Estimate of the norm of the biorthogonal sequence: T = ∞143

Appendix 2: Dynamical programming

2.6 Boundary controllability of the 1D heat equation

2.6.5 Estimate of the norm of the biorthogonal sequence: T = ∞143

T = ∞

Theorem 2.6.5 There exist two positive constants M and ω such that the biorthogonal of minimal norm(θm(∞, ·)m≥1 given by Theorem 2.6.4satisfies the following estimate

||θm(∞,·)||L²(0,∞)≤M πe^ωm, ∀m≥1. (2.168)

Proof. Let us introduce the following notations: Eⁿ := Eⁿ(Λ,∞) is the subspace generated by Λⁿ := e^−λ^k^t

1≤k≤ninL²(0, T) andE_mⁿ :=E²(m,Λ,∞) is the subspace generated by e^−λ^k^t

1≤k≤n k6=m

in L²(0, T).

Remark thatEⁿ andE_mⁿ are finite dimensional spaces and E(Λ,∞) =∪_n≥1Eⁿ, E(m,Λ,∞) =∪_n≥1E_mⁿ.

We have that, for each n ≥ 1, there exists a unique biorthogonal family (θⁿ_m)_1≤m≤n ⊂Eⁿ, to the family of exponentials e^−λ^k^t

1≤k≤n. More precisely, θ_mⁿ = pm−r_mⁿ

||pm−rⁿ_m||²_L2(0,∞)

, (2.169)

whererⁿ_mis the orthogonal projection ofpm overE_mⁿ. If

θⁿ_m=

k=1

c^m_kp_k (2.170)

then, by multiplying (2.170) bypl and by integrating in (0,∞), it follows that δ_m,l=X

k≥1

c^m_k Z T

p_l(t)p_k(t)dt, 1≤m, l≤n. (2.171) Moreover, by multiplying in (2.170) byθ_mⁿ and by integrating in (0,∞), we obtain that

||θ_mⁿ||²_L2(0,∞)=c^m_m. (2.172) IfGdenotes the Gramm matrix of the family Λ, i. e. the matrix of elements

g_k^l = Z ∞

p_k(t)p_l(t)dt, 1≤k, l≤n

we deduce from (2.171) thatc^m_k are the elements of the inverse ofG. Cramer’s rule implies that

c^m_m=|Gm|

|G| (2.173)

where |G|is the determinant of matrixGand |Gm| is the determinant of the matrixGmobtained by changing the m−th column ofGby them−th vector of the canonical basis.

It follows that

||θ_mⁿ||L²(0,∞)=

s|G_m|

|G| . (2.174)

The elements ofGmay be computed explicitly g_kⁿ=

Z ∞ 0

p_k(t)p_n(t) = Z ∞

e⁻⁽ⁿ²^+k²^)π²^tdt= 1 n²π²+k²π².

Remark 2.6.6 A formula, similar to (2.174), may be obtained for anyT >0.

Nevertheless, the determinants may be estimated only in the caseT =∞.

To compute the determinats|G|and|Gm|we use the following lemma (see [47]):

Lemma 2.6.2 If C= (cij)1≤i,j≤n is a matrix of coefficientscij = 1/(ai+bj) then

|C|= Q

1≤i<j≤n(a_i−a_j)(b_i−b_j) Q

1≤i,j≤n(ai+bj) . (2.175) It follows that

|G|= Q

1≤i<j≤n(i²π²−j²π²)² Q

1≤i,j≤n(i²π²+j²π²) , |Gm|= Q0

1≤i<j≤n(i²π²−j²π²)² Q0

1≤i,j≤n(i²π²+j²π²) where⁰ means that the index mhas been skipped in the product.

