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Local null-controllability of the 2-D Vlasov-Navier-Stokes system Iv an Moyano CMLS, Ecole Polytechnique June 22, 2016 Iv an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system The Vlasov-Navier-Stokes system

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Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Iv´ an Moyano

CMLS, ´ Ecole Polytechnique

June 22, 2016

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 5

The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles. Two essential magnitudes

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles. Two essential magnitudes the particles: the number of particles located at x in time t having velocity v is describe through the distribution function f (t, x, v) (Vlasov equation).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles. Two essential magnitudes the particles: the number of particles located at x in time t having velocity v is describe through the distribution function f (t, x, v) (Vlasov equation). the fluid: the velocity field of the fluid is described by a vector field u(t, x) (Navier-Stokes system).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles. Two essential magnitudes the particles: the number of particles located at x in time t having velocity v is describe through the distribution function f (t, x, v) (Vlasov equation). the fluid: the velocity field of the fluid is described by a vector field u(t, x) (Navier-Stokes system). The VNS system for (f , u) contains:

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles. Two essential magnitudes the particles: the number of particles located at x in time t having velocity v is describe through the distribution function f (t, x, v) (Vlasov equation). the fluid: the velocity field of the fluid is described by a vector field u(t, x) (Navier-Stokes system). The VNS system for (f , u) contains: a Vlasov (transport) equation for f ,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles. Two essential magnitudes the particles: the number of particles located at x in time t having velocity v is describe through the distribution function f (t, x, v) (Vlasov equation). the fluid: the velocity field of the fluid is described by a vector field u(t, x) (Navier-Stokes system). The VNS system for (f , u) contains: a Vlasov (transport) equation for f , a Navier-Stokes system for u,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Vlasov-Navier-Stokes (VNS): kinetic-fluid system describing the evolution of a large cloud of small particles interacting with a viscous incompressible fluid, under the effects of friction, without collisions between particles. Two essential magnitudes the particles: the number of particles located at x in time t having velocity v is describe through the distribution function f (t, x, v) (Vlasov equation). the fluid: the velocity field of the fluid is described by a vector field u(t, x) (Navier-Stokes system). The VNS system for (f , u) contains: a Vlasov (transport) equation for f , a Navier-Stokes system for u, coupling terms relying f and u.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Let T2 := R2/Z2, T > 0.            ∂tf + v · ∇xf + divv [(u − v)f ] = 0, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Let T2 := R2/Z2, T > 0.            ∂tf + v · ∇xf + divv [(u − v)f ] = 0, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Let T2 := R2/Z2, T > 0.            ∂tf + v · ∇xf + divv [(u − v)f ] = 0, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Let T2 := R2/Z2, T > 0.            ∂tf + v · ∇xf + divv [(u − v)f ] = 0, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. Analysis of the VNS system: Mean-field approach: L. Desvillettes, F. Golse and V. Ricci. (J. Stat. Phys., 2008) Weak solutions: L. Boudin, L. Desvillettes, C. Grandmont and

A. Moussa (DIE, 2008),
Hydrod. lim.: T. Goudon, P.E. Jabin, A. Vasseur (IUMJ,’08).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system

Let T2 := R2/Z2, T > 0.            ∂tf + v · ∇xf + divv [(u − v)f ] = 0, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. Goal: To control the dynamics of (f , u), both the particles and the fluid.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 17

The Vlasov-Navier-Stokes system

Let T2 := R2/Z2, T > 0.            ∂tf + v · ∇xf + divv [(u − v)f ] = 0, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. Goal: To control the dynamics of (f , u), both the particles and the fluid. How? We want to absorb particles from a subset of T2, ω ⊂ T2. This amounts to use an internal control in the Vlasov equation, located in the absorption region ω ⊂ T2.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system under control

