fractional heat equation under positivity constraints

C ONTROLLABILITY OF THE 1 ^D

FRACTIONAL HEAT EQUATION

UNDER POSITIVITY CONSTRAINTS

Umberto Biccari

[email protected] [email protected] cmc.deusto.es/umberto-biccari

DeustoTech, Universidad de Deusto, Bilbao, Spain

VIII Partial differential equations, optimal design and numerics Centro de Ciencias de Benasque Pedro Pascual, August 18^th-30^th2019

Joint work with Mahamadi Warma and Enrique Zuazua

1-d fractional heat equation







z

+ (−d

)

z = uχ

(x, t)

(−1, 1)

(0, T ) z

0 (x, t)

(−1, 1)

(0, T ), z(·, 0) = z

x

(−1, 1).

(F H)

• ω⊂

(−1, 1)

z

L

(−1, 1).

We are interested in analyzing controllability properties under positivity

state/control constraints.

The fractional Laplacian

Fractional Laplacian

(−d

)

z = C

P.V.

Z

R

z(x)

z(y)

|x−

y|

dy

C

=

R

1

cos(ζ

)

|ζ|^1+2s

dζ

= s2

Γ

Γ(1

s)

.

Fractional Sobolev spaces

Fractional Sobolev space

H^s(−1,1) :=

(

z∈L²(−1,1) : Z1

−1

Z 1

−1

|z(x)−z(y)|²

|x−y|^1+2s dxdy<∞ )

• IntermediateHilbertspace betweenL²(−1,1)andH¹(−1,1).

Scalar product

hz,wi_Hs(−1,1):=

Z 1

−1

zw dx+ Z 1

−1

Z1

−1

(z(x)−z(y))(w(x)−w(y))

|x−y|^1+2s dxdy

Fractional Sobolev norm

kzk_Hs(−1,1):=

Z 1

−1

|z|²dx+ Z1

−1

Z 1

−1

|z(x)−z(y)|²

|x−y|^1+2s dxdy

!¹₂

Known controllability results

Theorem (

)

There exists u ∈ L

(ω × (0, T )) such that the fractional heat equation (F H) is

• null-controllable at time T > 0 if and only if s > 1/2.

• approximately controllable at time T > 0 for all s ∈ (0, 1).

Remark: the equation being linear, by translation if s > 1/2 we have

controllability to trajectories b z.

Proof

Null controllability: through the moment method, based on the following behavior of the spectrum.

Eigenvalues (

)

λk= kπ

2 −(1−s)π 4

2s

+O 1

k

.

Approximate controllability: The result follows from the following property.

Parabolic unique continuation

Givens ∈ (0,1)andp^T ∈ L²(−1,1), letpbe the unique solution to the adjoint equation. Letω⊂(−1,1)be an arbitrary open set. Ifp=0 onω×(0,T), thenp=0 on(−1,1)×(0,T).

This, in turn, is a consequence of theunique continuationproperty for the Fractional Laplacian.

FallandFelli, Comm. Partial Differential Equations, 2014.

Constrained controllability

•

The fractional heat equation

z

is a given non-negative initial datum in L

(−1, 1) and u is a non-negative function, then so it is for the solution z of (F H).

B., Warma and Zuazua, 2019

Question

Can we control the fractional heat dynamics (F H) from any initial datum

z

L

(−1, 1) to any positive trajectory

z, under positivity constraints on the

control and/or the state?

Constrained controllability

Theorem (

)

Let s>1/2, z₀∈L²(−1,1)and letbz be a positive trajectory, i.e., a solution of(F H) with initial datum0<bz₀∈L²(−1,1)and right hand sidebu∈L^∞(ω×(0,T)). Assume that there existsν >0such thatbu≥νa.e inω×(0,T). Then, the following assertions hold.

1. There exist T>0and a non-negative control u∈L^∞(ω×(0,T))such that the corresponding solution z of(F H)satisfies z(x,T) =bz(x,T)a.e. in(−1,1).

Moreover, if z0≥0, we also have z(x,t)≥0for every(x,t)∈(−1,1)×(0,T).

2. Define the minimal controllability time by Tmin(z0,bz) := inf

n

T >0:∃ 0≤u∈L^∞(ω×(0,T))s.t.

z(·,0) =z₀and z(·,T) =bz(·,T)o . Then, Tmin>0.

