Pointwise Carleman estimates

and control theoretic implications

(Joint work with Professor Roberto Triggiani)

Xiangjin Xu

SUNY-Binghamton University

MSRI Summer Microprogram onNonlinear Partial Differential Equations

August 10th, 2007

Outline of Talk:

1. Qualitative statement of exact controllability of an evolution equation

2. Background and some history of E.C. of Schrodinger-type equations

3. Problem setting, geometric assumptions on triple M,Γ0,Γ1.

4. Pointwise Carleman Inequality.

5. Carleman estimates: first version: Global uniqueness.

6. Carleman estimates: second version: Continuous observability.

7. The Euler-Bernoulli Plate Equation.

Qualitative statement of exact controllability of an evolution equation.

Evolution y(t, x;u): a solution of differential equations on [0, T ]×M .

I. T arbitrary, infinite speed of propagation.

(Schrodinger equation, Euler-Bernoulli Plate Equation, · · · )

II. An initial condition y0 at t = 0 and a target state yT at t = T .

III. u a control function: acting on a portion of the boundary.

Exact controllability problem: Seek the control u such that:

(Initial condition y0)→[

evolution y(t, ·;u)

]→ (target state yT)

Example of NO exact control from boundary: the evolution equation on amanifold with a closed geodesic.

For Schrodinger equations with Dirichlet control on a bounded domain Ω:iyt +Ay = 0 on Ωy|Γ0

= 0, y|Γ1= u ∈ L2([0, T ]× Γ1), ∂Ω = Γ0 ∪ Γ1

Exact controllability at T on right space (H−1)

mSolution operator being ONTO target space (H−1)

mDual operator being bounded below

m

For the dual problem: Schrodinger equation with homogeneous Dirichlet B.C.:iwt +A∗w = 0 on Ωw|∂Ω = 0

PDE interpretation (Continuous Observability Inequality):

CTE(0) ≤∫ T

0

∫Γ1

∣∣∣∣∂w

∂ν

∣∣∣∣2dΓ1dt (GOAL)

(This is an inverse type inequality.)

Classic energy method for pure Schrodinger equations

Pure Schrodinger equation: iyt −∆y = 0:

• Dirichlet boundary control +optimal regularity. [Lasiecka-Triggiani ’91]

• Neumann boundary control. [Machtyngier ’90]

• Optimal regularity and exact controllability of wave, Schrodinger, plate-likeeq... [Ho, Lasiecka, Triggiani, Lions, Lagnese, etc].

Classic energy method failed.....

• ∆ replaced by a variable coefficient elliptic operator, A =∑

∂∂xi

(aij(x)∂

∂xj);

• Constant coefficient principal part with the presence of ”energy level” terms:

iyt −∆y = r(t, x) · ∇y + b(t, x)y.

Integral Carleman estimates with l.o.t.

• Geometric optics methods for pure Schrodinger equations with Dirichletcontrol. [Lebeau ’91]

• Integral Carleman estimates:

• general evolution equations in pseudo-differential setting; [Tataru’s Thesisat UVA ’92 and series papers later];

• hyperbolic equations, Schrodinger, plates, etc, with differential energy meth-ods [Lasiecka-Triggiani]

• Riemannian geometric versions of the above ”concrete” energy methods,[Lasiecka-Triggiani-Yao]

An additional obstacle:

Integral Carleman estimates: polluted by interior l.o.t. below the energy level.

CTE(0) ≤∫ T

0

∫Γ1

∣∣∣∣∂w

∂ν

∣∣∣∣2dΓ1dt + l.o.t.(||w||2L2([0,T ]×M))

Approach to absorbing l.o.t: Compactness/uniqueness method

• PDE theory (e.g. Hormander’s books)

• Control theory of PDE first by Littman,

Contradiction argument using a global uniqueness theorem for over-determinantproblems.

• For analytic coefficients, one can use Holmgren-John semi-global result.

