Averaged control Enrique Zuazua 1 Ikerbasque & BCAM & CIMI - - - PowerPoint PPT Presentation

Averaged control

Enrique Zuazua1

Ikerbasque & BCAM & CIMI - Toulouse enrique.zuazua@gmail.com enzuazua.net CIMI-Toulouse, April 2014

April 29, 2014

1Funded by the ERC Advanced Grant NUMERIWAVES

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Motivation

Motivation Often parameters of the control system under consideration are not fully known. It is then natural to look for robust control strategies, independent of the unknown parameters, and performing, overall, optimally. We introduce the notion of “averaged control” that consists, simply,

• n controlling the average of solutions with respect to the unknown

parameters. As we shall see, this leads to interesting new problems in the frame of

• bservability of parameter dependent systems.

The problems are linked, but different, to those arising in the context

• f simultaneous observation and control.
• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged control

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 4 / 50

jueves, 17 de julio de 14

Finite-dimensional systems Averaged control

Consider the finite dimensional linear control system ⇢ x0(t) = A(ν)x(t) + Bu(t), 0 < t < T, x(0) = x0. (1) In (1) the (column) vector valued function x(t, ν) =

• x1(t, ν), . . . , xN(t, ν)
• ∈ RN is the state of the system, A(ν) is

a N × N−matrix and u = u(t) is a M-component control vector in RM, M ≤ N. The matrix A is assumed to depend on a parameter ν in a continuous

• manner. To fix ideas we will assume that the parameter ν ranges

within the interval (0, 1). Note however that, to simplify the presentation, the control operator B has been taken to be independent of ν, the same as the initial datum x0 ∈ RN to be controlled. But the same analysis applies when both of them depend on ν.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged control

Given a control time T > 0 and a final target x1 ∈ RN we look for a control u such that the solution of (1) satisfies Z 1 x(T, ν)dν = x1. (2) This concept of averaged controllability differs from that of simultaneous controllability in which one is interested on controlling all states simultaneously and not only its average. When A is independent of the parameter ν, controllable systems can be fully characterized in algebraic terms by the rank condition rank h B, AB, . . . , AN−1B i = N. (3)

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged control

The following holds: Theorem Averaged controllability holds if and only the following rank condition is satisfied: rank h B, Z 1 [A(ν)]dνB, Z 1 [A(ν)]2dνB, . . . i = N. (4)

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged observation

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged observation

Let us now characterize the property of averaged control in terms of the

• bservability of the adjoint system.

The adjoint system depends also on the parameter ν: ⇢ −ϕ0(t) = A⇤(ν)ϕ(t), t ∈ (0, T) ϕ(T) = ϕ0. (5) Note that, for all values of the parameter ν, we take the same datum for ϕ at t = T. This is so because our analysis is limited to the problem of averaged controllability. The corresponding averaged observability property reads: |ϕ0|2 ≤ C Z T

• B⇤

Z 1 ϕ(t, ν)dν

• 2

dt. (6)

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged observation

The reason is the following duality identity: < Z x(T, ν)dν, ϕ0 > − < x0, Z ϕ(0, ν)dν >= Z T < u(t), B∗ Z ϕ(t, ν)dν In fact, once the averaged observability inequality above is satisfied, the control of minimal L2(0, T)-norm can be built by minimizing the quadratic functional below within the class of solutions of the parameter-dependent adjoint system, i.e. minimizing J

• ϕ0

= 1

2

R T

• R 1

0 B∗(ν)ϕ(t, ν)dν

• 2

dt − < x1, ϕ0 > + < x0, R 1

0 ϕ(0, ν)dν >

(7) in RN. The functional is continuous and convex, and its coercivity is guaranteed by the averaged observability inequality. The control is then u(t) = B∗ Z ˜ ϕ(t, ν)dν, where ˜ ϕ is the solution of the adjoint system associated to the minimizer ˜ ϕ0 of the functional J.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged observation

