IFCAMKick-off meeting

University of Nice,19-21 Novenber, 2012

Control ofCompressible Navier Stokes System

Mythily RamaswamyTIFR Centre for Applicable Mathematics, Bangalore, India

Mythily Ramaswamy () 21st November, 2012 1 / 26

T.I.F.R Center for Applicable Mathematics

TIFR Centre for Applicable Mathematics, Bangalore

Mythily Ramaswamy () 21st November, 2012 2 / 26

Collaborators and Support

Collaborators:

Jean-Pierre Raymond (IMT, Toulouse),

Shirshendu Chowdhury, Debayan Maiti, Debanjana Mitra (TIFR-CAM)

Financial Support :

Indo-French Center for Promotion of Advanced Research, New Delhi

Project : IFC 3701-1

Mythily Ramaswamy () 21st November, 2012 3 / 26

Introduction Compressible Navier Stokes System in 1D

Compressible Navier Stokes System in 1D

The Navier-Stokes system for viscous compressible fluid in an interval of Rfor density ρ(x, t) and velocity v(x, t) for x ∈ R, t > 0

ρt(x, t) + (ρv)x(x, t) = 0,

ρ(x, t)[vt(x, t) + v(x, t)vx(x, t)] + (p(ρ))x(x, t) − νvxx(x, t) = 0,(1)

ν > 0 is the fluid viscosity;p, the pressure satisfies the constitutive law

p(ρ) = a ργ for a > 0, γ ≥ 1. (2)

Mythily Ramaswamy () 21st November, 2012 4 / 26

Introduction Scope of our work

Scope of our work

Linearize the system around constant steady states

With suitable boundary conditions get the spectrum and and aFourier basis

Study controllability : interior and boundary null controllability andapproximate controllabilty

Study stabilizability

Using this study local stabilizability of the nonlinear system

Mythily Ramaswamy () 21st November, 2012 5 / 26

Introduction Scope of our work

Known Results

Local exact controllability

Amosova (Differential Equations, 2011) :For the system in (0, 1) with homogeneous boundary condition for thevelocity with an interior control only on the velocity local exactcontrollability to trajectories for the velocity, provided that the initialdensity is already on the targeted trajectory (initial conditionH1 ×H1

0 ).

Ervedoza, Glass, Guerrero and Puel (Arch. Rational Mech.Anal.,2012) :local exact controllability to constant states (ρ, v) with ρ > 0, v 6= 0in a bounded domain (0, L) using two boundary controls both fordensity and velocity when initial conditions are regular( H3(0, L)).

Mythily Ramaswamy () 21st November, 2012 6 / 26

Introduction Scope of our work

Known Results

Local exact controllability

Amosova (Differential Equations, 2011) :For the system in (0, 1) with homogeneous boundary condition for thevelocity with an interior control only on the velocity local exactcontrollability to trajectories for the velocity, provided that the initialdensity is already on the targeted trajectory (initial conditionH1 ×H1

0 ).

Ervedoza, Glass, Guerrero and Puel (Arch. Rational Mech.Anal.,2012) :local exact controllability to constant states (ρ, v) with ρ > 0, v 6= 0in a bounded domain (0, L) using two boundary controls both fordensity and velocity when initial conditions are regular( H3(0, L)).

Mythily Ramaswamy () 21st November, 2012 6 / 26

Introduction Linearization

Initial boundary value problem for the linearized system

Domain Ω = (0, π)(Q0, v0) : a constant steady state solution with Q0 > 0, v0 ≥ 0Linearized system around this solution :

∂tρ + v0ρx + Q0 ux = 0

∂tu − ν

Q0uxx + v0 ux + aγ Qγ−2

0 ρx = fχO

with O ⊂ ΩInitial, boundary conditions :

ρ(x, 0) = ρ0(x) ; u(x, 0) = u0(x),u(0, t) = q0(t) ; u(π, t) = q1(t) ∀ t > 0

Additional boundary conditions for ρ wherever v0 > 0Distributed (internal) control : f ; Boundary controls q0, q1

Mythily Ramaswamy () 21st November, 2012 7 / 26

Linearization around (Q0, 0)

Function space framework

Function space for the case v0 = 0 : Z = L2(Ω) × L2(Ω)Equip with innerproduct⟨(

ρu

),

(σv

)⟩z

= aγQγ−20

∫ π

0ρ(x)σ(x)dx+Q0

∫ π

0u(x)v(x)dx

A dense subspace :

D(A) = (ρ(x)u(x)

)∈ Z : u(x) ∈ H1

0 (Ω), (bρ(x)− cu′(x)) ∈ H1(Ω)

Define A : D(A)→ Z :

A =

[0 Q0

ddx

aγQγ−20

ddx

−νQ0

d2

dx2

]D(A) is dense in Z ; A is maximal monotone

(−A,D(A)) is the infinitesimal generator of C0 semigroup S(t) on Z.

