Infinite Chains of Vehicles

Avraham FeintuchMath Dept, Ben Gurion Univ

Bruce FrancisECE Dept, Univ of Toronto

(submitted to Automatica)

The problem

formation stability of an infinite chain

0 1−1 2

· · · · · ·

Figure 1: Infinite chain of cars

Let p(t) denote the infinite vector of displacements at time t, that is, the components of p(t) arepn(t), n ∈ Z. In the existing literature, e.g., [1], [5], the state space for p(t) has been the Hilbertspace �2 of square-summable sequences, the advantage of this setting being that Fourier transformscan be exploited. But this assumption requires that pn(t)→ 0 as n goes to ±∞, for every t. Thisseems to be an unjustified assumption to make at the start of a stability theory, before anythinghas been proved: If we want to know about the behaviour of p(t) as t→∞ there is no justificationin limiting p(0) to satisfy pn(0) → 0 as n → ±∞. Therefore we take the state space to be theBanach space �∞ of bounded sequences. Then pn(0) can all be of roughly equal magnitude, or theycan be randomly distributed in an interval, etc. The only requirement is that pn(0) lie in someinterval independent of n. The goal of this paper is to develop a stability theory in this context,an �∞ theory, and to show that it is different from the �2 theory. We illustrate with an example.

Example

Suppose each car heads toward the sum of the relative displacements to its two neighbours:

q̇n = (qn+1 − qn) + (qn−1 − qn)= qn+1 + qn−1 − 2qn.

It follows that pn satisfies the same equation:

ṗn = pn+1 + pn−1 − 2pn.

Let p denote the infinite vector of displacements. Thus p = 0 is an equilibrium. If p(0) ∈ �2, itturns out (proved in the paper) that p(t) converges to zero, that is, the cars return to their originalpositions! But why should the infinite chain behave in this way? After all, the cars are not fittedwith global sensors to know where the origin is. This anomaly is caused entirely by taking p(0)in �2. On the other hand, if pn(0) = c for some nonzero c and all n, then the cars will not move.This equilibrium is not captured by the �2 theory, because �2 does not contain nontrivial constantsequences; it is captured by the �∞ theory developed in this paper.

A literature review at this point is difficult because of lack of notation. It is therefore postponeduntil the end of the paper.

2 Preliminaries

The signals that we deal with are denoted, for example, by x(t), where t denotes time and x is avector with an infinite number of components, xn, n ∈ Z. The meaning is that xn(t) is the statevector of car n. For simplicity, the dimension of xn(t) is just 1. Thus for each t, x(t) is the statevector of the entire chain. The simplest situation is where the chain is spatially invariant, aproperty to be defined soon.

2

Scenario 1: nominally stationary masses or cars

Scenario 2: nominally constant speed cars

Motivation for studying this problem

We didn’t know the answer.

Existing formulations seem not right.

A seminar in our physics department.

Warning

This talk has mathematics.

Lots of it.

Problem formulation

0 1−1 2

· · · · · ·

Figure 1: Infinite chain of cars

Let p(t) denote the infinite vector of displacements at time t, that is, the components of p(t) arepn(t), n ∈ Z. In the existing literature, e.g., [1], [5], the state space for p(t) has been the Hilbertspace �2 of square-summable sequences, the advantage of this setting being that Fourier transformscan be exploited. But this assumption requires that pn(t)→ 0 as n goes to ±∞, for every t. Thisseems to be an unjustified assumption to make at the start of a stability theory, before anythinghas been proved: If we want to know about the behaviour of p(t) as t→∞ there is no justificationin limiting p(0) to satisfy pn(0) → 0 as n → ±∞. Therefore we take the state space to be theBanach space �∞ of bounded sequences. Then pn(0) can all be of roughly equal magnitude, or theycan be randomly distributed in an interval, etc. The only requirement is that pn(0) lie in someinterval independent of n. The goal of this paper is to develop a stability theory in this context,an �∞ theory, and to show that it is different from the �2 theory. We illustrate with an example.

