Sparse control and stabilization to consensus
of collective behavior models
E. Trélat
Univ. Paris 6, Labo. J.-L. Lions, et Institut Universitaire de France
Works with: M. Caponigro, M. Fornasier, B. Piccoli, F. Rossi
Graz, June 2015
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
Prototypical model for the interaction of N agents:
Cucker-Smale model (2007)
ẋi (t) = vi (t)
v̇i (t) =
N
vj (t) − vi (t)
1 X
N
(1 + kxj (t) − xi (t)k2 )β
i = 1, . . . , N
j=1
β > 0, xi ∈ Rd , vi ∈ Rd
xi : main state of the agent i
vi : consensus parameter of the agent i
• initially introduced to describe the formation and evolution of languages
• then for describing the flocking of a swarm of birds
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
Prototypical model for the interaction of N agents:
General (finite-dimensional) Cucker-Smale model
ẋi (t) = vi (t),
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)),
N
i = 1, . . . , N
j=1
xi ∈ Rd , vi ∈ Rd
a ∈ C 1 ([0, +∞)) nonincreasing positive function (interaction potential, rate of communication)
a(s) =
1
(1+s2 )β
in the classical Cucker-Smale model
There are other classes of models, cf social dynamics...
Bellomo, Bertozzi, Boudin, Carrillo, Couzin, Cristiani, Cucker, Degond, Desvillettes, Dong, Fornasier, Frasca,
Ha, Haskovec, Kim, Lee, Leonard, Levy, Mordecki, Motsch, Parrish, Perthame, Piccoli, Salvarani, Smale,
Tadmor, Toscani, Tosin, Vecil, Vicsek,........
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
Prototypical model for the interaction of N agents:
General (finite-dimensional) Cucker-Smale model
ẋi (t) = vi (t),
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)),
N
i = 1, . . . , N
j=1
xi ∈ Rd , vi ∈ Rd
a ∈ C 1 ([0, +∞)) nonincreasing positive function (interaction potential, rate of communication)
a(s) =
1
(1+s2 )β
in the classical Cucker-Smale model
Objectives:
• Model self-organization and consensus emergence in a group of agents
• Organization via intervention (social forces),
enforce or facilitate pattern formation or convergence to consensus
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t))
N
i = 1, . . . , N
j=1
Mean consensus vector: v̄ =
N
1 X
vi (t)
N
→ constant
i=1
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t))
N
i = 1, . . . , N
j=1
With matrix notations:
0
1
0
1
N
N
“X
”
1 @` ´
1 @X
A
aij (vj − vi ) =
Av i −
aij vi A
v̇i =
N
N
j=1
j=1
hence
ẋ = v ,
v̇ = −Lv
with
L=
1
(D − A) Laplacian,
N
D = diag
N
“X
aij
”
j=1
N.B.: L > 0 and L(v , . . . , v ) = 0
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t))
N
i = 1, . . . , N
j=1
d N
(R )
∀v ∈ (Rd )N ,
(
⊥
= Vf ⊕ V⊥ with
Vf
V⊥
=
=
{(v1 , . . . , vN ) ∈ (Rd )N
{(v1 , . . . , vN ) ∈ (Rd )N
|
|
v1 = · · · = vN ∈ Rd }
PN
i=1 vi = 0}
v = vf + v⊥ with vf = (v̄ , . . . , v̄ ) ∈ Vf and v⊥ ∈ V⊥ .
X (t) =
N
1 X
kxi (t) − xj (t)k2
2
2N
i,j=1
V (t) =
N
N
N
1 X
1 X
1 X
2
2
kv
(t)
−
v
(t)k
=
kv
(t)
−
v̄
k
=
kv (t)⊥i k2
i
j
i
2N 2
N
N
i,j=1
V̇ (t) = −
i=1
N
1 X
a(kxj (t) − xi (t)k)kvi (t) − vj (t)k2
N
i=1
60
i,j=1
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t))
N
i = 1, . . . , N
j=1
Definitions
Consensus point = steady configuration = (x, v ) ∈ (Rd )N × Vf .
Consensus manifold = (Rd )N × Vf .
(x(t), v (t)) tends to consensus (or flocking) if vi (t) −→ v̄
t→+∞
∀i = 1, . . . , N, or,
equivalently, if V (t) −→ 0.
t→+∞
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t))
N
i = 1, . . . , N
j=1
Note that
N
1 X
a(kxj (t) − xi (t)k)kvi (t) − vj (t)k2
N
i,j=1
“p
”
6 −2a
2NX (t) V (t)
V̇ (t) = −
and hence if X (t) remains bounded then V̇ 6 −αV , whence flocking.
|
{z
}
??
