Sparse Stabilization of Dynamical Systems driven by ...€¦Attraction and Avoidance Forces Mattia...

Sparse Stabilization of Dynamical Systems driven byAttraction and Avoidance Forces

Mattia Bongini

Technische Universitat Munchen,Department of Mathematics,

Chair of Applied Numerical Analysis

[email protected]

OCIP 2014TUM, Munich

March 3-5, 2014

June 15, 2014Mattia Bongini Sparse Stabilization of Dynamical Systems driven by Attraction and Avoidance Forces 1 of 22

Introduction

Large particle systems arise in many modern applications:

Large Facebook “friendship” network

Dynamical data analysis: R. palustris

protein-protein interaction network

Image halftoning via variational dithering

Computational chemistry: molecule simulation

Mattia Bongini Sparse Stabilization of Dynamical Systems driven by Attraction and Avoidance Forces 2 of 22

Split coherence in homophilious societies:government?

� A society is said to be homophilious whenever its agents aresharply more influenced by near agents than far ones;

� In homophilious societies, global self-organization can be expectedas soon as enough initial coherence is reached (Cucker and Smale2007 – consensus emergence);

� However, it is common experience that coherence in a homophilioussociety can be lost, leading sometimes to dramatic consequences,questioning strongly the role and the effectiveness of governments.

Question: can a government endowed with limited resourcesrescue/stabilize a society by minimal interventions? Which ones?

A framework for consensus emergence

The Cucker-Smale model is one of the most famous models for socialdynamics.

xi = vi ∈ Rd

N∑j=1

a(‖xi − xj‖2

)(vj − vi ) ∈ Rd

where a(t) := aβ(t) = 1(1+t2)β

, β > 0 models the exchange of

information between agents.

� β ≤ 12 heterophilious society ⇒ unconditional consensus,

� β > 12 homophilious society ⇒ consensus conditional to initial

coherence.

xi = vi ∈ Rd

N∑j=1

a(‖xi − xj‖2

)(vj − vi ) ∈ Rd

coherence.

xi = vi ∈ Rd

N∑j=1

a(‖xi − xj‖2

)(vj − vi ) ∈ Rd

coherence.

Homophilious societies are sparsely stabilizable

� The work Caponigro-Fornasier-Piccoli-Trelat shows that, in theregime of homophilious society (β > 1

2 ) the Cucker-Smale systemxi = vi

N∑j=1

a(‖xi − xj‖2

)(xj − xi ) + ui

can be stabilized to consensus by using only sparse controls, i.e.controls which are zero for almost every agent.

If, on the one side, the homophilious character of a society playsagainst its coherence, on the other side, it plays at its advantage

if we allow for sparse external intervention.

� Explains the effectiveness of parsimonious interventions ofgovernments in societies.

N∑j=1

a(‖xi − xj‖2

)(xj − xi ) + ui

N∑j=1

a(‖xi − xj‖2

)(xj − xi ) + ui

Dynamical systems driven by attraction and repulsionforces

The Cucker-Dong model: for every 1 ≤ i ≤ Nxi = vi ∈ Rd

vi = −bivi +N∑j=1

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1

f(‖xi − xj‖2

)(xi − xj) ∈ Rd

where� bi : [0,+∞)→ [0,Λ] is the friction acting on the system,� a : [0,+∞)→ [0,+∞) is the rate of communication,� f : (0,+∞)→ (0,+∞) such that∫ +∞

δf (r) dr <∞ for every δ > 0,

∫ +∞

0f (r) dr = +∞

models the repulsion between agents.