3. For T=T_min, there exists a non-negative control u∈ M(ω×(0,T_min)), the space of Radon measures onω×(0,T_min), such that the corresponding solution of(F H)satisfies z(x,T) =bz(x,T)a.e. in(−1,1).

Proof - main ingredients

The proof requires two main ingredients:

1.

Controllability through

observability inequality

kp(·,

0)k

C

0

Z

ω

|p(x,

t)| dxdt

2. Dissipativity

of the fractional heat semi-group.

Some preliminary results

Theorem

Let{µ_k}k≥1be a sequence of real numbers satisfying the conditions:

1. There existsγ >0such thatµ_k+1−µ_k≥γfor all k≥1.

2. P

k≥1µ⁻¹_k <+∞.

Then, for any T>0, there is a positive constantC=C(T)>0such that, for any finite sequence {c_k}k≥1it holds the inequality

X

k≥1

|c_k|²e^{−2µk T}≤C

X

k≥1

cke^{−µk t}

2

L1(0,T)

.

PROOF:

Under the above hypothesis on{µk}k≥1, the functionF(t) :=P

k≥1cke^{−µk t}satisfies

|ck| ≤CkFk_L1(0,T), X

k≥1

|ck|e^(µ1−µk)t≤C(t)kFk_L1(0,T), C(t)uniformly bounded∀t>0.

Then

X

k≥1

|ck|²e^{−2µk T}=X

k≥1

|ck|e^(µ1−µk)t

|ck|e^(µk−µ1)te^{−2µk T}

≤CkFk²_L1(0,T).

Some preliminary results

Lemma

Consider the eigenvalue problem for the Dirichlet fractional Laplacian in(−1,1):

((−d_x²)^sφ_k =λ_kφ_k, x∈(−1,1)

φk=0, x∈(−1,1)^c.

Then for any open subsetω⊂(−1,1), there is a positive constantβ >0such that kφ_kk_L1(ω)≥β >0.

PROOF (main idea):

The proof is based on the fact that

Z

ω

|φ_k(x)|dx≥ Z

ω

sin

µk(1+x) +(1−s)π 4

−c(1−s)

√s µ^−1−2s_k

dx

µk :=kπ

2 −(1−s)π 4

L ¹ observability

Proposition

For any T

0 and p

L

(−1, 1), let p

L

((0, T ); H

(−1, 1))

C([0, T ]; L

(−1, 1)) with p

L

((0, T ); H

(−1, 1)) be the weak solution of the adjoint system







−pt

+ (−d

)

p = 0, (x, t)

(−1, 1)

(0, T ) p = 0, (x, t)

(−1, 1)

(0, T ) p(·, T ) = p

(·), x

(−1, 1).

Then, for any s

1/2, there is a constant C = C(T )

0 such that

kp(·,

0)k

C

0

Z

ω

|p(x,

t)| dxdt

.

L ^∞ controllability

Theorem

For any z₀ ∈ L²(−1,1), s >1/2and T >0, there exists a control function u ∈ L^∞(ω×(0,T))such that the corresponding unique weak solution z of (F H)with initial datum z(x,0) =z0(x)satisfies z(x,T) =0a.e. in(−1,1). Moreover, there is a constant C>0(depending only on T ) such that

kuk_L∞(ω×(0,T))≤Ckz₀k_L2(−1,1). PROOF:

Classical duality argument.

Proof of the main result - 1: constrained controllability

• Subtractingbzin the equation, it is enough to show that there exist a timeT >0 and a controlv∈L^∞(ω×(0,T)),v>−νa.e. inω×(0,T), such that







ξt+ (−d_x²)^sξ=vχω, (x,t)∈(−1,1)×(0,T) ξ=0, (x,t)∈(−1,1)^c×(0,T) ξ(·,0) =z₀(·)−bz₀(·), x∈(−1,1)

⇒ ξ(x,T) =0.

• This is equivalent to the observability inequality kp(·, τ)k²_L2(−1,1)≤C(T−τ)

Z T τ

Z

ω

|p(x,t)|dxdt

!2

.

• Using that the eigenvalues{λ_k}_k≥1form a non-decreasing sequence, and the dissipativityof the fractional heat semi-group:

kp(·,0)k²_L2(−1,1)≤e^−2λ1τkp(·, τ)k²_L2(−1,1)

≤e^−2λ1τC(T−τ) Z T

0

Z

ω

|p(x,t)|dxdt

!2

.

