seminar ii: mean- eld sparse optimal...

IntroductionSpecial Examples

The finite dimensional systemThe coupled ODE and PDE system

Seminar II: Mean-field sparse optimal control

Yinglong Zhang

September 30, 2016

Yinglong Zhang Mean-field sparse optimal control



Outline

1 Introduction

2 Special ExamplesSparse optimal control of Cucker-Smale modelMean-field optimal controlOthers

3 The finite dimensional systemIntroductionDynamical system IOptimal control system II

4 The coupled ODE and PDE systemIntroductionDynamical system IIIOptimal control system IV




Introduction

1 Introduction







• Seminar I: {xi = vi , t > 0, i = 1, 2, · · · ,N,vi = H ∗ µN(xi , vi ) +f (t, xi , vi ),

where (xi , vi ) ∈ R2d . H is some social force, such that, attraction,repulsion, noise, · · ·




• Seminar II: {xi = vi , t > 0, i = 1, 2, · · · ,N,vi = H ∗ µN(xi , vi ).

(1)

• Difference: Sparse optimal control.




We rewrite the above system:yk = wk , t ∈ [0,T ],

wk = H ∗ (µN + µm)(yk ,wk) +control , k = 1, 2, · · · ,m,xi = vi ,

vi = H ∗ (µN + µm)(xi , vi ), i = 1, 2, · · · ,N,

(2)




• Examples in true life:

shepherd dog strategy: How many dogs we need?

Social system: Can a government endowed with limited resources /rescue / stabilize a society by minimal intervention?




In the following, we will study:

• Why we need to study sparse optimal control?

• How?




Sparse optimal control of Cucker-Smale modelMean-field optimal controlOthers

Introduction

1 Introduction








Optimal control system 1

We consider the Cucker-Smale modelxi = vi , t > 0, i = 1, 2, · · · ,N,

vi =1

N

N∑j=1

a(‖xj − xi‖)(vj − vi )+ui ,

where a(r) = 1(1+r2)β

, β > 0 is the communication weight.

• Cost function:

min{B(u, v) + γ(B(x , x))1

N

N∑i=1

‖ui‖} subject toN∑i=1

‖ui‖ ≤ M,





where

B(u, v) :=1

2N2

∑i,j

〈ui − uj , vi − vj〉

=1

N

N∑i=1

〈ui − u, vi − v〉.

And

γ(X ) =

∫ ∞√X

a(√

2Ns)ds.

• Reference: [Caponigro-F-Piccoli-Trelat]





Theorem 1.1 [Ha-Liu]

Let (x , v) ∈ C1([0,∞),R2d×N) be a solution to Cucker-Smale system.Then

(i) If β ≤ 12 , the solution tends to consensus for every initial data.

(ii) If β > 12 , let (x0, v0) satisfy

√V 0 <

∫ ∞√X 0

a(√

2Nr)dr , (3)

where

X (t) =1

N

N∑j=1

‖xj − x‖2, V (t) =1

N

N∑j=1

‖vj − v‖2.

Then, the solution with initial data (x0, v0) tends to consensus.





Definition 1.2 (Consensus region)

We call the set of (x , v) ∈ RN × RN satisfying (3) in Theorem 1 theconsensus region.

Remark A.1

X (t) = B(x , x), V (t) = B(v , v).

Aim: to find admissible controls steering the system to consensusregion in finite time.





• Set

Vf = {v ∈ (Rd)N∣∣v1 = · · · = vN},

V⊥ = {v ∈ (Rd)N∣∣ N∑i=1

vi = 0}.

=⇒(i) The whole space can be decomposed as

(Rd)N = Vf ⊕ V⊥.

(ii)

V (t) = B(v , v) =1

N

N∑i=1

‖v⊥i‖2.