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1

f(‖xi − xj‖2

)(xi − xj) ∈ Rd

where� bi : [0,+∞)→ [0,Λ] is the friction acting on the system,

� a : [0,+∞)→ [0,+∞) is the rate of communication,� f : (0,+∞)→ (0,+∞) such that∫ +∞

∫ +∞

0f (r) dr = +∞

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1

f(‖xi − xj‖2

)(xi − xj) ∈ Rd

where� bi : [0,+∞)→ [0,Λ] is the friction acting on the system,� a : [0,+∞)→ [0,+∞) is the rate of communication,

� f : (0,+∞)→ (0,+∞) such that∫ +∞

∫ +∞

0f (r) dr = +∞

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1

f(‖xi − xj‖2

)(xi − xj) ∈ Rd

where� bi : [0,+∞)→ [0,Λ] is the friction acting on the system,� a : [0,+∞)→ [0,+∞) is the rate of communication,� f : (0,+∞)→ (0,+∞) such that∫ +∞

∫ +∞

0f (r) dr = +∞

Example: Lennard-Jones potential

� It is the potential of the Van der Waals force.

� It can be seen as a Cucker-Dong system with

a(r) =σar7

and f (r) =σfr13

Difference f (r)− a(r) for Lennard-Jones potentials.

� It is the potential of the Van der Waals force.� It can be seen as a Cucker-Dong system with

a(r) =σar7

and f (r) =σfr13

� It is the potential of the Van der Waals force.� It can be seen as a Cucker-Dong system with

a(r) =σar7

and f (r) =σfr13

Total Energy of Cucker-Dong Systems

We introduce

� the kinetic energy K (t) := 12

∑Ni=1 ‖vi (t)‖2,

� the potential energy

P(t) :=1

N∑i,j=1

∫ ‖xi (t)−xj (t)‖2

0a(r)dr +

N∑i,j=1

∫ ∞‖xi (t)−xj (t)‖2

f (r)dr ,

� the total energy E (t) := K (t) + P(t).

Proposition

If the system is frictionless (bi ≡ 0) then for every t ≥ 0, ddtE (t) = 0.

We introduce

∑Ni=1 ‖vi (t)‖2,

P(t) :=1

N∑i,j=1

∫ ‖xi (t)−xj (t)‖2

0a(r)dr +

N∑i,j=1

∫ ∞‖xi (t)−xj (t)‖2

f (r)dr ,

Proposition

We introduce

∑Ni=1 ‖vi (t)‖2,

P(t) :=1

N∑i,j=1

∫ ‖xi (t)−xj (t)‖2

0a(r)dr +

N∑i,j=1

∫ ∞‖xi (t)−xj (t)‖2

f (r)dr ,

Proposition

We introduce

∑Ni=1 ‖vi (t)‖2,

P(t) :=1

N∑i,j=1

∫ ‖xi (t)−xj (t)‖2

0a(r)dr +

N∑i,j=1

∫ ∞‖xi (t)−xj (t)‖2

f (r)dr ,

Proposition

Conditional consensus emergence

Theorem (Cucker - Dong)Consider a population of N agents modeled by a Cucker-Dong systemwith a(t) := aβ(t) = 1

(1+t2)β, β > 0

‖xi (0)− xj(0)‖ > 0 for all i 6= j .

Then there exists a unique solution (x(t), v(t)) of the system withinitial state (x(0), v(0)). Moreover if one of the two followinghypotheses holds:

1. β ≤ 1

2. β > 1 and E (0) < ϑ := (N − 1)∫∞

0 a(r)dr

then the population is cohesive and collision-avoiding, i.e., there existtwo constants B0 and b0 > 0 such that, for every t ≥ 0

b0 ≤ ‖xi (t)− xj(t)‖ ≤ B0 for all 1 ≤ i 6= j ≤ N.

(1+t2)β, β > 0

‖xi (0)− xj(0)‖ > 0 for all i 6= j .

1. β ≤ 1

2. β > 1 and E (0) < ϑ := (N − 1)∫∞

0 a(r)dr

(1+t2)β, β > 0

‖xi (0)− xj(0)‖ > 0 for all i 6= j .

