Outline Why viscoelastic materials? The viscoelasticity model The - - PowerPoint PPT Presentation

Viscoelasticity with moving controls1

• E. Zuazua

BCAM-Ikerbasque & BCAM & CIMI - Toulouse

CIMI, Toulouse, March 2014

1Joint work in collaboration with F. Chaves, L. Rosier and X. Zhang

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 1 / 35

Outline

Why viscoelastic materials? The viscoelasticity model The null controllability problem A particular case Observability inequality Final comments

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 2 / 35

Why viscoelastic materials2?

Viscoelastic materials are those for which the behavior combines liquid-like and solid-like characteristics. Viscoelasticity is important in areas such as biomechanics, power industry or heavy construction: Synthetic polymers; Wood; Human tissue, cartilage; Metals at high temperature; Concrete, bitumen; ...

2See H. T. Banks, S. Hu and Z. R. Kenz, A Brief Review of Elasticity and

Viscoelasticity for Solids, Adv. Appl. Math. Mech., Vol. 3, No. 1, 1-51.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 3 / 35

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 4 / 35

Viscoelasticity

A wave equation with both viscous Kelvin-Voigt and frictional damping: ytt − ∆y − ∆yt + b(x)yt = 1ωh, x ∈ Ω, t ∈ (0, T), (1) y = 0, x ∈ ∂Ω, t ∈ (0, T), (2) y(x, 0) = y0(x), yt(x, 0) = y1(x) x ∈ Ω. (3) Here, Ω is a smooth, bounded open set in RN, b ∈ L∞(Ω) is a given function determining the frictional damping and h = h(x, t) is a control located in a open subset ω of Ω. We want to study the following problem: Given (y0, y1). Find a control h such that the associated solution to (1)-(3) satisfies y(T) = yt(T) = 0.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 5 / 35

A geometric obstruction

Standard results on unique continuation do not apply. The principal part of the operator is ∂t∆. Then characteristic hyperplanes are of the form t = t0 and x · e = 1. Vertical hyperplanes make it impossible to prove unique continuation from ω × (0, T) towards the whole domain Ω, even in the context of constant

• coefficients. Holmgren’s uniqueness Theorem cannot be applied.

This phenomenon was previously observed by S. Micu in the context of the Benjamin-Bona-Mahoni equation 3 4 In that context the underlying operator is ∂t − ∂3

xxt

but its principal part is the same ∂3

xxt.

• 3S. Micu, SIAM J. Control Optim., 39(2001), 1677–1696.
• 4X. Zhang and E. Z. Matematische Annalen, 325 (2003), 543-582.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 6 / 35

Viscoelasticity = Waves + Heat

ytt − ∆y − ∆yt = 0 = ytt − ∆y = 0 + ∂t[yt] − ∆yt = 0 Both equations are controllable. Should then the superposition be controllable as well?

Interesting open question: The role of splitting and alternating directions in the controllability of PDE.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 7 / 35

Viscoelasticity = Heat + ODE

yt − ∆y + (b(x) − 1)y = z, (4) zt + z = 1ωh + (b(x) − 1)y, (5) y(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), (6) z(x, 0) = z0(x), x ∈ Ω, (7) y(x, 0) = y0(x), x ∈ Ω. (8) The question now becomes: Given (y0, z0). Find a control h such that the associated solution to (9)-(13) satisfies y(T) = z(T) = 0.

In this form the controllability of the system is less clear. We are acting on the ODE variable z. But the control action does not allow to control the whole z. We are effectively acting on y through z. What is the overall impact of the control?

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 8 / 35

Viscoelasticity = Heat + ODE

yt − ∆y + (b(x) − 1)y = z, (4) zt + z = 1ωh + (b(x) − 1)y, (5) y(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), (6) z(x, 0) = z0(x), x ∈ Ω, (7) y(x, 0) = y0(x), x ∈ Ω. (8) The question now becomes: Given (y0, z0). Find a control h such that the associated solution to (9)-(13) satisfies y(T) = z(T) = 0.

In this form the controllability of the system is less clear. We are acting on the ODE variable z. But the control action does not allow to control the whole z. We are effectively acting on y through z. What is the overall impact of the control?

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 8 / 35

Viscoelasticity = Heat + ODE. Second version

Note that ytt − ∆y − ∆yt + yt = (∂t − ∆)(∂t + I). Then yt + y = v, (9) vt − ∆v = 1ωh + (1 − b(x))(v − y), (10) v(x, t) = y(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), (11) v(x, 0) = y1(x) + y0(x), x ∈ Ω, (12) y(x, 0) = y0(x), x ∈ Ω. (13) The question now becomes: Given (y0, z0). Find a control h such that the associated solution to (9)-(13) satisfies y(T) = v(T) = 0.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 9 / 35

Viscoelasticity = Heat + Memory

Note that ytt − ∆y − ∆yt = ∂t[yt − ∆y − ∆ t y]. The later, heat with memory, was addressed by Gurrero and Imanuvilov5, showing that the system is not null controllable.

