velocities are given by v i, ( t ) = v f ( t ). We observe that in order to obtain a unique solution, we need an initial condition for v f (0) which may not be the true initial velocities v i (0) of the agents. Intuitively, one expectsa rapid initial transient layer during which flocking occurs and the agents reach the flockingvelocity v f (0).In the next subsection, we scale time to investigate this transient layer. The given problem has initial condition, x (0) = ( x (0) . . . , x N (0)) and v (0) = ( v (0) , . . . , v N (0)).We zoom into the transient layer at t = 0 by introducing the variable τ = t/(cid:15) . We define X and V by X ( τ, (cid:15) ) = x ( t, (cid:15) ) = x ( (cid:15)τ, (cid:15) ) and V ( τ, (cid:15) ) = v ( t, (cid:15) ) = v ( (cid:15)τ, (cid:15) ) . Differentiating with respect to τ , we have that1 (cid:15) dX i ( τ, (cid:15) ) dτ = dx i ( t, (cid:15) ) dt and 1 (cid:15) dV i ( τ, (cid:15) ) dτ = dv i ( t, (cid:15) ) dt With the change of variable we have the following system of differential equations: X (cid:48) i = (cid:15)V i ,V (cid:48) i = α i ( V i − V i ) + (cid:15)β i , (4.20)where the prime denotes differentiation with respect to τ . The initial conditions to imposeare X i (0) = x i (0) ,V i (0) = v i (0) . (4.21)As before, we assume an (cid:15) -expansion for X i and V i of the following form: X i ( τ, (cid:15) ) = X i, ( τ, (cid:15) ) + (cid:15)X i, ( τ, (cid:15) ) + . . .V i ( τ, (cid:15) ) = V i, ( τ, (cid:15) ) + (cid:15)V i, ( τ, (cid:15) ) + . . . (4.22)Substituting this expansion in (4.20) we obtain X (cid:48) i, + (cid:15)X (cid:48) i, + . . . = (cid:15) ( V i, + (cid:15)V i, + . . . ) ,V (cid:48) i, + (cid:15)V (cid:48) i, + . . . = ( α i, + (cid:15)α i, + . . . )(( V i, − V i, ) + (cid:15) ( V i, − V i, ) . . . ) , + (cid:15) ( β i, + (cid:15)β i, + . . . ) . Balancing the (cid:15) terms, we find that X (cid:48) i, = 0 ,V (cid:48) i, = α i, ( V i, − V i, ) . (4.23)19t follows that X i, ( τ ) = X i (0) = x i (0). This means that during the initial transient theleading order positions do not change in time τ .The model (4.23) is similar to (2.11) without the steering terms, except that the positions X i, are constant. Hence the influence matrix a ij = φ ij ( X ) is constant and strictly positive.Defining d X ( τ ) = max i,j (cid:107) X i, ( τ ) − X j, ( τ ) (cid:107) , d V ( τ ) = max i,j (cid:107) V i, ( τ ) − V j, ( τ ) (cid:107) , to be the diameters in the position and the velocity spaces respectively, we see that theassumptions of Lemma (4) and Theorem (2) are satisfied since the diameter in the steeringspace is zero. Thus Theorem (2) can be invoked to conclude that d V ( τ ) → τ → ∞ .Now let us find lim τ →∞ V i, ( τ ). The second equation of (4.23) is V (cid:48) i, = α i, ( V i, − V i, ) = α i, (cid:32) N (cid:88) j =1 φ ij ( X ) V j, − V i, (cid:33) = N (cid:88) j =1 q ij V j, . Where Q = ( q ij ) is the same matrix that we have used in (4.13). Taking the k th componentsand letting Z ki = V ki, and Z k = ( Z k , . . . , Z kN ) we have Z (cid:48) k = Q Z k . That is Z (cid:48) ki = N (cid:88) j =1 q ij Z kj . Multiplying by π i and sum it from 1 to N , we have N (cid:88) i =1 π i Z (cid:48) ki = N (cid:88) i =1 N (cid:88) j =1 π i q ij Z kj = N (cid:88) j =1 (cid:32) N (cid:88) i =1 π i q ij (cid:33) Z kj = 0 . This implies that for t ≥ N (cid:88) i =1 π i Z ki ( t ) = N (cid:88) i =1 π i Z ki (0) . (4.24)However, all the eigenvalues of Q except for one zero eigenvalue have negative real parts.Thus Z k ( t ) → Z k where Z k is a multiple of (1 , . . . , t . That is Z k = c k (1 , . . . , t . To find c k , we take limits in (4.24):lim τ →∞ N (cid:88) i =1 π i Z ki ( t ) = c k = N (cid:88) i =1 π i Z ki (0) = N (cid:88) i =1 π i V ki, (0) . We deduce thatlim τ →∞ V i, ( τ ) = V f (0) = ( c , . . . , c d ) = (cid:32) N (cid:88) i =1 π i V i, (0) , . . . , N (cid:88) i =1 π i V di, (0) (cid:33) . (4.25)20 Numerical Examples
In this section, we present some numerical simulations to illustrate our theoretical analysis.We consider the collection of N = 7 agents in two dimensions. We shall choose the initialpositions and initial velocities randomly inside square regions [0 , × [0 ,
8] in position and[0 , × [0 ,
3] in velocity spaces respectively.We assume all agents wish to follow the same trajectory y ( t ) = (100 + 10 sin(0 . t ) ,
10 + 10 cos (0 . t )) t in the position space. We assume each agent i implements a feedback law for steeringaccording to β i ( t ) = γ ( ˙ y ( t ) − v i ( t )) + γ ( y ( t ) − x i ( t )) , Where γ and γ are two parameters. In these MATLAB simulations, we took γ = 2 and γ = 0 .