Since we are working in the finite-dimensional context, the observability inequality (6) is equivalent to the following uniqueness property: B∗ Z 1 ϕ(t, ν)dν = 0 ∀t ∈ [0, T] ⇒ ϕ0 ≡ 0. (8) To analyze this inequality we use the following representation of the adjoint state: ϕ(t, ν) = exp[A∗(ν)(T − t)]ϕ0. Then, the fact that B∗ Z 1 ϕ(t, ν)dν = 0 ∀t ∈ [0, T] is equivalent to B∗ Z 1 exp[A∗(ν)(t − T)]dν ϕ0 = 0 ∀t ∈ [0, T]. The result follows using the time analyticity of the matrix exponentials, and the classical argument consisting in taking consecutive derivatives at time t = T.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Averaged observation

B∗ Z 1 exp[A∗(ν)(t − T)]dν ϕ0 ≡ 0 ∼ B∗ Z 1 [A∗(ν)]kdν ϕ0 = 0, ∀k ≥ 1. Contrarily to the classical rank condition for the controllability of a given system, in the present context of averaged control, all moments, of an arbitrarily high order, need to be taken into account in the characterization. For instance, if the dependence of A(ν) with respect to ν is odd, we see that all the terms involving an odd power vanish, and only the even ones remain...

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Comparison with simultaneous controllability

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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jueves, 17 de julio de 14

Finite-dimensional systems Comparison with simultaneous controllability

There is an extensive literature on the simultaneous control of systems, mainly in the PDE setting. The question is then whether one can control different dynamics by means of the same control function. In the context we are working this would consist on considering the state equation ⇢ x0(t) = A(ν)x(t) + Bu(t), 0 < t < T, x(0) = x0(ν), (9) with initial data depending on ν, and then looking for a control u = u(t), such that the solution x(t, ν) satisfies x(T, ν) = x1, ∀ν. This, of course, requires much stronger observability inequalities as well.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Comparison with simultaneous controllability

To better clarify the situation we consider the case where ν takes two possible values, so that the system only has two possible modes, governed by the operators A1 and A2. The parameter dependent system then reads: ⇢ x0

j(t) = Ajxj(t) + Bu(t), 0 < t < T,

xj(0) = x0

j , j = 1, 2.

(10) Similarly, the adjoint system can be written as: ⇢ −ϕ0

j(t) = A⇤ j ϕj(t),

t ∈ (0, T) ϕj(T) = ϕ0

j , j = 1, 2.

(11) The simultaneous observability inequality reads |ϕ0

1|2 + |ϕ0 2|2 ≤ C

Z T |B⇤[ϕ1 + ϕ2]|2 dt, ∀ϕ0

j ∈ RN, j = 1, 2.

(12) For averaged controllability it is sufficient this to hold in the particular case where ϕ0

1 = ϕ0 2.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Comparison with simultaneous controllability

Summarizing Averaged control:

We control the average of the states. The observation is required only for the adjoint states such that each component departs from the same datum. Minimization is done over that subclass of solutions of the adjoint system.

Simultaneous control:

We control each component of the state. The observation is required for all adjoint states. Minimization is done over the whole class of solutions of the adjoint system.

One can connect one problem to the other by means of a continuation/penalization procedure. For instance, averaged control refers to the control of x1(T) + x2(T), letting x1(T) − x2(T) free. One can link this property to the more demanding one of simultaneous control by means of a penalization of the form |x1(T) − x2(T)| ≤ k, with k ranging in [0, ∞).

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Finite-dimensional systems Limitations

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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jueves, 17 de julio de 14

Simulations developed by J. Loheac

jueves, 17 de julio de 14

Finite-dimensional systems Limitations

Limitations Note that this notion of average control, which only ensures the control of part of the state, if we consider it as being the parameter-dependent family {xν}, is insufficient in the sense that: Even if the average is under control at time t = T, the property is lost for t ≥ T as soon as we stop controlling. This does not allow to handle LQR problems in infinite horizon. Average control is only one of the possibilities to deal with a more general problem of robust control for parameter-dependent systems. Following the lectures delivered by A. Cohen, for instance, one could look also at efficient methods for approximation. What would be the most representative choices of the parameter ν so to represent the control theoretical properties of the parameter-dependent system?