Let T2 := R2/Z2, T > 0, ω ⊂ T2,            ∂tf + v · ∇xf + divv [(u − v)f ] = 1ω(x)G, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. Goal: To control the dynamics of (f , u), both the particles and the fluid. How? We want to absorb particles from a subset of T2, ω ⊂ T2. This amounts to use an internal control in the Vlasov equation, located in the absorption region ω ⊂ T2.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system under control

Let T2 := R2/Z2, T > 0, ω ⊂ T2.            ∂tf + v · ∇xf + divv [(u − v)f ] = 1ω(x)G, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. This is a Nonlinear Control system:

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system under control

Let T2 := R2/Z2, T > 0, ω ⊂ T2.            ∂tf + v · ∇xf + divv [(u − v)f ] = 1ω(x)G, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. This is a Nonlinear Control system: state: (f , u), pair distribution function-velocity field,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system under control

Let T2 := R2/Z2, T > 0, ω ⊂ T2.            ∂tf + v · ∇xf + divv [(u − v)f ] = 1ω(x)G, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. This is a Nonlinear Control system: state: (f , u), pair distribution function-velocity field, control: source 1ω(x)G(t, x, v), located in [0, T] × ω × R2.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system under control

Let T2 := R2/Z2, T > 0, ω ⊂ T2.            ∂tf + v · ∇xf + divv [(u − v)f ] = 1ω(x)G, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. This is a Nonlinear Control system: state: (f , u), pair distribution function-velocity field, control: source 1ω(x)G(t, x, v), located in [0, T] × ω × R2. The control G should modify the dynamics of (f , u)

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system under control

Let T2 := R2/Z2, T > 0, ω ⊂ T2.            ∂tf + v · ∇xf + divv [(u − v)f ] = 1ω(x)G, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. This is a Nonlinear Control system: state: (f , u), pair distribution function-velocity field, control: source 1ω(x)G(t, x, v), located in [0, T] × ω × R2. The control G should modify the dynamics of (f , u) by absorbing particles from ω,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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The Vlasov-Navier-Stokes system under control

Let T2 := R2/Z2, T > 0, ω ⊂ T2.            ∂tf + v · ∇xf + divv [(u − v)f ] = 1ω(x)G, in (0, T) × T2 × R2, ∂tu + (u · ∇x) u − ∆xu + ∇xp = jf − ρf u, in (0, T) × T2, divx u(t, x) = 0, in (0, T) × T2, f |t=0 = f0(x, v), in T2 × R2, u|t=0 = u0, in T2, with jf (t, x) :=

R2 vf (t, x, v) dv and ρf (t, x) :=
R2 f (t, x, v) dv.

Two Interaction terms: Non-linearities 1.- Coupling divv [(u − v)f ], 2.- Brinkman force jf − ρf u. This is a Nonlinear Control system: state: (f , u), pair distribution function-velocity field, control: source 1ω(x)G(t, x, v), located in [0, T] × ω × R2. The control G should modify the dynamics of (f , u) by absorbing particles from ω, by acting on the velocity of the fluid.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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Goal: an exact controllability result

The question of controllability:

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 28

Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X. We search a control G ∈ U, supported in (0, T) × ω × R2, such that the solution of VNS satisfies

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 29

Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X. We search a control G ∈ U, supported in (0, T) × ω × R2, such that the solution of VNS satisfies t = 0 t = T (f0, u0) − →

G

(f1, u1)

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 30

Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X. We search a control G ∈ U, supported in (0, T) × ω × R2, such that the solution of VNS satisfies t = 0 t = T (f0, u0) − →

G

(f1, u1) We shall answer positively to this question for X, U spaces of regular functions (H¨

lder),

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X. We search a control G ∈ U, supported in (0, T) × ω × R2, such that the solution of VNS satisfies t = 0 t = T (f0, u0) − →

G

(f1, u1) We shall answer positively to this question for X, U spaces of regular functions (H¨

lder),

for f0 and u0 small in X (local result),

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 32

Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X. We search a control G ∈ U, supported in (0, T) × ω × R2, such that the solution of VNS satisfies t = 0 t = T (f0, u0) − →