Hence,

|Gm|

|G| = 2m²π²

k=1

0(m²+k²)²

(m²−k²)². (2.176) From (2.174) and (2.176) we deduce that

||θⁿ_m||L²(0,∞)=√ 2mπ

k=1

0 m²+k²

|m²−k²|. (2.177) Lemma 2.6.3 The norm of the biorthogonal sequence (θm(∞,·))m≥1 to the familyΛ in L²(0,∞)given by Theorem 2.6.4, verifies

||θm(∞,·)||L²(0,∞)=√ 2mπ

∞

k=1

0 m²+k²

|m²−k²|. (2.178) Proof. It consists in passing to the limit in (2.177) asn→ ∞. Remark first that, for eachm≥1, the product

∞

k=1

0 m²+k²

|m²−k²| is convergent since

1≤

∞

k=1

0 m²+k²

|m²−k²| = exp

∞

k=1 0ln

m²+k²

|m²−k²| !

≤

≤exp

∞

k=1 0ln

1 + 2m²

|m²−k²| !

≤exp 2m²

∞

k=1

0 1

|m²−k²|

<∞.

Consequently, the limit lim_n→∞||θ_mⁿ||_L2(0,∞) = L ≥ 1 exists. The proof ends if we prove that

n→∞lim ||θⁿ_m||_L2(0,∞)=||θm||_L2(0,∞). (2.179) Identity (2.169) implies that lim_n→∞||pm−r_mⁿ||_L2(0,∞)= 1/Land (2.179) is equivalent to

n→∞lim ||pm−rⁿ_m||L²(0,∞)=||pm−rm||L²(0,∞). (2.180) Let now ε > 0 be arbitrary. Sincerm ∈ E(m,Λ,∞) it follows that there existn(ε)∈N^∗ andr^ε_m∈Em^n(ε)with

||r_m−r^ε_m||_L2(0,∞)< ε.

For anyn≥n(ε) we have that

||pm−rm||= min

r∈E(m,Λ,∞)||pm−r|| ≤ ||pm−r_mⁿ||= min

r∈Eⁿ_m||pm−r|| ≤

≤ ||pm−r_m^ε|| ≤ ||pm−rm||+||rm−r_m^ε||<||pm−rm||+ε.

Thus, (2.180) holds and Lemma 2.6.3 is proved.

Finally, to evaluateθm(∞,·) we use the following estimate

Lemma 2.6.4 There exist two positive constants M andω such that for any m≥1,

∞

k=1

0 m²+k²

|m²−k²| ≤M e^ωm. (2.181) Proof. Remark that

Y⁰

m²+k²

|m²−k²| = exp

X⁰

kln

m²+k²

|m²−k²|

≤exp

X⁰

kln

1 + 2m²

|m²−k²|

. Now

X⁰

kln

1 + 2m²

|m²−k²|

≤ Z m

1 + 2m² m²−x²

dx+

+ Z 2m

1 + 2m² x²−m²

dx+

Z ∞ 2m

1 + 2m² x²−m²

dx=

=m Z 1

1 + 2 1−x²

dx+

Z 2 1

1 + 2 x²−1

dx+

+ Z ∞

1 + 2 x²−1

=m(I1+I2+I3). We evaluate now each one of these integrals.

I1= Z 1

1 + 2 1−x²

dx=

Z 1 0

1 + 2

(1−x)(1 +x)

dx≤

Z 1 0

1 + 2 1−x

dx=− Z 1

(1−x)⁰ln

1 + 2 1−x

dx=

=−(1−x) ln

1 + 2 1−x

+ Z 1

3−xdx=c₁<∞, I2=

Z 2 1

1 + 2 x²−1

dx≤

Z 2 1

1 + 2

(x−1)²

dx=

= Z 2

(x−1)⁰ln

1 + 2

(x−1)²

dx=

=−(x−1) ln

1 + 2

(x−1)²

+ Z 2

2 + (x−1)²dx=c₂<∞.

I3= Z ∞

1 + 2 x²−1

dx≤

Z ∞ 2

1 + 2

(x−1)²

dx≤

≤ Z ∞

(x−1)²dx=c3<∞.

The proof finishes by taking ω=c1+c2+c3.

The proof of Theorem 2.6.5 ends by taking into account relation (2.178) and Lemma 2.6.4.

2.6.6 Estimate of the norm of the biorthogonal sequence:

T < ∞

We consider nowT <∞. To evaluate the norm of the biorthogonal sequence (θm(T,·))_m≥1 inL²(0, T) the following result is necessary. The first version of this result may be found in [203] (see also [78] and [104]).

