Kenji Kashima (Kyoto Univ.)

Selective Pattern Formation in Reaction-Diffusion Systems

Joint work with: T. Ogawa (Dynamical system theory) T. Sakurai (Physics) Selective pattern formation control: Spatial spectrum consensus and Turing instability approach, Automatica 2015

Turing Pattern The Chemical Basis of Morphogenesis (1952)

A key mechanism for pattern formation ◦ Chemical reaction: Belousov-Zhabotinsky ◦ Biosystems: skin, gene expression

Specially uniform reaction-diffusion equation

2/26

Outline Preliminary: Diffusively coupled identical

(stable) systems ◦ Synchronization ◦ Turing Instability

Problem formulation: Nonlinear PDE with unknown target state

Approach: Spatial spectrum consensus Simulation and concluding remarks

3/26

Synchronization

Subsystem dynamics ◦ �̇�𝑖 𝑡 = 𝐴𝑥𝑖 𝑡 + 𝐷𝑢𝑖 𝑡

Rendezvous, synchronization ◦ 𝑥𝑖 − 𝑥𝑗 → 0, 𝑡 → ∞ ∀𝑖, 𝑗

Diffusive coupling ◦ 𝑢𝑖 𝑡 = ∑ 𝛾𝑖𝑗(𝑥𝑗 𝑡 − 𝑥𝑖 𝑡 )𝑖∈𝑁𝑖

𝛾𝑖𝑗 ≥ 0 𝑗 ≠ 𝑖 , connection topology, strength

𝑖

𝑵𝒊

4/26

Closed-loop dynamics �̇� = 𝐴⨂𝐼𝑛 + 𝐷⨂Γ 𝑥 ◦ 𝑥 ≔ 𝑥1 ⋯ 𝑥𝑛 T ◦ Γ: Graph Laplacian Eigenvalues: 0 = 𝜆1 > 𝜆2 ≥ 𝜆3 ≥ ⋯

eig 𝐴⨂𝐼𝑛 + 𝐷⨂Γ = ⋃ eig(𝐴 + 𝜆𝐷)𝜆∈eig(Γ)

Special case: 𝐷 = 𝑐𝐼, 𝑐 > 0 ◦ Stability/Instability preserving: 𝜆𝐴 + 𝑐𝜆Γ Consensus achieved

Other cases: Not necessarily true 5/26

Formulation by reaction-diffusion

Reaction (stable) 𝑑𝑑𝑑𝑑

= 𝑓(𝑧) 𝑧(𝑡)

◦ GAS equil. 𝑧 = 0

Diffusion 𝜕𝑑𝜕𝑑

= 𝑓 𝑧 + 𝐷∆𝑧, 𝜉 ∈ Ρ 𝑧(𝑡, 𝜉)

◦ Diagonal 𝐷 ⋟ 0 ◦ Is 𝑧𝑒𝑒(𝜉) ≡ 0 stable as well?

Example: ◦ �̇� = 𝑢 − 𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 ◦ �̇� = 3𝑢 − 2𝑣 + 𝑑𝑣∆𝑣

𝑢

𝑣

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Rectangular Domain Example ◦ �̇� = 𝑢 − 𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 ◦ �̇� = 3𝑢 − 2𝑣 + 𝑑𝑣∆𝑣 ◦ 𝑑𝑣/𝑑𝑢 ≫ 1

𝑢(𝑡, 𝑥, 𝑦)

𝑥

𝑦

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Mathematical model Reaction-Diffusion

◦ ��̇� = 𝑎11𝑢 + 𝑎12𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 �̇� = 𝑎12𝑢 + 𝑎22𝑣 + 𝑑𝑣∆𝑣