1. β ≤ 1

2. β > 1 and E (0) < ϑ := (N − 1)∫∞

0 a(r)dr

(1+t2)β, β > 0

‖xi (0)− xj(0)‖ > 0 for all i 6= j .

1. β ≤ 1

2. β > 1 and E (0) < ϑ := (N − 1)∫∞

0 a(r)dr

Non-consensus events are possible

� We call the conclusion of the Cucker-Dong Theorem the consensusstate for the system.

� The theorem says that if the system is heterophilious, theconsensus state naturally occurs.

� If β > 1 then the consensus state is not reached by all(x0, v0) ∈ (Rd)N × (Rd)N , as proved by Cucker and Dong.

� Indeed, the condition E (0) < ϑ can be violated in three cases:� the agents have too high initial speed ⇒ K explodes;� there are two or more very near agents ⇒ P explodes.� a big majority of the agents are very far from each other.

� Indeed, the condition E (0) < ϑ can be violated in three cases:

� the agents have too high initial speed ⇒ K explodes;� there are two or more very near agents ⇒ P explodes.� a big majority of the agents are very far from each other.

� Indeed, the condition E (0) < ϑ can be violated in three cases:� the agents have too high initial speed ⇒ K explodes;

� there are two or more very near agents ⇒ P explodes.� a big majority of the agents are very far from each other.

� Indeed, the condition E (0) < ϑ can be violated in three cases:� the agents have too high initial speed ⇒ K explodes;� there are two or more very near agents ⇒ P explodes.

� a big majority of the agents are very far from each other.

Non-consensus events need intervention

� Assume we are in the case β > 1 and E (0) ≥ ϑ. Can we againstabilize the society by external parsimonious intervention?

� We introduce a control term inside the modelxi = vi

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

where u1, . . . , uN : [0,+∞)→ (Rd)N are measurable functionssatisfying

N∑i=1

‖ui (t)‖ ≤ M

for every t ≥ 0, for a given constant M > 0.

Non-consensus events need intervention

� Assume we are in the case β > 1 and E (0) ≥ ϑ. Can we againstabilize the society by external parsimonious intervention?

� We introduce a control term inside the modelxi = vi

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

where u1, . . . , uN : [0,+∞)→ (Rd)N are measurable functionssatisfying

N∑i=1

‖ui (t)‖ ≤ M

for every t ≥ 0, for a given constant M > 0.

Consequences of the introduction of control

Proposition

Assume bi ≡ 0. The total energy is no more a conserved quantity. Inparticular

dtE (t) = 2 〈u(t), v(t)〉 .

� This form of the energy dissipation suggests controls only actingon the kinetic part of the energy:

ui (t) = −αivi (t)

‖vi (t)‖, αi ≥ 0.

� The `N1 − `d2 constraint maximizes the sparsity of ui , i.e. αi = 0 foralmost every i .

Proposition

dtE (t) = 2 〈u(t), v(t)〉 .

‖vi (t)‖, αi ≥ 0.

Proposition

dtE (t) = 2 〈u(t), v(t)〉 .

‖vi (t)‖, αi ≥ 0.

Introducing the sparse control

DefinitionLet 0 ≤ ε ≤ M

E(0) and t ≥ 0. We define the sparse feedback control

with strength ε u(t) ∈ (Rd)N as

ui (t) =

{−εE (t) vi (t)

‖vi (t)‖ if i = ι(t)

0 if i 6= ι(t)

where ι(t) ∈ {1, . . . ,N} is the minimum index such that

‖vι(t)(t)‖ = maxj=1,...,N ‖vj(t)‖.

Hence the control acts on the most“stubborn” agent at every time. We maycall this control the “shepherd dogstrategy”.

Introducing the sparse control

DefinitionLet 0 ≤ ε ≤ M

E(0) and t ≥ 0. We define the sparse feedback control

with strength ε u(t) ∈ (Rd)N as

ui (t) =

{−εE (t) vi (t)

‖vi (t)‖ if i = ι(t)

0 if i 6= ι(t)

where ι(t) ∈ {1, . . . ,N} is the minimum index such that

‖vι(t)(t)‖ = maxj=1,...,N ‖vj(t)‖.

