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

The Vlasov-Navier-Stokes system Let T 2 := R 2 / Z 2 , T > 0. in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 0 ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f 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 T 2 , ω ⊂ T 2 . This amounts to use an internal control in the Vlasov equation, located in the absorption region ω ⊂ T 2 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

The Vlasov-Navier-Stokes system under control Let T 2 := R 2 / Z 2 , T > 0, ω ⊂ T 2 , in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 1 ω ( x ) G ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f 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 T 2 , ω ⊂ T 2 . This amounts to use an internal control in the Vlasov equation, located in the absorption region ω ⊂ T 2 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

The Vlasov-Navier-Stokes system under control Let T 2 := R 2 / Z 2 , T > 0, ω ⊂ T 2 . in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 1 ω ( x ) G ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f u . This is a Nonlinear Control system: Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

The Vlasov-Navier-Stokes system under control Let T 2 := R 2 / Z 2 , T > 0, ω ⊂ T 2 . in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 1 ω ( x ) G ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f 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

The Vlasov-Navier-Stokes system under control Let T 2 := R 2 / Z 2 , T > 0, ω ⊂ T 2 . in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 1 ω ( x ) G ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f u . This is a Nonlinear Control system: state: ( f , u ), pair distribution function-velocity field, control: source 1 ω ( x ) G ( t , x , v ), located in [0 , T ] × ω × R 2 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

The Vlasov-Navier-Stokes system under control Let T 2 := R 2 / Z 2 , T > 0, ω ⊂ T 2 . in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 1 ω ( x ) G ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f u . This is a Nonlinear Control system: state: ( f , u ), pair distribution function-velocity field, control: source 1 ω ( x ) G ( t , x , v ), located in [0 , T ] × ω × R 2 . The control G should modify the dynamics of ( f , u ) Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

The Vlasov-Navier-Stokes system under control Let T 2 := R 2 / Z 2 , T > 0, ω ⊂ T 2 . in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 1 ω ( x ) G ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f u . This is a Nonlinear Control system: state: ( f , u ), pair distribution function-velocity field, control: source 1 ω ( x ) G ( t , x , v ), located in [0 , T ] × ω × R 2 . 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

The Vlasov-Navier-Stokes system under control Let T 2 := R 2 / Z 2 , T > 0, ω ⊂ T 2 . in (0 , T ) × T 2 × R 2 ,  ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 1 ω ( x ) G ,   in (0 , T ) × T 2 , ∂ t u + ( u · ∇ x ) u − ∆ x u + ∇ x p = j f − ρ f u ,    in (0 , T ) × T 2 , div x u ( t , x ) = 0 , in T 2 × R 2 , f | t =0 = f 0 ( x , v ) ,     in T 2 , u | t =0 = u 0 ,  � � with j f ( t , x ) := R 2 vf ( t , x , v ) d v and ρ f ( t , x ) := R 2 f ( t , x , v ) d v . Two Interaction terms: Non-linearities 1.- Coupling div v [( u − v ) f ], 2.- Brinkman force j f − ρ f u . This is a Nonlinear Control system: state: ( f , u ), pair distribution function-velocity field, control: source 1 ω ( x ) G ( t , x , v ), located in [0 , T ] × ω × R 2 . 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