• For partial analytic coefficients, recent works by Tataru, Hormander.

Could NOT do:

In the presence of low regularity coefficients, possibly also time-dependent,such uniqueness result was generally not available.

Pointwise Carleman estimates since late 90’s

• Inspired by Novosibirsk school for global uniqueness for 2nd order hyper-bolic equations on Euclidean domain with Dirichlet B.C. case. [ Lavrentev-Romanov-Shishataskii ’86]

• Global uniqueness, observability and stabilization for 2nd order hyperbolicequations on Euclidean domain with purely Neumann B.C. case and mixedB.C. case. [Lasiecka-Triggiani-Zhang ’00]

• Global uniqueness, observability and stabilization for 2nd order hyperbolicequations on Riemannian manifold with purely Neumann B.C. case and mixedB.C. case. [Triggiani-Yao ’02]

• Global uniqueness, observability and stabilization for Schrodinger equationson Euclidean domain with purely Neumann B.C. case and mixed B.C. case.[Lasiecka-Triggiani-Zhang ’04]

Problem setting

(M, g) Riemannian manifold; ∂M = Γ = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅. Given T > 0,

Pw = iwt + ∆gw = F (w, w,∇w,∇w) + f, in Q = (0, T ]×M (1)

Linear: F = (P (t, x),∇w) + p0(t, x)w with |P |, p0 ∈ L∞(Q), and f ∈ L2(Q).

Semilinear: |F (w, w,∇w,∇w)|2 ≤ C(|∇w|2 + |w|2p), with p < n/(n− 2), n ≥ 3,and p < ∞, n = 1,2.

For Σ = (0, T ]× Γ, Σ1 = (0, T ]× Γ1, consider the boundary conditions:

(i) Neumann B.C.: ∂w∂ν|Σ = 0, and w|Σ1

= u,

(ii) Dirichlet B.C.: w|Σ = 0, and ∂w∂ν|Σ1

= u.

Observation u in present problem (or control in the dual problem) takesplace only on a sub-portion Γ1 of the boundary.

Geometrical assumption on triple M,Γ0,Γ1

∃ a strictly convex (w.r.t. Riemannian metric) C3 function d : M → R+, s.t.for the conservative gradient field h(x) = ∇d(x):

(I) Neumann B.C.: ∂d∂ν

= ∇d · ν = 0, on Γ0.

Dirichlet B.C.: ∂d∂ν

= ∇d · ν ≤ 0, on Γ0.

(II) Hessian of d(x) is coercive:

D2d(X, X) = (DX(∇d), X) ≥ 2(X, X), ∀x ∈ M, X ∈ TxM

A temporary assumption: no critical point of d(x) on M (enough, near Γ0),

infM|∇d| = p > 0

To remove above assumption, splitting M as M = M1 ∪M2, for two suitablyoverlapping sub-manifolds M1 and M2, and working with two strictly convexfunctions d1 and d2, where di satisfies above assumption on Mi.

Examples

1. A bounded domain Ω ⊂ Rn satisfies

(i) convex (respectively, concave) on the side of the portion Γ0 of its boundary,

(ii) there exists a radial vector field (x−x0) for some x0 ∈ Rn which is entering(respectively, exiting) Ω through Γ0.

Convex function: d(x) = 12||x− x0||2 + · · ·

2. Riemannian manifolds:

• Portion of totally geodesic ball with partial geodesic flat boundary on non-negative curvature manifolds;

• Submanifold with partial geodesic flat boundary on negative curvature man-ifolds.

