TUM - How to steer high-dimensional Cucker-Smale systems to … · 2014. 7. 16. · [email protected] joint work with Mattia Bongini, Massimo Fornasier, Oliver Junge AIMS

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

[email protected]

joint work with Mattia Bongini, Massimo Fornasier, Oliver JungeAIMS Madrid 2014, Special Session 48, July 8, 2014

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?

Benjamin Scharf Control of high-dim. Cucker-Smale systems using low-dim. information 2 of 27

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.



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



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


Intro and known results – Pictures

The Cucker-Smale model - Explosion


Intro and known results – Pictures

The Cucker-Smale model - Consensus


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.


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.



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)



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.



An example of sparse control


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?


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


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


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



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 !



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 !



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.



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.


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.


Dimension reduction – Does the low-dimensional system stay close?

Low-dim. and projection of high-dim. stay close


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!



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).



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.



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).



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



Numerics: random N = 10, d = 500, θ = 20, β = 0.65



Numerics: outlier N = 9, d = 100, θ = 5, β = 0.6 (i)

0 2 4 6 8 10

0

2

4

6

8

10



Numerics: outlier N = 9, d = 100, θ = 5, β = 0.6 (ii)



The End

Room for improvement

Thank you for your attention

e-mail: [email protected]

web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf


[email protected]

http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf

Documents

TUM - How to steer high-dimensional Cucker-Smale systems to … · 2014. 7. 16. · [email protected] joint work with Mattia Bongini, Massimo Fornasier, Oliver Junge AIMS