• 5S. Guerrero, O. Yu. Imanuvilov, Remarks on non controllability of the heat

equation with memory, ESAIM: COCV, 19 (1)(2013), 288–300.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 10 / 35

The case b ≡ 1

When b ≡ 1, the system reads: vt − ∆v = 1ωh, yt + y = v. (14) Its controllability is unclear in this form. But we can consider the system with an added ficticious control: vt − ∆v = 1ωh, yt + y = v + 1ωk. (15) Control in two steps: Use the control h to control v to zero in time T/2. Then use the control k to control the ODE dynamics in the time-interval [T/2, T].

• Warning. The second step cannot be fulfilled since the ODE does not

propagate the action of the controller which is confined in ω. Possible solution: Make the control in the second equation move or, equivalently, replace the ODE by a transport equation.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 11 / 35

This strategy was introduced and found to be successful in

• P. Martin, L. Rosier, P. Rouchon, Null Controllability of the Structurally Damped

Wave Equation with Moving Control, SIAM J. Control Optim., 51 (1)(2013), 660–684.

• L. Rosier, B.-Y. Zhang, Unique continuation property and control for the

Benjamin-Bona-Mahony equation on a periodic domain, J. Differential Equations 254 (2013), 141-178. by using Fourier series decomposition. In the context of the example under consideration, if we make the control set ω move to ω(t) with a velocity field a(t), then the ODE becomes: yt + a(t) · ∇y = 1ωk. And it is sufficient that all characteristic lines pass by ω to ensure controllability

• r, in other words, that the set ω(t) covers the whole domain Ω in its motion.

Question: How to prove this kind of result in a more general setting where b = 1 so that the system does not decouple?

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 12 / 35

An example of moving support of the control

Ω 0 ≤ t < t1 t2 < t ≤ T Ω1(t) X(ω0, t, 0) X(ω0, t, 0) X(ω0, t, 0) Γ(t) t1 < t < t2 Γ(t) Γ(t) Ω2(t) Ω1(t) Ω2(t)

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 13 / 35

Other related systems

This issue of moving control is closely related to the works by J. M. Coron, S. Guerrero and G. Lebeau67 on the vanishing viscosity limit for the control of convection-diffusion equations. It is also linked to the recent work by S. Ervedoza,

• O. Glass, S. Guerrero & J.-P. Puel 8 on the control of 1 − d compressible

Navier-Stokes equations.

6J.-M. Coron and S. Guerrero, A singular optimal control: A linear 1-D parabolic

hyperbolic example, Asymp. Analisys, 44 (2005), pp. 237-257.

• 7S. Guerrero and G. Lebeau, Singular Optimal Control for a transport-diffusion

equation, Comm. Partial Differential Equations, 32 (2007), 1813-1836.

• 8S. Ervedoza, O. Glass, S. Guerrero, J.-P. Puel, Local exact controllability for the

1-D compressible Navier- Stokes equation, Archive for Rational Mechanics and Analysis, 206 (1)(2012), 189-238.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 14 / 35

Moving controls where also used successfully in various other contexts:

• A. Khapalov, Controllability of the wave equation with a moving point

control, Appl. Math. Optim., 31 (1995), pp. 155175.

• X. Zhang, Rapid exact controllability of the semi linear wave equation, Chin.
• Ann. of Math. 20B: 3 (1999), 377-384.
• C. Castro and E. Z Unique continuation and control for the heat equation

from an oscillating lower dimensional manifold. SIAM J. Cont. Optim., 43 (4) (2005), 1400-1434.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 15 / 35

Observability

We consider the dual problem of (16)-(20): −pt − ∆p + (b(x) − 1)p = (b(x) − 1)q, (x, t) ∈ Ω × (0, T), (16) −qt + q = p, (x, t) ∈ Ω × (0, T), (17) p(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), (18) p(x, T) = p0(x), x ∈ Ω, (19) q(x, T) = q0(x), x ∈ Ω. (20) The null controllability property i equivalent to the following observability one ||p(0)||2 + ||q(0)||2 ≤ C T

• ω

|q|2dxdt, (21) for all solutions of (16)-(20). But the structure of the underlying PDE operator and, in particular, the existence