1. We computed the solutions of the full model (2.11) for (cid:15) = 0 . (cid:15) = 0 .
01 and (cid:15) = 0 . v f (0)to be used in conjunction with (4.19), we use the equation (4.25).Finally, we computed the leading order approximation and compared it to the simulationwhen (cid:15) = 0 . (cid:15) = 0 .
01 and (cid:15) = 0 . (cid:15) we used the same randomly chosen initialconditions which we provide here. Initial positions were x (0) = (6 . , . t , x (0) = (1 . , . t , x (0) = (4 . , . t ,x (0) = (6 . , . t , x (0) = (2 . , . t , x (0) = (5 . , . t ,x (0) = (1 . , . t , and the initial velocities were v (0) = (2 . , . t , v (0) = (1 . , . t , v (0) = (2 . , . t ,v (0) = (1 . , . t , v (0) = (2 . , . t , v (0) = (1 . , . t ,v (0) = (0 . , t . In these simulations, we have used the following functions. φ ij ( x, u ) = φ ( r ij ) (cid:80) k φ ( r ik ) where, r ij = (cid:107) x j − x i (cid:107) φ ( r ) = 1(1 + r ) . α i ( t ) = ξ i ( u i ) = 10(0 . (cid:107) u i (cid:107) ) . where, u i = v i − v i . a) Positions (b) Velocities Figure 3: Trajectories in position and velocity spaces. Cyan circles represent initial valuesand red stars the final values. Case (cid:15) = 0 . (a) x i ( t ) and y ( t ) (b) x i ( t ) and y ( t ) Figure 4: Positions against time. Case (cid:15) = 0 .
1. Target trajectory in dash black.22 a) v i ( t ) and ˙ y ( t ) (b) v i ( t ) and ˙ y ( t ) Figure 5: Velocities against time. Case (cid:15) = 0 . (a) v i ( t ) and ˙ y ( t ) (b) v i ( t ) and ˙ y ( t ) Figure 6: Velocities against time for t close to zero. Case (cid:15) = 0 . a) Positions (b) Velocities Figure 7: Trajectories in position and velocity spaces. Cyan circles represent initial valuesand red stars the final values.. Case (cid:15) = 0 . (a) x i ( t ) and y ( t ) (b) x i ( t ) and y ( t ) Figure 8: Trajectories against time. Case (cid:15) = 0 . a) v i ( t ) and ˙ y ( t ) (b) v i ( t ) and ˙ y ( t ) Figure 9: Velocities against time. Case (cid:15) = 0 .
01. Target velocity in dash black (a) v i ( t ) and ˙ y ( t ) (b) v i ( t ) and ˙ y ( t ) Figure 10: Velocities against time for t close to zero. Case (cid:15) = 0 .
01. Target velocity in dashblack 25 a) Positions (b) Velocities
Figure 11: Trajectories in position and velocity spaces. Cyan circles represent initial valuesand red stars the final values.. Case (cid:15) = 0 . (a) x i ( t ) and y ( t ) (b) x i ( t ) and y ( t ) Figure 12: Positions against time. Case (cid:15) = 0 . a) i ( t ) and ˙ y ( t ) (b) v i ( t ) and ˙ y ( t ) Figure 13: Velocities against time. Case (cid:15) = 0 . (a) v i ( t ) and ˙ y ( t ) (b) v i ( t ) and ˙ y ( t ) Figure 14: Velocities against time for t close to zero. Case (cid:15) = 0 . V f (0) and then stay together and steer towards the target velocity. Our simulations alsoshow that after some time the trajectory of each agent is similar to the target trajectorywhich is a circle. Furthermore, Figures,4, 8 and 12 which show the plots of trajectoriescomponents against time we see that after some time, all the components follow the targettrajectory components (in dashed black). Figures 5, 9 and 13 is the plots of componentsvelocities against time. In conjunction with figures 6, 10 and 14 we read that velocitiescomponents converge very fast to a common velocity and then in the long run, align withthe target velocity (in dashed black). Comparison of the leading order and the cases (cid:15) = 0 . , (cid:15) = 0 . and (cid:15) = 0 . (cid:15) . In Figures 15 ,16 and 17, theapproximate model solution is in green color while the actual model solution is in blue. Thesmall red circles and red stars represent the initial and final positions for the actual model.Likewise, the small yellow circles and yellow stars represent the initial and final position forthe approximate model. Graphically, we see that the singular perturbation theory showsgood agreement with the exact model. (a) Positions (b) Velocities Figure 15: Comparison between the reduced model (in green) and exact solution (in blue)for (cid:15) = 0 . a) Positions (b) Velocities Figure 16: Comparison between the reduced model (green) and exact solution (in blue) for (cid:15) = 0 . (a) Positions (b) Velocities Figure 17: Comparison between the reduced model (in green) and exact solution (in blue)for (cid:15) = 0 .