G

(f1, u1) We shall answer positively to this question for X, U spaces of regular functions (H¨

lder),

for f0 and u0 small in X (local result), for f1 = 0 and u1 = 0, (null-controllability),

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 33

Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X. We search a control G ∈ U, supported in (0, T) × ω × R2, such that the solution of VNS satisfies t = 0 t = T (f0, u0) − →

G

(f1, u1) We shall answer positively to this question for X, U spaces of regular functions (H¨

lder),

for f0 and u0 small in X (local result), for f1 = 0 and u1 = 0, (null-controllability), for a time T > 0 large enough,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 34

Goal: an exact controllability result

The question of controllability: Let X, U functional spaces. Let T > 0, ω ⊂ T2. Pick an initial state (f0, u0) ∈ X and a final state (f1, u1) ∈ X. We search a control G ∈ U, supported in (0, T) × ω × R2, such that the solution of VNS satisfies t = 0 t = T (f0, u0) − →

G

(f1, u1) We shall answer positively to this question for X, U spaces of regular functions (H¨

lder),

for f0 and u0 small in X (local result), for f1 = 0 and u1 = 0, (null-controllability), for a time T > 0 large enough, for ω ⊂ T2 satisfying a geometric assumption.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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Main result: Local null-controllability result

THEOREM (IM, 2016) Let γ > 2 and let ω ⊂ T2 satisfy the strip assumption. ∃ ǫ > 0, M > 0, T0 > 0 such that ∀T ≥ T0, and for every f0 ∈ C 1(T2 × R2) ∩ W 1,∞(T2 × R2) and u0 satisfying that u0 ∈ C 1(T2; R2) ∩ H2(T2; R2), divx u0 = 0, u0H

1 2 (T2) ≤ M,

f0C 1(T2×R2) + (1 + |v|)γ+2f0C 0(T2×R2) ≤ ǫ, ∃κ > 0, sup

T2×R2(1 + |v|)γ (|∇xf0| + |∇vf0|) (x, v) ≤ κ,

there exists a control G ∈ C 0([0, T] × T2 × R2) such that a strong solution of (VNS) with f |t=0 = f0 and u|t=0 = u0 exists, is unique and satisfies f |t=T = 0, u|t=T = 0.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 36

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Nonlinear non-collisional models (C (f ) ≡ 0)

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 37

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Nonlinear non-collisional models (C (f ) ≡ 0) Vlasov-Poisson (F is an electric field). Local and global exact controllability (O. Glass, JDE, 2003),

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 38

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Nonlinear non-collisional models (C (f ) ≡ 0) Vlasov-Poisson (F is an electric field). Local and global exact controllability (O. Glass, JDE, 2003), Vlasov-Poisson (F is an electric field + Lorentz/magnetic forces). Local and global exact controllability (O. Glass and

D. Han-Kwan, JDE, 2011),

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 39

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Nonlinear non-collisional models (C (f ) ≡ 0) Vlasov-Poisson (F is an electric field). Local and global exact controllability (O. Glass, JDE, 2003), Vlasov-Poisson (F is an electric field + Lorentz/magnetic forces). Local and global exact controllability (O. Glass and

D. Han-Kwan, JDE, 2011),

Vlasov-Maxwell relativistic (F is given by relativistic Maxwell equations). Local exact controllability (O. Glass and D. Han-Kwan, JMPA, 2015),

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 40

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Nonlinear non-collisional models (C (f ) ≡ 0) Vlasov-Poisson (F is an electric field). Local and global exact controllability (O. Glass, JDE, 2003), Vlasov-Poisson (F is an electric field + Lorentz/magnetic forces). Local and global exact controllability (O. Glass and