Theorem 2.6.6 Let Λ be the family of exponential functions e^−λⁿ^t

n≥1 and letT be arbitrary in(0,∞). The restriction operator

RT :E(Λ,∞)→E(Λ, T), RT(v) =v_|_[0,T]

is invertible and there exists a constant C >0, which only depends onT, such that

||R⁻¹_T || ≤C. (2.182)

Proof. Suppose that, on the contrary, for someT >0 there exists a sequence of exponential polynomials

Pk(t) =

N(k)

n=1

akne^−λⁿ^t⊂E(Λ, T) such that

k→∞lim ||Pk||_L2(0,T)= 0 (2.183) and

||Pk||L²(0,∞)= 1, ∀k≥1. (2.184) By using the estimates from Theorem 2.6.5 we obtain that

|amn|=

Z ∞ 0

Pk(t)θm(∞, t)dt

≤ ||Pk||L²(0,∞)||θm(∞,·)||L²(0,∞)≤M e^ωm. Thus

|Pk(z)| ≤

N(k)

n=1

|akn| e^−λⁿ^z

≤M

∞

n=1

e^ωn−n²^π²^Re(z). (2.185) If r > 0 is given let ∆_r ={z∈C : Re(z)> r}. For all z ∈ ∆_r, we have that

|Pk(z)| ≤M

∞

n=1

e^ωn−n²^π²^r≤M(ω, r). (2.186) Hence, the family (Pk)_k≥1 consists of uniformly bounded entire functions.

From Montel’s Theorem (see [43]) it follows that there exists a subsequence,

denoted in the same way, which converges uniformly on compact sets of ∆_rto an analytic functionP.

Choose r < T. From (2.183) it follows that lim_k→∞||P_k||_L2(r,T) = 0 and therefore P(t) = 0 for all t ∈ (r, T). Since P is analytic in ∆_r, P must be identically zero in ∆r.

Hence, (Pk)_k≥1converges uniformly to zero on compact sets of ∆r. Let us now return to (2.185). There existsr0>0 such that

|P_k(z)| ≤M e^−Re(z), ∀z∈∆_r₀. (2.187) Indeed, there exists r₀>0 such that

ωn−n²π²Re(z)≤ −Re(z)−n, ∀z∈∆_r₀ and therefore, for anyz∈∆r₀,

|Pk(z)| ≤MX

n≥1

e^ωn−n²^π²^Re(z)≤M e^−Re(z)X

n≥1

e⁻ⁿ= M

e−1e^−Re(z). Lebesgue’s Theorem implies that

k→∞lim ||Pk||_L2(r,∞)= 0 and consequently

lim

k→∞||Pk||L²(0,r)= 1.

If we take r < T the last relation contradicts (2.184) and the proof ends.

We can now evaluate the norm of the biorthogonal sequence.

Theorem 2.6.7 There exist two positive constantsM andωwith the property that

||θ_m(T,·)||_L2(0,T)≤M e^ωm, ∀m≥1 (2.188) where (θ_m(T,·))_m≥1 is the biorthogonal sequence to the family Λ in L²(0, T) which belongs toE(Λ, T)and it is given in Theorem2.6.4.

Proof. Let (R⁻¹_T )^∗ : E(Λ,∞) → E(Λ, T) be the adjoint of the bounded operatorR⁻¹_T . We have that

δkj= Z ∞

pk(t)θj(∞, t)dt= Z ∞

(R⁻¹_T RT)(pk(t))θj(∞, t)dt=

= Z T

RT(pk(t))(R⁻¹_T )^∗(θj(∞, t))dt.

Since (R⁻¹_T )^∗(θ_j(∞,·))∈E(Λ, T), from the uniqueness of the biorthogonal sequence inE(Λ, T), we finally obtain that

(R⁻¹_T )^∗(θ_j(∞,·)) =θ_j(T,·), ∀j≥1.

Hence

||θj(T,·)||_L2(0,T)=

(R⁻¹_T )^∗(θj(∞,·))

_L₂_(0,T)≤ ||R⁻¹_T || ||θj(∞,·)||_L2(0,∞), since||(R⁻¹_T )^∗||=||R⁻¹_T ||.

