Singular Optimal Control : a Degenerate Parabolic-Hyperbolic example - PowerPoint PPT Presentation

Singular Optimal Control : a Degenerate Parabolic-Hyperbolic example Mamadou Gueye, Universidad Federico Santa María Nonlinear Partial Differential Equations and Applications A conference in the honor of Jean-Michel Coron for his 60th birthday Paris June, 20 1

Introduction Let ( ε, T, L, M ) ∈ (0 , + ∞ ) 3 × R . We consider  y t − εy xx + My x = 0 in (0 , L ) × (0 , L ) ,    ( TD ) y (0 , t ) = u ( t ) , y ( L, t ) = 0 on (0 , T ) ,  y ( x, 0) = y 0 ( x )  in (0 , L ) .  For y 0 ∈ H − 1 (0 , L ) we denote by U ( ε, T, L, M, y 0 ) the set of controls u ∈ L 2 (0 , T ) such that the corresponding solution of ( TD ) satisfies y ( · , T ) ≡ 0 . We can define the quantity which measures the cost of the null controllability of ( TD ): � min {� u � L 2 (0 ,T ) : u ∈ U ( ε, T, L, M, y 0 ) } � K ( ε, T, L, M ) := sup . � y 0 � H − 1(0 ,L ) ≤ 1 The underlying transport equation is controllable if and only if T > L/ | M | . J.-M. Coron and S. Guerrero Singular optimal control: A linear 1-D parabolic-hyperbolic example, 2005. Paris June, 20 2

State of the art � O. Glass 2009, Uniform controllability � P. Lissy 2012, Link with cost of controllability in small time � S. Guerrero, G. Lebeau 2007, Higher dimension and Lipschitz transport coefficient � O. Glass, S Guerrero 2007, Uniform controllability of the Burgers equation � M. Léautaud 2010, Uniform controllability of scalar conservation laws Paris June, 20 3

Degenerate transport diffusion equations Let ( ε, T, L, M, α ) ∈ (0 , + ∞ ) 4 × (0 , 1) . We consider  y t − ε ( x α +1 y x ) x + Mx α y x = 0 in (0 , L ) × (0 , L ) ,    ( TD ) y (0 , t ) = u ( t ) , y ( L, t ) = 0 on (0 , T ) ,  y ( x, 0) = y 0 ( x )  in (0 , L ) .  Assume that ( TD ) is null controllable in some space H , we denote by U ( ε, T, L, M, α, y 0 ) the set of controls u ∈ L 2 (0 , T ) such that the corresponding solution of ( TD ) satisfies y ( · , T ) ≡ 0 . � We are interested in the behaviour of the cost of the null controllability: � min {� u � L 2 (0 ,T ) : u ∈ U ( ε, T, L, M, α, y 0 ) } � K ( ε, T, L, M, α ) := sup . � y 0 � H ≤ 1 � For every ( ε, T, L, M, α ) ∈ (0 , + ∞ ) 4 × (0 , 1) such that M/ε > α and any y 0 ∈ L 2 ((0 , L ); x − M/ε d x ) , there exits a control u ∈ L 2 (0 , T ) sush that the associated solution to ( TD ) satisfies y ( · , T ) ≡ 0 . Paris June, 20 4

A singular Sturm-Liouville Problem Consider the differential expression defined by A [ y ]( x ) := − ε ( x α +1 y ′ ) ′ + Mx α y ′ = λy ( x ) , x ∈ (0 , L ) , λ ∈ R . � Particular case of Bessel differential equation x 2 y ′′ + axy ′ + ( bx ℓ + c ) y = 0 , x ∈ (0 , ∞ ) , ℓ � = 0 . The solutions can be written in terms of Bessel functions: 2 (1 − a ) Z ν ( κ − 1 √ ν := 1 κ := ℓ 1 � (1 − a ) 2 − 4 c , bx κ ) , b � = 0 : y ( x ) = x 2 , ℓ � Then, under the structural assumption M/ε > α , we can prove that ( A , D ( A )) is self- adjoint on L 2 ((0 , L ); x − M/ε d x ) and generates an analytic semigroup of bounded linear operators S ( t ) t ≥ 0 . Let ν := ( M/ε − α ) / (1 − α ) and κ := 1 2 (1 − α ) , we have 1 (2 κ ) 2 1 Φ n ( x ) := 2 ( M/ε − α ) J ν ( j ν,n x κ ) , λ n := ε ( κj ν,n ) 2 ν ( j ν,n ) | x x ∈ (0 , 1) . | J ′ Paris June, 20 5