Hence the control acts on the most“stubborn” agent at every time. We maycall this control the “shepherd dogstrategy”.

Aims of the work

We want to show that

� if E (0) > ϑ and E (0) ≈ ϑ =⇒ there is a sampled sparse strategyas before which steers the system to consensus in finite time,

� the sparse control minimizes ddtE (t) in a very large set U of

controls satisfying the `N1 − `d2 constraint,� for some u ∈ U, there exists a solution of the system

xi = vi

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + u.

Aims of the work

controls satisfying the `N1 − `d2 constraint,

� for some u ∈ U, there exists a solution of the systemxi = vi

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + u.

Aims of the work

controls satisfying the `N1 − `d2 constraint,� for some u ∈ U, there exists a solution of the system

xi = vi

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + u.

Sampling and hold

Strategy of proof: we will follow a sampling-and-hold approach as inCaponigro-Fornasier-Piccoli-Trelat.

� We will take a sampling time τ and consider the systemxi = vi

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

such that the control satisfies ui (t) = ui (kτ) for everyt ∈ [kτ, (k + 1)τ ], k ∈ N, where u is the sparse control;

� if τ is sufficiently small we avoid chattering phenomena;

� if the control is sufficiently strong (i.e. the parameter ε issufficiently large) the system is steered to satisfy E (t) < ϑ in finitetime.

Sampling and hold

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

Sampling and hold

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

Sampling and hold

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

Sampled sparse strategies drives the system toconsensus

Main Theorem (B. - Fornasier)

Fix M > 0. Let (x0, v0) ∈ (Rd)N × (Rd)N be such that the followinghold:

1. ‖x0i − x0j‖ > 0 for all i 6= j ,

2. ‖ 1N

∑Ni=1 vi (0)‖ > 0,

3. E (0) ≥ ϑ > E (0) exp

(−2√

M‖ 1N

∑Ni=1 vi (0)‖3

E(0)√

E(0)+MN

Then there exist τ0 > 0, L > 0 and T > 0 such that the samplingsolution of the Cucker-Dong system associated with the sparse controlu with strength ε ≥ L, the sampling time τ ≤ τ0 and initial datum(x0, v0) reaches the consensus region in finite time T .

1. ‖x0i − x0j‖ > 0 for all i 6= j ,

2. ‖ 1N

∑Ni=1 vi (0)‖ > 0,

3. E (0) ≥ ϑ > E (0) exp

(−2√

M‖ 1N

∑Ni=1 vi (0)‖3

E(0)√

E(0)+MN

1. ‖x0i − x0j‖ > 0 for all i 6= j ,

2. ‖ 1N

∑Ni=1 vi (0)‖ > 0,

3. E (0) ≥ ϑ > E (0) exp

(−2√

M‖ 1N

∑Ni=1 vi (0)‖3

E(0)√

E(0)+MN

1. ‖x0i − x0j‖ > 0 for all i 6= j ,

2. ‖ 1N

∑Ni=1 vi (0)‖ > 0,

3. E (0) ≥ ϑ > E (0) exp

(−2√

M‖ 1N

∑Ni=1 vi (0)‖3

E(0)√

E(0)+MN

Enlarging the set of admissible controls

� The above result cannot be used to prove directly the existence ofa solution for controlled Cucker-Dong systems, because if we let τin the Main Theorem go to 0 we usually do not obtain a sparsecontrol.

� We thus need to enlarge the set of admissible control to obtain anexistence result with this argument.

� Define for every t > 0 the set

K(t) :=

{u ∈ (Rd)N |

N∑i=1

‖ui‖ ≤ M · E(t)E(0)

and for every t > 0 and q > 0 the functional Jt,q : (Rd)N → R

Jt,q(u) = 〈v(t), u〉+

∥∥∥ 1N

∑Ni=1 vi (0)

∥∥∥q

N∑i=1

‖ui‖ .