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Cucker-Smale consensus model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t))
N
i = 1, . . . , N
j=1
Proposition (Ha-Ha-Kim, 2010)
Let (x0 , v0 ) ∈ (Rd )N × (Rd )N be such that
Z
+∞
√
√
p
a( 2Nr ) dr > V (0).
X (0)
Then the solution with initial data (x0 , v0 ) tends to
consensus.
→ self-organization of the group in this so-called consensus region (region of natural
asymptotic stability to consensus). Sharp estimate.
Example: a(x) =
1
(1+x 2 )β
with 0 < β 6 1/2 (initial model by Cucker-Smale, 2007)
→ globally stable
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Kinetic Cucker-Smale model
By passing to the limit (hydrodynamic, or mean-field limit):
Kinetic Cucker-Smale model
∂t µ + hv , ∇x µi + divv (ξ[µ] µ) = 0
µ(t): probability measure on Rd × Rd
(if µ(t, x, v ) = f (t, x, v ) dx dv , then f is the density of the crowd)
ξ[µ]: interaction kernel
Z
a(kx − yk)(w − v ) dµ(y, w)
ξ[µ](x, v ) =
Rd ×Rd
Link with finite dimension:
µ(t) =
N
1 X
δ(xi (t),vi (t))
N
i=1
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Kinetic Cucker-Smale model
Kinetic Cucker-Smale model
∂t µ + hv , ∇x µi + divv (ξ[µ] µ) = 0
Space and velocity barycenters:
Z
Z
x̄(t) =
x dµ(t)(x, v ),
IRd ×IRd
v̄ (t) =
v dµ(t)(x, v )
IRd ×IRd
Then:
˙
x̄(t)
= v̄
Variance:
Z
V (t) =
v̄ = Cst
|v − v̄ |2 dµ(t)
IRd ×IRd
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Kinetic Cucker-Smale model
Kinetic Cucker-Smale model
∂t µ + hv , ∇x µi + divv (ξ[µ] µ) = 0
A solution µ ∈ C 0 (IR, Pc (IRd × IRd )) converges to flocking if:
∃XM > 0 | supp(µ(t)) ⊆ B(x̄(t), XM ) × IRd
V (t) =
R
IRd ×IRd
∀t > 0
|v − v̄ |2 dµ(t) −→ 0
t→+∞
→ similar results: consensus region...
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Control of the Cucker-Smale model
When consensus is not achieved by self-organization:
is it possible to control the group to consensus by
means of an external action?
→ organization via intervention
Controlled Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
N
X
kui (t)k 6 M
i=1
for M > 0 fixed.
(Caponigro Fornasier Piccoli Trélat, M3AS 2015)
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Control of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
Objectives:
- Design a feedback control steering ”optimally” the system to consensus, with:
(i) a minimal amount of components active at each time
(ii) a minimal amount of switchings in time
- Control the system to any prescribed consensus
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Control of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
Objectives:
- Design a feedback control steering ”optimally” the system to consensus, with:
(i) a minimal amount of components active at each time
(ii) a minimal amount of switchings in time
- Control the system to any prescribed consensus
→ concept of sparse control
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Control of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
Objectives:
- Design a feedback control steering ”optimally” the system to consensus, with:
(i) a minimal amount of components active at each time → componentwise sparse
control
(ii) a minimal amount of switchings in time → time sparse control
- Control the system to any prescribed consensus
→ concept of sparse (feedback) control
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Control of the kinetic Cucker-Smale model
Controlled kinetic Cucker-Smale model
∂t µ + hv , ∇x µi + divv ((ξ[µ]+χω u)µ) = 0
with u ∈ L∞ (IR × IRd × IRd ) and ω(t) ⊂ IRd measurable, s.t.
ku(t, ·, ·)kL∞ (IRd ×IRd ) 6 1
(bounded external action)
and
Z
dµ(t)(x, v ) 6 c
µ(t)(ω(t)) =
ω(t)
(one is allowed to act only on a given proportion c of the crowd)
Variant:
Z
|ω(t)| =
dx dv 6 c
ω(t)
(limit on the space of configurations)
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse stabilization of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
A first remark
The feedback control defined by
u(t) = −αv⊥ (t),
with α > 0 small enough, stabilizes the system to consensus (in infinite time)
Indeed:
V̇ 6 − N2
P
i hv⊥i , ui i
= −2αV
BUT this control is far from sparse!
