Controllability of parabolic systems: the moment method Evolution - PowerPoint PPT Presentation

Controllability of parabolic systems: the moment method Evolution Equations: long time behavior and control Farid Ammar Khodja Laboratoire de Mathématiques de Besançon Mathematics in Savoie: 15-18 june 2015 Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

Introduction The main goal of this course is to give a review of results relating controllability issues for parabolic systems obtained via the moment method (as used by Fattorini and Russell in the seventies). Other powerfull techniques have been developped during these last 20 years: Carleman inequalities: Fursikov-Imanuvilov. The Lebeau-Robbiano method. The return method: J-M Coron. The transmutation method: Miller. Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

The control system Consider the following system: 8 ( ∂ t � D ∆ � A ) y = Bu 1 ω , Q T : = ( 0 , T ) � Ω , < y = Cv 1 Γ 0 , Σ T : = ( 0 , T ) � ∂ Ω , : y ( 0 , � ) = y 0 , Ω , where: Ω � R N is a smooth bounded domain, ω � Ω is an open set, Γ 0 � ∂ Ω is a relatively open subset; D = diag ( d 1 , ..., d n ) , A = ( a ij ) 1 � i , j � N 2 L ∞ ( Q T ; L ( R n )) , B = ( b ij ) , C = ( c ij ) 2 L ∞ ( Q T ; L ( R m , R n )) : control matrices. Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

Concepts of controllability Approximate controllability at time T > 0 : for all ε > 0 , for all � y 0 , y 1 � 2 X � X , there exists ( u , v ) 2 L 2 ( Q T ) � L 2 ( Σ T ) such that � � y ( T ) � y 1 � � X � ε . Null controllability at time T > 0 : for all y 0 2 X , there exists ( u , v ) 2 L 2 ( Q T ) � L 2 ( Σ T ) such that y ( T ) = 0 in Ω . Here X is a space where the system is well-posed: for example X = L 2 ( Ω ; R n ) when C = 0 or X = H � 1 ( Ω ; R n ) when C 6 = 0 leads to a solution y 2 C ([ 0 , T ] , X ) if the initial datum y 0 2 X . Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

Dual concepts: adjoint system and observability Consider the dual system: 8 ( ∂ t + D ∆ + A � ) ϕ = 0 , Q T : = ( 0 , T ) � Ω , < ϕ = 0 , Σ T : = ( 0 , T ) � ∂ Ω , : ϕ ( T , � ) = ϕ 0 , Ω , If ϕ 0 2 X = L 2 ( Ω ) , then there is a unique solution � [ 0 , T ] ; L 2 � Ω ; R N �� \ L 2 � � Ω , R N �� 0 , T ; H 1 ϕ 2 C . 0 If ϕ 0 2 X = H 1 0 ( Ω ) , then there is a unique solution � � Ω , R N �� \ L 2 � � Ω , R N �� 0 , T ; H 2 \ H 1 [ 0 , T ] ; H 1 ϕ 2 C . 0 0 Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

If C = 0 ( distributed control ): Theorem The system is approximately controllable in X = L 2 � Ω ; R N � if, and only if, the unique continuation property holds true for any solution � [ 0 , T ] ; L 2 � Ω ; R N �� ϕ 2 C of the adjoint problem: � ) ϕ 0 = 0 . 1 ω B � ϕ = 0 , in Q T = Theorem The system is null controllable in X = L 2 � Ω ; R N � if, and only if there exists C T > 0 such that Z T Z k ϕ ( 0 ) k 2 ω j B � ϕ j 2 X � C T 0 for any ϕ 0 2 X . Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

If B = 0 ( boundary control ): Theorem The system is approximately controllable in X = H � 1 � Ω ; R N � if, and only if, the unique continuation property holds true for any solution � [ 0 , T ] ; H 1 � ϕ 2 C 0 ( Ω ) of the adjoint problem: � ) ϕ 0 = 0 . 1 Γ 0 C � ϕ = 0 , in Σ T = Theorem The system is null controllable in X = H � 1 � Ω ; R N � if, and only if, there exists C T > 0 such that � � Z T Z 2 � � � C � ∂ϕ k ϕ ( 0 ) k 2 � � 0 ( Ω ; R N ) � C T � H 1 ∂ν Γ 0 0 � Ω ; R N � for any ϕ 0 2 H 1 . 0 Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

Results for the scalar case Theorem The problem 8 ( ∂ t � ∆ � a ) y = u 1 ω , Q T : = ( 0 , T ) � Ω , < Σ T : = ( 0 , T ) � ∂ Ω , y = 0 , : y ( 0 , � ) = y 0 , Ω , is null and approximately controllable in X = L 2 ( Ω ) for any open set ω � Ω , provided that a 2 L ∞ ( Q T ) . As a consequence, the problem 8 ( ∂ t � ∆ � a ) y = 0 , Q T : = ( 0 , T ) � Ω , < Σ T : = ( 0 , T ) � ∂ Ω , y = v 1 Γ 0 , : y ( 0 , � ) = y 0 , Ω , is null and approximately controllable in X = H � 1 ( Ω ) for any relatively open set Γ 0 � ∂ Ω . Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

Remarks on the scalar case Approximate and null controllability are equivalent concepts in the scalar case. Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