Definition 1.3

We select the sparse feedback control u0 = u0(x , v) ∈ Uad(x , v) (the setof admissible control) according to the following criterion:

(i) If max1≤i≤N

‖v⊥i‖ ≤ γ(B(x , x)), then u0 = 0,

(ii) If max1≤i≤N

‖v⊥i‖ > γ(B(x , x)), let j ∈ {1, · · · ,N} be the smallest index

such that‖v⊥j‖ = max

1≤i≤N‖v⊥i‖.

then

u0j = −M v⊥j

‖v⊥j‖, u0

i = 0 for every i 6= j . (4)





Theorem 1.4

For every M > 0, there exists a solution of the optimal control system 1associated with the sparse control u0 as in Definition 1.4. Fix M > 0 andconsider the control u0 law given by Definition 1.3. Then for every initialcondition (x0, v0) ∈ RN × RN , there exists τ0 > 0 small enough, suchthat for all τ ∈ (0, τ0] the sampling solution of optimal control system 1associated with the control u0, the sampling time τ , and initial pair(x0, v0) reaches the consensus region in finite time.

• The choice of u0 is the best!

We can refer to [Bongini-F-Frohlich-Haghverdi], [Bongini-F-Kalise] forother results about the general Cucker-Smale model.





[F-Solombrino]

A natural question: What if N →∞? ”Curse of dimensionality”

Particle Cucker-Smale model ⇒ u → δ, as N →∞.

Thus, we consider (Smooth controls)

∂tµ+ v · ∇xµ = ∇v · [(H ∗ µ+ f )µ],

with f has some good property. (Xiongtao)

Question: What is f∞? It is an open problem.

Thus, we need to study: (Sparse optimal control)





For the other results, for example, use JohnsonõLindenstrauss matrix tolower dimension, PMP, · · · , the history is written by you!




IntroductionDynamical system IOptimal control system II

Introduction

1 Introduction








We add controls on the m leaders to system (7)yk = wk , t ∈ [0,T ],

wk = H ∗ (µN + µm)(yk ,wk)+uk , k = 1, 2, · · · ,m,xi = vi ,

vi = H ∗ (µN + µm)(xi , vi ), i = 1, 2, · · · ,N,

(5)

where H is a locally Lipschitz interaction kernel with sublinear growth,u = (u1, · · · , um) ∈ L1([0,T ];Uad) with Uad a fixed non-empty compactsubset of Rd×m and Uad ⊆ B(0,R) for R > 0.





For given initial datum

(y1(0), · · · , ym(0),w1(0), · · · ,wm(0), x1(0), · · · , xN(0), v1(0), · · · , vN(0))

∈ (Rd)m × (Rd)m × (Rd)N × (Rd)N ,

We consider the following optimal control problem:

minu∈L1([0,T ];Uad )

∫ T

0

{L(y(t),w(t), µN(t)) +1

m

m∑k=1

|uk |}dt, (6)





where

µN =1

N

N∑i=1

δ(xi , vi ), µm =1

m

m∑k=1

δ(yk ,wk),

are the time-dependent atomic measures supported on the phase spacetrajectories (yk(t),wk(t)) ∈ R2d , for k = 1, · · · ,m and(xi (t), vi (t)) ∈ R2d , for i = 1, · · · ,N, respectively, constrained by beingthe solution of the system (9) with initial data ζ0 = (y0,w0, x0, v0)





Remark A.2

(i) We use the 1- Wasserstein distance for µ.

(ii) If

µm =1

m

m∑k=1

δξk , µ′m =1

m

m∑k=1

δξ′k .

are two atomic measures, then we immediately have that

W1(µm, µ′m) ≤ 1

m

m∑k=1

|ξk − ξ′k |.





Assumptions:

(H) Let H : R2d → Rd be a locally Lipschitz function such that, forconstant C > 0

|H(ξ)| ≤ C (1 + |ξ|), for all ξ ∈ R2d .

(L) Let L : X := R2d×m × P1(R2d → R+ be a continuous function withrespect to the distance induced on X by the norm ‖ · ‖X .