D. Han-Kwan, JDE, 2011),

Vlasov-Maxwell relativistic (F is given by relativistic Maxwell equations). Local exact controllability (O. Glass and D. Han-Kwan, JMPA, 2015), Vlasov-Stokes (F is given by a Stokes system). Local exact controllability (IM, 2015).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 41

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Collisional Linear models: C (f ) is a collision operator.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 42

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Collisional Linear models: C (f ) is a collision operator. Kolmogorov equation: C (f ) = −∆v

K. Beauchard and E. Zuazua, Ann. IH, 2009. K. Beauchard,

MCSS, 2014

J. Le Rousseau and I.M, JDE, 2016.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 43

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Collisional Linear models: C (f ) is a collision operator. Kolmogorov equation: C (f ) = −∆v

K. Beauchard and E. Zuazua, Ann. IH, 2009. K. Beauchard,

MCSS, 2014

J. Le Rousseau and I.M, JDE, 2016.

Orstein-Uhlenbeck operators (including Fokker-Planck)

K. Beauchard and K. Pravda-Starov, 2016.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 44

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Collisional Linear models: C (f ) is a collision operator. Kolmogorov equation: C (f ) = −∆v

K. Beauchard and E. Zuazua, Ann. IH, 2009. K. Beauchard,

MCSS, 2014

J. Le Rousseau and I.M, JDE, 2016.

Orstein-Uhlenbeck operators (including Fokker-Planck)

K. Beauchard and K. Pravda-Starov, 2016.

Linear Boltzmann equation.

D. Han-Kwan and M. L´

eautaud, Ann. PDE, 2015.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 45

Controllability of kinetic equations

Vlasov equations: describe the dynamics of a cloud of particles f (t, x, v) undergoing macroscopic forces F or collisions C , ∂tf + v · ∇xf + F(t, x) · ∇vf = C (f ). Collisional Linear models: C (f ) is a collision operator. Kolmogorov equation: C (f ) = −∆v

K. Beauchard and E. Zuazua, Ann. IH, 2009. K. Beauchard,

MCSS, 2014

J. Le Rousseau and I.M, JDE, 2016.

Orstein-Uhlenbeck operators (including Fokker-Planck)

K. Beauchard and K. Pravda-Starov, 2016.

Linear Boltzmann equation.

D. Han-Kwan and M. L´

eautaud, Ann. PDE, 2015. Fokker-Planck and linear Boltzmann.

C. Bardos and K.D. Phung, 2016.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 46

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 47

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why?

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 48

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 49

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 50

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|. What to do, then?

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 51

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|. What to do, then? There is still hope: Coron’s Return method

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 52

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|. What to do, then? There is still hope: Coron’s Return method Essential idea: to construct a trajectory of the nonlinear problem such that the linearised problem around it is controllable.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 53

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|. What to do, then? There is still hope: Coron’s Return method Essential idea: to construct a trajectory of the nonlinear problem such that the linearised problem around it is controllable. Scheme for non-collisional kinetic equations: (O. Glass and D. Han-Kwan, VP, VM, ’03,’15)

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 54

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|. What to do, then? There is still hope: Coron’s Return method Essential idea: to construct a trajectory of the nonlinear problem such that the linearised problem around it is controllable. Scheme for non-collisional kinetic equations: (O. Glass and D. Han-Kwan, VP, VM, ’03,’15) return method + Leray-Schauder fixed-point theorem.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 55

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|. What to do, then? There is still hope: Coron’s Return method Essential idea: to construct a trajectory of the nonlinear problem such that the linearised problem around it is controllable. Scheme for non-collisional kinetic equations: (O. Glass and D. Han-Kwan, VP, VM, ’03,’15) return method + Leray-Schauder fixed-point theorem. we construct a reference trajectory (f , u) eliminating the bad directions and the slow velocities.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 56