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Finite averages of wave equations

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 19 / 50

jueves, 17 de julio de 14

Averaged control of PDEs : The finite case Finite averages of wave equations

The same analysis can be developed for systems of wave equations depending on parameters2 ∂ttui (ci(t, x)rui) = 0, ; uj|Σ = 0, ; ui(0, ·) = γ, ∂tui(0, ·) = β, i = 1, 2 . We assume the coefficients to be C 1,1 so that the bicharacteristic rays are well-defined. Our analysis, based on previous works on the propagation of microlocal defect measures, applies, both for manifolds without boundary or for bounded domains with, say, Dirichlet boundary conditions.

• 2M. Lazar and E. Zuazua, Averaged control and observation of

parameter-depending wave equations, preprint, 2013.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Finite averages of wave equations

In what concerns the problem of observability, the question can be formulated as follows: Under which conditions we can get the following observability estimate E(0)  C Z T Z

ω

|ru1 + ru2|2dxdt? where E(0) := ||rxγ||2

L2 + ||β||2 L2.

Note that the main difference with respect to previous works on simultaneous control is that the initial data of both wave equations are taken to be the same.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Finite averages of wave equations

Theorem (M. Lazar & E. Z.) Assume (0T) ⇥ ω ✓ R ⇥ Ω satisfies GCC for the Wave Equation # 1. c1(t, x) > c2(t, x) > 0, (t, x) 2 (0, T) ⇥ ω. Then there exists a constant C such that E(0)  C Z T Z

ω

|ru1 + ru2|2dxdt. The time of control is that of the fast velocity of propagation.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Finite averages of wave equations

The proof Goes by contradiction and uses microlocal defect or H-measures. The different velocities of propagation make the support of the measures to be disjoint. Each measure propagates along the corresponding bicharacteristics. The contradiction is reached as soon as one of the measures gets to the initial time.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Finite averages of wave equations

Other tools The proof does not seem to be easy to be performed using other tools. For instance: Multipliers do not seem to work, due to the difficulties of dealing with the cross terms. Ingham like inequalities, even in 1-d, would lead to longer time intervals for observation. Carleman inequalities would find the same difficulties as multipliers. The classical argument of D. Russell’s about “stabilization implies controllability” does not seem to work since we do not have tools for averaged stabilization...

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Averaged control of PDEs : The finite case Finite averages of wave equations

Some references (non exhaustive...)

• F. Alabau-Boussouira, M. L´

eautaud, Journal de Math´ ematiques Pures et Appliqu´ ees, 99(5) (2013) 544–576.

• N. Burq, Asymptot. Anal., 14(2) (1997) 157–191.
• N. Burq and P. G´

erard, C. R. Acad. Sci. Paris S´

• er. I Math.,

325(7) (1997) 749–752.

• N. Burq and G. Lebeau, Annales Scientifiques de l˜

O´ Ecole Normale Sup´ erieure, 34 (6) (2001), pp. 817 D870.

• B. Dehman, M. L´

eautaud, J. Le Rousseau, Arch. Rational

• Mech. Anal., in print.
• P. G´

erard, Comm. Partial Differential Equations 16 (1991) 1761–1794.

• L. Rosier, L. de Teresa, Comptes Rendus Math´

ematique, 349 (5

• 6) (2011), pp. 291 - 296.
• L. Tartar, Proc. Roy. Soc. Edinburgh. Sect. A 115 (1990)

193–230.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Finite averages of wave equations

Comment on stabilization For one single wave equation it is easy to see that the observability property implies stabilization. The simplest way to see this is a classical argument by A. Haraux. This is not the case for the average because it is not governed by a semigroup.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Additive perturbations of PDE

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 27 / 50

jueves, 17 de julio de 14

Averaged control of PDEs : The finite case Additive perturbations of PDE

Consider the following more complex system u1,t − ∆u1 = 0, (t, x) ∈ R × Ω u2,tt − ∆u2 = 0, (t, x) ∈ R × Ω ui = 0, (t, x) ∈ R × ∂Ω, i = 1, 2 u1(0, ·) = ϕ(·) x ∈ Ω. We assume that the main dynamics, the one we want to observe, is given by u1, governed by the heat equation. The wave solution, u2, is then adding an additive perturbation. Note that no information is given on u2, other than being the solution of the wave equation. In particular, nothing is known on its initial data.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Additive perturbations of PDE

We know that there exists a constant C such that the following estimate holds3 X

k

e−c√λk| ˆ ϕk|2 ≤ C Z T Z

ω

|u1|2dxdt. (13) We claim that X

k

e−c√λk| ˆ ϕk|2 ≤ C Z T Z

ω

|u1 + u2|2dxdt , (14) for all solutions (u1, u2) of the above system.

