Lecture 2: Controllability of parabolic equations Enrique FERN - - PowerPoint PPT Presentation

Lecture 2: Controllability of parabolic equations

Enrique FERN ´ ANDEZ-CARA

• Dpto. E.D.A.N. - Univ. of Sevilla

Controllability concepts Results for heat-like equations and systems Further comments and applications

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Outline

1

Generalities

2

Basic results for the heat equation Controllability concepts The main results Additional comments

3

Controllability of other parabolic systems Non-scalar coupled systems Stochastic controllability

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Generalities

Controllability of time-dependent systems

An abstract problem: yt − Ay = Bv, t ∈ (0, T) y(0) = y0

• A : D(A) ⊂ H → H, B : D(B) ⊂ U → H are linear
• v = v(t) is the control, y = y(t) is the state

Exact controllability problem: Choose y0, y1 ∈ H (the space of states) ∃v such the state associated to y0 satisfies y(T) = y1? Relaxing y(T) = y1: other controllability problems Solvability can depend on time reversibility, regularity, structure of the control set, size

• f T, etc.

Many contributions: [Fattorini, Russell, J-L Lions, . . . ] ; more recently, [Fursikov, Imanuvilov, Lasiecka, Triggiani, Lebeau, Zuazua, Coron, . . . ]

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

Controllability concepts

Ω ⊂ RN, T > 0 (N ≥ 1), regular Γ = ∂Ω; ω ⊂ Ω (small) The state equation:    yt − ∆y = v1ω in Ω × (0, T) y = 0

• n Γ × (0, T)

y(0) = y0 in Ω (1) We assume: y0 ∈ L2(Ω), v ∈ L2(ω × (0, T)) ∃! solution y ∈ C0([0, T]; L2(Ω)) Set R(T; y0) = { y(·, T) : v ∈ L2(ω × (0, T)) }. Then:

• (1) is approximately controllable if R(T; y0) = L2(Ω) for all y0
• It is exactly controllable if R(T; y0) = L2(Ω) for all y0
• It is null controllable if R(T; y0) ∋ 0 for all y0
• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

Controllability concepts

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

• EC cannot hold, except possibly if ω = Ω:

y(T) is always smooth in Ω \ ω

• NC is equivalent to the EC to the states in S(T)(L2(Ω)), i.e.

R(T; y0) ∋ 0 ∀y0 ⇔ R(T; y0) ∋ S(T)(L2(Ω)) ∀y0 (write y = S(t)y0 + z and work with z) Consequence: NC ⇒ AC

• We will see: AC and NC hold. Thus, AC ⇔ EC for PDEs!

(contrarily to ODEs)

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

The main results

Theorem 1 AC holds for (1) for any ω and any T > 0 SKETCH OF THE PROOF: Fix ω and T > 0 (1) is AC ⇔ R(T; 0) is dense in L2(Ω) ⇔ R(T; 0)⊥ = {0} (R(L) is dense iff L∗ is one-to-one, uniqueness) Assume ϕ0 ∈ R(T; 0)⊥ and introduce −ϕt − ∆ϕ = 0 in Ω × (0, T) ϕ(T) = ϕ0 in Ω, etc. (2) Then:

ω×(0,T)

ϕv dx dt =

• Ω

ϕ0y(T) dx = 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: the solutions to (2) are analytic in space! ✷

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

The main results

Also, 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) = 1 2

ω×(0,T)

|ϕ|2 dx dt + εϕ0L2 −

• Ω

ϕ0y1 dx, (3) QUESTIONS: A general linear system: yt − ∇ · (D(x, t)∇y) + B(x, t) · ∇y + a(x, t)y = v1ω y|t=0 = y0, 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 T? Can we do again the same? (unknown for N ≥ 2)

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

The main results

Theorem 2 [Fursikov-Imanuvilov, 1991 . . . ] NC holds for (1) for any ω and any T > 0 SKETCH OF THE PROOF: Fix again ω and T > 0 (1) is NC ⇔ R(T; y0) ∋ 0 for all y0 ⇔ Observability of (2), i.e. ϕ(0)2