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse stabilization of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
Idea:
1
0
1
N
N
X
X
1
1
@
@
hvi − vj , ui − uj iA = P min
hv⊥i , u⊥i iA
PN min
N ku k6M
2N 2
N
kui k6M
i
i=1
i=1
0
i,j=1
i=1
⇒ u = −αv⊥
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse stabilization of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
Idea:
0
1
N
N
1 X
1 X
@
hvi − vj , ui − uj i+γ(X )
kui kA ,
PN min
2N 2
N
kui k6M
i=1
i,j=1
i=1
where
Z
γ(X ) =
+∞
√
√
a( 2Nr ) dr
X
`1 norm ⇒ enforce sparsity
γ(X ) threshold ⇒ the control switches off when entering the consensus region
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse stabilization of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
This leads to the componentwise sparse feedback control u:
if max kv⊥i (t)k 6 γ(X (t))2 , then u(t) = 0
16i6N
if kv⊥j (t)k = max kv⊥i (t)k > γ(X (t))2 (with j be the smallest index) then
16i6N
uj (t) = −M
v⊥j (t)
kv⊥j (t)k
,
and
ui (t) = 0
for every i 6= j
Theorem
This feedback control stabilizes the system to consensus
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse stabilization of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
Theorem
This feedback control stabilizes the system to consensus
Indeed:
V̇ 6
2 X
M
hv⊥i , ui i = −2 kv⊥j k
N
N
i
with
kv⊥j k = max kv⊥i k >
16i6N
√
V
⇒
V̇ 6 −2
M√
V
N
hence in finite time we enter the consensus region, and
then u = 0 (forever)
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse stabilization of the Cucker-Smale model
ẋi (t) = vi (t)
v̇i (t) =
N
1 X
a(kxj (t) − xi (t)k)(vj (t) − vi (t)) + ui (t)
N
i = 1, . . . , N
j=1
with
PN
i=1
kui (t)k 6 M
Remarks
This feedback control is componentwise sparse.
This feedback control is however not necessarily time sparse: it may chatter
→ To avoid possible chattering in time: sampling in time (sample-and-hold)
⇒ we obtain a time sparse and componentwise sparse feedback control
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse is better
Proposition
d
V (t) over all possible feedbacks controls.
dt
In other words, the feedback control u(t) is the best choice in terms of the rate of
convergence to consensus
For every time t, u(t) minimizes
→ ”Sparse is better”:
A policy maker, who is not allowed to have prediction on future developments,
should always consider more favorable to intervene with stronger actions on the
fewest possible instantaneous optimal leaders than trying to control more agents
with minor strength.
E. Trélat
Sparse control and stabilization of Cucker-Smale models
3
3 agents in flocking position:
2
x1 = (2, 0),
1
0
v1 = (5, 0)
x2 = (1, −2),
v2 = (5, 0)
x3 = (0, 1),
v3 = (5, 0)
−1
plus one agent to be controlled:
−2
x4 = (5, −5),
−3
v4 = (0, 5)
−4
−5
−6
−1
0
1
2
3
4
5
E. Trélat
6
—– uncontrolled trajectories
—– controlled trajectories
**** active control
Sparse control and stabilization of Cucker-Smale models
3 agents in flocking position:
3
x1 = (2, 0),
2
1
v1 = (5, 0)
x2 = (1, −2),
v2 = (5, 0)
x3 = (0, 1),
v3 = (5, 0)
0
−1
plus 2 agents to be controlled:
−2
−3
x4 = (5, −5),
v4 = (0, 5)
x5 = (5, −1),
v5 = (−5, 0)
−4
−5
−6
−1
0
1
2
3
4
5
E. Trélat
6
—– uncontrolled trajectories
—– controlled trajectories
**** active control
Sparse control and stabilization of Cucker-Smale models
1.5
1
20 agents
random initial positions (on a circle)
0.5
random initial speeds (sufficiently
large so that the uncontrolled system
does not flock)
0
−0.5
—– uncontrolled trajectories
—– controlled trajectories
**** active control
−1
−1.5
−1.5
−1
−0.5
0
0.5
1
1.5