Remarks on the scalar case Approximate and null controllability are equivalent concepts in the scalar case. Boundary and distributed controllability are also equivalent. Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

Remarks on the scalar case Approximate and null controllability are equivalent concepts in the scalar case. Boundary and distributed controllability are also equivalent. Unlike the hyperbolic case, the minimal time of control is T min = 0 . Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

The techniques used for the proof Fursikov and Imanuvilov (95’) have proved a global Carleman inequality: Let β 0 ( x ) ( t , x ) 2 Q T = Ω � ( 0 , T ) , η ( t , x ) : = s t ( T � t ) , s ρ ( t , x ) : = t ( T � t ) , ( t , x ) 2 Q T . and Z ρ τ � 1 e � 2 η � j ϕ t j 2 + j ∆ ϕ j 2 + ρ 2 jr ϕ j 2 + ρ 4 j ϕ j 2 � I ( τ , ϕ ) = . Q T Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

The techniques used for the proof Fursikov and Imanuvilov (95’) have proved a global Carleman inequality: Let β 0 ( x ) ( t , x ) 2 Q T = Ω � ( 0 , T ) , η ( t , x ) : = s t ( T � t ) , s ρ ( t , x ) : = t ( T � t ) , ( t , x ) 2 Q T . and Z ρ τ � 1 e � 2 η � j ϕ t j 2 + j ∆ ϕ j 2 + ρ 2 jr ϕ j 2 + ρ 4 j ϕ j 2 � I ( τ , ϕ ) = . Q T Carleman inequality : there exist a positive function β 0 2 C 2 ( Ω ) , s 0 > 0 and C > 0 such that 8 s � s 0 and 8 τ 2 R : � Z � Z T Z ρ τ e � 2 η j ϕ t � c ∆ ϕ j 2 + ω ρ τ + 3 e � 2 η j ϕ j 2 I ( τ , ϕ ) � C , Q T 0 for any function ϕ satisfying ϕ = 0 on Σ T and for which the right Mathematics in Savoie: 15-18 june 2015 hand-side is de…ned. (LMB) Control parabolic / 72

For 8 ( ∂ t + ∆ + a ) ϕ = 0 , Q T : = ( 0 , T ) � Ω , < Σ T : = ( 0 , T ) � ∂ Ω , ϕ = 0 , : ϕ ( T , � ) = ϕ 0 , Ω , this inequality gives in particular � Z � Z Z T Z ρ e � 2 η j a ϕ j 2 + e � 2 η ρ 4 j ϕ j 2 ω ρ 4 e � 2 η j ϕ j 2 � C Q T Q T 0 + Z Z T Z ω ρ 4 e � 2 η j ϕ j 2 , 8 s � s 0 ( k a k ∞ ) . e � 2 η ρ 4 j ϕ j 2 � C Q T 0 Using that there exists α 2 R such that t 7! R Ω e α t ϕ 2 is increasing, we get the observability inequality Z T Z k ϕ ( 0 ) k 2 ω j ϕ j 2 L 2 ( Ω ) � C 0 and thus the null controllability result. Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

A …rst application to a parabolic system 8 ( ∂ t � ∆ ) y 1 = a 11 y 1 + a 12 y 2 > > Q T : = ( 0 , T ) � Ω , < ( ∂ t � d ∆ ) y 2 = a 21 y 1 + a 22 y 2 + u 1 ω , Σ T : = ( 0 , T ) � ∂ Ω , > y = ( y 1 , y 2 ) = 0 , > : y ( 0 , � ) = y 0 , Ω , Theorem If there exists ω 0 � ω such that a 12 � σ > 0 on ( 0 , T ) � ω 0 then the system is null (and approximately) controllable for any d > 0 . Proof. Carleman inequality for each equation of the adjoint system and then use the assumption a 12 � σ > 0 on ( 0 , T ) � ω 0 . This result generalizes to n � n cascade systems. Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

A natural question arises at this level: what happens if supp ( a 12 ) \ ω = ? ? Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

A natural question arises at this level: what happens if supp ( a 12 ) \ ω = ? ? What happens for the boundary control system: 8 ( ∂ t � ∆ ) y 1 = a 11 y 1 + a 12 y 2 > > Q T : = ( 0 , T ) � Ω , > > ( ∂ t � d ∆ ) y 2 = a 21 y 1 + a 22 y 2 , < � y 1 � � 0 � ? > Σ T : = ( 0 , T ) � ∂ Ω , y = = v , > > y 2 1 > : y ( 0 , � ) = y 0 , Ω , Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72

A natural question arises at this level: what happens if supp ( a 12 ) \ ω = ? ? What happens for the boundary control system: 8 ( ∂ t � ∆ ) y 1 = a 11 y 1 + a 12 y 2 > > Q T : = ( 0 , T ) � Ω , > > ( ∂ t � d ∆ ) y 2 = a 21 y 1 + a 22 y 2 , < � y 1 � � 0 � ? > Σ T : = ( 0 , T ) � ∂ Ω , y = = v , > > y 2 1 > : y ( 0 , � ) = y 0 , Ω , There exist only partial answers to these two questions... even in the one-dimensional case. Mathematics in Savoie: 15-18 june 2015 (LMB) Control parabolic / 72