TUM - How to steer high-dimensional Cucker-Smale systems to … · 2014. 7. 16. ·...
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How to steer high-dimensional Cucker-Smale systems toconsensus using low-dimensional information only
Benjamin Scharf
Technische Universitat Munchen,Department of Mathematics,Applied Numerical Analysis
joint work with Mattia Bongini, Massimo Fornasier, Oliver JungeAIMS Madrid 2014, Special Session 48, July 8, 2014
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Overview
Introduction and known results of the Cucker-Smale modelThe general Cucker-Smale modelPicturesResults on consensusSteering Cucker-Smale model to consensus using sparse control
Dimension reduction of the Cucker-Smale modelHigh dimensional dynamical systemsDimension reduction and Johnson-Lindenstrauss matrices (JLM)Does the low-dimensional system stay close?Can we use low-dimension information to find the right control?
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Intro and known results – The general Cucker-Smale model
The Cucker-Smale model - Introdruction
. . . a dynamical system used to describe the nature of a group ofmoving agents, i.e. birds, formation/evolution of languages etc.
xi (t) = vi (t)
vi (t) = 1N
N∑j=1
a(‖xj(t)− xi (t)‖) · (vj(t)− vi (t)) ,
where x1, . . . , xN , v1, . . . , vN ∈ Rd with given initial values at time 0and a is a non-increasing pos. Lipschitz function.
Cucker and Smale: a(x) =K
(σ2 + x2)β, K , σ > 0, β ≥ 0
Ref.: F. Cucker and S. Smale. Emergent behavior in flocks. IEEE Trans. Automat.Control 52(5):852–862, 2007.F. Cucker and S. Smale. On the mathematics of emergence. Jpn. J. Math. 2(1):197–227,2007.
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Intro and known results – The general Cucker-Smale model
The Cucker-Smale model - Main parameters
To measure the distances of the particles as well as their velocities weintroduce:
X (t) :=1
2N2
N∑i ,j=1
‖xi (t)− xj(t)‖2 =1
N
N∑i=1
‖xi (t)− x(t)‖2 = x2 − x2,
V (t) :=1
2N2
N∑i ,j=1
‖vi (t)− vj(t)‖2 =1
N
N∑i=1
‖vi (t)− v(t)‖2 = v2 − v2
The main question is: Does the system tend to consensus?
? limt→∞
vi (t) = v or equivalently limt→∞
V (t) = 0 ?
This would imply: The system moves as a swarm, i.e.
x(t) ≈ x(t0) + (t − t0)v
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Intro and known results – The general Cucker-Smale model
The Cucker-Smale model - Main parameters
To measure the distances of the particles as well as their velocities weintroduce:
X (t) :=1
2N2
N∑i ,j=1
‖xi (t)− xj(t)‖2 =1
N
N∑i=1
‖xi (t)− x(t)‖2 = x2 − x2,
V (t) :=1
2N2
N∑i ,j=1
‖vi (t)− vj(t)‖2 =1
N
N∑i=1
‖vi (t)− v(t)‖2 = v2 − v2
The main question is: Does the system tend to consensus?
? limt→∞
vi (t) = v or equivalently limt→∞
V (t) = 0 ?
This would imply: The system moves as a swarm, i.e.
x(t) ≈ x(t0) + (t − t0)v
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Intro and known results – Pictures
The Cucker-Smale model - Explosion
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Intro and known results – Pictures
The Cucker-Smale model - Consensus
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Intro and known results – Results on consensus
The Cucker-Smale model - Consensus
Lemma (Lyapunov functional behaviour)
It holds
d
dtV (t) ≤ a
(√2NX (t)
)√V (t) as long as V(t) > 0.
Hence: If X (t) is bounded, the system tends to consensus.
Theorem (Ha, Ha, Kim 2010)
If
∫ ∞√
X (0)a(√
2Nr)
dr ≥√V (0), then lim
t→∞V (t) = 0.
Ref.: S.-Y. Ha, T. Ha, and J.-H. Kim. Emergent behavior of a Cucker-Smale type particlemodel with nonlinear velocity couplings. IEEE Trans. Automat. Control 55(7):1679–1683,2010.
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Intro and known results – Steering Cucker-Smale model to consensus using sparse control
The consensus manifold
Idea: If we are not in the consensus manifold, use (sparse) control
Ref.: M. Caponigro, M. Fornasier, B. Piccoli, and E. Trelat. Sparse stabilization andcontrol of the Cucker-Smale model. Math. Contr. Relat. Field. 3(4):447–466, 2013.