• f time-like characteristic hyperplanes, makes impossible the propagation of

information in the space-like directions, thus making the observability inequality (21) also impossible.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 16 / 35

Lack of observability for b ≡ 1

−pt − ∆p = 0 , (x, t) ∈ Ω × (0, T), (22) −qt + q = p, (x, t) ∈ Ω × (0, T), (23) It is impossible that ||p(0)||2 + ||q(0)||2 ≤ C T

• ω

|q|2dxdt, (24)

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 17 / 35

Those negative results are well-known in a number of other models: Benjamin-Bona-Mahoni (S. Micu, X. Zhang & E. Z.); Heat equations with memory (closely related to the coupled systems under consideration ”heat + ODE”) ( S. Guerrero & O. Yu. Imanuvilov) In both cases the controllability fails because of the presence of accumulation points in the spectrum. A similar situation can be encountered in:

• F. Ammar Khodja, K. Mauffrey and A. M¨

unch, Exact boundary controllability of a system of mixed order with essential spectrum, , SIAM J. Cont. Optim. 49 (4) (2011), 18571879. In the context of the system of viscoelasticity under consideration the accumulation point in the spectrum is due to the ODE component of the system. In the BBM case is due to the compactness of the generator of the dynamics.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 18 / 35

Remedy: Moving control

Let us assume that ω ≡ ω(t). The controllable system under consideration then reads: yt − ∆y + (b(x) − 1)y = z, (25) zt + z = 1ω(t)h + (b(x) − 1)y, (26) y(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), (27) z(x, 0) = z0(x), x ∈ Ω, (28) y(x, 0) = y0(x), x ∈ Ω. (29)

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 19 / 35

Motion of the support of the control

In practice, the trajectory of the control can be taken to be determined by the flow X(x, t, t0) generated by some vector field f ∈ C([0, T]; W 2,∞(RN; RN)), i.e. X solves    ∂X ∂t (x, t, t0) = f (X(x, t, t0), t), X(x, t0, t0) = x. (30) Admissible trajectories: There exist a bounded, smooth, open set ω0 ⊂ RN, a curve Γ ∈ C ∞([0, T]; RN), and two times t1, t2 with 0 ≤ t1 < t2 ≤ T such that: Γ(t) ∈ X(ω0, t, 0) ∩ Ω, ∀t ∈ [0, T]; (31) Ω ⊂ ∪t∈[0,T]X(ω0, t, 0) = {X(x, t, 0); x ∈ ω0, t ∈ [0, T]}; (32) Ω \ X(ω0, t, 0) is nonempty and connected for t ∈ [0, t1] ∪ [t2, T]; (33) Ω \ X(ω0, t, 0) has two connected components for t ∈ (t1, t2); (34) ∀γ ∈ C([0, T]; Ω), ∃t ∈ [0, T], γ(t) ∈ X(ω0, t, 0). (35)

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 20 / 35

Motion of the support of the control

In practice, the trajectory of the control can be taken to be determined by the flow X(x, t, t0) generated by some vector field f ∈ C([0, T]; W 2,∞(RN; RN)), i.e. X solves    ∂X ∂t (x, t, t0) = f (X(x, t, t0), t), X(x, t0, t0) = x. (30) Admissible trajectories: There exist a bounded, smooth, open set ω0 ⊂ RN, a curve Γ ∈ C ∞([0, T]; RN), and two times t1, t2 with 0 ≤ t1 < t2 ≤ T such that: Γ(t) ∈ X(ω0, t, 0) ∩ Ω, ∀t ∈ [0, T]; (31) Ω ⊂ ∪t∈[0,T]X(ω0, t, 0) = {X(x, t, 0); x ∈ ω0, t ∈ [0, T]}; (32) Ω \ X(ω0, t, 0) is nonempty and connected for t ∈ [0, t1] ∪ [t2, T]; (33) Ω \ X(ω0, t, 0) has two connected components for t ∈ (t1, t2); (34) ∀γ ∈ C([0, T]; Ω), ∃t ∈ [0, T], γ(t) ∈ X(ω0, t, 0). (35)

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 20 / 35

A failing moving support

X(ω0, T, 0) Ω X(ω0, t, 0) ω0

Figure: Example for which condition (34) fails.

Remark: Note that it would be OK for b ≡ 1.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 21 / 35

A successful motion

Ω 0 ≤ t < t1 t2 < t ≤ T Ω1(t) X(ω0, t, 0) X(ω0, t, 0) X(ω0, t, 0) Γ(t) t1 < t < t2 Γ(t) Γ(t) Ω2(t) Ω1(t) Ω2(t)

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 22 / 35

Observability inequality

The system is null controllable under the assumptions above on the moving

• support. But the proof cannot be done as when b ≡ 1 by decoupling. One rather

needs to employ Carleman inequalities to prove the observability one.