Mythily Ramaswamy () 21st November, 2012 8 / 26

Linearization around (Q0, 0)

Operator Equation

Call U(x, t) =(ρ(x, t)u(x, t)

)System without controls :

dU(t)dt

+ AU(t) = 0, t > 0

U(0) = U0 ∈ Z.

For every U0 ∈ Z, there is a unique solution U in C([0,∞),Z)

Mythily Ramaswamy () 21st November, 2012 9 / 26

Linearization around (Q0, 0) Existence via semigroup theory

Spectrum of A when v0 = 0

The point spectrum of A consists of

a finite number of pairs of complex eigenvalues :

Real(λk) ≥ ν

2Q0, |Im(λk)| ≤ 2aγQ0

ν.

an infinite number of pairs of real eigenvalues :

limn→∞

λn =aγQ0

ν:= ω0, µn →∞ as n→∞.

Eigenfunctions coresponding to λn and µn respectively :

ξn(x) =(

cos(nx)λnQ0n

sin(nx)

), ζn(x) =

(cos(nx)µnQ0n

sin(nx)

).

Mythily Ramaswamy () 21st November, 2012 10 / 26

Linearization around (Q0, 0) Spectral analysis

Orthonormal basis

Define a Fourier basis Φn in Z :

Φ0(x) =1√bπ

(10

);

Φ2n(x) =

√2bπ

(cos(nx)

0

), Φ2n−1(x) =

√2

Q0π

(0

sin(nx)

)for n ≥ 1.Define the subspaces :

V0 = span Φ0; Vn = span Φ2n,Φ2n−1, n ≥ 1

Z is the orthogonal sums of the subspaces Vnn≥0.

Z0, is the orthogonal sum of Vnn≥1 :

Z0 := (ρu

)∈ Z :

∫ π

0ρ(x)dx = 0.

Mythily Ramaswamy () 21st November, 2012 11 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Interior control

Qn : Can we bring the linearized system to rest in time T bycontrolling the velocity in Ω ?

The system :

U(t) +AU(t) = F(t)U(0) = U0

F(t) =(

0f(., t)

), f ∈ L2((0,∞), L2(Ω))

Using the Fourier basis in Z0,

F(t) =∞∑n=1

fn(t)Φ2n−1.

Mythily Ramaswamy () 21st November, 2012 12 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Controllability

When a basis is available, one can try building a control :

Project the system on finite dimensional spaces

If the finite dimensional system is controllable, find a minimum normcontrol

Sum up these finite dimensional control to build a control for the fullspace

Mythily Ramaswamy () 21st November, 2012 13 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Projected system

The system projected on Vn is

Un(t) +AnUn(t) = pn(t)B, B =[

01

]Un(0) = U0,n

Un = U |Vn and U0,n = U(0) |Vn .

For any given T > 0, the finite dimensional system is controllable.

Using optimal control theory, can find minimal norm control :

fn(t) = −(B∗e−(T−t)A∗nW−1T e−TAn)U0,n,

WT =

T∫0

e−tAnBB∗e−tA∗ndt

Mythily Ramaswamy () 21st November, 2012 14 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Projected system

The system projected on Vn is

Un(t) +AnUn(t) = pn(t)B, B =[

01

]Un(0) = U0,n

Un = U |Vn and U0,n = U(0) |Vn .

For any given T > 0, the finite dimensional system is controllable.

Using optimal control theory, can find minimal norm control :

fn(t) = −(B∗e−(T−t)A∗nW−1T e−TAn)U0,n,

WT =

T∫0

e−tAnBB∗e−tA∗ndt

Mythily Ramaswamy () 21st November, 2012 14 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Estimates for minimum norm control

DenoteCn(t) = −(B∗e−(T−t)A∗nW−1

T e−TAn)

Minimal norm control: fn(t) = Cn(t)U0,n

Cn(t) = [c1(t), c2(t)]

For large n, using the behaviour of eigenvalues

c1(t) = O(n e−α(2T−t)

2 ); c2(t) = O(e−α(2T−t)

c )

uniformly with respect to t ∈ [0, T ].