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Intro and known results – Steering Cucker-Smale model to consensus using sparse control
Sparsely Controlled Cucker-Smale system
Goal: Stear the system (using sparse control) to the consensusmanifold and then stop the control.Minimize the time to consensus and the number of agents to act on:
xi (t) = vi (t)
vi (t) = 1N
N∑j=1
a(‖xj(t)− xi (t)‖) · (vj(t)− vi (t)) + ui (t)
using `N1 − `d2 -norm constraint (compare to compressed sensing)
N∑i=1
‖ui (t)‖2 ≤ Θ.
Observe: The control only acts on the most stubborn guy! (shepherddog/Schaferhund strategy)
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Intro and known results – Steering Cucker-Smale model to consensus using sparse control
How to construct controls - Sample and Hold
Sample and Hold idea: First construct solutions with controlsconstant on intervals [kτ, (k + 1)τ ] - time-sparse controls. The`N1 -optimization leads to the sparse control
ui =
−Θv⊥ι
‖v⊥ι ‖
if ι is first i : ‖v⊥i (kτ)‖ = maxj=1,...,N
‖v⊥j (kτ)‖
0 otherwise
Theorem (Caponigro, Fornasier, Piccoli, Trelat 2012)
For every Θ (constraint size) there exists τ0 > 0 such that for all sam-pling times τ ∈ (0, τ0] the sampling solution of the controlled Cucker-Smale system reaches the consensus region in finite time.
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Intro and known results – Steering Cucker-Smale model to consensus using sparse control
An example of sparse control
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Dimension reduction
Table of contents
Introduction and known results of the Cucker-Smale modelThe general Cucker-Smale modelPicturesResults on consensusSteering Cucker-Smale model to consensus using sparse control
Dimension reduction of the Cucker-Smale modelHigh dimensional dynamical systemsDimension reduction and Johnson-Lindenstrauss matrices (JLM)Does the low-dimensional system stay close?Can we use low-dimension information to find the right control?
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Dimension reduction – High dimensional dynamical systems
Introduction
The case N →∞ (large number of agents) is widely considered in theliterature:
� locations, velocities ⇒ density distributions
� dynamical system, ODE ⇒ PDE
Ref.: M. Fornasier and F. Solombrino. Mean-field optimal control, to appear in ESAIM,Control Optim. Calc. Var., arXiv:1306.5913, 2013.
The case d →∞ (high dimension/many coordinates/variables) is ofour interest.
Example: Social movement (panic), Financial movement
Here: x not locations, rather variables/state of the system(Health, pulse, strength; situation of the market, IFO-Index etc.)v describes the movement towards consensus
⇒ Goal: Panic prevention, Black Swan prevention
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Dimension reduction – High dimensional dynamical systems
Introduction
The case N →∞ (large number of agents) is widely considered in theliterature:
� locations, velocities ⇒ density distributions
� dynamical system, ODE ⇒ PDE
Ref.: M. Fornasier and F. Solombrino. Mean-field optimal control, to appear in ESAIM,Control Optim. Calc. Var., arXiv:1306.5913, 2013.
The case d →∞ (high dimension/many coordinates/variables) is ofour interest.
Example: Social movement (panic), Financial movement
Here: x not locations, rather variables/state of the system(Health, pulse, strength; situation of the market, IFO-Index etc.)v describes the movement towards consensus
⇒ Goal: Panic prevention, Black Swan prevention
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Dimension reduction – Dimension reduction and Johnson-Lindenstrauss matrices (JLM)
Reduction of the dimension of Cucker-Smale
Idea: Consider reduction of the system by M ∈ Rk×d (first no control)
∂
∂tMvi (t) =
1
N
N∑j=1
a(‖xj(t)− xi (t)‖) · (Mvj(t)−Mvi (t))
wish∼ 1
N
N∑j=1
a(‖Mxj(t)−Mxi (t)‖) · (Mvj(t)−Mvi (t)) .
Idea: Consider low-dimensional Cucker-Smale system in Rk with(y0,w0) = (Mx0,Mv0) as initial values.