• 3E. Fern´

andez-Cara and E.Z. The cost of approximate controllability for heat equations: The linear case. Advances in Differential Equations, 5 (4-6) (2000), 465–514.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Additive perturbations of PDE

Proof P1 = ∂t − ∆, P2 = ∂tt − ∆. Observe that v1 = P2(u1 + u2) = P2u1. (15) solves the heat equation with the same Dirichlet boundary conditions: v1,t − ∆v1 = 0, ; v1|Σ = 0, ; v1(0) = [u1,tt − ∆u1](0) = [∆2 − ∆](ϕ). The proof follows, using X

k

e−c√λk| ˆ ϕk|2 ≤ Cs||v1||2

H−s(ω×(0,T)).

(16) and ||v1||H−2(ω×(0,T)) = ||P2(u1)||H−2(ω×(0,T)) = ||P2(u1 + u2)||H−2(ω×(0,T)) ≤ C||u1 + u2||L2(ω×(0,T)).

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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jueves, 17 de julio de 14

Averaged control of PDEs : The finite case Systems of heat equations

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 31 / 50

jueves, 17 de julio de 14

Averaged control of PDEs : The finite case Systems of heat equations

For simplicity we state the result in the case of two equations:4 u1,t ∆u1 = 0, ; u2,t c2∆u2 = 0, ui|Σ = 0, i = 1, 2, u1(0, ·) = ϕ 2 L2(Ω). We underline that no information is provided on the initial datum of u2. Theorem Let ω an open non-empty subset of Ω, c2 > 0, c2 6= 1 and T > 0. Then, ||u1(T)||2

L2(Ω)  C

Z T Z

ω

|u1 + u2|2ddt (17) for all solutions (u1, u2) of the above system.

4To be compared with the existing wide literature on the control of systems

• f heat equations, Stokes equaltions, often with a limited number of

controllers...

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Ingham like inequalities

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 33 / 50

jueves, 17 de julio de 14

Averaged control of PDEs : The finite case Ingham like inequalities

The proof we have applied here is inspired in the techniques we employed in 5 to deal with wave equations on graphs. There the time-periodicity

• f solutions of the 1 − d equation was used to annihilate some of the

components of the system under consideration, constituted by the deformation of each string of the graph. Consider two non-harmonic Fourier series, representing the values at a given point of the solutions of the 1 − d wave and heat equations, respectively: uh(t) = X

k

ake−k2t, uw(t) = X

k

bkeikt, (18) and their superposition u(t) = uh(t) + uw(t) = X

k

[ake−k2t + bkeikt]. (19) We claim that for all T ≥ 2 there exist positive constants c = c(T), C > 0 such that X

k

e−c(T)k2|ak|2 ≤ C Z T |uh(t) + uw(t)|2dt, (20) { } { }

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs : The finite case Ingham like inequalities

Using that uw is time-periodic of time-period 2, we have u(t + 2) − u(t) = uh(t + 2) − uh(t) = X

k

[(e−2k2 − 1)ake−k2t]. Applying well-known inequalities for series of real exponentials 6 we get X

k

|e−2k2−1|2a2

ke−(T−2)k2 ≤ C

Z T−2 |u(t+2)−u(t)|2dt ≤ C Z T u2(t)dt, and this easily leads to the result. Note that, the most striking aspect of the estimate is that, it is totally independent of the Fourier coefficients {bk} governing the wave-like

• perturbation. It would be interesting to make a more systematic analysis
• f these issues for the interaction of more general wave and heat spectraa

aNull Controllability of the Linearized Compressible Navier Stokes System in

• ne dimension. S. Chowdhury, D. Mitra, M.Ramaswamy, M. Renardy.
• 6A. L´
• pez and E. Zuazua, Uniform null controllability for the one

dimensional heat equation with rapidly oscillating periodic density. Annales IHP. Analyse non lin´ eaire, 19 (5) (2002), 543-580.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

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Averaged control of PDEs: Continuous averages Continuous averages of heat equations

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 36 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of heat equations

Motivation So far we have analyzed the property of averaged control in, mainly, two situations: Continuous averages of finite-dimensional systems; Finitely many PDE. What about continuous averages of PDE?