The proof finishes by taking into account the estimates from Theorem 2.6.5.

Remark 2.6.7 From the proof of Theorem 2.6.7 it follows that the constant ω does not depend ofT.

Chapter 3

Propagation, Observation, Control and

Finite-Difference Numerical Approximation of Waves

published in Bol. Soc. Esp. Mat. Apl. No. 25 (2003), 55–126.

3.1 Introduction

In recent years important progress has been made on problems of observation and control of wave phenomena. There is now a well established theory for wave equations with sufficiently smooth coefficients for a number of systems and variants: Lam´e and Maxwell systems, Schr¨odinger and plate equations, etc. However, when waves propagate in highly heterogeneous or discrete media much less is known.

These problems of observability and controllability can be stated as follows:

• Observability. Assuming that waves propagate according to a given wave equation and with suitable boundary conditions, can one guarantee that their whole energy can be estimated (independently of the solution) in terms of the energy concentrated on a given subregion of the domain (or its boundary) where propagation occurs in a given time interval ?

• Controllability. Can solutions be driven to a given state at a given final time by means of a control acting on the system on that subregion?

151

It is well known that the two problems are equivalent provided one chooses an appropriate functional setting, which depends on the equation (see for, instance, [142, 143],[241]). It is also well known that in order for the observation and/or control property to hold, aGeometric Control Condition (GCC)should be satisfied [14]. According to the GCC all rays of Geometric Optics should intersect the observation/control region in the time given to observe/control.

In this work we shall mainly focus on the issue of how these two properties behave under numerical approximation schemes. More precisely, we shall discuss the problem of whether, when the continuous wave model under consideration has an observation and/or control property, it is preserved for numerical approximations, and whether this holds uniformly with respect to the mesh size so that, in the limit as the mesh size tends to zero, one recovers the observation/control property of the continuous model.

But, before getting into the matter, let us briefly indicate some of the in- dustrial and/or applied contexts in which this kind of problems arises. The interested reader on learning more on this matter is refered to the SIAM Re- port [204], or, for more historical and engineering oriented applications, to [138]. The problem of controllability is classical in the field of Mathematical Control Theory and Control Engineering. We refer to the books by Lee and Marcus [136] and Sontag [206] for a classical and more modern, respectively, account of the theory for finite-dimensional systems with applications. The book by Fattorini [76] provides an updated account of theory in the context of semigroups which is therefore more adapted to the models governed by partial differential equations (PDE) and provides also some interesting examples of applications.

The problems of controllability and/or observability are in fact only some of those arising in the applications of control theory nowadays. In fact, an important part of the modelling effort needs to be devoted to defining the appropriate control problem to be addressed. But, whatever the control ques- tion we address is, the deep mathematical issues that arise when facing these problems of observability and controllability end up entering in one way or another. Indeed, understanding the properties of observation and controllability for a given system requires, first, analyzing the fine dynamical properties of the system and, second, the effect of the controllers on its dynamics.

In the context of control for PDE one needs to distinguish, necessarily, the elliptic, parabolic and hyperbolic ones since their distinguished qualitative properties make them to behave also very differently from a control point of view. The issue of controllability being typically of dynamic nature (although it also makes senses for elliptic or stationary problems) it is natural to address parabolic and hyperbolic equations, and, in particular, the heat and the wave equation.

Most of this article is devoted to the wave equation (although we shall

also discuss briefly the beam equation, the Schr¨odinger equation and the heat equation). The wave equation is a simplified hyperbolic problem arising in many areas of Mechanics, Engineering and Technology. It is indeed, a model for describing the vibrations of structures, the propagation of acoustic or seismic waves, etc. Therefore, the control of the wave equation enters in a way or another in problems related with control mechanisms for structures, buildings in the presence of earthquakes, for noise reduction in cavities and vehicles, etc.

We refer to [11], and [192] for insteresting practical applications in these areas.