Goal: an exact controllability result The question of controllability: Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . We search a control G ∈ U , supported in (0 , T ) × ω × R 2 , such that the solution of VNS satisfies Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . We search a control G ∈ U , supported in (0 , T ) × ω × R 2 , such that the solution of VNS satisfies t = 0 t = T − → ( f 0 , u 0 ) ( f 1 , u 1 ) G Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . We search a control G ∈ U , supported in (0 , T ) × ω × R 2 , such that the solution of VNS satisfies t = 0 t = T − → ( f 0 , u 0 ) ( f 1 , u 1 ) G We shall answer positively to this question for X , U spaces of regular functions (H¨ older), Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . We search a control G ∈ U , supported in (0 , T ) × ω × R 2 , such that the solution of VNS satisfies t = 0 t = T − → ( f 0 , u 0 ) ( f 1 , u 1 ) G We shall answer positively to this question for X , U spaces of regular functions (H¨ older), for f 0 and u 0 small in X (local result), Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . We search a control G ∈ U , supported in (0 , T ) × ω × R 2 , such that the solution of VNS satisfies t = 0 t = T − → ( f 0 , u 0 ) ( f 1 , u 1 ) G We shall answer positively to this question for X , U spaces of regular functions (H¨ older), for f 0 and u 0 small in X (local result), for f 1 = 0 and u 1 = 0, (null-controllability), Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . We search a control G ∈ U , supported in (0 , T ) × ω × R 2 , such that the solution of VNS satisfies t = 0 t = T − → ( f 0 , u 0 ) ( f 1 , u 1 ) G We shall answer positively to this question for X , U spaces of regular functions (H¨ older), for f 0 and u 0 small in X (local result), for f 1 = 0 and u 1 = 0, (null-controllability), for a time T > 0 large enough, Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Goal: an exact controllability result The question of controllability: Let X , U functional spaces. Let T > 0, ω ⊂ T 2 . Pick an initial state ( f 0 , u 0 ) ∈ X and a final state ( f 1 , u 1 ) ∈ X . We search a control G ∈ U , supported in (0 , T ) × ω × R 2 , such that the solution of VNS satisfies t = 0 t = T − → ( f 0 , u 0 ) ( f 1 , u 1 ) G We shall answer positively to this question for X , U spaces of regular functions (H¨ older), for f 0 and u 0 small in X (local result), for f 1 = 0 and u 1 = 0, (null-controllability), for a time T > 0 large enough, for ω ⊂ T 2 satisfying a geometric assumption. Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Main result: Local null-controllability result THEOREM (IM, 2016) Let γ > 2 and let ω ⊂ T 2 satisfy the strip assumption . ∃ ǫ > 0 , M > 0 , T 0 > 0 such that ∀ T ≥ T 0 , and for every f 0 ∈ C 1 ( T 2 × R 2 ) ∩ W 1 , ∞ ( T 2 × R 2 ) and u 0 satisfying that u 0 ∈ C 1 ( T 2 ; R 2 ) ∩ H 2 ( T 2 ; R 2 ) , div x u 0 = 0 , � u 0 � H 2 ( T 2 ) ≤ M , 1 � f 0 � C 1 ( T 2 × R 2 ) + � (1 + | v | ) γ +2 f 0 � C 0 ( T 2 × R 2 ) ≤ ǫ, T 2 × R 2 (1 + | v | ) γ ( |∇ x f 0 | + |∇ v f 0 | ) ( x , v ) ≤ κ, ∃ κ > 0 , sup there exists a control G ∈ C 0 ([0 , T ] × T 2 × R 2 ) such that a strong solution of (VNS) with f | t =0 = f 0 and u | t =0 = u 0 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = C ( f ) . Nonlinear non-collisional models ( C ( f ) ≡ 0) Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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 , ∂ t f + v · ∇ x f + F ( t , x ) · ∇ v f = 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

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

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

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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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

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 = ∅ � � x + t v ∩ ω , it Low velocities: Even if ( x , v ) is such that | v | 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 ( f 0 , u 0 ) and satisfying ( f , u ) | t = T = (0 , 0). Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

How to eliminate the obstructions? Method of characteristics Given f 0 and u regular, the transport equation � ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 0 , (0 , T ) × T 2 × R 2 , T 2 × R 2 , f | t =0 = f 0 ( x , v ) , has the explicit solution f ( t , x , v ) = e 2 t f 0 (( 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

How to eliminate the obstructions? Method of characteristics Given f 0 and u regular, the transport equation � ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 0 , (0 , T ) × T 2 × R 2 , T 2 × R 2 , f | t =0 = f 0 ( x , v ) , has the explicit solution f ( t , x , v ) = e 2 t f 0 (( 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

How to eliminate the obstructions? Method of characteristics Given f 0 and u regular, the transport equation � ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 0 , (0 , T ) × T 2 × R 2 , T 2 × R 2 , f | t =0 = f 0 ( x , v ) , has the explicit solution f ( t , x , v ) = e 2 t f 0 (( 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

How to eliminate the obstructions? Method of characteristics Given f 0 and u regular, the transport equation � ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 0 , (0 , T ) × T 2 × R 2 , T 2 × R 2 , f | t =0 = f 0 ( x , v ) , has the explicit solution f ( t , x , v ) = e 2 t f 0 (( 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

How to eliminate the obstructions? Method of characteristics Given f 0 and u regular, the transport equation � ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 0 , (0 , T ) × T 2 × R 2 , T 2 × R 2 , f | t =0 = f 0 ( x , v ) , has the explicit solution f ( t , x , v ) = e 2 t f 0 (( 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

How to eliminate the obstructions? Method of characteristics Given f 0 and u regular, the transport equation � ∂ t f + v · ∇ x f + div v [( u − v ) f ] = 0 , (0 , T ) × T 2 × R 2 , T 2 × R 2 , f | t =0 = f 0 ( x , v ) , has the explicit solution f ( t , x , v ) = e 2 t f 0 (( 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 ) ∈ T 2 × R 2 , ∃ t > 0 s.t. X ( t , 0 , x , v ) ∈ ω, . ∂ t u + u · ∇ x u − ∆ x u + ∇ x p = j f − ρ f u . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: velocity field We exploit the strip assumption on ω ⊂ T 2 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: velocity field We exploit the strip assumption on ω ⊂ T 2 . 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

