pointwise carleman estimates and control theoretic...
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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
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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.
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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.
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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.)
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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.
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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))
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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.
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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]
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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.
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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.
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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.
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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).
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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.
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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.
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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)].
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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 .
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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).
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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.
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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).
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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.
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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).
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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.
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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).
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Thanks!
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