But the wave equation, as we said, is also a prototype of infinite-dimensional, purely conservative dynamical system. As we shall see, most of the theory can be adapted to deal also with Schr¨odinger equation which opens the frame of applications to the challenging areas of Quantum computing and control (see [25]). It is well known that the interaction of waves with a numerical mesh produces dispersion phenomena and spurious¹high frequency oscillations ([224], [217]). In particular, because of this nonphysical interaction of waves with the discrete medium, the velocity of propagation of numerical waves and, more precisely, the so called group velocity² may converge to zero when the wavelength of solutions is of the order of the size of the mesh and the latter tends to zero. As a consequence of this fact, the time needed to uniformly (with respect to the mesh size) observe (or control) the numerical waves from the boundary or from a subset of the medium in which they propagate may tend to infinity as the mesh becomes finer. Thus, the observation and control properties of the discrete model may eventually disappear.

This effect is compatible and not in contradiction (as one’s first intuition might suggest) with the convergence of the numerical scheme in the classical sense and with the fact that the observation and control properties of the continuous model do hold. Indeed, convergent numerical schemes may have an unstable behavior in what concerns observability. In fact, we shall only discuss classical and well known convergent semi-discrete and fully discrete approximations of the wave equation, but we shall see that, despite the schemes under consideration are convergent, the failure of convergence occurs at the level of observation and control properties. As we said above, this is due to the fact that most numerical schemes exhibit dispersion diagrams (we shall

1The adjective spurious will be used to designate any component of the numerical solution that does not correspond to a solution of the underlying PDE. In the context of the wave equation, this happens at the high frequencies and, consequently, these spurious solutions weakly converge to zero as the mesh size tends to zero. Consequently, the existence of these spurious oscillations is compatible with the convergence (in the classical sense) of the numerical scheme, which does indeed hold for fixed initial data.

2At the numerical level it is important to distinguish phase velocity and group velocity.

Phase velocity refers to the velocity of propagation of individual monocromatic waves, while group velocity corresponds to the velocity of propagation of wave packets, that may signif- icantly differ from the phase velocity when waves of similar frequencies are combined. See, for instance, [217].

give a few examples below) showing that the group velocity of high frequency numerical solutions tends to zero.

The main objectives of this paper are:

• To explain how numerical dispersion and spurious high frequency oscillations occur;

• To describe their consequences for observation/control problems;

• To describe possible remedies for these pathologies;

• To consider to what extent these phenomena occur for other models like plate or heat-like equations.

The previous discussion can be summarized by saying that discretization and observation or control do not commute:

Continuous M odel+Observation/Control+N umerics 6=

Continuous M odel+N umerics+Observation/Control.

Indeed, here are mainly two alternative approaches to follow. The PDE approach consists on approximating the control of the underlying PDE through its corresponding optimality system or Euler-Lagrange equations. This provides convergent algorithms that produce good numerical approximations of the true control of the continuous PDE but, certainly one needs to go through PDE theory to develop it. But we may also first discretize the continuous model, then compute the control of the discrete system and use it as a numerical approximation of the continuous one. One of the main goals of this article is to explain that this second procedure, which is often used in the lit- erature without comment, may diverge. We shall describe how this divergence may occur and develop some numerical remedies. In other words, the topic of the manuscript may also be viewed as a particular instance ofblack-boxversus problem specific control. In the black box approach, when willing to control a PDE we make a finite-dimensional model approximating the PDE and control it. The other approach is to develop the theory of control for the PDE and discretize the control obtained that way. It is often considered that the black-box method is more robust. In this article we show that it may fail dramatically for wave-like problems. ³

Summarizing, controlling a discrete version of a continuous wave model is often a bad way of controlling the continuous wave model itself: stable solvers

3There are however some other situations in which it works. We refer to E. Casas [29] for the analysis of finite-element approximations of elliptic optimal control problems and to [46]

for an optimal shape design problem for the Laplace operator.

for solving the initial-boundary value problem for the wave equation do not need to yield stable control solvers.

It is also worth underlying that the instabilities we shall describe have a rather catastrophic nature. Indeed the divergence rate of controls is not polynomial on the number of nodes of the mesh but rather exponential. This shows that the stability can not restablished by simply changing norms on the observed quantities or relaxing the regularity of controls by a finite number of derivatives.