◦ Square domain, periodic boundary conditions This toy models is… ◦ an activator-inhibitor RD systems with cubic

nonlinearities ◦ able to captures the essential mathematical

structure of spatial symmetry ◦ essentially equivalent to phase-field model, the Cahn-Hilliard model,

the Ginzburg-Landau equation, the Swift-Hohenberg equation, and so on 8/26

Outline Preliminary: Diffusively coupled identical

(stable) systems ◦ Multi-agent systems ◦ Turing Instability

Problem formulation: Nonlinear PDE with unknown target state

Approach: Spatial spectrum consensus Simulation and concluding remarks

9/26

Control of pattern formation Cardiac arrhythmias

Chemical reaction 𝜕𝑧(𝑡,𝑝)𝜕𝑡

= 𝑓 𝑧(𝑡,𝑝) + 𝐷∆𝑧 𝑡, 𝑝

Controller

Video projector

Photo-sensitive excitable media

Half mirror

Sakurai et al., Design and control of wave propagation patterns in excitable media, Science (2001)

+𝒘(𝒕,𝒑)

S. Luther et al., Low-energy control of electrical turbulence in the heart, Nature (2011)

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What we attempt to do is … not to generate an arbitrarily given pattern by

forcing some inputs

but to assist a (possibly hidden) pattern generation

11/26

Assisted pattern formation Example revisited: ◦ Initial pattern dependent convergence to one of:

Can we generate their superposition as well?

? 12/26

Spatial Fourier Transformation Basis functions are stripes labeled by 𝑚 ∈ ℤ2 ◦ 𝑝𝑚 𝑥,𝑦 = 𝑒2𝜋𝑗(𝑚𝑥𝑥+𝑚𝑦𝑦) ◦ 𝑢 𝑡, 𝑥,𝑦 = ∑ 𝑢�𝑚 𝑡 𝑝𝑚 𝑥,𝑦𝑚 , 𝑣�𝑚 𝑡

Reaction-Diffusion

◦ ��̇� = 𝑎11𝑢 + 𝑎12𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 �̇� = 𝑎12𝑢 + 𝑎22𝑣 + 𝑑𝑣∆𝑣

◦ �𝑢�̇𝑚 = (𝑎11 −𝑠𝑚𝑑𝑢)𝑢�𝑚 +𝑎12𝑣�𝑚 − 𝑢�𝑚(𝑐𝑐𝑢𝑝𝑐𝑒) 𝑣�̇𝑚 = 𝑎12𝑢�𝑚 + (𝑎22−𝑠𝑚𝑑𝑣)𝑣�𝑚

◦ 𝑠𝑚 ≔ 2𝜋𝑚𝑥𝐿𝑥

2+ 2𝜋𝑚𝑦

𝐿𝑦

2

(𝐴 − 𝑠𝑚𝐷) 𝑢�𝑚𝑣�𝑚

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Problem Formulation Controlled Reaction-Diffusion

◦ ��̇� = 𝑎11𝑢 + 𝑎12𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 + 𝑤 �̇� = 𝑎12𝑢 + 𝑎22𝑣 + 𝑑𝑣∆𝑣

◦ ℳ: = 𝑚:𝐴 − 𝑠𝑚𝐷 is unstable

Control objective: in large 𝑡 limit ◦ 𝑢(𝑡, 𝑥,𝑦) → 𝑢�(𝑡)∑ 𝑝𝑚 𝑥,𝑦𝑚∈ℳ ◦ 𝑢� 𝑡 ↛ 0,∞ ◦ 𝑤(𝑡, 𝑥,𝑦) → 0

◦ Remarks Not to maximize 𝑢�(𝑡) Not to regulate 𝑢(𝑡, 𝑥,𝑦) → 𝑢∘ 𝑥,𝑦

?

4,0 2,2 (2,−2)

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Outline Preliminary: Diffusively coupled identical