K(t) :=

{u ∈ (Rd)N |

N∑i=1

‖ui‖ ≤ M · E(t)E(0)

Jt,q(u) = 〈v(t), u〉+

∥∥∥ 1N

∑Ni=1 vi (0)

∥∥∥q

N∑i=1

‖ui‖ .

K(t) :=

{u ∈ (Rd)N |

N∑i=1

‖ui‖ ≤ M · E(t)E(0)

Jt,q(u) = 〈v(t), u〉+

∥∥∥ 1N

∑Ni=1 vi (0)

∥∥∥q

N∑i=1

‖ui‖ .

Existence of solutions

Theorem (B. - Fornasier)

If the hypotheses of the Main Theorem are satisfied, then there existT > 0 and q > 0 such that

� the sparse feedback control belongs to the set argminu∈K(t) Jt,q(u)for every t ≤ T ;

� there exists a solution of the systemxi = vi

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

associated to a control u ∈ argminu∈K(t) Jt,q(u) for every t ≤ T .

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

a(‖xi − xj‖2

)(xj − xi ) +

N∑j=1j 6=i

f(‖xi − xj‖2

)(xi − xj) + ui

Exponential decay rate of the energy

Theorem (B. - Fornasier)Suppose we are under the assumptions of the Main Theorem. Thesparse feedback control is then an instantaneous minimizer of thefunctional

D(t, u) =d

dtE (t)

over all possible feedback controls in argminu∈K(t) Jt,q(u).

Moreover for the sparse feedback control strategy we have for everyt ≥ 0,

E (t) ≤ E (0)e−

2‖ 1N

∑Ni=1 vi (0)‖E(0)

Exponential decay rate of the energy

Theorem (B. - Fornasier)Suppose we are under the assumptions of the Main Theorem. Thesparse feedback control is then an instantaneous minimizer of thefunctional

D(t, u) =d

dtE (t)

over all possible feedback controls in argminu∈K(t) Jt,q(u).Moreover for the sparse feedback control strategy we have for everyt ≥ 0,

E (t) ≤ E (0)e−

2‖ 1N

∑Ni=1 vi (0)‖E(0)

Summing up our results

� Again we have proven that an homophilious society can bestabilized by parsimonious intervention;

� the sparse control strategy is the most efficient: we pay ourattention solely to the “most stubborn” agent while leaving theother free to adjust themselves;

� in contrast to what happen with the Cucker-Smale model, ourresult is conditional (it depends on the initial conditions of thesystem)⇒ we don’t know if the conditions are necessary.

� in contrast to what happen with the Cucker-Smale model, ourresult is conditional (it depends on the initial conditions of thesystem)

⇒ we don’t know if the conditions are necessary.

A numerical experiment

Consider a frictionless Cucker-Dong system with 8 agents, d = 2,β = 1.02, and f (r) = 1/r1.1.

No control active

Sparse control with M = 1

No control active

A few info

� WWW: http://www-m15.ma.tum.de/

� References:� M. Bongini and M. Fornasier, Sparse stabilization of dynamical

systems driven by attraction and avoidance forces, to appear inNetworks and Heterogeneous Media, pp. 32

� M. Caponigro, M. Fornasier, B. Piccoli, and E. Trelat, Sparsestabilization and control of alignment models, to appear inMathematical Models and Methods in Applied Sciences, 2014, pp. 33

� F.Cucker, J. Dong, A conditional, collision-avoiding, model forswarming, IEEE Trans. Automat. Control, 56(5):1124–1129, 2011

� F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans.Automat. Control, 52(5):852–862, 2007

� M. Fornasier and F. Solombrino, Mean-field optimal control, to appearin ESAIM: Control, Optimization, and Calculus of Variations, 2013,pp. 31

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