Pseudo-convex function φ(t, x) on Q

Define function φ(t, x) on Q = [0, T )×M as

φ(t, x) = d(x)− c(t− T2)2, 0 ≤ t ≤ T, x ∈ M

where c = cT large enough s.t. cT 2 > 4maxM d(x) + 4δ

for sufficiently small δ > 0, and fixed. φ(t, x) satisfies:

(I) φ(0, x) = φ(T, x) = d(x)− cT 2

4≤ −δ, uniformly for x ∈ M

(II) there are 0 < t0 < T2

< t1 < T , such that minx∈M,t∈[t0,t1] φ(t, x) > − δ2

E(t) =

∫M

|∇w(t)|2dx;

E(t) =

∫M

[|∇w(t)|2 + |w(t)|2]dx = ||w(t)||2H1(M).

Fundamental technical Lemma

Assume w(t, x) ∈ C2(Q,C), l(t, x) ∈ C3(Q,R), Φ(t, x), Ψ(t, x) ∈ C1(Q,R),with ∇x(lt) ≡ 0; θ(t, x) = el(t,x); v(t, x) = θ(t, x)w(t, x).

Let ε > 0 arbitrary, the following pointwise inequality holds true:

(1 +1

ε)e2l(t,x)|iwt + ∆w|2 −

dM

dt+ divV

≥−2(Ψ + ∆l)∆l + 4D2l(∇l,∇l) + 2(∇(Φ−Ψ),∇l)− ε|Ψ + ∆l|2

−1

ε|∇(∆l)|2 − 4(∇l,∇(∆l))−Ψ2 −Φ2 + 2Φ∆l + ltt

|v|2

+2

D2l(∇v,∇v) + D2l(∇v,∇v)− (Ψ + ∆l)|∇v|2

− ε|∇v|2 (2)

where M(w) and V (w) have explicit formula. Let ξ ≡ Rew and η ≡ Imw:

M(w) ≡ θ[2(∇l,∇ξ)η − ξ(∇l,∇η)− lt|w|2];

V (w) ≡ θ2(2|∇l|2 −∆l −Ψ + Φ)∇l|w|2 + lt(η∇ξ − ξ∇η)−∇l(ξtη − ξηt)

+1

2(2|∇l|2 −∆l)∇|w|2 + (∇l,∇w)∇w + (∇l,∇w)∇w −∇l|∇w|2.

Pointwise Carleman Inequality

Choose l(t, x) = τφ(t, x), Φ(t, x) = −∆l(t, x), either Ψ(t, x) = ∆l(t, x) orΨ(t, x) = 0 in above Lemma, we have

(1 +

1

ε

)e2τφ(t,x)|iwt + ∆w|2 −

dM

dt+ divV

≥2τ

[D2d(

∇v

|∇v|,∇v

|∇v|) + D2d(

∇v

|∇v|,∇v

|∇v|)

]− ε

|∇v|2

+

[4τ3D2d(∇d,∇d) + O(τ2)

]|v|2

≥ 4τρ− ε|∇v|2 + [4τ3p2 + O(τ2)]|v|2

≥ δ04τρ− εθ2|∇w|2 + [4τ3p2(1− δ0) + O(τ2)]θ2|w|2 (3)

for some 0 < δ0 < 1. Note that:

• D2d(·, ·) positive definite ⇒ for small ε > 0, the coefficient of |∇v|2 positive;

• τ > 0 large enough and d(x) no critical point ⇒ the coefficient of v2 positive.

Carleman estimates: first version

Integrate (3) over Q = [0, T ]×M , applying the assumption on F (w), we havethe following first version Carleman estimates:

BΣ(w) + (1 +1

ε)

∫ T

0

∫M

e2τφ(t,x)[|F|2 + |f |2]dxdt

≥ mτ

∫ T

0

∫M

e2τφ(t,x)[|∇w|2 + |w|2]dxdt− cτe−2δτ [E(T ) + E(0)]

≥ mτe−δτ

∫ t1

t0

E(t)dt− Cτe−2δτ [E(T ) + E(0)]. (4)

where mτ →∞ as τ →∞ at the growth rare as τ , the boundary term

BΣ(w) =

∫ T

0

∫M

divV dxdt =

∫ T

0

∫Γ

V · νdΓdt

∫Q

∂M

∂tdtdx =

[∫M

Mdx

]T

0

≤ τC

[∫M

e2τφ(|∇w|2 + |w|2)dx

]T

0

≤ Cτe−2τδ[E(T ) + E(0)].