Example

Suppose each car heads toward the sum of the relative displacements to its two neighbours:

q̇n = (qn+1 − qn) + (qn−1 − qn)= qn+1 + qn−1 − 2qn.

It follows that pn satisfies the same equation:

ṗn = pn+1 + pn−1 − 2pn.

Let p denote the infinite vector of displacements. Thus p = 0 is an equilibrium. If p(0) ∈ �2, itturns out (proved in the paper) that p(t) converges to zero, that is, the cars return to their originalpositions! But why should the infinite chain behave in this way? After all, the cars are not fittedwith global sensors to know where the origin is. This anomaly is caused entirely by taking p(0)in �2. On the other hand, if pn(0) = c for some nonzero c and all n, then the cars will not move.This equilibrium is not captured by the �2 theory, because �2 does not contain nontrivial constantsequences; it is captured by the �∞ theory developed in this paper.

A literature review at this point is difficult because of lack of notation. It is therefore postponeduntil the end of the paper.

2 Preliminaries

The signals that we deal with are denoted, for example, by x(t), where t denotes time and x is avector with an infinite number of components, xn, n ∈ Z. The meaning is that xn(t) is the statevector of car n. For simplicity, the dimension of xn(t) is just 1. Thus for each t, x(t) is the statevector of the entire chain. The simplest situation is where the chain is spatially invariant, aproperty to be defined soon.

2

positions qn(t)

nominally qn = n

N cars on R, N very large.

Model by N =∞.

0 1−1 2

· · · · · ·

Figure 1: Infinite chain of cars

Let p(t) denote the infinite vector of displacements at time t, that is, the components of p(t) arepn(t), n ∈ Z. In the existing literature, e.g., [1], [5], the state space for p(t) has been the Hilbertspace �2 of square-summable sequences, the advantage of this setting being that Fourier transformscan be exploited. But this assumption requires that pn(t)→ 0 as n goes to ±∞, for every t. Thisseems to be an unjustified assumption to make at the start of a stability theory, before anythinghas been proved: If we want to know about the behaviour of p(t) as t→∞ there is no justificationin limiting p(0) to satisfy pn(0) → 0 as n → ±∞. Therefore we take the state space to be theBanach space �∞ of bounded sequences. Then pn(0) can all be of roughly equal magnitude, or theycan be randomly distributed in an interval, etc. The only requirement is that pn(0) lie in someinterval independent of n. The goal of this paper is to develop a stability theory in this context,an �∞ theory, and to show that it is different from the �2 theory. We illustrate with an example.

Example

Suppose each car heads toward the sum of the relative displacements to its two neighbours:

q̇n = (qn+1 − qn) + (qn−1 − qn)= qn+1 + qn−1 − 2qn.

It follows that pn satisfies the same equation:

ṗn = pn+1 + pn−1 − 2pn.

Let p denote the infinite vector of displacements. Thus p = 0 is an equilibrium. If p(0) ∈ �2, itturns out (proved in the paper) that p(t) converges to zero, that is, the cars return to their originalpositions! But why should the infinite chain behave in this way? After all, the cars are not fittedwith global sensors to know where the origin is. This anomaly is caused entirely by taking p(0)in �2. On the other hand, if pn(0) = c for some nonzero c and all n, then the cars will not move.This equilibrium is not captured by the �2 theory, because �2 does not contain nontrivial constantsequences; it is captured by the �∞ theory developed in this paper.

A literature review at this point is difficult because of lack of notation. It is therefore postponeduntil the end of the paper.

2 Preliminaries

The signals that we deal with are denoted, for example, by x(t), where t denotes time and x is avector with an infinite number of components, xn, n ∈ Z. The meaning is that xn(t) is the statevector of car n. For simplicity, the dimension of xn(t) is just 1. Thus for each t, x(t) is the statevector of the entire chain. The simplest situation is where the chain is spatially invariant, aproperty to be defined soon.