Strategy for the non-collisional cases: local results

Obstructions for controllability: the linearised system around zero is not controllable (transport equation, solved through characteristics). Why? Bad directions: (x, v) is such that {x + tv} ∩ ω = ∅ Low velocities: Even if (x, v) is such that

x + t v

|v|

∩ ω, it

can take a very large time to get there at speed |v|. What to do, then? There is still hope: Coron’s Return method Essential idea: to construct a trajectory of the nonlinear problem such that the linearised problem around it is controllable. Scheme for non-collisional kinetic equations: (O. Glass and D. Han-Kwan, VP, VM, ’03,’15) return method + Leray-Schauder fixed-point theorem. we construct a reference trajectory (f , u) eliminating the bad directions and the slow velocities. we construct a solution of VNS close to (f , u) beginning at (f0, u0) and satisfying (f , u)|t=T = (0, 0).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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How to eliminate the obstructions? Method of characteristics

Given f0 and u regular, the transport equation ∂tf + v · ∇xf + divv [(u − v)f ] = 0, (0, T) × T2 × R2, f |t=0 = f0(x, v), T2 × R2, has the explicit solution f (t, x, v) = e2tf0((X, V )(0, t, x, v)), where (X, V ) are the characteristics associated to −v + u.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 58

How to eliminate the obstructions? Method of characteristics

Given f0 and u regular, the transport equation ∂tf + v · ∇xf + divv [(u − v)f ] = 0, (0, T) × T2 × R2, f |t=0 = f0(x, v), T2 × R2, has the explicit solution f (t, x, v) = e2tf0((X, V )(0, t, x, v)), where (X, V ) are the characteristics associated to −v + u. Moral: the particles follow the characteristic flow.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 59

How to eliminate the obstructions? Method of characteristics

Given f0 and u regular, the transport equation ∂tf + v · ∇xf + divv [(u − v)f ] = 0, (0, T) × T2 × R2, f |t=0 = f0(x, v), T2 × R2, has the explicit solution f (t, x, v) = e2tf0((X, V )(0, t, x, v)), where (X, V ) are the characteristics associated to −v + u. Moral: the particles follow the characteristic flow. To absorb the particles, we have to get into the control region ω.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 60

How to eliminate the obstructions? Method of characteristics

Given f0 and u regular, the transport equation ∂tf + v · ∇xf + divv [(u − v)f ] = 0, (0, T) × T2 × R2, f |t=0 = f0(x, v), T2 × R2, has the explicit solution f (t, x, v) = e2tf0((X, V )(0, t, x, v)), where (X, V ) are the characteristics associated to −v + u. Moral: the particles follow the characteristic flow. To absorb the particles, we have to get into the control region ω. Strategy to avoid obstructions (Return Method):

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 61

How to eliminate the obstructions? Method of characteristics

Given f0 and u regular, the transport equation ∂tf + v · ∇xf + divv [(u − v)f ] = 0, (0, T) × T2 × R2, f |t=0 = f0(x, v), T2 × R2, has the explicit solution f (t, x, v) = e2tf0((X, V )(0, t, x, v)), where (X, V ) are the characteristics associated to −v + u. Moral: the particles follow the characteristic flow. To absorb the particles, we have to get into the control region ω. Strategy to avoid obstructions (Return Method): to construct a reference vector field u such that

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 62

How to eliminate the obstructions? Method of characteristics

Given f0 and u regular, the transport equation ∂tf + v · ∇xf + divv [(u − v)f ] = 0, (0, T) × T2 × R2, f |t=0 = f0(x, v), T2 × R2, has the explicit solution f (t, x, v) = e2tf0((X, V )(0, t, x, v)), where (X, V ) are the characteristics associated to −v + u. Moral: the particles follow the characteristic flow. To absorb the particles, we have to get into the control region ω. Strategy to avoid obstructions (Return Method): to construct a reference vector field u such that ∀(x, v) ∈ T2 × R2, ∃t > 0 s.t. X(t, 0, x, v) ∈ ω, . ∂tu + u · ∇xu − ∆xu + ∇xp = jf − ρf u.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 63