Other videos: 10 agents, 50 agents
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Kinetic Cucker-Smale model
Theorem
∀µ0 ∈ Pcac (IRd × IRd ) ∃χω u satisfying ku(t, ·, ·)kL∞ (IRd ×IRd ) 6 1 and
Z
Z
dµ(t)(x, v ) 6 c
µ(t)(ω(t)) =
or
|ω(t)| =
ω(t)
dx dv 6 c
ω(t)
and ∃!µ ∈ C 0 (IR; Pcac (IRd × IRd )) solution, s.t. µ(0) = µ0 , and converging to flocking
Remarks
µ remains AC and of compact support (but becomes singular in infinite time)
the control χω u is designed in an explicit way, and
ω(t) piecewise constant (p.c.) in t
u(t, x, v ) p.c. in t for (x, v ) fixed, C 0 and piecewise linear in (x, v ) for t fixed
∀µ0 ∈ Pcac (IRd × IRd )
∃T (µ0 ) > 0
E. Trélat
|
∀t > T (µ0 )
u(t, x, v ) = 0
Sparse control and stabilization of Cucker-Smale models
Kinetic Cucker-Smale model
Theorem
∀µ0 ∈ Pcac (IRd × IRd ) ∃χω u satisfying ku(t, ·, ·)kL∞ (IRd ×IRd ) 6 1 and
Z
Z
dµ(t)(x, v ) 6 c
µ(t)(ω(t)) =
or
|ω(t)| =
ω(t)
dx dv 6 c
ω(t)
and ∃!µ ∈ C 0 (IR; Pcac (IRd × IRd )) solution, s.t. µ(0) = µ0 , and converging to flocking
„
∂t µ + div(x,v ) (Vω,u [µ]µ) = 0
with
Vω,u [µ] =
v
ξ[µ] + χω u
«
Controlled particle flow Φω,u (t) generated by Vω,u [µ(t)]: characteristics
ẋ(t) = v (t),
v̇ (t) = ξ[µ(t)](x(t), v (t)) + χω(t) u(t, x(t), v (t))
⇒ µ(t) = Φω,u (t)#µ0
Shepherd control design strategy: choose ω(t) and u(t) such that Vω,u [µ(t)] points
inwards the invariant domain (confining the population)
i.e., the size of suppv (µ(t)) decreases (exponentially) in time
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Controllability near the consensus manifold
In finite dimension only:
Proposition
Local controllability holds at almost every point of the consensus manifold. Moreover,
controllability can be realized with time sparse and componentwise sparse controls
Corollary
Any point of (Rd )N × (Rd )N can be steered to almost any point of the consensus
manifold in finite time by means of a time sparse and componentwise sparse control
(by stabilization and then iterated local controllability along a path of consensus points)
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Sparse optimal control
Another way of designing sparse controls: by optimal control
Optimal control problem with a fixed initial point and free final point, where the cost to
be minimized is
T
Z
0
N “
N
N Z T
”2
X
X
1 X
vi (t) −
vj (t) dt + γ
kui (t)kdt
N
0
i=1
j=1
where γ > 0 is fixed, under the constraint
i=1
PN
i=1
kui (t)k 6 M.
The `1 -norm in the red term implies componentwise sparsity features of the optimal
control.
(proof by applying the Pontryagin maximum principle + genericity arguments)
E. Trélat
Sparse control and stabilization of Cucker-Smale models
Generalizations
There are many generalizations of the Cucker-Smale model:
general potentials (friction, attraction/repulsion, ...):
Cucker, Dong, Ha, Ha, Kim, Leonard, Motsch, Slemrod, Tadmor
stochastic aspects (adding noise):
delay:
Carrillo, Cucker, Fornasier, Ha, Lee, Levy, Mordecki, Toscani
ongoing work with Cristina Pignotti
Models in infinite dimension (hydrodynamic, kinetic, mean field limit):
Carrillo, Degond, Fornasier, Ha, Hascovek, Motsch, Rosado, Tadmor, Toscani
Other open problems:
cluster control, control of opinion formation, black swan, cancerology
M. Caponigro, M. Fornasier, B. Piccoli, E. Trélat, Sparse stabilization and control of alignment models,
Math. Models Methods Applied Sciences 25 (2015), no. 3, 521–564.
B. Piccoli, F. Rossi, E. Trélat, Control to flocking of the kinetic Cucker-Smale model,
to appear in SIAM J. Math. Analysis.
E. Trélat
Sparse control and stabilization of Cucker-Smale models