2

dynamically coupled

q̇n = f(qn+1 − qn, qn−1 − qn)

stable or not?

no boundary conditions

0 1−1 2

· · · · · ·

Figure 1: Infinite chain of cars

Let p(t) denote the infinite vector of displacements at time t, that is, the components of p(t) arepn(t), n ∈ Z. In the existing literature, e.g., [1], [5], the state space for p(t) has been the Hilbertspace �2 of square-summable sequences, the advantage of this setting being that Fourier transformscan be exploited. But this assumption requires that pn(t)→ 0 as n goes to ±∞, for every t. Thisseems to be an unjustified assumption to make at the start of a stability theory, before anythinghas been proved: If we want to know about the behaviour of p(t) as t→∞ there is no justificationin limiting p(0) to satisfy pn(0) → 0 as n → ±∞. Therefore we take the state space to be theBanach space �∞ of bounded sequences. Then pn(0) can all be of roughly equal magnitude, or theycan be randomly distributed in an interval, etc. The only requirement is that pn(0) lie in someinterval independent of n. The goal of this paper is to develop a stability theory in this context,an �∞ theory, and to show that it is different from the �2 theory. We illustrate with an example.

Example

Suppose each car heads toward the sum of the relative displacements to its two neighbours:

q̇n = (qn+1 − qn) + (qn−1 − qn)= qn+1 + qn−1 − 2qn.

It follows that pn satisfies the same equation:

ṗn = pn+1 + pn−1 − 2pn.

Let p denote the infinite vector of displacements. Thus p = 0 is an equilibrium. If p(0) ∈ �2, itturns out (proved in the paper) that p(t) converges to zero, that is, the cars return to their originalpositions! But why should the infinite chain behave in this way? After all, the cars are not fittedwith global sensors to know where the origin is. This anomaly is caused entirely by taking p(0)in �2. On the other hand, if pn(0) = c for some nonzero c and all n, then the cars will not move.This equilibrium is not captured by the �2 theory, because �2 does not contain nontrivial constantsequences; it is captured by the �∞ theory developed in this paper.

A literature review at this point is difficult because of lack of notation. It is therefore postponeduntil the end of the paper.

2 Preliminaries

The signals that we deal with are denoted, for example, by x(t), where t denotes time and x is avector with an infinite number of components, xn, n ∈ Z. The meaning is that xn(t) is the statevector of car n. For simplicity, the dimension of xn(t) is just 1. Thus for each t, x(t) is the statevector of the entire chain. The simplest situation is where the chain is spatially invariant, aproperty to be defined soon.

2

equilibria qn+1 − qn = 1

state q = (. . . , q−1|q0, q1, , . . . )

q ∈ R∞

R∞ is a weak topological vector space

For stability analysis,look at displacements awayfrom nominal locations.

0 1−1 2

· · · · · ·

Figure 1: Infinite chain of cars

Let p(t) denote the infinite vector of displacements at time t, that is, the components of p(t) arepn(t), n ∈ Z. In the existing literature, e.g., [1], [5], the state space for p(t) has been the Hilbertspace �2 of square-summable sequences, the advantage of this setting being that Fourier transformscan be exploited. But this assumption requires that pn(t)→ 0 as n goes to ±∞, for every t. Thisseems to be an unjustified assumption to make at the start of a stability theory, before anythinghas been proved: If we want to know about the behaviour of p(t) as t→∞ there is no justificationin limiting p(0) to satisfy pn(0) → 0 as n → ±∞. Therefore we take the state space to be theBanach space �∞ of bounded sequences. Then pn(0) can all be of roughly equal magnitude, or theycan be randomly distributed in an interval, etc. The only requirement is that pn(0) lie in someinterval independent of n. The goal of this paper is to develop a stability theory in this context,an �∞ theory, and to show that it is different from the �2 theory. We illustrate with an example.

Example

Suppose each car heads toward the sum of the relative displacements to its two neighbours:

q̇n = (qn+1 − qn) + (qn−1 − qn)= qn+1 + qn−1 − 2qn.

