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

      

yt − εyxx + Myx = 0 in (0, L) × (0, L), y(0, t) = u(t), y(L, t) = 0

• n (0, T),

y(x, 0) = y0(x) in (0, L).

(TD) For y0 ∈ H−1(0, L) we denote by U(ε, T, L, M, y0) the set of controls u ∈ L2(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):

K(ε, T, L, M) := sup

y0H−1(0,L)≤1

• min{uL2(0,T) : u ∈ U(ε, T, L, M, y0)}
• .

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

      

yt − ε(xα+1yx)x + Mxαyx = 0 in (0, L) × (0, L), y(0, t) = u(t), y(L, t) = 0

• n (0, T),

y(x, 0) = y0(x) in (0, L).

(TD) Assume that (TD) is null controllable in some space H, we denote by U(ε, T, L, M, α, y0) the set of controls u ∈ L2(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: K(ε, T, L, M, α) := sup

y0H≤1

• min{uL2(0,T) : u ∈ U(ε, T, L, M, α, y0)}
• .

For every (ε, T, L, M, α) ∈ (0, +∞)4 × (0, 1) such that M/ε > α and any y0 ∈ L2((0, L); x−M/εdx), there exits a control u ∈ L2(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(xα+1y′)′ + Mxαy′ = λy(x), x ∈ (0, L), λ ∈ R. Particular case of Bessel differential equation x2y′′ + axy′ + (bxℓ + c)y = 0, x ∈ (0, ∞), ℓ = 0.

The solutions can be written in terms of Bessel functions:

b = 0 : y(x) = x

1 2(1−a)Zν(κ−1√

bxκ), ν := 1 ℓ

• (1 − a)2 − 4c ,

κ := ℓ 2, Then, under the structural assumption M/ε > α, we can prove that (A, D(A)) is self-

adjoint on L2((0, L); x−M/εdx) and generates an analytic semigroup of bounded linear

• perators S(t)t≥0. Let ν := (M/ε − α)/(1 − α) and κ := 1

2(1 − α), we have

Φn(x) := (2κ)

1 2

|J′

ν(jν,n)|x

1 2(M/ε−α)Jν (jν,n xκ) ,

λn := ε(κjν,n)2 x ∈ (0, 1).

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 K(ε, α, T, L, M) ≤ exp

• −Q(T, L, M, α)

ε

• if

T > (2 √ 6)L1−α M(1 − α)

and

K(ε, α, T, L, M) ≥ exp

• C(T, L, M, α)

ε

• if

T < (0, 98)L1−α M(1 − α) . 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 y0 ∈ L2((0, L); x−M/εdx) := H. Then u ∈ U(ε, T, L, M, α, y0) if and only if

• y0, Φn

H = −(M/ε − α)(2jν,n)ν

2νΓ(ν + 1)

T

0 u(t) exp (−λn(T − t)) dt,

∀n ∈ N\{0}. Find a biorthogonal family {Ψk(t)}k∈N\{0}

T

0 Ψk(t)e−λℓ(T−t)dt = δkℓ,

k, ℓ ∈ N\{0}. Construct a family {Jk(z)}k∈N\{0} of entire functions of exponential type satisfying Jk(−iλℓ) = δkℓ, 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(κjν,k)2, k ∈ N\{0}} as simple zeros is

Λ(z) :=

+∞

• k=1
• 1 −

iz ε(κjν,k)2

• = Γ(ν + 1)
• 2√εκ

√ iz

ν

Jν

√

iz √εκ

• .

Moreover, Λ(·) is of exponential type and

|Λ(z)| ≤ exp

  

• |z|

κ√ε

  

as

|z| → +∞. Now, we consider ˜ Jk(z) := Λ(z) Λ′(−iλk)(z + iλk),

• ne easily deduces that

˜ Jk(−iλl) = δkl. ˜ Jk(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 ˜

Jk(·) bounded on the real line and of

relevant exponential type. Let us set

Jk(z) := ˜ Jk(z) H(z) H(−iλk). Where, H is constructed to satisfy

      

H(−iλk) ≥ Cβ ∀ k ≥ 1, |H(z)| ≤ eβ|ℑ(z)| ∀ z ∈ C, H(ix) ≥ Ceγ|x| ∀ x ∈ R.

(Mult)

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 f(s) :=

−T/2

−T/2 u

• t + T

2

• e

−it

• s−iδ

ε

• dt

s ∈ C.

for a δ > 0, that will be choosen later. Then, f is an entire function satifying

f(ak) = 0, k ∈ N\{0, 1},

with

ak := i

• (εκjν,k)2 + δ
• ,

k ∈ N\{0}. Moreover, f satisfies log |f(s)| ≤ T|ℑ(s) − δ| 2ε + log

• KT 1/2|J′

ν(jν,1)|

√ 2κ

• .

Classical representation of entire functions of exponential type A in C+ log |f(z)| = Aℑ(z) +

∞

• ℓ=1

log

• z − aℓ

z − aℓ

• + ℑ(z)

π

+∞

−∞

log |f(s)| |s − z|2 ds.

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