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 37 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of heat equations

Let us consider the heat equation, depending on a diffusivity coefficient σ ranging on a bounded interval 0 < σ1 ≤ σ ≤ σ2 < ∞: ut − σ∆u = 0, (t, x) ∈ R × Ω. All equations are assumed to take u0 ∈ L2(Ω) as initial datum, and the same boundary conditions (Dirichlet ones to fix ideas). Define the average v(x, t) = Z σ2

σ1

u(x, t; σ)dσ. (21) Can we identify the dynamical system governing the evolution of v?

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 38 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of heat equations

Note that u(x, t; σ) = U(x, σt), where U solves the same heat equation with diffusivity σ = 1. Thus, v(x, t) = Z σ2

σ1

u(x, t; σ)dx = Z σ2

σ1

U(x, σt)dσ = 1 t [V (x, σ2t) − V (x, σ1t)], (22) where V (x, t) = Z t U(x, s)ds + ξ(x), −∆ξ = u0, x ∈ Ω; ξ = 0, x ∈ ∂Ω, and Vt − ∆V = 0; V (0) = ξ ∈ L2(Ω).

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 39 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of heat equations

Thus, tv(x, t) = W2(x, t) − W1(x, t) where Wj(x, t) = V (x, σjt) solves Wj,t − σj∆Wj = 0. We can then apply the results above on finite combinations of heat equations with different diffusivities and get X

k

e−c√λk|ˆ u0,k|2 ≤ C Z T Z

ω

|W1 − W2|2ddt = C Z T Z

ω

|tv(x, t)|2ddt = C||t Z σ2

σ1

u(x, t; σ)dσ||2

L2(ω×(0,T)) ≤ CT 2||

Z σ2

σ1

u(x, t; σ)dσ||2

L2(ω×(0,T))

Note that we do not perceive the effect of the weight t that would introduce some polynomial changes on the weights appearing in the Fourier decomposition, which is negligible with respect to the fact that weights degenerate exponentially for heat-like equations.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 40 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of heat equations

Accordingly, the following holds: Theorem Let ω be an open non-empty subset of Ω and T > 0. Assume the unknown diffusivity σ ranges over the set 0 < σ1 ≤ σ ≤ σ2 < ∞. Then, the initial datum can be observed through the averages of solutions with resct to σ in ω × (0, T) on an exponentially weighted Fourier space. Theorem Let ω be an open non-empty subset of Ω and T > 0. Assume the unknown diffusivity σ of the controlled system yt − σ∆y = f 1ω, y|Σ = 0, y(0, ·) = y0 ∈ L2(Ω), (23) ranges over the set 0 < σ1 ≤ σ ≤ σ2 < ∞. Then, for all y0 ∈ L2(Ω) there exists a control f ∈ L2(ω × (0, T)) such that Z σ2

σ1

y(x, T; σ)dσ = 0.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 41 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages An abstract setting

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 42 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages An abstract setting

The same result can be developed in an abstract setting7: ut − σAu = 0, t > 0; , u(0) = u0, (24) where A is the generator of a continuous semigroup in a Hilbert or Banach space. Then u(t, σ) = U(σt), where U(·, σ) solves Ut − AU = 0, t > 0; , U(0) = u0(σ). (25) And Z u(t, σ)dσ = Z U(σt)dσ = 1 t [V (σ2t) − V (σ1t)], where V = V (t) solves the same abstract parabolic equation by with datum ξ = A−1

• 7M. Tucsnak, G. Weiss, Observation and control for operator

semigroups, Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher, Birkh¨ auser Verlag, Basel, 2009.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 43 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of wave equations

Outline

1

Motivation

2

Finite-dimensional systems Averaged control Averaged observation Comparison with simultaneous controllability Limitations

3

Averaged control of PDEs : The finite case Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities

4

Averaged control of PDEs: Continuous averages Continuous averages of heat equations An abstract setting Continuous averages of wave equations

5

Perspectives and open problems

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 44 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of wave equations

Let us now consider the case of wave equations with an unknown velocity

• f propagation parameter 0 < σ1 ≤ σ ≤ σ2 < ∞:

utt − σ2∆u = 0, ; u|Σ = 0 ; u(0) = u0, ut(x, 0) = u1. (26) Consider v(x, t) = Z σ2

σ1

u(x, t; σ)dσ, (27) and note that Uσ(x, t; σ) = u(x, t/σ; σ), solves the same wave equation with σ = 1. Note however that, this time, the initial velocity of U depend on the parameter σ so that U(x, 0; σ) = U0(x); Ut(x, 0; σ) = 1 σu1(x).