Proof of the main result - 1: constrained controllability

• By duality, the controlvcan be chosen such that kvk²_L∞(ω×(0,T))≤e^−2λ1τC(T−τ)

z0−bz0

2 L²(−1,1).

• Takingτ=T/2, we obtain

kvk²_L∞(ω×(0,T))≤e^−λ1TC(T) z0−bz0

2 L²(−1,1).

• The observability constantC(T)isuniformly boundedfor anyT>0. Hence, for T large enough we have

kvk²_L∞(ω×(0,T))< ν.

• This implies thatv>−ν. Therefore, the controlv>−νsteersξfromz₀−bz₀to zero in timeT >0, providedT is large enough. Consequently,zis controllable to the trajectorybzin timeT.

• Ifz0≥0, thanks to themaximum principle, we also havez(x,t)≥0 for every (x,t)∈(−1,1)×(0,T).

Proof of the main result - 2: positivity of T _min

• Solution of (F H) in the basis of the eigenfunctions{φk}k≥1: z(x,t) =X

k≥1

z_k(t)φ_k(x).

zk(t) =z_k⁰e^−λkt+ Z t

0

e^{−λk(t−τ)}uk(τ)dτ, uk(t) :=

Z

ω

u(x,t)φk(x)dx.

• z(x,T) =bz(x,T)a.e. in(−1,1):

zk(T) = Z 1

−1

bz(x,T)φk(x)dx=:ζk ⇒ ζk−z_k⁰e^−λkT = Z T

0

e^{−λk(T−τ)}uk(τ)dτ.

• For every 0≤τ≤T:

ζ_k−z_k⁰e^−λkT ≤ Z T

0

u_k(τ)dτ≤ζ_ke^λkT −z_k⁰.

• Assume by contradiction that, for everyT>0, there exists a non-negative control functionu^T steeringz0tobz(·,T)in timeT, and thatbz(·,T)6=z0. Then:

lim

T→0⁺

Z T 0

u^T_k(τ)dτ=ζ_k−z_k⁰=:γ =⇒ z_k⁰=ζ_k−γ.

Proof of the main result - 2: positivity of T _min

• z0∈L²(−1,1):

X

k≥1

|z_k⁰|²=X

k≥1

ζ_k²−2γζk+γ²

<+∞ ⇒ lim

k→+∞

ζ_k²−2γζk+γ²

=0.

• Since{φ_k}k≥1is an orthonormal complete system inL²(−1,1),φ_k*0in L²(−1,1)ask→+∞. Hence:

k→+∞lim (bz(·,T), φk)_L2(−1,1)= lim

k→+∞

Z1

−1

bz(x,T)φk(x)dx

= lim

k→+∞ζk =0 ⇒ γ=0.

Consequently

0=z_k⁰−ζk = Z 1

−1

(z0(x)−bz(x,T))φk(x)dx, for all k≥1.

• This is possible if and only ifz0(x) =bz(x,T)a.e. in(−1,1), which contradicts our previous assumption.

Proof of the main result - 3: minimal-time control

Constrained controllability of the system (F H) holds in the minimal timeTminwith controls in the (Banach) space of the Radon measuresM(ω×(0,Tmin))endowed with the norm

kµkM(ω×(0,Tmin))= sup (Z

ω×(0,Tmin)

ϕ(x,t)dµ(x,t) :

ϕ∈C(ω×[0,Tmin],R), max

ω×[0,Tmin]|ϕ|=1 )

.

Solutions of (F H) with controls inM(ω×(0,Tmin))are defined by transposition

Transposition solution

Givenz0∈L²(−1,1),T>0, andu∈ M(ω×(0,T)), the functionz∈L¹((−1,1)×(0,T))is a solution of (F H) defined by transposition if

Z

ω×(0,T)

p(x,t)du(x,t) =hz(·,T),pTi − Z1

−1

z0(x)p(x,0)dx,

where, for everypT∈L^∞(−1,1), the functionp∈L^∞(Q)is the unique solution of







−pt+ (−d_x²)^sp=0, (x,t)∈(−1,1)×(0,T) p=0, (x,t)∈(−1,1)^c×(0,T) p(·,T) =pT, x∈(−1,1).

Proof of the main result - 3: minimal-time control

• DenoteTk:=Tmin+¹_k,k≥1.