Mythily Ramaswamy () 21st November, 2012 15 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Estimates for minimum norm control

DenoteCn(t) = −(B∗e−(T−t)A∗nW−1

T e−TAn)

Minimal norm control: fn(t) = Cn(t)U0,n

Cn(t) = [c1(t), c2(t)]

For large n, using the behaviour of eigenvalues

c1(t) = O(n e−α(2T−t)

2 ); c2(t) = O(e−α(2T−t)

c )

uniformly with respect to t ∈ [0, T ].

Mythily Ramaswamy () 21st November, 2012 15 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Estimates for minimum norm control

DenoteCn(t) = −(B∗e−(T−t)A∗nW−1

T e−TAn)

Minimal norm control: fn(t) = Cn(t)U0,n

Cn(t) = [c1(t), c2(t)]

For large n, using the behaviour of eigenvalues

c1(t) = O(n e−α(2T−t)

2 ); c2(t) = O(e−α(2T−t)

c )

uniformly with respect to t ∈ [0, T ].

Mythily Ramaswamy () 21st November, 2012 15 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Null controllability

f ∈ L2((0,∞), L2(Ω)) iff∑‖fn‖2L2(0,∞) <∞, iff∑∞n=1

∫ π0 |c1(t)U1

0,n + c2(t)U20,n|2dt < ∞

if and only if the initial value U0 ∈ H1m(Ω)× L2(Ω) .

Theorem

For T > 0, the system is null controllable in time T , using interior control

f ∈ L2((0,∞), L2(Ω)) if and only if U0 =(ρ0

u0

)∈ H1

m(Ω)× L2(Ω),

H1m(Ω) = ρ ∈ H1(Ω) :

∫ π0 ρ(x)dx = 0.

This is optimal in the sense that the system is not null controllable usingany boundary control or an interior control acting on a subset of Ω.

Mythily Ramaswamy () 21st November, 2012 16 / 26

Linearization around (Q0, 0) Controllability of Linearized CNS System

Null controllability

f ∈ L2((0,∞), L2(Ω)) iff∑‖fn‖2L2(0,∞) <∞, iff∑∞n=1

∫ π0 |c1(t)U1

0,n + c2(t)U20,n|2dt < ∞

if and only if the initial value U0 ∈ H1m(Ω)× L2(Ω) .

Theorem

For T > 0, the system is null controllable in time T , using interior control

f ∈ L2((0,∞), L2(Ω)) if and only if U0 =(ρ0

u0

)∈ H1

m(Ω)× L2(Ω),

H1m(Ω) = ρ ∈ H1(Ω) :

∫ π0 ρ(x)dx = 0.

This is optimal in the sense that the system is not null controllable usingany boundary control or an interior control acting on a subset of Ω.

Mythily Ramaswamy () 21st November, 2012 16 / 26

Linearization at (Q0, v0)

Linearized system at (Q0, v0)

For the linearized system around (Q0, v0) withPeriodic boundary conditions for ρ, u and ux in (0, 2π)

The spectrum lies on the left side of the complex plane, infinite number oftwo pairs of complex eigenvalues and no accumulation point in thespectrum. Absolute value of the eigenvalues goes to infinity.Can work with Fourier basis and Moment method.

ResultsApproximately controllable for T sufficiently large.Null controllable with regular interior control and also regular boundarycontrol if T > 2π

v0.

Mythily Ramaswamy () 21st November, 2012 17 / 26

Linearization at (Q0, v0)

Dirichlet Boundary Conditions

For the linearized system around (Q0, v0) withDirichlet boundary conditions for ρ, uNo knowledge of spectrum and eigenfunctions.

Theorem

Let O = (0, l), 0 < l < L. System is approximately controllable at time T,by the localized interior control f ∈ L2(0, T, L2(O)) for the velocity if

T > (L−l)V0

.

Proof follows by showing that the adjoint problem satisfies this uniquecontinuation property if T > (L−l)

V0.