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 45 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of wave equations

So, for the argument of the previous section to be applied directly, we rather take the initial velocity in the original equation satisfied by u to be σ u1, so that the corresponding U is independent of σ and satisfies Utt − ∆U = 0 ; U|Σ = 0 ; U(0, ·) = u0, Ut(x, 0) = u1. (28) Then, u(x, t; σ) = U(x, σt), and therefore v(x, t) = Z σ2

σ1

u(x, t; σ)dσ = Z σ2

σ1

U(x, σt)dσ = 1 t [V (x, σ2t)−V (x, σ1t)], where V (x, t) = Z t U(x, s)ds + ξ(x) −∆ξ = u1, ξ|Σ = 0, which is a solution of Vtt − ∆V = 0 ; V |Σ = 0 ; V (0, ·) = ξ ∈ H1

0(Ω), Vt(x, 0) = u0(x) ∈ L2(Ω).

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 46 / 50

jueves, 17 de julio de 14

Averaged control of PDEs: Continuous averages Continuous averages of wave equations

Consequently tv(x, t) = [V (x, σ2t) − V (x, σ1t)]. Note that there is a regularizing effect so that tv gains one derivative with respect to the Sobolev regularity of u for each value of σ. Applying the previous result in collaboration with M. Lazar, the following holds: Theorem Let ω be an open non-empty subset of Ω and assume that (ω, T) satisfies the GCC with the fastest velocity of propagation σ2. Then, the averages of the solutions of the parameters-depending wave equation with respect to the unknown velocity of propagation σ is such that ||u0||2

H−1(Ω) + ||u1||2 H−2(Ω) ≤ C||t

Z σ2

σ1

u(x, t; σ)dσ||2

L2(ω×(0,T)).

(29)

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 47 / 50

jueves, 17 de julio de 14

Perspectives and open problems

Additive perturbations The method of proof applied to deal with additive perturbations, which consists in applying the PDE operators involved in a recursive manner, does not apply for systems of PDEs depending on a countable or continuous parameter. This seems to be the case even for heat like equations, despite of the regularizing effect. The proof used to deal with additive perturbations of PDE does not apply in the context of boundary control, since the PDE constraint cannot be restricted to the boundary manifold. The same can be said about the recent results8 about the observation

• f the heat equation from sets of positive measure.

The method does not apply in a straightforward manner when the coefficients in the equation are variable because of the lack of commutativity.

• 8J. Apraiz, L. Escauriaza, G. Wang and C. Zhang, Observability

inequalities and measurable sets, to appear in J. Europ. Math. Soc.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 48 / 50

jueves, 17 de julio de 14

Perspectives and open problems

Continuous averages In the case of wave equations, can we treat general initial data without taking the velocity to be of the form σu1? More general dependences on parameters, for instance for PDE involving elliptic operators of the form, div(a(x, σ)r·). Can we treat the average itself, without removing the weight 1/t which leads, in some cases, to the loss of derivatives in the

• bservation.

Possible connections on the theory of averaging of PDE, homogenization, singular limits...? Possibility of using asymptotic expansions exploiting analytic dependence.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 49 / 50

jueves, 17 de julio de 14

Perspectives and open problems

Random dependence: So far we considered PDE depending on deterministic unknown parameters. The same issues arise in the context of uncertain parameters governed by some probabilistic law. Fourier series: We have seen a surprising result on the possibility of extending Ingham-like inequalities to a setting where several series

• verlap, and all this in a robust manner. But our argument used in an

essential manner the time periodicity of the wave component. The extension of these inequalities to more general hyperbolic-parabolic spectra would be of interest.

• E. Zuazua (Ikerbasque-BCAM-CIMI)

Averaged control-CIMI2014 April 29, 2014 50 / 50

jueves, 17 de julio de 14