Dynamical system I

Firstly, we consider the existence and uniqueness of solutions for system:yk = wk , t ∈ [0,T ],


vi = H ∗ (µN + µm)(xi , vi ), i = 1, 2, · · · ,N,

(7)

with u ∈ L1([0,T ];Uad), Uad ⊆ B(0,R), R > 0.





Denote ζ(t) = (y ,w , x , v) and ξ(t) = (y ,w) or ξ(t) = (x , v), then thesystem (1.8) can be written

ζ(t) = g(t, ζ(t)),

where

g(t, ζ(t)) =(w ,H∗(µN+µm)(yk ,wk)mk=1+uk , v ,H∗(µN+µm)(xi , vi )

Ni=1

).





The existence and uniqueness of solution

Proposition 2.2

Let H be a map satisfying (H). Then, given a control u ∈ L1([0,T ],Uad)and an initial datum ζ0 = (y0,w0, x0, v0), there exists a uniqueCaratheodory solution ζ(t) = (y(t),w(t), x(t), v(t)) of (9) such that

‖ζ(t)‖ ≤ (‖ζ0‖+ CT )eCT ,

for all t ∈ [0,T ], where C > 0 is a constant depending on C > 0, R > 0but not depending on N. Moreover, the trajectory is Lipschitz continuousin time, i.e.

‖ζ(t1)− ζ(t2)‖ ≤ L|t1 − t2|, t1, t2 ∈ [0,T ],

for the Lipschitz constant L = C (1 + (‖ζ0‖+ CT )eCT ).





Proof.

We use the condition (H) and Lemma 3.1 to get

‖g(t, ζ)‖ ≤ C (1 + ‖ζ‖), for all ζ ∈ R2d . (8)

By the standard theory about solutions to Caratheodory differentialequations (Theorem A.2), the existence and the uniqueness of solutionsfollows. We use Gronwall’s inequality yields that

‖ζ(t)‖ ≤ (‖ζ0‖+ CT )eCT .

Moreover, we use (13) to obtain

‖ζ(t1)− ζ(t2)‖ = ‖∫ t2

t1

g(s, ζ(s))ds‖ ≤∫ t2

t1

C (1 + ‖ζ(s)‖)ds

≤ C (1 + (‖ζ0‖+ CT )eCT )|t1 − t2|.





Optimal control system II

Now we consider the optimal control system:


∫ T

0

{L(y(t),w(t), µN(t)) +1

m

m∑k=1

|uk |}dt, (9)

with ζ(t) := (y(t),w(t), x(t), v(t)) the solution of the following system:yk = wk , t ∈ [0,T ],


vi = H ∗ (µN + µm)(xi , vi ), i = 1, 2, · · · ,N,

(10)





Theorem 4.1

Under the assumption (H) and (L), the finite horizon optimal controlproblem II with initial datum ζ0 = (y0,w0, x0, v0) ∈ R4d has solutions.




IntroductionDynamical system IIIOptimal control system IV

Introduction

1 Introduction








The coupled ODE and PDE system

The dynamics reads as (N →∞)yk = wk , t > 0,

wk = H ∗ (µ+ µm)(yk ,wk)+uk , k = 1, 2, · · · ,m,∂tµ+ v · ∇xµ = ∇v · [H ∗ (µ+ µm)µ],

(11)

where

µm(yk ,wk) =1

m

m∑k=1

δ(yk ,wk).





For given initial data (y0,w0, µ0), we consider the following optimalcontrol problem:


∫ T

0

{L(y(t),w(t), µ(t)) +1

m

m∑k=1

|uk |}dt.