It follows that pn satisfies the same equation:

ṗn = pn+1 + pn−1 − 2pn.

Let p denote the infinite vector of displacements. Thus p = 0 is an equilibrium. If p(0) ∈ �2, itturns out (proved in the paper) that p(t) converges to zero, that is, the cars return to their originalpositions! But why should the infinite chain behave in this way? After all, the cars are not fittedwith global sensors to know where the origin is. This anomaly is caused entirely by taking p(0)in �2. On the other hand, if pn(0) = c for some nonzero c and all n, then the cars will not move.This equilibrium is not captured by the �2 theory, because �2 does not contain nontrivial constantsequences; it is captured by the �∞ theory developed in this paper.

A literature review at this point is difficult because of lack of notation. It is therefore postponeduntil the end of the paper.

2 Preliminaries

The signals that we deal with are denoted, for example, by x(t), where t denotes time and x is avector with an infinite number of components, xn, n ∈ Z. The meaning is that xn(t) is the statevector of car n. For simplicity, the dimension of xn(t) is just 1. Thus for each t, x(t) is the statevector of the entire chain. The simplest situation is where the chain is spatially invariant, aproperty to be defined soon.

2

pn = qn − n

If p(0) �= 0, what happens to p(t)?

p = (. . . , p−1|p0, p1, . . . )

What’s the proper state space?

It depends.

We’ll look at two, a Hilbert space and a Banach space.

x, xn ∈ R, n ∈ Z

�x, y� =�

xnyn

�x�2 =��

x2n

�1/2

xn → 0, n→ ±∞

�2, Hilbert space

0 1−1 2

· · · · · ·

Figure 1: Infinite chain of cars

Let p(t) denote the infinite vector of displacements at time t, that is, the components of p(t) arepn(t), n ∈ Z. In the existing literature, e.g., [1], [5], the state space for p(t) has been the Hilbertspace �2 of square-summable sequences, the advantage of this setting being that Fourier transformscan be exploited. But this assumption requires that pn(t)→ 0 as n goes to ±∞, for every t. Thisseems to be an unjustified assumption to make at the start of a stability theory, before anythinghas been proved: If we want to know about the behaviour of p(t) as t→∞ there is no justificationin limiting p(0) to satisfy pn(0) → 0 as n → ±∞. Therefore we take the state space to be theBanach space �∞ of bounded sequences. Then pn(0) can all be of roughly equal magnitude, or theycan be randomly distributed in an interval, etc. The only requirement is that pn(0) lie in someinterval independent of n. The goal of this paper is to develop a stability theory in this context,an �∞ theory, and to show that it is different from the �2 theory. We illustrate with an example.

Example

Suppose each car heads toward the sum of the relative displacements to its two neighbours:

q̇n = (qn+1 − qn) + (qn−1 − qn)= qn+1 + qn−1 − 2qn.

It follows that pn satisfies the same equation:

ṗn = pn+1 + pn−1 − 2pn.

Let p denote the infinite vector of displacements. Thus p = 0 is an equilibrium. If p(0) ∈ �2, itturns out (proved in the paper) that p(t) converges to zero, that is, the cars return to their originalpositions! But why should the infinite chain behave in this way? After all, the cars are not fittedwith global sensors to know where the origin is. This anomaly is caused entirely by taking p(0)in �2. On the other hand, if pn(0) = c for some nonzero c and all n, then the cars will not move.This equilibrium is not captured by the �2 theory, because �2 does not contain nontrivial constantsequences; it is captured by the �∞ theory developed in this paper.

A literature review at this point is difficult because of lack of notation. It is therefore postponeduntil the end of the paper.

2 Preliminaries

The signals that we deal with are denoted, for example, by x(t), where t denotes time and x is avector with an infinite number of components, xn, n ∈ Z. The meaning is that xn(t) is the statevector of car n. For simplicity, the dimension of xn(t) is just 1. Thus for each t, x(t) is the statevector of the entire chain. The simplest situation is where the chain is spatially invariant, aproperty to be defined soon.