Reference solution: velocity field We exploit the strip assumption on ω ⊂ T 2 . There exist a straight line H = span ( n ⊥ H ) contained in ω . We want to accelerate all the characteristics in the direction of n H . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: velocity field We exploit the strip assumption on ω ⊂ T 2 . There exist a straight line H = span ( n ⊥ H ) contained in ω . We want to accelerate all the characteristics in the direction of n H . We use a controllability result for the NS system in T 2 (Coron-Fursikov, 1996): we can modify the fluid to pass from 0 to the stationary solution n H . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: velocity field We exploit the strip assumption on ω ⊂ T 2 . There exist a straight line H = span ( n ⊥ H ) contained in ω . We want to accelerate all the characteristics in the direction of n H . We use a controllability result for the NS system in T 2 (Coron-Fursikov, 1996): we can modify the fluid to pass from 0 to the stationary solution n H . 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

Reference solution: velocity field We exploit the strip assumption on ω ⊂ T 2 . There exist a straight line H = span ( n ⊥ H ) contained in ω . We want to accelerate all the characteristics in the direction of n H . We use a controllability result for the NS system in T 2 (Coron-Fursikov, 1996): we can modify the fluid to pass from 0 to the stationary solution n H . 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

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

Reference solution: distribution function We have found u satisfying (Coron-Fursikov, 1996): � ∂ t u + ( u · ∇ ) u − ∆ x u + ∇ x p = w , (0 , T ) × T 2 , (0 , T ) × T 2 , div x u = 0 , Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: distribution function We have found u satisfying (Coron-Fursikov, 1996): � ∂ t u + ( u · ∇ ) u − ∆ x u + ∇ x p = w , (0 , T ) × T 2 , (0 , T ) × T 2 , div x u = 0 , for a control w supported in (0 , T ) × ω . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: distribution function We have found u satisfying (Coron-Fursikov, 1996): � ∂ t u + ( u · ∇ ) u − ∆ x u + ∇ x p = w , (0 , T ) × T 2 , (0 , T ) × T 2 , div x u = 0 , for a control w supported in (0 , T ) × ω . Associated distribution function: let Z 1 , Z 2 ∈ S ( R 2 ) such that � � R 2 v i Z j ( v ) d v = δ ij , R 2 Z i ( v ) d v = 0 , i , j = 1 , 2 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: distribution function We have found u satisfying (Coron-Fursikov, 1996): � ∂ t u + ( u · ∇ ) u − ∆ x u + ∇ x p = w , (0 , T ) × T 2 , (0 , T ) × T 2 , div x u = 0 , for a control w supported in (0 , T ) × ω . Associated distribution function: let Z 1 , Z 2 ∈ S ( R 2 ) such that � � R 2 v i Z j ( v ) d v = δ ij , R 2 Z i ( v ) d v = 0 , i , j = 1 , 2 . Then, define f ( t , x , v ) := ( Z 1 , Z 2 )( v ) · w ( t , x ), which gives Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Reference solution: distribution function We have found u satisfying (Coron-Fursikov, 1996): � ∂ t u + ( u · ∇ ) u − ∆ x u + ∇ x p = w , (0 , T ) × T 2 , (0 , T ) × T 2 , div x u = 0 , for a control w supported in (0 , T ) × ω . Associated distribution function: let Z 1 , Z 2 ∈ S ( R 2 ) such that � � R 2 v i Z j ( v ) d v = δ ij , R 2 Z i ( v ) d v = 0 , i , j = 1 , 2 . Then, define f ( t , x , v ) := ( Z 1 , Z 2 )( v ) · w ( t , x ), which gives (0 , T ) × T 2 , w = j f − ρ f u , (0 , T ) × ( T 2 \ ω ) × R 2 , � � ∂ t f + v · ∇ x f + div v ( u − v ) f = 0 , f | t =0 = 0 , f | t = T = 0 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

PROPOSITION Let ω ⊂ T 2 satisfy the strip assumption. There exists T 0 > 0 such that for any T ≥ T 0 , there exists a reference solution ( f , u ) of the VNS system such that f ∈ C ∞ ([0 , T ] × T 2 ; S ( R 2 )) , u ∈ C ∞ ([0 , T ] × T 2 ; R 2 ) , ( f , u ) | t =0 = (0 , 0) , ( f , u ) | t = T = (0 , 0) , supp ( f ) ⊂ (0 , T ) × ω × R 2 , and such that the characteristics associated to u satisfy � T 12 , 11 T � ∀ ( x , v ) ∈ T 2 × R 2 , ∃ t ∈ such that 12 X ( t , 0 , x , v ) ∈ ω, with | V ( t , 0 , x , v ) · n H | ≥ 5 . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , in such a way V ǫ has a fixed point g ∗ satisfying: Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , in such a way V ǫ 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