Proposition

Let T, X, ω0 and ω be as above. Then there exists a constant C > 0 such that for all (p0, q0) ∈ L2(Ω)2, the solution (p, q) of (16)-(20) satisfies

• Ω

[|p(x, 0)|2 + |q(x, 0)|2]dx ≤ C T

• ω(t)

|q(x, t)|2 dxdt. (36)

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 23 / 35

Proof of the observability inequality

Our strategy to prove (36) is based on the use of Carleman inequalities for the heat and the ODE. Two main difficulties appear:

1

Carleman inequalities for heat and ODE equations with a moving control region;

2

We must have the same weight functions in the Carleman for both equations9. Fortunately, we can handle both difficulties. Note that similar strategies were implemented successfully for the system of thermoelasticity in

• P. Albano, D. Tataru, Carleman estimates and boundary observability for a coupled

parabolic-hyperbolic system, Electron. J. Differential Equations, 22 (2000), 1–15.

9see [1], [4] for the case of a heat and wave equation

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 24 / 35

Lemma

There exist some constants λ0 > 0, s0 > 0 and C0 > 0 such that for all λ ≥ λ0, all s ≥ s0 and all p ∈ C([0, T]; L2(Ω)) with pt + ∆p ∈ L2(0, T; L2(Ω)), the following holds T

• Ω

[(sθ)−1(|∆p|2 + |pt|2) + λ2(sθ)|∇p|2 + λ4(sθ)3|p|2]e−2sϕdxdt ≤ C0 T

• Ω

|pt + ∆p|2e−2sϕdxdt + T

• ω1(t)

λ4(sθ)3|p|2e−2sϕdxdt

• ,

(37) for all ω0 ⊂ ω1.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 25 / 35

Lemma

There exist some numbers λ1 ≥ λ0, s1 ≥ s0 and C1 > 0 such that for all λ ≥ λ1, all s ≥ s1 and all q ∈ H1(0, T; L2(Ω)), the following holds T

• Ω

(λ2sθ)|q|2e−2sϕdxdt ≤ C1 T

• Ω

|qt|2e−2sϕdxdt + T

• ω(t)

λ2(sθ)2|q|2e−2sϕdxdt

• ,

(38) for all ω0 ⊂ ω.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 26 / 35

In the previous Lemmas, the weights have the form: ϕ(x, t) = g(t)(e

3 2 λ||ψ||L∞ − eλψ(x,t)) ∼ (e 3 2 λ||ψ||L∞ − eλψ(x,t))

t(T − t) , (39) θ(x, t) = g(t)eλψ(x,t) ∼ eλψ(x,t) t(T − t), (40) where ψ ∈ C ∞(Ω × [0, T]) is a weight having the following properties:

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 27 / 35

There exist a number δ ∈ (0, T/2) such that ∇ψ(x, t) = 0, t ∈ [0, T], x ∈ Ω \ X(ω1, t, 0), (41) ψt(x, t) = 0, t ∈ [0, T], x ∈ Ω \ X(ω1, t, 0), (42) ψt(x, t) > 0, t ∈ [0, δ], x ∈ Ω \ X(ω1, t, 0), (43) ψt(x, t) < 0, t ∈ [T − δ, T], x ∈ Ω \ X(ω1, t, 0), (44) ∂ψ ∂n (x, t) ≤ 0, t ∈ [0, T], x ∈ ∂Ω, (45) ψ(x, t) > 3 4||ψ||L∞(Ω×(0,T)), t ∈ [0, T], x ∈ Ω. (46) for all ω0 ⊂ ω1, ω1 ⊂ ω. Remark: Basically, ψ drags the critical points of ψ(x, 0) inside the control region during the evolution of the flow.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 28 / 35

There exist a number δ ∈ (0, T/2) such that ∇ψ(x, t) = 0, t ∈ [0, T], x ∈ Ω \ X(ω1, t, 0), (41) ψt(x, t) = 0, t ∈ [0, T], x ∈ Ω \ X(ω1, t, 0), (42) ψt(x, t) > 0, t ∈ [0, δ], x ∈ Ω \ X(ω1, t, 0), (43) ψt(x, t) < 0, t ∈ [T − δ, T], x ∈ Ω \ X(ω1, t, 0), (44) ∂ψ ∂n (x, t) ≤ 0, t ∈ [0, T], x ∈ ∂Ω, (45) ψ(x, t) > 3 4||ψ||L∞(Ω×(0,T)), t ∈ [0, T], x ∈ Ω. (46) for all ω0 ⊂ ω1, ω1 ⊂ ω. Remark: Basically, ψ drags the critical points of ψ(x, 0) inside the control region during the evolution of the flow.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 28 / 35