2

pn = qn − n

If p(0) ∈ �2, then p(t)→ 0.

Also, 1 �∈ �2.

As if there were GPS!

0 1−1

0 1−1

�2 models this

instead of this

So �2 is not right to model N = large but finite.

x, xn ∈ R, n ∈ Z

Alternative

�∞, Banach space

�x, y�, none

�x�∞ = supn |xn|

0 1−1 2

· · · · · ·

Figure 1: Infinite chain of cars

Let p(t) denote the infinite vector of displacements at time t, that is, the components of p(t) arepn(t), n ∈ Z. In the existing literature, e.g., [1], [5], the state space for p(t) has been the Hilbertspace �2 of square-summable sequences, the advantage of this setting being that Fourier transformscan be exploited. But this assumption requires that pn(t)→ 0 as n goes to ±∞, for every t. Thisseems to be an unjustified assumption to make at the start of a stability theory, before anythinghas been proved: If we want to know about the behaviour of p(t) as t→∞ there is no justificationin limiting p(0) to satisfy pn(0) → 0 as n → ±∞. Therefore we take the state space to be theBanach space �∞ of bounded sequences. Then pn(0) can all be of roughly equal magnitude, or theycan be randomly distributed in an interval, etc. The only requirement is that pn(0) lie in someinterval independent of n. The goal of this paper is to develop a stability theory in this context,an �∞ theory, and to show that it is different from the �2 theory. We illustrate with an example.

Example

Suppose each car heads toward the sum of the relative displacements to its two neighbours:

q̇n = (qn+1 − qn) + (qn−1 − qn)= qn+1 + qn−1 − 2qn.

It follows that pn satisfies the same equation:

ṗn = pn+1 + pn−1 − 2pn.

Let p denote the infinite vector of displacements. Thus p = 0 is an equilibrium. If p(0) ∈ �2, itturns out (proved in the paper) that p(t) converges to zero, that is, the cars return to their originalpositions! But why should the infinite chain behave in this way? After all, the cars are not fittedwith global sensors to know where the origin is. This anomaly is caused entirely by taking p(0)in �2. On the other hand, if pn(0) = c for some nonzero c and all n, then the cars will not move.This equilibrium is not captured by the �2 theory, because �2 does not contain nontrivial constantsequences; it is captured by the �∞ theory developed in this paper.

A literature review at this point is difficult because of lack of notation. It is therefore postponeduntil the end of the paper.

2 Preliminaries

The signals that we deal with are denoted, for example, by x(t), where t denotes time and x is avector with an infinite number of components, xn, n ∈ Z. The meaning is that xn(t) is the statevector of car n. For simplicity, the dimension of xn(t) is just 1. Thus for each t, x(t) is the statevector of the entire chain. The simplest situation is where the chain is spatially invariant, aproperty to be defined soon.

2

pn = qn − n

If p(0) ∈ �∞, does p(t) converge?

In general, no.

�∞ right, at least better

�2 wrong, but easy

Conclusion

but hard

Details

A simpler problem: rendezvous

Rendezvous: meet at the same point

Jadbabaie, Lin, Morse, 2002: heading angles “rendezvous”

Lin, Morse, Anderson, 2005

Cortés, Martínez, Bullo, 2005, ...

Lin, Broucke, Maggiore, Francis, 2004 - 2007

Ji, Egerstedt, 2007

Serial pursuit

In �2, rendezvous at the origin.

q̇n = qn−1 − qn

y = Ux, yn = xn−1

q̇ = (U − I)q

U ∈ B(�2)

q(t) = e(U−I)tq(0)

U : (. . . , x−1|x0, x1, . . . ) �→ (. . . , x−2|x−1, x0, . . . )

resolvent of A: λ such that (λI −A)−1 ∈ B(�2)

spectrum of A

eigenvalue of A

spectrum of U : unit circle, no eigenvalues, zero kernel

Ax = λx, x �= 0

U : x �→ y, y(t) = x(t− T )

L2(R) −→ L2(R)

L2(jR) −→ L2(jR)Û : X �→ Y, Y (jω) = e−jωT X(jω)

Proof for continuous time

Show that the spectrum of Û is the unit circle.

