Lecture 3: Controllability of some hyperbolic equations Enrique - - PowerPoint PPT Presentation

Lecture 3: Controllability of some hyperbolic equations

Enrique FERN ´ ANDEZ-CARA

• Dpto. E.D.A.N. - Univ. of Sevilla

Results for the wave equation Other hyperbolic equations and systems Applications . . .

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Outline

1

Basic results for the wave equation Controllability concepts The main results

2

Other hyperbolic equations and systems Semilinear wave equations Linear elasticity and Lam´ e systems Visco-elasticity and Maxwell fluids

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Basic results for the wave equation

Controllability concepts

Consider the controlled linear wave equation:    ytt − ∆y = v1ω in Q y = 0

• n Σ

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω (1) (same notation as in Lecture 2: Q = ω × (0, T), ω ⊂ Ω, etc.) ∀(y0, y1) ∈ H1

0 × L2, ∀v ∈ L2(ω × (0, T)) ∃! solution

y ∈ C0([0, T]; H1

0) ∩ C1([0, T]; L2)

Now: the control is v and the state is (y, yt) R(T; y0, y1) := {(y(·, T), yt(·, T)) : v ∈ L2(ω × (0, T))}. Then:

• (1) is AC if R(T; y0, y1) = H1

0 × L2 for all (y0, y1)

• It is EC if R(T; y0, y1) = H1

0 × L2 for all (y0, y1)

• It is NC if R(T; y0, y1) ∋ (0, 0) for all (y0, y1)

Other spaces where (1) is well posed can also be used

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Basic results for the wave equation

Controllability concepts

ytt − ∆y = v1ω in Ω × (0, T) y(0) = y0, yt(0) = y1 in Ω, etc. (1) First results:

• For AC, EC or NC to hold, T has to be large (finite speed of propagation)
• NC and EC are equivalent; (1) is linear and reversible in time
• EC ⇒ AC, but the reciprocal is false (conditions on T and ω)

Very different properties to those satisfied by the heat equation!

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Basic results for the wave equation

The main results

Theorem 1 AC holds for (1) for any ω and any T > T∗(ω) SKETCH OF THE PROOF: Fix ω and T > 0 (1) is AC ⇔ R(T; 0, 0) is dense ⇔ R(T; 0, 0)⊥ = {0, 0} Assume (ϕ0, ϕ1) ∈ R(T; 0, 0)⊥ and introduce ϕtt − ∆ϕ = 0 in Ω × (0, T) ϕ(T) = ϕ0, ϕt(T) = ϕ1 in Ω, etc. (2) Then:

ω×(0,T)

ϕv dx dt = (ϕ0, ϕ1), (y(T), yt(T)) = 0 ∀v Hence: AC holds iff the following uniqueness property is true: ϕ solves (2), ϕ = 0 in ω × (0, T) ⇒ ϕ ≡ 0 But this is true if T > T∗(ω) (Holmgren’s Theorem) ✷

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Basic results for the wave equation

The main results

Again, possible to construct the “best” control: u = ˆ ϕ|ω×(0,T) , with ˆ ϕ associated to ˆ ϕ0 and ˆ ϕ0 minimizing Jε(ϕ0, ϕ1) = 1 2

ω×(0,T)

|ϕ|2 + ε(ϕ0, ϕ1)L2×H−1 − (ϕ0, ϕ1), (y0, y1) QUESTIONS: A more general system: ytt − ∇ · (D(x, t)∇y) + B(x, t) · ∇y + a(x, t)y = v1ω (y, yt)|t=0 = (y0, y1), etc. with Dij, Bi, a ∈ L∞, D(x, t)ξ · ξ ≥ α0|ξ|2 a.e. Minimal regularity hypotheses to have AC for all ω and large T? Can we then repeat this construction? Unknown; even for D ≡ Id, B ≡ 0

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Basic results for the wave equation

The main results

What about EC? (1) ytt − ∆y = v1ω (y, yt)(0) = (y0, y1), . . . (2) ϕtt − ∆ϕ = 0 (ϕ, ϕt)(T) = (ϕ0, ϕ1), . . . Proposition 1 [J-L Lions, H.U.M. method, 1988] EC holds for (1) with v ∈ L2(ω × (0, T)) iff (2) is observable: (ϕ, ϕt)(0)2

L2×H−1 ≤ C ω×(0,T)

|ϕ|2 dx dt (3) When (3) holds, one can minimize in L2 × H−1 W(ϕ0, ϕ1) = 1 2

ω×(0,T)

|ϕ|2 + (ϕ, ϕt)(0), (y0, y1) Then: v = ˆ ϕ1ω , where ˆ ϕ corresponds to the minimizer ( ˆ ϕ0, ˆ ϕ1), is the null control with minimal L2 norm

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Basic results for the wave equation

The main results

The EC problem is reduced to the analysis of (3). We have Theorem 2 Assume: x0 ∈ RN, ω ⊃ Γ(x0) := {x ∈ Γ : (x − x0) · n(x) > 0}, T > T(x0) := 2x − x0L∞. Then: (1) is EC at time T SKETCH OF THE PROOF: First, boundary observability for T > T(x0) (ϕ, ϕt)(0)2