Reference solution: velocity field

We exploit the strip assumption on ω ⊂ T2.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 64

Reference solution: velocity field

We exploit the strip assumption on ω ⊂ T2. There exist a straight line H = span(n⊥

H) contained in ω.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 65

Reference solution: velocity field

We exploit the strip assumption on ω ⊂ T2. There exist a straight line H = span(n⊥

H) contained in ω. We want to

accelerate all the characteristics in the direction of nH.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 66

Reference solution: velocity field

We exploit the strip assumption on ω ⊂ T2. There exist a straight line H = span(n⊥

H) contained in ω. We want to

accelerate all the characteristics in the direction of nH. We use a controllability result for the NS system in T2 (Coron-Fursikov, 1996): we can modify the fluid to pass from 0 to the stationary solution nH.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 67

Reference solution: velocity field

We exploit the strip assumption on ω ⊂ T2. There exist a straight line H = span(n⊥

H) contained in ω. We want to

accelerate all the characteristics in the direction of nH. We use a controllability result for the NS system in T2 (Coron-Fursikov, 1996): we can modify the fluid to pass from 0 to the stationary solution nH. We wait enough time until all the characteristics have met ω.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 68

Reference solution: velocity field

We exploit the strip assumption on ω ⊂ T2. There exist a straight line H = span(n⊥

H) contained in ω. We want to

accelerate all the characteristics in the direction of nH. We use a controllability result for the NS system in T2 (Coron-Fursikov, 1996): we can modify the fluid to pass from 0 to the stationary solution nH. We wait enough time until all the characteristics have met ω. We put the velocity field back to zero thanks to the Coron-Fursikov’s theorem.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 69

Reference solution: distribution function

We have found u satisfying (Coron-Fursikov, 1996):

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 70

Reference solution: distribution function

We have found u satisfying (Coron-Fursikov, 1996): ∂tu + (u · ∇) u − ∆xu + ∇xp = w, (0, T) × T2, divx u = 0, (0, T) × T2,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 71

Reference solution: distribution function

We have found u satisfying (Coron-Fursikov, 1996): ∂tu + (u · ∇) u − ∆xu + ∇xp = w, (0, T) × T2, divx u = 0, (0, T) × T2, for a control w supported in (0, T) × ω.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 72

Reference solution: distribution function

We have found u satisfying (Coron-Fursikov, 1996): ∂tu + (u · ∇) u − ∆xu + ∇xp = w, (0, T) × T2, divx u = 0, (0, T) × T2, for a control w supported in (0, T) × ω. Associated distribution function: let Z1, Z2 ∈ S (R2) such that

R2 viZj(v) dv = δij,
R2 Zi(v) dv = 0,

i, j = 1, 2.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 73

Reference solution: distribution function

We have found u satisfying (Coron-Fursikov, 1996): ∂tu + (u · ∇) u − ∆xu + ∇xp = w, (0, T) × T2, divx u = 0, (0, T) × T2, for a control w supported in (0, T) × ω. Associated distribution function: let Z1, Z2 ∈ S (R2) such that

R2 viZj(v) dv = δij,
R2 Zi(v) dv = 0,

i, j = 1, 2. Then, define f (t, x, v) := (Z1, Z2)(v) · w(t, x), which gives

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 74

Reference solution: distribution function

We have found u satisfying (Coron-Fursikov, 1996): ∂tu + (u · ∇) u − ∆xu + ∇xp = w, (0, T) × T2, divx u = 0, (0, T) × T2, for a control w supported in (0, T) × ω. Associated distribution function: let Z1, Z2 ∈ S (R2) such that

R2 viZj(v) dv = δij,
R2 Zi(v) dv = 0,

i, j = 1, 2. Then, define f (t, x, v) := (Z1, Z2)(v) · w(t, x), which gives w = jf − ρf u, (0, T) × T2, ∂tf + v · ∇xf + divv