Ref.: M. Fornasier, J. Haskovec and J. Vybiral. Particle systems and kinetic equationsmodeling interacting agents in high dimension. SIAM J. Multiscale Mod. Sim.9(4):1727-1764,2011
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Dimension reduction – Dimension reduction and Johnson-Lindenstrauss matrices (JLM)
Johnson-Lindenstrauss matrices
Lemma (Johnson-Lindenstrauss matrices (JLM))
Let x1, . . . , xN be points in Rd . Given ε > 0, there exists a constant
k0 = O(ε−2 logN), (1)
such that for all integers k ≥ k0 there is a k × d matrix M for which
(1− ε)‖xi‖2 ≤ ‖Mxi‖2 ≤ (1 + ε)‖xi‖2, for all i = 1, . . . ,N.
How to construct a Johnson-Lindenstrauss matrix for N points and ε?
Only stochastic constructions are known (for k0 as in (1))!
� entries aij = ±1 (Bernoulli) or N (0, 1)
� partial Fourier matrix (k of d random rows, mult. columns by ±1)
� ...
These (correctly normalized) matrices work for given points with highprobability. One does not need to know xi !
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Dimension reduction – Dimension reduction and Johnson-Lindenstrauss matrices (JLM)
Johnson-Lindenstrauss matrices
Lemma (Johnson-Lindenstrauss matrices (JLM))
Let x1, . . . , xN be points in Rd . Given ε > 0, there exists a constant
k0 = O(ε−2 logN), (1)
such that for all integers k ≥ k0 there is a k × d matrix M for which
(1− ε)‖xi‖2 ≤ ‖Mxi‖2 ≤ (1 + ε)‖xi‖2, for all i = 1, . . . ,N.
How to construct a Johnson-Lindenstrauss matrix for N points and ε?Only stochastic constructions are known (for k0 as in (1))!
� entries aij = ±1 (Bernoulli) or N (0, 1)
� partial Fourier matrix (k of d random rows, mult. columns by ±1)
� ...
These (correctly normalized) matrices work for given points with highprobability. One does not need to know xi !
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Dimension reduction – Dimension reduction and Johnson-Lindenstrauss matrices (JLM)
A continuous Johnson-Lindenstrauss Lemma
Need the Johnson-Lindenstrauss property for continuous trajectories!
Lemma (Bongini, Fornasier, Scharf 2014 (BFS))
Let ϕ : [0, 1]→ Rd be a Lipschitz function (bound Lϕ), 0 < ε′ < 1/2,δ > 0 and M ∈ Rk×d be a Johnson-Lindenstrauss matrix for
ε =ε′
2and N ≥ 4 · Lϕ ·
√d + 2
δε′
points with high probability. Then for every t ∈ [0, 1] one of thefollowing holds (with the same high probability):
(1− ε′)‖ϕ(t)‖ ≤ ‖Mϕ(t)‖ ≤ (1 + ε′)‖ϕ(t)‖or
‖ϕ(t)‖ ≤ δ and ‖Mϕ(t)‖ ≤ δ.
Remark: Cucker-Smale traj. need not to have bounded curvature.
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Dimension reduction – Dimension reduction and Johnson-Lindenstrauss matrices (JLM)
A continuous Johnson-Lindenstrauss Lemma
Need the Johnson-Lindenstrauss property for continuous trajectories!
Lemma (Bongini, Fornasier, Scharf 2014 (BFS))
Let ϕ : [0, 1]→ Rd be a Lipschitz function (bound Lϕ), 0 < ε′ < 1/2,δ > 0 and M ∈ Rk×d be a Johnson-Lindenstrauss matrix for
ε =ε′
2and N ≥ 4 · Lϕ ·
√d + 2
δε′
points with high probability. Then for every t ∈ [0, 1] one of thefollowing holds (with the same high probability):
(1− ε′)‖ϕ(t)‖ ≤ ‖Mϕ(t)‖ ≤ (1 + ε′)‖ϕ(t)‖or
‖ϕ(t)‖ ≤ δ and ‖Mϕ(t)‖ ≤ δ.
Remark: Cucker-Smale traj. need not to have bounded curvature.
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Dimension reduction – Does the low-dimensional system stay close?
Dimension reduction without control
Theorem (BFS 2014)
� (x , v) = (x1, . . . , xN , v1, . . . , vN): solution of the d-dimensionalCucker-Smale system for given initial values x(0), v(0) ∈ RN×d ,
� M ∈ Rk×d continuous Johnson-Lindenstrauss matrix for‖xi (t)− xj(t)‖ for given ε, δ > 0 and all t ∈ [0,T ]
Then the k(low)-dimensional Cucker-Smale system (y ,w) with initialvalues (y(0),w(0)) = (Mx(0),Mv(0)) stays close to the projectedd(high)-dimensional Cucker-Smale system, i.e.,
‖y(t)−Mx(t)‖+ ‖w(t)−Mv(t)‖ . (ε+ δ)eCt , t ≤ T .
