A. S. Morse Yale University University of Minnesota June 2, 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A AA A A AAAA A IMA Short Course Distributed Optimization and Control Flocking Asynchronously in Continuous Time

Each agent’s heading is updated at the same time as the rest using a local rule based on the average of its own current heading plus the headings of its “neighbors.”  i = heading ii s = speed s Vicsek et al. simulated a flock of n agents {particles} all moving in the plane at the same speed s, but with different headings  1,  2, ….  n at the same time as the rest Suppose agent’s clocks are not synchronized – what happens?

an d up d a t es i t s h ea di ngmono t on i ca ll yon ( t i k ; t i ( k + 1 ) ] f rom µ i ( t i k ) t o w i ( t i k ) : A t t i k agen t i compu t es i t s k t h way-po i n t A gen t i 0 seven tt i mes = 0 ; t i 1 ; t i 2 ;::: assume d t osa t i s f y U p d a t e R u l e f or A gen t i 0 s H ea di ng µ i E ven tt i mesno t necessar il yeven l yspace d orsync h ron i ze d w i t h o t h eragen t s even tt i mes. Indices of neighbors of agent i at time t ik

ii ti1ti1 ti2ti2 ti3ti3 ti4ti4 ti5ti5 waypoint

T = {0, t 1, t 2,... } ordered set of event times of all n agents. N i (t ik ) = set of labels of agent i’s neighbors at time t ik Thus each agent’s neighbors are defined at all of its own event times. To state the convergence result, we stipulate that each agent i has only itself as a neighbor at each time in T which is not at event time of agent i. N i (t) = {i} for any t 2 T which is not an event time of agent i. Thus each N i (t) is well defined for all t 2 T Extended neighbor graph E(t) is the neighbor graph of index sets N 1 (t), N 2 (t),..., N n (t) t 2 T. This has no effect on the update rules.

Extended neighbor graph E(t) at a time t which is an event time of only agents 1 and 3. Note that agents 2 and 4 have only themselves as neighbors.

along which the sequence of neighbor graphs N(0), N(1), …. is repeatedly jointly rooted, there is a constant  ss to which each agent’s heading  i converges exponentially fast. For any trajectory of the synchronous system Synchronous Case: CONVERGENCE Asynchronous Case: How can one prove this? along which the sequence of extended neighbor graphs E(0), E(t 1 ), …. is repeatedly jointly rooted, there is a constant  ss to which each agent’s heading  i converges exponentially fast. For any trajectory on T of the asynchronous system i 2 {1, 2, …,n}

ii ti1ti1 ti2ti2 ti3ti3 ti4ti4 ti5ti5 0 1  First develop a more explicit model

ii ti1ti1 ti2ti2 ti3ti3 ti4ti4 ti5ti5 0 1 

ii ti1ti1 ti2ti2 ti3ti3 ti4ti4 ti5ti5 0 1 

Can combine agent i’s two update equations to get the familiar update equation This formula tells how  i evolves only on agent i’s event time set. But to use this formula we need to know values of the  j at agent i’s event times In the synchronous case where event times are the same for all agents, the t ik are independent of i, and the preceding update equations are sufficient. For the asynchronous case a common time scale is needed …..

A Common Time Scale T = set of all event times t ik of all n agents Re-label the elements of T as t 0, t 1, t 2, … where t 0 = 0 and t  < t  +1 for  2 {0, 1, 2, …}

t 11 t 12 t 15 t 16 t 13 t 14 t 26 t 25 t 24 t 21 t 27 t 23 t 22 agent 2 agent 1 interacting

agent 2 agent 1 t 11 t 12 t 15 t 16 t 13 t 14 t 26 t 25 t 24 t 21 t 27 t 23 t 22 t3t3 t5t5 t 10 t 13 t7t7 t8t8 t 11 t9t9 t6t6 t1t1 t 12 t4t4 t2t2 T =

The n mutually unsynchronized processes below, P 1, P 2, …P n together constitute the asynchronous system to be analyzed via “analytic synchronization.” Merge all event time sequences into a single ordered sequence T. Analytic Synchronization At times in T between two successive event times in T i, define the state of P i to be constant at the same value as at the first of these two event times. Define the “synchronized state” of P i at event times t  2 to be the original unsynchronized state of P i at these times plus possibly some additional variables. Analyze the synchronous system S comprised of the n synchronized P i i 2 {1, 2, …,n}

Synchronizing P i Can show that these variable evolve on all of T as i 2 {1, 2, …,n} For all times t k 2 T = {t 0, t 1,.... } between agent i’s qth and (q +1)th event times t iq and t i(q+1) respectively, including time t iq, define Can you do this? where T i is the set of event times of agent i

Defining the Synchronous System S Comprised of the n Synchronized P i stochastic matrix S Asynchronous flocking matrix

R = set of all lists of n real numbers r = {r 1, r 2, …., r n } where r i 2 [0, 1] B = set of all lists of n integers b = {b 1, b 2, …., b n } where b i 2 {0, 1} Asynchronous Flocking Matrices Note that the set of all asynchronous flocking matrices, namely image of F  is compact because R is closed. G sa = set of all self arced directed graphs with n vertices F : G sa £ R £ B ! set of all 2n £ 2n stochastic matrices where It is possible to construct a function, which is continuous on R, such that

Example Suppose t k = T is an event time of agents 2 and 3 in a 4 agent network Suppose the extended neighbor graph E(T) is

µ 1 µ 2 µ 3 µ 4 w 1 w 2 w 3 w 4 1 & 4: 2 & 3:

Not necessarily rooted Vertices without self arcs = ° (F)

For all times t k 2 T = {t 0, t 1,.... } between agent i’s qth and (q +1)th event times t iq and t i(q+1) respectively, including time t iq, we defined Then we defined and asserted that To prove that all  i converge to a common heading  ss can be shown to be equivalent to proving that call 2n entries x i in x converge to  ss. Check this! Summary

To prove that all  i converge to a common heading  ss can be shown to be equivalent to proving that call 2n entries x i in x converge to  ss. Thus as before the problem reduces to determining conditions under which But the graphs of the F(k) do not necessarily have self arcs at all vertices. So the preceding facts about compositions of self-arced graphs do not apply Moreover the convergence condition is stated in terms of sequences of extended neighbor graphs, not sequences of asynchronous flocking matrix graphs.