Global uniqueness

Theorem (Global uniqueness) [Triggiani-Xu ’07]

Let w ∈ H2,2(Q) = L2(0, T ;H2(M))∩H2(0, T ;L2(M)) be a solution of (1) andf = 0, with Σ = [0, T ]× Γ, Σ1 = [0, T ]× Γ1,

(I) Neumann case:w satisfies the B.C.:

∂w∂ν|Σ = 0, and w|Σ1

= 0, where h · ν = 0 on Γ0.

then such a solution must vanish: w = 0 in [0, T )×M .

(II) Dirichlet case:w satisfies the B.C.:

w|Σ = 0, and ∂w∂ν|Σ1

= 0, where h · ν ≤ 0 on Γ0.

then such a solution must vanish: w = 0 in [0, T )×M .

Sketch of proof: Carleman estimates ⇒ Global uniqueness

• Step 1. From definition of BΣ(w), one hasNeumann B.C. ⇒ BΣ(w) = 0

Dirichlet B.C. ⇒ BΣ(w) = 2τ∫ T

0

∫Γ0

e2τφ|∂w∂ν|2h · ν ≤ 0.

• Step 2. With BΣ(w) ≤ 0 and f = 0, from Carleman estimate (4), one has

0 ≥ mτe−δτ

∫ t1

t0

E(t)dt− Cτe−2δτ [E(T ) + E(0)]

i.e.

∫ t1

t0

E(t)dt ≤Cτe−2δτ [E(T ) + E(0)]

mτe−δτ

Let τ ∞, one has m(τ) ∞ at the rate of τ , we conclude that∫ t1t0E(t)dt = 0 ⇒ w = 0 on (t0, t1)×M .

• Step 3. Extend (t0, t1) → [0, T ]: replace (t0, t1) by a large time intervalwhere φ(t, x) ≥ σ > −δ uniformly in M , with σ → −δ. ⇒ w = 0 on (0, T )×M ;

w = 0 on [0, T ]×M from w ∈ C([0, T ];L2(M)), a-fortiori from w ∈ H2,2(Q).

Carleman estimates: second version

• Assume P (t, x) is purely imaginary (as in the case of magnetic potential).(while dim = 1, no need to assume P (t, x) purely imaginary)

• Energy method: multiply (1) by i[∆w−w], take real parts, ⇒ |E(t)−E(s)| ≤G(T ) + cT

∫ t

sE(σ)dσ. where G(T ) = C||f ||2

L2(0,T ;H1(M)) + boundary terms.

• Apply Gronwall inequality ⇒ E(t) ≥ E(T )+E(0)2

e−cTT −G(T ); 0 ≤ t ≤ T.

First version Carleman estimates⇒second version Carleman estimates:

BΣ(w) + (1 +1

ε)

∫Q

e2τφ|f |2dxdt + C||f ||2L2(0,T ;H1(M))

≥

mτe−δτ t1 − t0

2e−cTT − cτe−2δτ

[E(T ) + E(0)]

≥ kφ,τ [E(T ) + E(0)]. (5)

where the new boundary term:

BΣ(w) = BΣ(w) + boundary terms from Gronwall inequality.

Extension of Carleman estimates to finite energy solutions

Approximation by smooth solutions to extend all previous estimates fromH2,2(Q) solutions to finite energy solutions in the the class

w ∈ C(0, T ;H1(M)), wt,∂w

∂ν∈ L2(0, T ;L2(Γ)).