Contrasting Behaviour

Model 1 : v0 = 0, system - parabolic

Model 2 : v0 > 0, system - hyperbolic

Mythily Ramaswamy () 21st November, 2012 18 / 26

Linearization at (Q0, v0)

Controllability results

Both Systems are approximately controllable.

Boundary null controllable ?:Model 1 : No ;Model 2 : Yes with regular control.

Null controllable with localized interior control?Model 1 : No ;Model 2 : yes with regular control

Qn Is the system stabilizable?

Yes, the spectrum is in the negative half space.Solution decays like e−σt for o < σ < minν0, ω0

Mythily Ramaswamy () 21st November, 2012 19 / 26

Stabilizability

Stabilizability for v0 = 0

Qn : The linearized system is stabilizable at what rate of decay?

Stabilizable with decay rate e−σt for o < σ < ω0, the accumulationpoint for the real eigenvalues of A.

Qn : Can we stabilize with decay rate e−σt for σ ≥ ω0 ?

∂t

[ρu

]+[

0 Q0ddx

b ddx −c d2dx2

] [ρu

]= 0,

ρ(x, 0) = ρ0(x); u(x, 0) = u0(x),u(0, t) = 0; u(π, t) = q(t) ∀ t > 0.

Difficulty : Infinitely many eigenvalues will be greater than σ.

Mythily Ramaswamy () 21st November, 2012 20 / 26

Stabilizability

Stabilizability for v0 = 0

Qn : The linearized system is stabilizable at what rate of decay?

Stabilizable with decay rate e−σt for o < σ < ω0, the accumulationpoint for the real eigenvalues of A.

Qn : Can we stabilize with decay rate e−σt for σ ≥ ω0 ?

∂t

[ρu

]+[

0 Q0ddx

b ddx −c d2dx2

] [ρu

]= 0,

ρ(x, 0) = ρ0(x); u(x, 0) = u0(x),u(0, t) = 0; u(π, t) = q(t) ∀ t > 0.

Difficulty : Infinitely many eigenvalues will be greater than σ.

Mythily Ramaswamy () 21st November, 2012 20 / 26

Stabilizability

Stabilizability for v0 = 0

Qn : The linearized system is stabilizable at what rate of decay?

Stabilizable with decay rate e−σt for o < σ < ω0, the accumulationpoint for the real eigenvalues of A.

Qn : Can we stabilize with decay rate e−σt for σ ≥ ω0 ?

∂t

[ρu

]+[

0 Q0ddx

b ddx −c d2dx2

] [ρu

]= 0,

ρ(x, 0) = ρ0(x); u(x, 0) = u0(x),u(0, t) = 0; u(π, t) = q(t) ∀ t > 0.

Difficulty : Infinitely many eigenvalues will be greater than σ.

Mythily Ramaswamy () 21st November, 2012 20 / 26

Stabilizability

Stabilizability for v0 = 0

Qn : The linearized system is stabilizable at what rate of decay?

Stabilizable with decay rate e−σt for o < σ < ω0, the accumulationpoint for the real eigenvalues of A.

Qn : Can we stabilize with decay rate e−σt for σ ≥ ω0 ?

∂t

[ρu

]+[

0 Q0ddx

b ddx −c d2dx2

] [ρu

]= 0,

ρ(x, 0) = ρ0(x); u(x, 0) = u0(x),u(0, t) = 0; u(π, t) = q(t) ∀ t > 0.

Difficulty : Infinitely many eigenvalues will be greater than σ.

Mythily Ramaswamy () 21st November, 2012 20 / 26

Stabilizability

Stabilizability for v0 = 0

Qn : The linearized system is stabilizable at what rate of decay?

Stabilizable with decay rate e−σt for o < σ < ω0, the accumulationpoint for the real eigenvalues of A.

Qn : Can we stabilize with decay rate e−σt for σ ≥ ω0 ?

∂t

[ρu

]+[

0 Q0ddx

b ddx −c d2dx2

] [ρu

]= 0,

ρ(x, 0) = ρ0(x); u(x, 0) = u0(x),u(0, t) = 0; u(π, t) = q(t) ∀ t > 0.

Difficulty : Infinitely many eigenvalues will be greater than σ.