(u − v)f
= 0,

(0, T) × (T2 \ ω) × R2, f |t=0 = 0, f |t=T = 0.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 75

PROPOSITION Let ω ⊂ T2 satisfy the strip assumption. There exists T0 > 0 such that for any T ≥ T0, there exists a reference solution (f , u) of the VNS system such that f ∈ C ∞([0, T] × T2; S (R2)), u ∈ C ∞([0, T] × T2; R2), (f , u)|t=0 = (0, 0), (f , u)|t=T = (0, 0), supp(f ) ⊂ (0, T) × ω × R2, and such that the characteristics associated to u satisfy ∀(x, v) ∈ T2 × R2, ∃t ∈ T 12, 11T 12

such that

X(t, 0, x, v) ∈ ω, with |V (t, 0, x, v) · nH| ≥ 5.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 76

Construction of (f , u) close to (f , u): fixed-point scheme

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 77

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 78

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2),

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 79

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), in such a way Vǫ has a fixed point g∗ satisfying:

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 80

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), in such a way Vǫ has a fixed point g∗ satisfying: g∗ solves (VNS) for a certain control G ∗,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 81

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), in such a way Vǫ has a fixed point g∗ satisfying: g∗ solves (VNS) for a certain control G ∗, ∀(x, v) ∈ T2 × R2, ∃t such that X g∗(t, 0, x, v) ∈ ω.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), in such a way Vǫ has a fixed point g∗ satisfying: g∗ solves (VNS) for a certain control G ∗, ∀(x, v) ∈ T2 × R2, ∃t such that X g∗(t, 0, x, v) ∈ ω. We define Vǫ in three steps (NS-absorption-extension):

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 83

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), in such a way Vǫ has a fixed point g∗ satisfying: g∗ solves (VNS) for a certain control G ∗, ∀(x, v) ∈ T2 × R2, ∃t such that X g∗(t, 0, x, v) ∈ ω. We define Vǫ in three steps (NS-absorption-extension):

1 g ∈ Sǫ −

→ ug, solution of NS with Brinkman force jg − ρgug,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 84

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), in such a way Vǫ has a fixed point g∗ satisfying: g∗ solves (VNS) for a certain control G ∗, ∀(x, v) ∈ T2 × R2, ∃t such that X g∗(t, 0, x, v) ∈ ω. We define Vǫ in three steps (NS-absorption-extension):

1 g ∈ Sǫ −

→ ug, solution of NS with Brinkman force jg − ρgug,

2 ug

− →

characteristics

˜ Vǫ[g], solution of Vlasov with absorption in ω,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 85

Construction of (f , u) close to (f , u): fixed-point scheme

We define a domain, Sǫ

f ; ǫ − close to f
⊂ C 0([0, T] × T2 × R2)

depending on ǫ > 0 small, and a continuous operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), in such a way Vǫ has a fixed point g∗ satisfying: g∗ solves (VNS) for a certain control G ∗, ∀(x, v) ∈ T2 × R2, ∃t such that X g∗(t, 0, x, v) ∈ ω. We define Vǫ in three steps (NS-absorption-extension):

1 g ∈ Sǫ −

→ ug, solution of NS with Brinkman force jg − ρgug,

2 ug

− →

characteristics

˜ Vǫ[g], solution of Vlasov with absorption in ω,

˜ Vǫ[g] − →

Π−extension Vǫ[g] = f + Π(˜

Vǫ[g]).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 86

End of the proof the controllability theorem I

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 87

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 88

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point. We prove that Vǫ(Sǫ) ⊂ Sǫ,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 89

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point. We prove that Vǫ(Sǫ) ⊂ Sǫ, Vǫ is continuous.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 90

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point. We prove that Vǫ(Sǫ) ⊂ Sǫ, Vǫ is continuous. Thus, (Leray-Schauder) ∃g∗ ∈ Sǫ such that Vǫ[g∗] = g∗.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 91