Up to now, we have discussed control problems in quite a vague way. In fact, rigorously speaking, the discussion above concerns the problem of exact controllabilityin which the goal is to drive the solution of an evolution problem to a given final state exactly in a given time. It is in this setting where the pathological numerical high frequency waves may lead to lack of convergence.

But this lack of convergence does not occur if the control problem is relaxed to an approximate or optimal control problem. In this paper we shall illus- trate this fact by showing that, although controls may diverge when we impose an exact control condition requiring the whole solution to vanish at the final time, when relaxing this condition (by simply requiring the solution to become smaller in norm than a given arbitrarily small numberε(approximate control) or to minimize the norm of the solution within a class of bounded controls (optimal control)) then the controls are bounded and converge ash→0 to the corresponding control of the continuous wave equation.

However, even if one is interested on those weakened versions of the control problem, taking into account that the exact controllability one can be obtained as a suitable limit of them,the previous discussion indicates the instability and extreme sensitivity of all control problems for waves under numerical discretiza- tions.

As a consequence of this, computing efficiently the control of the continuous wave model may be a difficult task, which has has been undertaken in a number of works by Glowinski, Li, and Lions [98], Glowinski [95], and Asch and Lebeau [4], among others. The effort that has been carried out in these papers is closely related to the existing work on developing numerical schemes with suitable dispersion properties ([224], [217]), based on the classical notion ofgroup velocity. But a full understanding of these connections in the context of control and observation of numerical waves requires an additional effort to which this paper is devoted.

In this paper, avoiding unnecessary technical difficulties, we shall present the main advances in this field, explaining the problems under consideration, the existing results and methods and also some open problems that, in our opin- ion, are particularly important. We shall describe some possible alternatives for avoiding these high frequency spurious oscillations, including Tychonoff reg- ularization, multigrid methods, mixed finite elements, numerical viscosity, and

filtering of high frequencies. All these methods, although they may look very different one from another in a first view, are in fact different ways of taking care of the spurious high frequency oscillations that common numerical approximation methods introduce. Despite the fact that the proofs of convergence may be lengthy and technically difficult (and often constitute open problems), once the high frequency numerical pathologies under consideration are well understood, it is easy to believe that they are indeed appropriate methods for computing the controls.

Our analysis is mainly based on the Fourier decomposition of solutions and classical results on the theory of non-harmonic Fourier series. In recent works by F. Maci`a [157], [158] tools of discrete Wigner measures (in the spirit of G´erard [91] and Lions and Paul [152]) have been developed to show that, as in the continuous wave equation, in the absence of boundary effects, one can characterize the observability property in terms of geometric properties related to the propagation of bicharacteristic rays. In this respect it is important to observe that the bicharacteristic rays associated with the discrete model do not obey the same Hamiltonian system as the continuous ones but have their own dynamics (as was pointed out before in [217]). As a consequence, numerical solutions develop quite different dynamical properties at high frequencies since both velocity and direction of propagation of energy may differ from those of the continuous model. Ray analysis allows one to be very precise when filtering the high frequencies and to do this filtering microlocally⁴. In this article we shall briefly comment on this discrete ray theory but shall mainly focus on the Fourier point of view, which is sufficient to understand the main issues under consideration. This ray theory provides a rigorous justification of a fact that can be formally analyzed and understood through the notion of group velocity of propagation of numerical waves [217].

All we have said up to now concerning the wave equation can be applied with minor changes to several other models that are purely conservative. However, many models from physics and mechanics have some damping mechanism built in. When the damping term is “mild” the qualitative properties are the same as those we discussed above. That is for instance the case for the dissipative wave equation

u_tt−∆u+ku_t= 0

that, under the change of variables v =e^−kt/2u, can be transformed into the

4Microlocal analysis deals, roughly speaking, with the possibility of localizing functions and its singularities not only in the physical space but also in the frequency domain. In particular, one can localize in the frequency domain not only according to the size of frequencies but also to sectors in the euclidean space in which they belong to. This allows introducing the notion of microlocal regularity, see for instance ([109])

No documento Controllability of Partial Differential Equations (páginas 143-159)