(stable) systems ◦ Multi-agent systems ◦ Turing Instability

Problem formulation: nonlinear PDE with unknown target state

Approach: Spatial spectrum consensus Concluding remarks

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Assumptions Controlled RD

◦ �𝑢�̇𝑚 = (𝑎11 −𝑠𝑚𝑑𝑢)𝑢�𝑚 +𝑎12𝑣�𝑚 − 𝑢�𝑚(𝑐𝑐𝑢𝑝𝑐𝑒) + 𝑤�𝑚 𝑣�̇𝑚 = 𝑎12𝑢�𝑚 + (𝑎22−𝑠𝑚𝑑𝑣)𝑣�𝑚

for 𝑚 ∈ ℤ2 ◦ ℳ = 𝑚:𝐴 − 𝑠𝑚𝐷 is unstable ◦ Approximation: 𝑢�𝑚, 𝑣�𝑚 ≈ 0 for 𝑚 ∉ ℳ

Assumption ◦ 𝑠𝑚 = �̅� for all 𝑚 ∈ ℳ

(𝐴 − 𝑠𝑚𝐷) 𝑢�𝑚𝑣�𝑚

4,0 2,2 (2,−2)

𝑠𝑚 ≔2𝜋𝑚𝑥

𝐿𝑥

2

+2𝜋𝑚𝑦

𝐿𝑦

2

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Theorem: Spatial spectrum consensus Consider the truncated coupled ℂ2-valued dynamics

�𝑢�̇𝑚 = (𝑎11 −�̅�𝑑𝑢)𝑢�𝑚 +𝑎12𝑣�𝑚 − 𝑢�𝑚(𝑐𝑐𝑢𝑝𝑐𝑒) + 𝑤�𝑚 𝑣�̇𝑚 = 𝑎12𝑢�𝑚 + (𝑎22−�̅�𝑑𝑣)𝑣�𝑚

for 𝑚 ∈ ℳ. Then, 𝑤�𝑚 = 𝜎� 𝛾𝑛𝑚(𝑢�𝑚 − 𝑢�𝑛)

𝑛

yields ◦ 𝑢�𝑚 𝑚∈ℳ is bounded and achieves consensus ◦ 𝑤�𝑚 → 0 for 𝑚 ∈ ℳ ◦ the origin is unstable 𝑢(𝑡, 𝑥,𝑦) → 𝑢�(𝑡)� 𝒑𝒎 𝑥,𝑦

𝑚∈ℳ

𝑢� 𝑡 ↛ 0,∞ 𝑤(𝑡, 𝑥, 𝑦) → 0

(𝐴 − �̅�𝐷) 𝑢�𝑚𝑣�𝑚

𝐴 − �̅�𝐷 is unstable 17/26

Embedding onto PDE Controller implementation ◦ 𝑤 𝑡, 𝑥,𝑦 = ∑ 𝑤�𝑚 𝑡 𝑝𝑚 𝑥,𝑦𝑚∈ℳ ◦ 𝑤�𝑚 = 𝜎∑ 𝛾𝑛𝑚(𝑢�𝑚 − 𝑢�𝑛)𝑛∈ℳ

Controlled (𝒘(𝒕,𝒑) → 𝟎)

��̇� = 𝑎11𝑢 + 𝑎12𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 + 𝑤 �̇� = 𝑎12𝑢 + 𝑎22𝑣 + 𝑑𝑣∆𝑣

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Center manifold theorem �̇�𝑠 = 𝑓𝑠

𝑒 𝑥𝑠, 𝑥𝑢 , �̇�𝑢 = 𝑓𝑢𝑒(𝑥𝑠, 𝑥𝑢)

◦ Local stability: 𝑥𝑠 stable, 𝑥𝑢 unstable for 𝑞 > 𝑞∗ When 𝑞 is slightly larger than 𝑞∗, ◦ Existence of an invariant set ◦ If the dynamics on the invariant set is stable, then

this set is attractive.

𝑥𝑠

𝑥𝑢

𝑥𝑠 = 𝜊( 𝑥𝑢 2)

19/26

Application to pattern formation Reaction-Diffusion

◦ ��̇� = 𝑎11𝑢 + 𝑎12𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 �̇� = 𝑎12𝑢 + 𝑎22𝑣 + 𝑑𝑣∆𝑣