H1

0 ×L2 ≤ C

Γ(x0)×(0,T)

• ∂ϕ

∂n

• 2

dΓ dt (4) [Ho, 1986; J-L Lions, 1988], multipliers techniques This gives EC with controls on the boundary Then: (4) ⇒ (3) when ω is a neighborhood of Γ(x0) ✷ The class of sets ω can be enlarged, [Osses, 2001]

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Basic results for the wave equation

The main results

More generally: Theorem 3 [Bardos-Lebeau-Rauch, 1992; Burq, 1997] Assume: Ω is of class C3. Then: (3) holds iff (ω, T) satisfies the GCC: Every ray that begins to propagate in Ω at time t = 0 and is reflected on Γ enters ω at a time t < T The proof uses microlocal defect measures, [Gerard, 1991] The result also holds for ytt − ∇ · (D(x, t)∇y) + B(x, t) · ∇y + a(x, t)y = v1ω (y, yt)|t=0 = (y0, y1), etc. when Dij, Bi, a ∈ C2, D(x, t)ξ · ξ ≥ α0|ξ|2 QUESTION: Minimal regularity hypotheses for Theorem 3? Other methods exist, in particular Carleman estimates [Zhang, 2000; Puel, 2004; etc.]

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Other hyperbolic equations and systems

Semilinear wave equations

The semilinear wave equation:    ytt − ∆y + f(y) = v1ω in Q y = 0

• n Σ

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω (5) Theorem 4 [Zuazua, 1993] Assume: N = 1, f ∈ C1(R), lim|s|→∞

f(s) s log2(1+|s|) = 0

Then: EC holds for (5) for any ω = (a, b), T > max (a, 1 − b) Theorem 5 [Zhang, 2000] Assume: f ∈ C1(R), lim|s|→∞

f(s) s log1/2(1+|s|) = 0

Then: EC holds for (5) for ω ⊃ Γ(x0), T > T∗(Ω, x0) QUESTION: N ≥ 2, f ∈ C1(R), (ω, T) satisfying GCC Minimal hypotheses on f to have EC?

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Other hyperbolic equations and systems

Linear elasticity and Lam´ e systems

The Lam´ e system (linear elasticity + isotropy)    ytt − λ∆y − µ∇(∇ · y) = v1ω in Q y = 0

• n Σ

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω (6) with λ, µ > 0 (for instance) y = (y1, . . . , yN): the displacement v1ω: a force field Theorem 6 [Imanuvilov-Yamamoto, 2005] EC holds for (6) for any (ω, T) satisfying GCC

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Other hyperbolic equations and systems

Visco-elasticity and Maxwell fluids

The linearized Maxwell system:        yt + ∇π = ∇ · τ + v1ω, ∇ · y = 0 in Q τt + a τ = 2bDy in Q y = 0

• n Σ

y(0) = y0, τ(0) = τ 0 in Ω (7) y, π, τ: the velocity, pressure, extra-stress elastic tensor, resp. v1ω: a force field The linearization at zero of the true Maxwell model:        yt + (y · ∇)y + ∇π = ∇ · τ + v1ω, ∇ · y = 0 in Q τt + (y · ∇)τ + a τ + g(τ, ∇y) = 2bDy in Q y = 0

• n Σ

y(0) = y0, τ(0) = τ 0 in Ω (8) Very difficult to solve and analyze . . .

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Other hyperbolic equations and systems

Visco-elasticity and Maxwell fluids

yt + ∇π = ∇ · τ + v1ω, ∇ · y = 0 τt + a τ = 2bDy, etc. (7) Observe: (7) can be rewritten in the form    ztt − azt − b∆z + ∇Z = u1ω + e−at∇ · τ 0, ∇ · z = 0 in Q z = 0

• n Σ

(z, zt)(0) = (0, y0) in Ω with z = t

0 easy(s) ds, Z = eatπ

Results from [EFC, Boldrini, Doubova, Gonz´ alez-Burgos]: Theorem 7 AC holds for (7) for any ω and any T > T∗(Ω, ω, a, b) Theorem 8 Assume: x0 ∈ RN, ω ⊃ Γ(x0) := {x ∈ Γ : (x − x0) · n(x) > 0}, 0 < a < 2

• λ1b, T > T∗(Ω, x0, a, b)

Then: (7) is EC at time T

• E. Fern´

andez-Cara Control and hyperbolic PDEs

Other hyperbolic equations and systems

Visco-elasticity and Maxwell fluids

yt + ∇π = ∇ · τ + v1ω, ∇ · y = 0 τt + a τ = 2bDy, etc. (7) QUESTIONS: EC for 0 < a < 2

• λ1b if (ω, T) satisfies GCC?

EC for general a, b > 0, ω ⊃ Γ(x0) and large T?

• E. Fern´

andez-Cara Control and hyperbolic PDEs

THANK YOU VERY MUCH . . .

• E. Fern´

andez-Cara Control and hyperbolic PDEs