Dynamical system III

Now we consider the existence and stability of solutions to the followingdynamical system.

yk = wk , t > 0,

wk = H ∗ (µ+ µm)(yk ,wk) + uk , k = 1, 2, · · · ,m,∂tµ+ v · ∇xµ = ∇v · [H ∗ (µ+ µm)µ],

µm(yk ,wk) =1

m

m∑k=1

δ(yk ,wk),

(12)





Definition of solution

Definition 3.1

Let u ∈ L1([0,T ],Uad) be given. We say that a map(y ,w ,∆) : [0,T ]→ X := R2d×m × P1(R2d) is a solution of thecontrolled system (11) with control u, if

(i) the measure µ is equi-compactly supported in time, i.e. there existsR > 0 such that supp(µ(t)) ∈ B(0,R) for all t ∈ [0,T ];

(ii) the solution is continuous in time with respect to the following metricin X

‖(y ,w , µ)− (y ′,w ′, µ′)‖X :=1

m

m∑k=1

(|yk − y ′k |+ |wk − w ′k |) +W1(µ, µ′),

where W1(µ, µ′) is the 1-Wasserstein distance in P1(R2d);





(iii) the (y ,w) coordinates define a Caratheodory solution of thefollowing controlled problem with interaction kernel H, control u(·), andthe external field H ∗ µ:{

yk = wk , t > 0,

wk = H ∗ (µ+ µm)(yk ,wk), k = 1, 2, · · · ,m.

(iv) the µ component satisfies

d

dt

∫R2d

φdµ =

∫R2d

∇φ · ωdµ,

for every φ ∈ C∞c (Rd × Rd), in the sense of distributions, whereω : [0,T ]× Rd × Rd → Rd × Rd :

ω = (v ,H ∗ µ+ H ∗ µm).





Let, moreover, (y0,w0, µ0) ∈ X be given, with µ0 ∈ P1(R2d) of compactsupport. We say that (y ,w , µ) : [0,T ]→ X is a solution of (9) withinitial data (y0,w0, µ0) and control u if it is a solution of (9) with controlu and it satisfies (y(0),w(0), µ(0)) = (y0,w0, µ0).

Remark A.3

Similar with [F- Solombrino], once µm is a fixed, a measure µ is a weakequi-compactly supported solution of

∂tµ+ v · ∇xµ = ∇v · [H ∗ (µ+ µm)µ].

in the sense of (iv) in the above definition if and only if it satisfies (i) andthe measureõtheoretical fixed point equation

µ(t) = Φµ,µmt µ0,

where Φµ,µmt denotes push-forward of µ0 to µ(t).





The stability

Proposition 3.2

Let u ∈ L1([0,T ],Uad) be a given fixed control for (9) and two solutions(y1,w1, µ1) and (y2,w2, µ2) of (9) relative to the control u and givenrespective initial data (y i,0,w i,0, µi,0) ∈ X , with µ0,i , compactlysupported, i = 1, 2. Then, there exists a constant CT > 0 such that

‖(y1,w1, µ1)−(y2,w2, µ2)‖X ≤ CT‖(y1,0,w1,0, µ1,0)−(y2,0,w2,0, µ2,0)‖X ,

for all t ∈ [0,T ].





Existence of solution

Theorem 3.3

Let (y0,w0, µ0) ∈ X be given, with µ0 of bounded support in B(0,R),for R > 0. Define a sequence (µ0

N)N∈N of atomic probability measuresequi-compactly supported in B(0,R) such that each µ0

N is given by

µ0N =

N∑i=1

δx0i,N ,v

0i,N

and limN→∞W1(µ0N , µ

0) = 0. Fix now a weakly

convergent sequence (uN)N∈N ∈ L1([0,T ];Uad) of control functions, i.e.uN ⇀ u∗ in L1([0,T ];Uad). For each initial datum ζ0 = (y0,w0, µ0)depending on N, denote withζN(t) = (yN(t),wN(t), µN(t)) := (yN(t),wN(t), xN(t), vN(t)), the uniquesolution of the finite-dimensional control problem (1.9) with control uN .Then, the sequence (yN(t),wN(t), µN(t)) converges in C0([0,T ],X ) tosome (y∗,w∗, µ∗), which is a solution of dynamical system III with initialdata (y0,w0, µ0) and control u∗, in the sense of definition 3.1.





