infinite chains of vehicles avraham feintuch bruce francisscg.utoronto.ca/~francis/yale_12.pdf ·...
TRANSCRIPT
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Infinite Chains of Vehicles
Avraham FeintuchMath Dept, Ben Gurion Univ
Bruce FrancisECE Dept, Univ of Toronto
(submitted to Automatica)
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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
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Motivation for studying this problem
We didn’t know the answer.
Existing formulations seem not right.
A seminar in our physics department.
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Warning
This talk has mathematics.
Lots of it.
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Problem formulation
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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
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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, . . . )
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What’s the proper state space?
It depends.
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We’ll look at two, a Hilbert space and a Banach space.
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x, xn ∈ R, n ∈ Z
�x, y� =�
xnyn
�x�2 =��
x2n
�1/2
xn → 0, n→ ±∞
�2, Hilbert space
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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!
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0 1−1
0 1−1
�2 models this
instead of this
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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.
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�∞ right, at least better
�2 wrong, but easy
Conclusion
but hard
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Details
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A simpler problem: rendezvous
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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
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Serial pursuit
In �2, rendezvous at the origin.
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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, . . . )
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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
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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.
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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
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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
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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.