L2(jR) −→ L2(jR)Û : X �→ Y, Y (jω) = e−jωT X(jω)

(λI − Û)−1 : Y �→ X, X(jω) = 1λ− e−jωT Y (jω)

(λI − Û)−1 is bounded iff |λ| �= 1

q̇ = (U − I)q

q(t) = e(U−I)tq(0)

spectrum of U − I

e(U−I)t �→ 0

�e(U−I)t� = 1, ∀t

Nevertheless, e(U−I)tq(0) converges to 0.

The proof

x ∈ �2

X ∈ L2(S1)

F : x �→ X, X(ejω) =�

n

xne−jωn

Q(ejω, t) =�

n

qn(t)e−jωn

q̇ = (U − I)q, q(t) ∈ �2, ∀t

F q̇ = F (U − I)q

= (e−jω − 1)Fq

= (cos ω − 1− j sin ω)Fq

Q̇ = (cos ω − 1− j sin ω)Q

Q(t) = e(cos ω−1)te(−j sin ω)tQ(0)

L2(S1) ⊂ L1(S1)

�Q(t)�1 ≤ �e(cos ω−1)t�2�Q(0)�2

Q(t) = e(cos ω−1)te(−j sin ω)tQ(0)

≤�

12π

e−2tI0(2t)�1/2

�Q(0)�2

qn(t) =12π

� π

−πQ(ejω, t)ejωndω

�q(t)�∞ ≤ �Q(t)�1

→ 0QED

Thx to Wolfram Alpha

Turn to serial pursuit in �∞

Every point is a possible rendezvous point.

Diaconis and Stein, 1978

A ⊂ Z+toss a coin n times

Sn is the number of heads

When does Pr(Sn ∈ A) converge as n→∞?

Tauberian theory

Consider serial pursuit among qn, n ≥ 1.

Fix a set A ⊂ Z+.

Initialize qn(0) = 1 if n ∈ A, qn(0) = 0 else.

q1(t) converges as t→∞Then

iff

Pr(Sn ∈ A) converges as n→∞.

From this theory, we can get a q(0) ∈ �∞

for which q(t) does not converge.

12

4

8

16

0

But ...

q̇ = (U − I)q

q(t) = e(U−I)tq(0)

spectrum of U − I

�(U − I)e(U−I)t� → 0

B(�∞)-norm

q(t)→ span(1)

Proof that the Hilbert space infinite chain is stable, but the tails are as though they were tethered.

Example that the Banach space infinite chain may not converge as desired, but does converge in a weak sense.

Summary of results

History

Newton Principia 1687

speed of sound in an elastic medium

spring-mass model of medium

Mq̈n = K[qn+1 − (qn + d)]−K[qn − (qn−1 + d)]

pn = qn − nd

Mp̈n = K(pn+1 + pn−1 − 2pn)

pn(t) = p(nd, t)

Mp̈n = K(pn+1 + pn−1 − 2pn)

pn(t) = p(nd, t)

M∂2p

∂t2(x, t) = Kd2

∂2p

∂x2(x, t)

V =

�Kd

ρ

∂2p

∂t2= V 2

∂2p

∂x2V 2 =

Kd

M/d

ρ = mass density

Brillouin: Wave Propagation in Periodic Structures

But it is not rigorous.

“Rigorous discussion for the case of interactionsbetween nearest neighbours only”

Physics: derivations but not proofs.

1. Levine and Athans 19672. Melzer and Kuo 19713. Bamieh, Paganini, Dahleh 20024. D’Andrea and Dullerud 20035. Motee and Jadbabaie 20086. Curtain 2009 (errors in 5)7. Curtain, Iftime, Zwart 2010 (errors in 3)

p ∈ �2In all these, .

There are lots of open questions in .�∞

Thanks for your attention.