Some controllability results with a reduced number of controls - - PowerPoint PPT Presentation

Some controllability results with a reduced number of controls Quelques r´ esultats de contrˆ

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Matin´ ee Contrˆ

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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

1

Local null controllability of Navier-Stokes

2

Extensions to other systems

3

Perspectives and some references

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 R3 T > 0 ω ⊂ Ω (control set), Q := Ω × (0, T), Σ := ∂Ω × (0, T)    yt − ∆y + (y · ∇)y + ∇p = (v1, v2, 0)1ω, ∇ · y = 0 in Q, y = 0

• n Σ,

y(0) = y 0 in Ω, where v1 and v2 stand for the controls which act over the set ω. Local null controllability problem: If y 0 is small enough, can we find controls v1 and v2 in L2(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 yt − ∆y + ∇p = (v1, v2, 0)1ω, ∇ · y = 0, y|Σ = 0 Recently, Lissy (2012), Local null controllability of Navier-Stokes with (v1, 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 (v1, v2, 0) ∈ L2(ω × (0, T))3 (or (v1, 0, v3), or (0, v2, v3)).

Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references

Method of proof

Linearization around zero yt − ∆y + ∇p = f + (v1, v2, 0)1ω, ∇ · y = 0, y|Σ = 0 Null controllability of the linearized system (Main part of the proof). Main tool: Carleman estimate for the adjoint system −ϕt − ∆ϕ + ∇π = g, ∇ · ϕ = 0, ϕ|Σ = 0 There exists a constant C > 0 (depending on Ω, ω, T)

• Q

ρ1(t)|ϕ|2 ≤ C

Q

ρ2(t)|g|2 +

• ω×(0,T)

ρ3(t)(|ϕ1|2 + |ϕ2|2)

• Inverse mapping theorem for the nonlinear system

A = (yt − ∆y + (y · ∇)y + ∇p − (v1, v2, 0)1ω, y(0))

Local null controllability of Navier-Stokes Extensions to other systems Perspectives and some references

Boussinesq system

       yt − ∆y + (y · ∇)y + ∇p = (v1, 0, 0)1ω + (0, 0, θ) , ∇ · y = 0 in Q, θt − ∆θ + y · ∇θ = v01ω in Q, y = 0, θ = 0

• n Σ,

y(0) = y 0, θ(0) = θ0 in Ω, y : Velocity, θ : Temperature Same method applied to N-S to control with (v1, 0, v3) and v0 We use θ to control the third equation and set v3 ≡ 0 Theorem For every T > 0 and ω ⊂ Ω, the Boussinesq system is locally null controllable by controls v0 ∈ L2(ω × (0, T)) and (v1, 0, 0) ∈ L2(ω × (0, T))3 (or (0, v2, 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 = (v1, v2, 0) such that z(0) = 0, where        wt − ∆w + (w · ∇)w + ∇p0 = f + (v1, v2, 0)1ω, ∇ · w = 0 in Q, −zt − ∆z + (z · ∇t)w − (w · ∇)z + ∇q = w1O, ∇ · z = 0 in Q, w = z = 0

• n Σ,

w(0) = y 0, z(T) = 0 in Ω. z is controlled by w1O Application to insensitizing controls for Navier-Stokes Theorem (joint work with M. Gueye) Assume y 0 = 0, eK/t10f L2(Q)3 < ∞ and O ∩ ω = ∅. The previous system is null controllable by a control (v1, v2, 0) ∈ L2(ω × (0, T))3 (or (v1, 0, v3), or (0, v2, v3)).

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 −ϕt − ∆ϕ + ¯ y · Dϕ + ∇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., v0 ≡ 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.