Uniform Controllablity There exist Q ( T, L, M, α ) > 0 and C ( T, L, M, α ) > 0 such that, for every ( ε, T, L, M, α ) ∈ (0 , + ∞ ) 4 × (0 , 1) such that M/ε > α , we have √ 6) L 1 − α � � −Q ( T, L, M, α ) T > (2 K ( ε, α, T, L, M ) ≤ exp if ε M (1 − α ) and T < (0 , 98) L 1 − α � � C ( T, L, M, α ) if K ( ε, α, T, L, M ) ≥ exp M (1 − α ) . ε � G., P. Lissy, Singular optimal control of a 1 − D Parabolic-Hyperbolic Degenerate equation, accepted in ESAIM- Control Optim. Calc. Var. Paris June, 20 6

The moment method � Let y 0 ∈ L 2 ((0 , L ); x − M/ε d x ) := H . Then u ∈ U ( ε, T, L, M, α, y 0 ) if and only if � T H = − ( M/ε − α )(2 j ν,n ) ν � y 0 , Φ n � 0 u ( t ) exp ( − λ n ( T − t )) d t, ∀ n ∈ N \{ 0 } . 2 ν Γ( ν + 1) � Find a biorthogonal family { Ψ k ( t ) } k ∈ N \{ 0 } � T 0 Ψ k ( t ) e − λ ℓ ( T − t ) dt = δ kℓ , k, ℓ ∈ N \{ 0 } . � Construct a family { J k ( z ) } k ∈ N \{ 0 } of entire functions of exponential type satisfying J k ( − iλ ℓ ) = δ kℓ , k, ℓ ∈ N \{ 0 } . � Then using Paley-Wiener theorem to construct the biorthogonal family by inverse Fourier transform. Paris June, 20 7

Some elements of proof I An entire function having {− iε ( κj ν,k ) 2 , k ∈ N \{ 0 }} as simple zeros is � √ 2 √ εκ � ν + ∞ � � � � iz iz � Λ( z ) := 1 − = Γ( ν + 1) √ J ν √ εκ . ε ( κj ν,k ) 2 iz k =1 Moreover, Λ( · ) is of exponential type and   � | z | as | Λ( z ) | ≤ exp | z | → + ∞ . κ √ ε     � Now, we consider Λ( z ) ˜ J k ( z ) := Λ ′ ( − iλ k )( z + iλ k ) , one easily deduces that ˜ J k ( − iλ l ) = δ kl . � ˜ J k ( z ) cannot be bounded on the real line. Paris June, 20 8

Some elements of proof I We must use a multiplier to make the functions ˜ J k ( · ) bounded on the real line and of relevant exponential type. Let us set H ( z ) J k ( z ) := ˜ J k ( z ) H ( − iλ k ) . � Where, H is constructed to satisfy  H ( − iλ k ) ≥ C β ∀ k ≥ 1 ,    | H ( z ) | ≤ e β |ℑ ( z ) | ( Mult ) ∀ z ∈ C ,  H ( ix ) ≥ Ce γ | x |  ∀ x ∈ R .  C , β and γ to be chosen in terms of ε, T, M, α . We follow G. Tenenbaum and M. Tucsnak. � Precise asymptotics estimates for Bessel functions and their zeros. Paris June, 20 9

Some elements of proof II � Let u the optimal control assiociated to the first eigenvalue. Let us introduce the function f : C → C define by � − T/ 2 � � s − iδ t + T � � − it ε f ( s ) := − T/ 2 u e d t s ∈ C . 2 for a δ > 0 , that will be choosen later. Then, f is an entire function satifying ( εκj ν,k ) 2 + δ � � f ( a k ) = 0 , k ∈ N \{ 0 , 1 } , with a k := i k ∈ N \{ 0 } . , � Moreover, f satisfies K T 1 / 2 | J ′ � � ν ( j ν, 1 ) | log | f ( s ) | ≤ T |ℑ ( s ) − δ | √ + log . 2 ε 2 κ � Classical representation of entire functions of exponential type A in C + � + ∞ ∞ � � � + ℑ ( z ) log | f ( s ) | z − a ℓ � � � log | f ( z ) | = Aℑ ( z ) + log | s − z | 2 d s. � � � z − a ℓ � π −∞ � ℓ =1 Paris June, 20 10

Some open problems � What to do if M < 0 . � Diffusion with constant coefficient. � BV coefficients. � Higher dimension Paris June, 20 11

Happy Birthday Jean-Michel Paris June, 20 12