001 29
Concluding remarks
We introduced and analyzed a generalized model of flocking with steering. In our model, theacceleration of each agent has flocking and steering components. The flocking componentis a generalization of many existing models and takes into account real world factors suchas apriori bound on acceleration, masking effects and orientation bias. We proved that thegeneralized model with steering flocks under certain sufficient conditions which naturallyinclude assumptions on the steering components β i ( t ) of the accelerations of the agents.We also studied the case where flocking is much faster than steering using formal singularperturbation theory and showed that the leading order behavior is one where the agents flocktogether with velocity v f which evolves in time, see 4.15. Our simulations showed that theleading order approximation was very similar to the real solution for small values of (cid:15) a scaleparameter indicating the magnitude difference between flocking and steering accelerations.While this supports our formal derivation via singular perturbation theory, in future wewould like to derive rigorous results that support the formal theory.We also observe that the influence functions φ ij were assumed to be nonvanishing for all i, j ∈ , . . . , N in our flocking results. This implies that the communication graph formedby the agents is strongly connected. In the case of the robotic systems, this will be compu-tationally expensive. Even in the case of biological agents, all to all communication amongagents may not be a reasonable assumption. This raises the question whether one couldrelax the strict positivity condition and still obtain flocking results.Our flocking results assumed that the steering components β i ( t ) of the agents were asymp-totically in agreement ( β i ( t ) − β j ( t ) → t → ∞ ). A related natural question is if theagents form subgroups within which this condition holds but fails across these subgroups,then can we obtain clusters of agents such that agents within each cluster flock together. A Useful lemmas
Lemma 5
Let F : R n → R be locally Lipschitz and u : [0 , T ] → R be absolutely continuous.Then F ◦ u : [0 , T ] → R is absolutely continuous. Lemma 6
The function f i : R → R be absolutely continuous on [0 , T ] for i = 1 , . . . , n andlet f : R → R be defined by f ( t ) = max { f i ( t ) | i = 1 . . . n } . Suppose i ∗ : R → R satisfies f i ∗ ( t ) ( t ) ≥ f j ( t ) for all t and j = 1 , . . . , n . Then f is absolutelycontinuous and f (cid:48) ( t ) = f (cid:48) i ∗ ( t ) ( t ) for almost all t . Lemma 7
The forward maximal interval of existence of the model 2.10 is [0 , ∞ ) where weassume that α i , β i , a ij are all continuous functions on [0 , ∞ ) . Proof
Let us suppose that the forward maximal interval of existence is the interval [0 , T ∗ ),with T ∗ < ∞ . We define the energy of the system by E ( t ) = max i E i ( t ) = max i (cid:107) v i ( t ) (cid:107) . dE ( t ) dt = (cid:104) v i , ˙ v i (cid:105) = (cid:104) v i , α i ( v i − v i ) + β i (cid:105) = α i (cid:104) v i , v i (cid:105) − α i (cid:104) v i , v i (cid:105) + α i (cid:104) v i , β i (cid:105)≤ α i (cid:107) v i (cid:107) (cid:32)(cid:88) j a ij (cid:107) v j (cid:107) (cid:33) − α i (cid:107) v i (cid:107) + α i (cid:107) v i (cid:107)(cid:107) β i (cid:107)≤ α i (cid:107) v i (cid:107)(cid:107) β i (cid:107) We have used the Cauchy-Schwartz inequality and the conditions (cid:107) v j (cid:107) ≤ (cid:107) v i (cid:107) and (cid:80) j a ij = 1.We rearrange this inequality to dE ( t ) dt ≤ α i (2 E ( t )) (cid:107) β i (cid:107) ≤ α (2 E ( t )) (cid:107) β i (cid:107) . Where α is the maximum of α i ( t ) over i and t ∈ [0 , T ∗ ]. Multiplying this inequality by(2 E ( t )) − we may obtain dE ( t ) dt ≤ α (cid:107) β i (cid:107) . Let M β > (cid:107) β i ( t ) (cid:107) ≤ M β for all i and t ∈ [0 , T ∗ ]. We obtain (cid:16) E ( t ) − E (0) (cid:17) ≤ − αM β T ∗ . And we deduce that (cid:107) v i (cid:107) ≤ (cid:16) E (0) + 2 − αM β T ∗ (cid:17) < ∞ . We then deduce the upper bound of the vector position x i ( t ) as (cid:107) x i ( t ) (cid:107) ≤ (cid:16) E (0) + 2 − αM β T ∗ (cid:17) T ∗ + (cid:107) x i (0) (cid:107) . Since the solution remains in a compact set for t ∈ [0 , T ∗ ) we obtain a contradiction. References [1] Shin Mi Ahn and Seung-Yeal Ha. “Stochastic flocking dynamics of the Cucker–Smalemodel with multiplicative white noises”. In:
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