Remark A.4

In view of the uniqueness of the solution of the dynamical system III, wedo not need to restrict ourselves to a subsequence, but we can infer theconvergence of the entire sequence ζN to the solution of dynamicalsystem III.





Optimal control system IV

The dynamics reads as

yk = wk , t > 0,

wk = H ∗ (µ+ µm)(yk ,wk) + uk , k = 1, 2, · · · ,m,∂tµ+ v · ∇xµ = ∇v · [H ∗ (µ+ µm)µ],

µm(yk ,wk) =1

m

m∑k=1

δ(yk ,wk),

(13)

• The cost function:


∫ T

0

{L(y(t),w(t), µ(t)) +1

m

m∑k=1

|uk |}dt, (14)





The Γ-limit to the infinite-dimensional optimal control

Definition 5.1 (Γ-convergence)

Let X be a metrizable separable space and FN : X → (−∞,∞], N ∈ Nbe a sequence of functionals. Then, we say that FN Γ-converges to F ,

written as FNΓ−→ F , for an F : X → (−∞,∞], if

(i) lim inf-condition: for every u ∈ X and every sequence uN → u,

F (u) ≤ lim infN→∞

FN(uN);

(ii) lim sup-condition: for every u ∈ X there exists a sequence uN → u,called recovery sequence, such that

F (u) ≥ lim supN→∞

FN(uN).





• Fix now an initial datum (y0,w0, µ0) ∈ X , with µ0 compactlysupported, supp(µ0) ⊆ B(0,R), R > 0.

• Choose a sequence of equi-compactly supported atomic measures µ0N ,

supp(µ0N) ⊆ B(0,R), µ0

N = 1N

∑Ni=1 δx0

i ,v0i

such that

W1(µ0N , µ

0)→ 0, N → +∞.





• We define

FN(u) =

∫ T

0

{L(y(t),w(t), µN(t)) +1

m

m∑k=1

|uk |}dt, (15)

where (y ,w , x , v) is the solution ofyk = wk ,

wk = H ∗ (µN + µm)(yk ,wk), k = 1, 2, · · · ,m,xi = vi ,

vi = H ∗ (µN + µm)(xi , vi ), i = 1, 2, · · · ,N,

(16)

with initial data (y0,w0, x0N , v

0N).





Similarly, we define

F (u) =

∫ T

0

{L(y(t),w(t), µ(t)) +1

m

m∑k=1

|uk |}dt. (17)

Theorem 5.3

Let H and L be maps satisfying conditions (H) and (L) respectively.Given an initial datum (y0,w0, µ0) ∈ X and an approximating sequenceµ0N , with µ0, µ0

N equi-compactly supported, i.e. supp(µ0)⋃

supp(µ0N) ⊆ B(0,R), R > 0, for all N ∈ N, then the sequence of

functionals (FN)N∈N on L1([0,T ],Uad) defined above Γ-converges to thecost functional F (u).





Corollary 5.4

Let H and L be maps satisfying conditions (H) and (L) respectively.Given an initial datum (y0,w0, µ0) ∈ X , with µ0 compactly supported,supp(µ0

N) ⊆ B(0,R), R > 0, the optimal control problem


∫ T

0

{L(y ,w , µ) +1

m

m∑k=1

|uk |}dt,

has solutions, where the triplet (y ,w , µ) defines the unique solution ofthe dynamical system III with initial datum (y0,w0, µ0) and control u inthe sense of definition 3.1.





Moreover, solutions to the optimal control system IV can be constructedas weak limits u∗ of sequences of optimal controls u∗N of thefinite-dimensional problems


∫ T

0

{L(y(t),w(t), µN(t)) +1

m

m∑k=1

|uk |}dt,

where µN and µm,N are the time-dependent atomic measures supportedon the trajectories defining the solution of the dynamical system I withinitial datum (y0,w0, x0

N , v0N) ∈ X and control u, and

µ0N = 1

N

∑Ni=1 δx0

i ,v0i

such that

W1(µ0N , µ

0)→ 0, N → +∞.





Thank you!


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