I. Purely Dirichlet case: ∂w∂ν∈ L2(Γ) for w0 ∈ H1

0(M) from the optimal regularityresult [Lasiecka-Triggiani ’91]

II. Purely Neumann case:

• Difficult: no H1-traces on boundary for finite energy solutions

• Strategy: regularization procedure as [First by Lasiecka-Tataru ’93, Lasiecka-Triggiani-Zhang ’00](2nd order hyperbolic equations) + an unbounded per-turbation of the basic generator on the state space H1(M).

Continuous observability for Dirichlet B.C.

Consider the purely Dirichlet B.C. problem: iwt + ∆gw = F (w) + f, in Q,w(0, x) = w0(x), in M,w|Σ = 0, in Σ = [0, T ]× Γ.

(6)

Theorem:[Triggiani-Xu ’07] Let w be a solution of (6) with I.C. w0 ∈ H10(M)

and with f ∈ L2(0, T ;H1(M)), under assumptions on F (w) and geometricalconditions on d(x) (here only need ∇d · ν ≤ 0, on Γ0). Then there exists aconstant CT > 0, the continuous observability inequality is true:

CTE(0) ≤∫ T

0

∫Γ1

|∂w

∂ν|2dΓ1dt + ||f ||2L2(0,T ;H1(M)).

where CT has explicit formula, useful for nonlinear problems. And CT is oforder Ce−CL2

, where L is the appropriate norm of the coefficients P (t, x) andp0(t, x), for n ≥ 3, one has

L = |p0|L∞(M) + |p0|L1(0,T ;W 1,n(M)) + |P |L∞(0,T ;W 1,∞(Mn)). (7)

and analogously for n = 1,2.

Continuous observability for Neumann B.C.

Consider the purely Neumann B.C. problem: iwt + ∆gw = F (w) + f, in Q,w(0, x) = w0(x), in M,∂w∂ν|Σ = 0, in Σ = [0, T ]× Γ.

(8)

Theorem:[Triggiani-Xu ’07] Let w be a solution of (8) with I.C. w0 ∈ H1(M)and with f ∈ L2(0, T ;H1(M)), under assumptions on F (w) and geometricalconditions on d(x). Then there exists a constant CT > 0, the continuousobservability inequality is true:

CTE(0) ≤∫ T

0

∫Γ1

[|w|2 + |wt|2]dΓ1dt + ||f ||2L2(0,T ;H1(M)).

where CT has explicit formula as the purely Dirichlet B.C. case (7).

The Euler-Bernoulli Equation

Consider the following Euler-Bernoulli plate problem:

wtt + ∆2gw = F (w,∇w,∆w) + f, in Q = (0, T ]×M,

w(0, x) = w0(x), wt(t, x) = w1(x), in M,w|Σ = 0, ∆w|Σ = 0 in Σ = [0, T ]× Γ.

(9)

Writing the E-B equation as an iteration of two Schrodinger equations:

wtt + ∆2w = (∆ + i∂t)(∆− i∂t)w.

Setting v = iwt −∆w, rewrite problem (9) as

ivt + ∆v = F + f, in Q = (0, T ]×M,v(0, x) = iw1(x)−∆w0(x), in M,v|Σ = 0, in Σ = [0, T ]× Γ.

(10)

Setting Ew(t) =∫

M[|∇∆w(t)|2 + |∇wt(t)|2]dx = Ev(t) =

∫M|∇v(t)|2dx.

Theorem:[Triggiani-Xu ’07] Let w be the solution of problem (9) withw0, w1 ∈ H3(M)×H1(M). Let f ∈ L2(0, T ;H1(M)). Let d(x) be the strictlyconvex function satisfying above geometric assumptions. Then there exists aconstant CT > 0, the continuous observability inequality holds true:

CTEw(0) ≤∫ T

0

∫Γ1

[(∂∆w

∂ν

)2

+

(∂wt

∂ν

)2]dΓ1dt + ||f ||2L2(0,T ;H1(M)).

where CT has explicit formula as the purely Dirichlet B.C. case (7).

Thanks!