◦ �𝑢�̇𝑚 = (𝑎11 −𝑠𝑚𝑑𝑢)𝑢�𝑚 +𝑎12𝑣�𝑚 − 𝑢�𝑚(𝑐𝑐𝑢𝑝𝑐𝑒) 𝑣�̇𝑚 = 𝑎12𝑢�𝑚 + (𝑎22−𝑠𝑚𝑑𝑣)𝑣�𝑚

◦ 𝑠𝑚 ≔ 2𝜋𝑚𝑥𝐿𝑥

2+ 2𝜋𝑚𝑦

𝐿𝑦

2

(𝐴 − 𝑠𝑚𝐷) 𝑢�𝑚𝑣�𝑚

𝑥𝑠

𝑥𝑢

𝑥𝑠 = 𝜊( 𝑥𝑢 2)

ℳ: = 𝑚:𝐴 − 𝑠𝑚𝐷 is unstable (𝑢𝑚, 𝑣𝑚)𝑚∈ℳ 20/26

Spillover

RD system

Controller

Distribution profile domain Spatial spectrum domain

Controller

Locally stable modes

Locally unstable modes

Locally stable modes

Mismatch between locally unstable modes

Intensity of generated pattern

Technical assumptions guaranteed by suitable Lyapunov functionals

21/26

Center manifold for closed-loop PDE

��̇� = 𝑎11𝑢 + 𝑎12𝑣 − 𝑢3 + 𝑑𝑢∆𝑢 + 𝑤 �̇� = 𝑎12𝑢 + 𝑎22𝑣 + 𝑑𝑣∆𝑣

Notation: ◦ Orthogonal decomposition 𝑢 𝑡, 𝑥,𝑦 = 𝑢�(𝑡)∑ 𝒑𝒎 𝑥,𝑦𝑚∈ℳ + 𝑢𝑟𝑒𝑠 𝑡, 𝑥,𝑦

◦ Autonomous pattern formation for 𝑑𝑣 > 𝑑∗ Theorem: For any 𝜖 > 0, there exist δ,𝑇 > 0 such that ◦ 𝑢(0) < δ,𝑑𝑣 < 𝑑∗ + δ ⟹ 𝑢𝑟𝑒𝑠 𝑡 + 𝑤 𝑡 < 𝜖 𝑢�(𝑡) for 𝑡 > 𝑇

22/26

Intensity of generated pattern Input Residual component

𝜀1: Dynamic attractor Stabilization of standing waves ◦ Autonomous traveling waves

Controlled (𝒘(𝒕,𝒑) → 𝟎)

23/26

𝜀2: Delay robustness 𝑤 𝑡, 𝑥,𝑦 = 𝜎∑ 𝑤�𝑚 𝑡 − 𝑇 𝑝𝑚 𝑥,𝑦𝑚∈ℳ 𝑤�𝑚 = 𝜎∑ 𝛾𝑛𝑚(𝑢�𝑚 − 𝑢�𝑛)𝑛 Pattern formation condition for (𝜎,𝑇)

𝜕𝑧(𝑡,𝑝)𝜕𝑡

= 𝑓 𝑧(𝑡,𝑝) + 𝐷∆𝑧 𝑡, 𝑝

Controller

Video projector

Photo-sensitive excitable media

Half mirror

+𝒘(𝒕,𝒙,𝒚)

24/26

No More Fourier !!!

𝜀3: Enhancing rotational symmetry Utilizing the inherent spatial symmetry

𝑤(𝑡, 𝑥,𝑦) = 𝜎(ℛ3𝜋/2𝑢 𝑡, 𝑥,𝑦 − 𝑢(𝑡, 𝑥,𝑦)) ◦ 𝑤�𝑚 = 𝜎(𝑢�𝑚+1 − 𝑢�𝑚) ◦ Stability for residual modes guaranteed by the

small- gain theorem

あ

Controlled (𝒘(𝒕,𝒑) → 𝟎)

uncontrolled

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No Distributed Actuation !!!

Conclusion Formulated and solved feedback stabilization

of hidden Turing patterns Unavailability of the target profile was

managed by synchronization formulation Guided self-organization ◦ Semi-conductor engineering

Acknowledgement ◦ Toshiyuki Ogawa (Dynamical system theory) ◦ Tatsunari Sakurai (Physics)

◦ Selective pattern formation control: Spatial spectrum consensus and

Turing instability approach, Automatica 2015

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