L2 ≤ C

• ω×(0,T)

|ϕ|2 dx dt ∀ϕ0 ∈ L2(Ω) (4) (R(M) ⊂ R(L) iff M∗ϕ0 ≤ CL∗ϕ0 for all ϕ0) (4) is a consequence of Carleman:

• Ω×(0,T)

ρ−2 |ϕ|2 dx dt ≤ C

• ω×(0,T)

ρ−2 |ϕ|2 dx dt (5) This holds for appropriate ρ = ρ(x, t), with ρ ∼ e1/(T−t) The proof of (5) is complicate

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

The main results

From (5) we get: ϕ(0)2

L2 ≤ C

3T/4

T/4

ϕ(t)2

L2 ≤ C

• Ω×(T/4,3T/4) ρ−2 |ϕ|2 dx dt

≤ C

• ω×(0,T) ρ−2 |ϕ|2 dx dt ≤ C
• ω×(0,T) |ϕ|2 dx dt

✷ OTHER PROOF : Construct directly v such that y(T) = 0: y(t) =

i yi(t)ϕi, with −∆ϕi = λiϕi in Ω, ϕi = 0 on ∂Ω, etc.

• First, v|(0,T/2) such that yi(T/2) = 0 for λi ≤ µ; then v|(T/2,3T/4) ≡ 0
• Then, v|(3T/4,7T/8) such that yi(7T/8) = 0 for λi ≤ 2µ and v|(7T/8,15T/16) ≡ 0, etc.

. . . The key point: a finite dimensional observability inequality

• λi ≤µ

a2

i ≤ eC0 √µ

• ω
• λi ≤µ

aiϕi

• 2

dx (LR) [Lebeau-Robbiano, 1995]

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

The main results

QUESTIONS: Again the general linear system: yt − ∇ · (D(x, t)∇y) + B(x, t) · ∇y + a(x, t)y = v1ω y|t=0 = y0, etc. with Dij, Bi, a ∈ L∞, D(x, t)ξ · ξ ≥ α0|ξ|2 a.e. Minimal regularity hypotheses to have NC for all ω and T? For time-independent D, B, a, minimal hypotheses for (LR)? (again unknown for N ≥ 2) Carleman, observability and NC holds if D ∈ W 1,∞, but . . .

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

Additional comments

Boundary NC: Ω, T as before; γ ⊂ ∂Ω open, non-empty (small) The state equation:    yt − ∆y = 0 in Ω × (0, T) y = h1ω

• n ∂Ω × (0, T)

y(0) = y0 in Ω (6) Theorem 3 AC and NC hold for (6) for any γ and any T > 0 Very easy: “extend” Ω and (6) to a problem    yt − ∆y = v1ω in ˜ Ω × (0, T) y = 0

• n ∂ ˜

Ω × (0, T) y(0) = ˜ y0 in ˜ Ω with ω ⊂ ˜ Ω \ Ω, apply Theorems 1 and 2 and then restrict to Ω

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

Additional comments

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

• n Σ

y(0) = y0 in Ω (7) Theorem 4 [EFC-Zuazua, 2000] Assume: f ∈ C1(R), lim|s|→∞

f(s) s log3/2(1+|s|) = 0

Then: NC and AC hold for (7) for any ω and any T > 0 QUESTION: Same results with 3/2 replaced by some p ∈ (3/2, 2)?

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Basic results for the heat equation

Additional comments

Algorithms devised to construct “good” controls: Optimal control + penalty [Glowinski et al. 1995 . . . ] Fix y0 = 0 and y1 ∈ L2(Ω) and set Fk(v) = 1 2

ω×(0,T)

|v|2 dx dt + k 2 y(·, T) − y12

L2

Results from [J-L Lions, 1990]: Fk has a unique minimizer vk for all k > 0 and the yk satisfy yk(T) → y1 in L2(Ω) as k → ∞ Hence: it suffices to take v = vk for large k = k(ε) Theorem 5 [EFC-Zuazua, 2000] yk(T) − y1L2 ≤