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , in such a way V ǫ has a fixed point g ∗ satisfying: g ∗ solves (VNS) for a certain control G ∗ , ∀ ( x , v ) ∈ T 2 × R 2 , ∃ t such that X g ∗ ( t , 0 , x , v ) ∈ ω. Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , in such a way V ǫ has a fixed point g ∗ satisfying: g ∗ solves (VNS) for a certain control G ∗ , ∀ ( x , v ) ∈ T 2 × R 2 , ∃ t such that X g ∗ ( t , 0 , x , v ) ∈ ω. We define V ǫ in three steps (NS-absorption-extension): Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , in such a way V ǫ has a fixed point g ∗ satisfying: g ∗ solves (VNS) for a certain control G ∗ , ∀ ( x , v ) ∈ T 2 × R 2 , ∃ t such that X g ∗ ( t , 0 , x , v ) ∈ ω. We define V ǫ in three steps (NS-absorption-extension): 1 g ∈ S ǫ �− → u g , solution of NS with Brinkman force j g − ρ g u g , Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , in such a way V ǫ has a fixed point g ∗ satisfying: g ∗ solves (VNS) for a certain control G ∗ , ∀ ( x , v ) ∈ T 2 × R 2 , ∃ t such that X g ∗ ( t , 0 , x , v ) ∈ ω. We define V ǫ in three steps (NS-absorption-extension): 1 g ∈ S ǫ �− → u g , solution of NS with Brinkman force j g − ρ g u g , ˜ 2 u g �− → V ǫ [ g ], solution of Vlasov with absorption in ω , characteristics Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

Construction of ( f , u ) close to ( f , u ): fixed-point scheme We define a domain, ⊂ C 0 ([0 , T ] × T 2 × R 2 ) � � f ; ǫ − close to f S ǫ � depending on ǫ > 0 small, and a continuous operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , in such a way V ǫ has a fixed point g ∗ satisfying: g ∗ solves (VNS) for a certain control G ∗ , ∀ ( x , v ) ∈ T 2 × R 2 , ∃ t such that X g ∗ ( t , 0 , x , v ) ∈ ω. We define V ǫ in three steps (NS-absorption-extension): 1 g ∈ S ǫ �− → u g , solution of NS with Brinkman force j g − ρ g u g , ˜ 2 u g �− → V ǫ [ g ], solution of Vlasov with absorption in ω , characteristics ˜ Π − extension V ǫ [ g ] = f + Π(˜ V ǫ [ g ] �− → V ǫ [ g ]) . 3 Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. We prove that V ǫ ( S ǫ ) ⊂ S ǫ , Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. We prove that V ǫ ( S ǫ ) ⊂ S ǫ , V ǫ is continuous. Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. We prove that V ǫ ( S ǫ ) ⊂ S ǫ , V ǫ is continuous. Thus, (Leray-Schauder) ∃ g ∗ ∈ S ǫ such that V ǫ [ g ∗ ] = g ∗ . Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. We prove that V ǫ ( S ǫ ) ⊂ S ǫ , V ǫ is continuous. Thus, (Leray-Schauder) ∃ g ∗ ∈ S ǫ such that V ǫ [ g ∗ ] = g ∗ . By construction, the fixed point satisfies Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. We prove that V ǫ ( S ǫ ) ⊂ S ǫ , V ǫ is continuous. Thus, (Leray-Schauder) ∃ g ∗ ∈ S ǫ such that V ǫ [ 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

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. We prove that V ǫ ( S ǫ ) ⊂ S ǫ , V ǫ is continuous. Thus, (Leray-Schauder) ∃ g ∗ ∈ S ǫ such that V ǫ [ 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

End of the proof the controllability theorem I We have to show that the operator V ǫ : S ǫ → C 0 ([0 , T ] × T 2 × R 2 ) , has a fixed point. We prove that V ǫ ( S ǫ ) ⊂ S ǫ , V ǫ is continuous. Thus, (Leray-Schauder) ∃ g ∗ ∈ S ǫ such that V ǫ [ 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

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

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

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

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

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 + T 1 ≡ 0 everywhere, Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system

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 + T 1 ≡ 0 everywhere, We pass from u g ∗ | t = T + T 1 to u | t = T + T 1 + T 2 ≡ 0, thanks to the controllability of NS (Coron-Fursikov). Iv´ an Moyano Local null-controllability of the 2-D Vlasov-Navier-Stokes system