Remark: Fornasier, Haskovec, and Vybiral showed an analog for ageneral class of Cucker-Smale-like systems with bounded curvature.
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Dimension reduction – Does the low-dimensional system stay close?
Low-dim. and projection of high-dim. stay close
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Dimension reduction – Can we use low-dimension information to find the right control?
The plan – dimension reduction with control (i)
Consider the high-dimensional system, given (x(0), v(0)),xi (t) = vi (t)
vi (t) = 1N
N∑j=1
a(‖xj(t)− xi (t)‖) · (vj(t)− vi (t)) + uhi (t)
and the low-dimensional system, (y(0),w(0)) = (Mx(0),Mv(0)),yi (t) = yi (t)
wi (t) = 1N
N∑j=1
a(‖yj(t)− yi (t)‖) · (wj(t)− wi (t)) + u`i (t)
with analog constraints as before.
We want to act sparse and we want to compute the controlled agentonly with low-dimensional information!
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Dimension reduction – Can we use low-dimension information to find the right control?
The plan – dimension reduction with control (ii)
1. project the high-dimensional initial values with JLM M tolow-dimension, (y(0),w(0)) = (Mx(0),Mv(0))
2. choose the index of sparse control (one agent) from thelow-dimensional system, apply it to both systems!
u`i =
−Θw⊥ι
‖w⊥ι ‖
if ι is first i : ‖w⊥i (kτ)‖ = maxj=1,...,N
‖w⊥j (kτ)‖
0 otherwise
uhi =
−Θv⊥ι
‖v⊥ι ‖
if i = ι
0 otherwise
3. show: if M is a suitable (continuous) JLM, then� the systems stay close to each other (as before in the projected sense)
and� the control is reasonable for both systems (decays of W (t) and V (t)
are fast enough).
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Dimension reduction – Can we use low-dimension information to find the right control?
The main result
Theorem (BFS 2014)
Let M ∈ Rk×d and Θ > 0 (arbitrary constraint)
� (x , v) and (y ,w): solutions of the d-dimensional andk-dimensional (so) controlled Cucker-Smale systems, initial values(x(0), v(0)) and (Mx(0),Mv(0)), resp. ,
� assume M is continuous Johnson-Lindenstrauss matrix for‖xi (t)− xj(t)‖ and ‖vi (t)− vj(t)‖ sufficiently long, ε and δsufficiently (very) small (depending on the initial values and Θ)
Then for every τ ∈ (0, τ0] both so controlled Cucker-Smale systems
1. stay close to each other (in the projected sense),
2. reach consensus manifold in finite time, and
3. have reached consensus manifold, when a certain parameter of thelow-dimensional systems falls below a known threshold.
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Dimension reduction – Can we use low-dimension information to find the right control?
Remarks and open problems
� One can compute the index of the agent to control from lowdimension, but the direction of its control needs to be observedfrom high dimension.
� ε and δ (and everything else) do not depend on d , but heavily on N
� If Θ >> 0 and is constant with N, then ε has to be chosen s.t.
ε ∼ N−µ1 · e−µ2N , µ1, µ2 > 0.
� In the theorem we suppose M is a JLM for the trajectories, but:the initial values of the low-dimensional system and hence thecontrols depend on M⇒ as a worst case scenario we take all possible trajectories intoaccount (exponentially in the number of control switches).
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Dimension reduction – Can we use low-dimension information to find the right control?
Numerics: random N = 10, d = 500, θ = 20, β = 0.65
−20 −15 −10 −5 0 5 10 15 20 25 30−25
−20
−15
−10
−5
0
5
10
15
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Dimension reduction – Can we use low-dimension information to find the right control?
Numerics: random N = 10, d = 500, θ = 20, β = 0.65
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Dimension reduction – Can we use low-dimension information to find the right control?
Numerics: outlier N = 9, d = 100, θ = 5, β = 0.6 (i)
0 2 4 6 8 10
0
2
4
6
8
10
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Dimension reduction – Can we use low-dimension information to find the right control?
Numerics: outlier N = 9, d = 100, θ = 5, β = 0.6 (ii)
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Dimension reduction – Can we use low-dimension information to find the right control?
The End
Room for improvement
Thank you for your attention
e-mail: [email protected]
web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf
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