C log k and vkL2(Q) ≤ C √ k log k

Logarithmic (and therefore very slow) convergence rates In agreement with the high cost of AC: O(e1/ε)

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Controllability of other parabolic systems

Non-scalar coupled systems

∃ many generalizations and variants of these arguments: Time and space dependent (regular) coefficients Stokes-like systems: yt − ∆y + (a · ∇)y + (y · ∇)b + ∇p = v1ω ∇ · y = 0 and yt − ∆y + (a · ∇)y + (y · ∇)b + ∇p = θk + v1ω ∇ · y = 0 θt − κ∆θ + c · ∇θ = w1ω where a, b, c ∈ L∞ Other boundary conditions Other linear parabolic non-scalar systems

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Controllability of other parabolic systems

Non-scalar coupled systems

Consider:    yt − D∆y = My + Bv1ω, (x, t) ∈ Ω × (0, T), y(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), y(x, 0) = y0(x), x ∈ Ω, (8) Here: y = (y1, . . . , yn), v = (v1, . . . , vm), D, M and B are constant matrices (n ≥ 2) and D is definite positive Notation: [H; B] := [B|HB| · · · |Hn−1B] Theorem 6 [Ammar-Khodja et al. 2008] Assume: D is diagonal. Then: (8) is NC (for all ω and T) iff Rank [(−λiD + M); B] = n for all i (Kalman-like condition) QUESTIONS: Conditions on D, M and B that ensure NC in the general case? What happens when M = M(x, t)?

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Controllability of other parabolic systems

Stochastic controllability

The data:

• {Λ, F, P}: a complete probability space

Ef :=

• Λ f dP for any f ∈ L1(Λ, F; X) (X: Banach)
• Wt: a real Wiener process, i.e. (λ, t) → Wt(λ) is measurable W0 = 0 and the Wt

are normally distributed and independent, with EWt = 0 and E|Wt|2 = t Notation: Ft := σ( Ws, 0 ≤ s ≤ t ) I2(X) = {Φ ∈ L2(dP ⊗ dt; X) : Φ(·, t) is Ft-measurable a.e.} B ∈ I2(L2(Ω)), y0 ∈ L2(Ω) The state system (additive noise; to be satisfied P-a.e.):    yt − ∆y = v1ω + B(t) ˙ Wt in Ω × (0, T) y = 0

• n Γ × (0, T)

y(0) = y0 in Ω (9) For any v ∈ I2(L2(ω)), ∃!y ∈ I2(H1

0) ∩ L2(Λ; C0([0, T]; L2))

• E. Fern´

andez-Cara Controllability and parabolic PDEs

Controllability of other parabolic systems

Stochastic controllability

Theorem 7 YT = { y(T) : v ∈ I2(H) } is dense in L2(Λ, FT ; H), i.e. we have AC in quadratic mean. Theorem 8 Assume: B ∈ C1([0, T]; L2(Ω)), Supp B(t) ∩ ω = ∅ for all t and

Q

t

• γ(t)−1|B|2 + γ(t)3|Bt|2

e2β(x)/γ(t) < +∞ (10) for γ(t) ≡ t(T − t) and some β = β(x). Then: NC The proofs: similar to those above:

• A unique continuation property for Theorem 6
• An observability estimate for Theorem 7
• E. Fern´

andez-Cara Controllability and parabolic PDEs

Controllability of other parabolic systems

Stochastic controllability

More interesting: multiplicative noise    yt − ∆y = v1ω + ay ˙ Wt in Ω × (0, T) y = 0

• n Γ × (0, T)

y(0) = y0 in Ω (for instance a ∈ R) Now, AC is equivalent to unique continuation for    −ϕt − ∆ϕ = aq − q ˙ Wt in Ω × (0, T) ϕ ∈ I2(H1

0), q ∈ I2(L2)

ϕ(T) = ϕ0 in Ω Results by [Barbu et al., 2002], [Zhang, 2010]

• E. Fern´

andez-Cara Controllability and parabolic PDEs

THANK YOU VERY MUCH . . .

• E. Fern´

andez-Cara Controllability and parabolic PDEs