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point. We prove that Vǫ(Sǫ) ⊂ Sǫ, Vǫ is continuous. Thus, (Leray-Schauder) ∃g∗ ∈ Sǫ such that Vǫ[g∗] = g∗. By construction, the fixed point satisfies

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 92

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point. We prove that Vǫ(Sǫ) ⊂ Sǫ, Vǫ is continuous. Thus, (Leray-Schauder) ∃g∗ ∈ Sǫ such that Vǫ[g∗] = g∗. By construction, the fixed point satisfies the VNS system, for a certain control,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 93

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point. We prove that Vǫ(Sǫ) ⊂ Sǫ, Vǫ is continuous. Thus, (Leray-Schauder) ∃g∗ ∈ Sǫ such that Vǫ[g∗] = g∗. By construction, the fixed point satisfies the VNS system, for a certain control, that the characteristics are close to the reference characteristics, i.e., (X, V ) − (X g∗, V g∗) ǫ + M (thanks to stability estimates for NS).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 94

End of the proof the controllability theorem I

We have to show that the operator Vǫ : Sǫ → C 0([0, T] × T2 × R2), has a fixed point. We prove that Vǫ(Sǫ) ⊂ Sǫ, Vǫ is continuous. Thus, (Leray-Schauder) ∃g∗ ∈ Sǫ such that Vǫ[g∗] = g∗. By construction, the fixed point satisfies the VNS system, for a certain control, that the characteristics are close to the reference characteristics, i.e., (X, V ) − (X g∗, V g∗) ǫ + M (thanks to stability estimates for NS). Thus, choosing ǫ and M small enough, (X g∗, V g∗) meet ω.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 95

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 96

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that g∗|t=T = 0 outside ω.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 97

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that g∗|t=T = 0 outside ω. We have confined all the particles in ω!

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 98

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that g∗|t=T = 0 outside ω. We have confined all the particles in ω! Next step: How to attain the equilibrium (f , u) = (0, 0)?

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 99

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that g∗|t=T = 0 outside ω. We have confined all the particles in ω! Next step: How to attain the equilibrium (f , u) = (0, 0)? We pass from g∗|t=T = 0 outside ω to g∗|t=T+T1 ≡ 0 everywhere,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 100

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that g∗|t=T = 0 outside ω. We have confined all the particles in ω! Next step: How to attain the equilibrium (f , u) = (0, 0)? We pass from g∗|t=T = 0 outside ω to g∗|t=T+T1 ≡ 0 everywhere, We pass from ug∗|t=T+T1 to u|t=T+T1+T2 ≡ 0, thanks to the controllability of NS (Coron-Fursikov).

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 101

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that g∗|t=T = 0 outside ω. We have confined all the particles in ω! Next step: How to attain the equilibrium (f , u) = (0, 0)? We pass from g∗|t=T = 0 outside ω to g∗|t=T+T1 ≡ 0 everywhere, We pass from ug∗|t=T+T1 to u|t=T+T1+T2 ≡ 0, thanks to the controllability of NS (Coron-Fursikov). We thus get u|t=Tlarge = 0, f |t=Tlarge = 0,

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 102

End of the proof the controllability theorem II

Since (X g, V g) meet ω, thanks to the absorption procedure, we deduce that g∗|t=T = 0 outside ω. We have confined all the particles in ω! Next step: How to attain the equilibrium (f , u) = (0, 0)? We pass from g∗|t=T = 0 outside ω to g∗|t=T+T1 ≡ 0 everywhere, We pass from ug∗|t=T+T1 to u|t=T+T1+T2 ≡ 0, thanks to the controllability of NS (Coron-Fursikov). We thus get u|t=Tlarge = 0, f |t=Tlarge = 0, which ends the proof.

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

SLIDE 103

Thank you very much for your attention! Joyeux anniversaire, Jean-Michel !

Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system