Some controllability results with a reduced number of controls Quelques r´ esultats de contrˆ olabilit´ e avec un nombre r´ eduit de contrˆ oles Matin´ ee Contrˆ ole Nicol´ as Carre˜ no Laboratoire Jacques-Louis Lions UPMC (Thesis advisor: Sergio Guerrero) March 21th, 2013
Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references Outline Local null controllability of Navier-Stokes 1 Extensions to other systems 2 Perspectives and some references 3
Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references Local null controllability of Navier-Stokes Ω bounded connected regular open subset of R 3 T > 0 ω ⊂ Ω (control set), Q := Ω × (0 , T ), Σ := ∂ Ω × (0 , T ) y t − ∆ y + ( y · ∇ ) y + ∇ p = ( v 1 , v 2 , 0) 1 ω , ∇ · y = 0 in Q , y = 0 on Σ , y (0) = y 0 in Ω , where v 1 and v 2 stand for the controls which act over the set ω . Local null controllability problem: If � y 0 � is small enough, can we find controls v 1 and v 2 in L 2 ( ω × (0 , T )) such that y ( T ) = 0 ?
Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references Some results First results by Fern´ andez-Cara, Guerrero, Imanuvilov, Puel (2006) when ω ∩ ∂ Ω � = ∅ (We are interested in removing this geometric condition) Coron, Guerrero (2009) Null controllability of the Stokes system for a general ω ⊂ Ω y t − ∆ y + ∇ p = ( v 1 , v 2 , 0) 1 ω , ∇ · y = 0 , y | Σ = 0 Recently, Lissy (2012), Local null controllability of Navier-Stokes with ( v 1 , 0 , 0) (Return method) Our result Theorem (C., Guerrero) Local null controllability for general ω For every T > 0 and ω ⊂ Ω, the N-S system is locally null controllable by a control ( v 1 , v 2 , 0) ∈ L 2 ( ω × (0 , T )) 3 (or ( v 1 , 0 , v 3 ), or (0 , v 2 , v 3 )).
Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references Method of proof Linearization around zero y t − ∆ y + ∇ p = f + ( v 1 , v 2 , 0) 1 ω , ∇ · y = 0 , y | Σ = 0 Null controllability of the linearized system (Main part of the proof). Main tool: Carleman estimate for the adjoint system − ϕ t − ∆ ϕ + ∇ π = g , ∇ · ϕ = 0 , ϕ | Σ = 0 There exists a constant C > 0 (depending on Ω, ω , T ) � � � � � ρ 1 ( t ) | ϕ | 2 ≤ C ρ 2 ( t ) | g | 2 + ρ 3 ( t )( | ϕ 1 | 2 + | ϕ 2 | 2 ) ω × (0 , T ) Q Q Inverse mapping theorem for the nonlinear system A = ( y t − ∆ y + ( y · ∇ ) y + ∇ p − ( v 1 , v 2 , 0) 1 ω , y (0))
Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references Boussinesq system y t − ∆ y + ( y · ∇ ) y + ∇ p = ( v 1 , 0 , 0) 1 ω + (0 , 0 , θ ) , ∇ · y = 0 in Q , θ t − ∆ θ + y · ∇ θ = v 0 1 ω in Q , y = 0 , θ = 0 on Σ , y (0) = y 0 , θ (0) = θ 0 in Ω , y : Velocity, θ : Temperature Same method applied to N-S to control with ( v 1 , 0 , v 3 ) and v 0 We use θ to control the third equation and set v 3 ≡ 0 Theorem For every T > 0 and ω ⊂ Ω, the Boussinesq system is locally null controllable by controls v 0 ∈ L 2 ( ω × (0 , T )) and ( v 1 , 0 , 0) ∈ L 2 ( ω × (0 , T )) 3 (or (0 , v 2 , 0)).
Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references A coupled Navier-Stokes system We consider the following null controllability problem: To find a control v = ( v 1 , v 2 , 0) such that z (0) = 0, where w t − ∆ w + ( w · ∇ ) w + ∇ p 0 = f + ( v 1 , v 2 , 0) 1 ω , ∇ · w = 0 in Q , − z t − ∆ z + ( z · ∇ t ) w − ( w · ∇ ) z + ∇ q = w 1 O , ∇ · z = 0 in Q , w = z = 0 on Σ , w (0) = y 0 , z ( T ) = 0 in Ω . z is controlled by w 1 O Application to insensitizing controls for Navier-Stokes Theorem (joint work with M. Gueye) Assume y 0 = 0, � e K / t 10 f � L 2 ( Q ) 3 < ∞ and O ∩ ω � = ∅ . The previous system is null controllable by a control ( v 1 , v 2 , 0) ∈ L 2 ( ω × (0 , T )) 3 (or ( v 1 , 0 , v 3 ), or (0 , v 2 , v 3 )).
Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references Perspectives and some references Local exact controllability to the trajectories for Navier-Stokes − ϕ t − ∆ ϕ + ¯ y · D ϕ + ∇ π = g Boundary controllability with one vanishing component (taking the trace of an extended controlled solution does not work) No control in the heat equation for the Boussinesq system (i.e., v 0 ≡ 0) N. C. and S. Guerrero, Local null controllability of the N -dimensional Navier-Stokes system with N − 1 scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), no. 1, 139–153. N. C., Local controllability of the N -dimensional Boussinesq system with N − 1 scalar controls in an arbitrary control domain, Math. Control Relat. Fields, 2 (2012), no. 4, 361–382. N. C. and M. Gueye, Insensitizing controls with one vanishing component for the Navier-Stokes system, to appear in J. Math. Pures Appl.. N. C., S. Guerrero and M. Gueye, Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system, submitted.
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