Mythily Ramaswamy () 21st November, 2012 20 / 26

Stabilizability

Projections of the system on eigenspaces

Outline of the method

Write the system as an operator equation

Project it onto one dimensional eigenspaces corresponding toeigenvalues accumulating at ω0

Find the expression for minimum norm control stabilizing this onedimensional projections

Find its limit as n tends to infinity

Mythily Ramaswamy () 21st November, 2012 21 / 26

Stabilizability

Projections of the system on eigenspaces

Outline of the method

Write the system as an operator equation

Project it onto one dimensional eigenspaces corresponding toeigenvalues accumulating at ω0

Find the expression for minimum norm control stabilizing this onedimensional projections

Find its limit as n tends to infinity

Mythily Ramaswamy () 21st November, 2012 21 / 26

Stabilizability

Projections of the system on eigenspaces

Outline of the method

Write the system as an operator equation

Project it onto one dimensional eigenspaces corresponding toeigenvalues accumulating at ω0

Find the expression for minimum norm control stabilizing this onedimensional projections

Find its limit as n tends to infinity

Mythily Ramaswamy () 21st November, 2012 21 / 26

Stabilizability

Projections of the system on eigenspaces

Outline of the method

Write the system as an operator equation

Project it onto one dimensional eigenspaces corresponding toeigenvalues accumulating at ω0

Find the expression for minimum norm control stabilizing this onedimensional projections

Find its limit as n tends to infinity

Mythily Ramaswamy () 21st November, 2012 21 / 26

Stabilizability

Projections of the system on eigenspaces

Outline of the method

Write the system as an operator equation

Project it onto one dimensional eigenspaces corresponding toeigenvalues accumulating at ω0

Find the expression for minimum norm control stabilizing this onedimensional projections

Find its limit as n tends to infinity

Mythily Ramaswamy () 21st November, 2012 21 / 26

Stabilizability

stabilizability

Theorem

The system is not boundary stabilizable in Z0 with decay rate e−ωt forω > ω0, the accumulation point of the eigenvalues of A, using boundarycontrol q(t) at x = π for the velocity component u, for arbitrary initial

condition U0 =(ρ0

u0

)∈ Z0.

Mythily Ramaswamy () 21st November, 2012 22 / 26

Extensions

System of equations including temperature

Equations for density, velocity and temperature

system of three equations for ρ, v, θ

Linearization around (Q0, 0, θ0)

Behaviour like parabolic system

Mythily Ramaswamy () 21st November, 2012 23 / 26

Extensions

Nonlinear system

Full nonlinear system near a stationary solution (ρ0, 0)

Can we stabilize the nonlinear system with the control found for thelinearized system?

Mythily Ramaswamy () 21st November, 2012 24 / 26

Extensions

Linearized system in a rectangle

Linearized system in 2 dimension in a rectangle

Fourier Basis for suitable boundary conditions

Null controllability for Dirichlet boundary conditions

Optimal control problem

Mythily Ramaswamy () 21st November, 2012 25 / 26

Extensions

Summary and Future work

Linearized Compressible Navier Stokes SystemCoupled system of transport and parabolic equations.

linearization at (Q0, 0) parabolic

linearization at (Q0, v0) hyperbolic

Questions

Computational aspects : Can we compute the control ?

Feedback control, observers?

Analysis of linearized system at nonconstant steady states?

Nonlinear system behaviour?

Other coupled systems?

Mythily Ramaswamy () 21st November, 2012 26 / 26

Extensions

Summary and Future work

Linearized Compressible Navier Stokes SystemCoupled system of transport and parabolic equations.

linearization at (Q0, 0) parabolic

linearization at (Q0, v0) hyperbolic

Questions

Computational aspects : Can we compute the control ?

Feedback control, observers?

Analysis of linearized system at nonconstant steady states?

Nonlinear system behaviour?

Other coupled systems?

Mythily Ramaswamy () 21st November, 2012 26 / 26

Extensions

S. Chowdhury and M. Ramaswamy, Optimal Control of LinearizedCompressible Navier-Stokes Equations, To appear in ESIAM : COCV -Control, Optimization and Calculus of Variations.

S. Chowdhury, M. Ramaswamy and J-P. Raymond, Controllability andstabilizability of the linearized compressible Navier-Stokes system inone dimension, To appear in SICON.

S. Chowdhury, D. Mitra, Controllability Results for LinearizedCompressible Navier Stokes System in One Dimension, Submitted

Mythily Ramaswamy () 21st November, 2012 26 / 26