There exists a sequence of non-negative controls{u^Tk}_k≥1⊂L^∞(ω×(0,T_k)) such that the corresponding solutionz^kof (F H) withz^k(x,0) =z0(x)a.e. in (−1,1)satisfiesz^k(x,Tk) =bz(x,Tk)a.e. in(−1,1).

• Extend these controls bybuon(T_k,T_min+1)to get a new sequence in L^∞(ω×(0,T_min+1)).

• pT >0 ⇒ p(x,t)≥θ >0 for all(x,t)∈(−1,1)×(0,Tmin+1). Then, θ

u^Tk

L¹(ω×(0,T_min+1))=θ Z T_min+1

0

Z

ω

u^Tk(x,t)dxdt

≤ ZT_min+1

0

Z 1

−1

p(x,t)u^Tk(x,t)dxdt

=hz(·,T),pTi − Z1

−1

z0(x)p(x,0)dx≤M.

• {u^Tk}k≥1isboundedinL¹(ω×(0,T_min+1)), hence, it is bounded in the space M(ω×(0,T_min+1)). Thus, extracting a sub-sequence, we have:

u^Tk *^∗ eu weakly−∗inM(ω×(0,Tmin+1))as k→+∞.

Proof of the main result - 3: minimal-time control (cont.)

• For anyklarge enough andT_min<T₀<T_min+1, we have Z

ω×(0,T₀)

p(x,t)du^Tk(x,t) =hbz(·,T0),pTi − Z1

−1

z0(x)p(x,0)dx.

• p_T: firstnon-negativeeigenfunction of(−d_x²)^s

p∈C([0,T];D((−d_x²)^s)),→C([0,T]×[−1,1]).

• By definition ofweak^∗limit, lettingk→+∞, we obtain Z

ω×(0,T₀)

p(x,t)deu(x,t) =hbz(·,T₀),pTi − Z 1

−1

z0(x)p(x,0)dx, which implies thatz(x,T₀) =bz(x,T₀)a.e. in(−1,1).

• Taking the limit asT₀→T_minand using the fact that

|eu|(ω×(Tmin,T0)) =|ˆu|(ω×(Tmin,T0)) =0, as T0→Tmin

we deduce thatz(x,Tmin) =bz(x,Tmin)a.e. in(−1,1).

Numerical simulations

• We consider the problem of steering the initial datumz0(x) =¹₂cos ^π₂x to the target trajectorybzsolution ofF Hwith initial datumbz₀(x) =6cos ^π₂x

and right-hand sidebu≡1.

• We chooses=0.8andω= (−0.3,0.8)⊂(−1,1)as the control region.

• The approximation of the minimal controllability time is obtained by solving the following constrained minimization problem:

minimize T













 T >0

zt+ (−d_x²)^sz=uχω, a.e.in(−1,1)×(0,T) z(·,0) =z₀≥0, a.e.in(−1,1) z≥0, a.e.in(−1,1)×(0,T) u≥0, a.e.inω×(0,T).

Numerical simulations

To perform the simulations, we apply a FE method for the space discretization of the fractional Laplacian on a uniform space-grid

x

=

+ 2i N

,

i = 1, . . . , N

with

. Moreover, we use an explicit Euler scheme for the time integration on the time-grid

t

= Tj N

,

j = 0, . . . , N

with

satisfying the

condition. In particular, we

choose here

Numerical simulations

•

We obtain the minimal time

•

In this time horizon, the fractional heat equation

is controllable from

the initial datum z

to the desired trajectory

z(·, T ) by maintaining the

positivity of the solution.

Numerical simulations

• The impulsional behavior of the control is lost when extending the time horizon beyondTmin.

This control has been computed by solving the minimization problem:

min kz(·,T)−bz(·,T)k_L2(−1,1)













 T>0

zt+ (−d_x²)^sz=uχω, a.e.in(−1,1)×(0,T) z(·,0) =z0≥0, a.e.in(−1,1) z≥0, a.e.in(−1,1)×(0,T)

u≥0, a.e.inω×(0,T).

Numerical simulations

•

Finally, when considering a time horizon T

T

, constrained

controllability fails.

THANK YOU FOR YOUR ATTENTION!

This project has received funding from the European Research Coun-

cil (ERC) under the European Union’s Horizon 2020 research and

innovation program (grant agreement No 694126-DYCON).