Obstruction for the weight function

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 29 / 35

Proof of Proposition 2.1

The proof of Proposition 2.1 can now be performed as follows: Step 1. We apply the Carleman estimate (37) for the parabolic equation (16) and the Carleman estimate (38) for the ODE (17). Adding both inequalities we get, after absorbing the lower order terms, a global estimation of p and q in terms of “local” integrals of p and q. Step 2. We estimate the local integral of p in terms of a local integral of q and some small order terms. Finally, we combine all the estimates and derive global estimation of p and q in terms of a local integral of q. Finally, the observability inequality (36) follows from classical semigroup estimates.

Remark

Observe that from steps 1 and 2, we control (y, z) solution of (9)-(13) at the same time. That is, we control the parabolic and ODE part of (1)-(3) at the same time and with the same argument, i.e., Carleman inequality for both equations.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 30 / 35

As a consequence, we have the null controllability of (1)-(3):

Theorem

Let T > 0, X(x, t, t0) and ω0 be as in (31)-(35), and let ω be any open set in Ω such that ω0 ⊂ ω. Then for all (y0, y1) ∈ L2(Ω)2 with y1 − ∆y0 ∈ L2(Ω), there exists a function h ∈ L2(0, T; L2(Ω)) for which the solution of ytt − ∆y − ∆yt + b(x)yt = 1ω(t)(x)h, (x, t) ∈ Ω × (0, T), (47) y(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), (48) y(., 0) = y0, yt(., 0) = y1, (49) fulfills y(., T) = yt(., T) = 0.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 31 / 35

Final comments

Can the technical geometric assumptions on the moving control be removed? Can one derive similar results by simply assuming that the support of the control covers the whole domain? To which extent this methodology can be applied in problems where there are vertical characteristic hyperplanes (BBM, heat with memory,...)? Other models with memory. Nonlinear versions.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 32 / 35

Thank you!

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 33 / 35

References

• P. Albano, D. Tataru, Carleman estimates and boundary observability for a

coupled parabolic-hyperbolic system, Electron. J. Differential Equations, 22 (2000), 1–15.

• C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the
• bservation, control and stabilization of waves from the boundary , SIAM J.
• Cont. Optim., 30 (1992), 1024–1065.

J.-M. Coron and S. Guerrero, A singular optimal control: A linear 1-D parabolic hyperbolic example, Asymp. Analisys, 44 (2005), pp. 237-257.

• S. Ervedoza, O. Glass, S. Guerrero, J.-P. Puel, Local exact controllability for

the 1-D compressible Navier-Stokes equation, Archive for Rational Mechanics and Analysis, 206 (1)(2012), 189-238.

• S. Ervedoza, E. Zuazua, A systematic method for building smooth controls

for smooth data, Discrete and Continuous Dynamical Systems Series B, 14 (4)(2010), 1375–1401.

• S. Guerrero, O. Yu. Imanuvilov, Remarks on non controllability of the heat

equation with memory, ESAIM: COCV, 19 (1)(2013), 288–300.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 34 / 35

• H. T. Banks, S. Hu and Z. R. Kenz, A Brief Review of Elasticity and

Viscoelasticity for Solids, Adv. Appl. Math. Mech., Vol. 3, No. 1, 1-51.

• S. Guerrero and G. Lebeau, Singular Optimal Control for a transport-diffusion

equation, Comm. Partial Differential Equations, 32 (2007), 1813-1836.

• A. Khapalov, Controllability of the wave equation with moving point control,
• Appl. Math. Optim., 31 (2)(1995), 155–175.
• G. Lebeau, E. Zuazua, Null controllability of a system of linear
• thermoelasticity. Archives for Rational Mechanics and Analysis, 141(4)(1998),

297-329.

• L. Rosier, B.-Y. Zhang, Unique continuation property and control for the

Benjamin-Bona-Mahony equation on a periodic domain, J. Differential Equations 254 (2013), 141–178.

• P. Martin, L. Rosier, P. Rouchon, Null Controllability of the Structurally

Damped Wave Equation with Moving Control, SIAM J. Control Optim., 51 (1)(2013), 660–684.

• L. Rosier, B.-Y. Zhang, Unique continuation property and control for the

Benjamin-Bona-Mahony equation on a periodic domain, J. Differential Equations 254 (2013), 141–178.

(BCAM & Ikerbasque) Viscoelasticity with moving controls CIMI, Toulouse, March 2014 35 / 35