Approximation of the exact controls for the beam equation Sorin - - PowerPoint PPT Presentation

Approximation of the exact controls for the beam equation

Sorin Micu

University of Craiova (Romania)

Graz, June 25, 2015 Joint works with Florin Bugariu, Nicolae Cˆ ındea, Ionel Rovent ¸a and Laurent ¸iu Temereanc˘ a

Controlled hinged beam equation

Given any time T > 0 and initial data (u0, u1) ∈ H := H1

0(0, π) × H−1(0, π),

the exact controllability in time T of the linear beam equation with hinged (simply-supported) ends,            u′′(t, x) + uxxxx(t, x) = 0, x ∈ (0, π), t > 0 u(t, 0) = u(t, π) = uxx(t, 0) = 0, t > 0 uxx(t, π) = v(t), t > 0 u(0, x) = u0(x), u′(0, x) = u1(x), x ∈ (0, π) (1) consists of finding a scalar function v ∈ L2(0, T), called control, such that the corresponding solution (u, u′) of (1) verifies u(T, · ) = u′(T, · ) = 0. (2)

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method)

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method)

Multipliers

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method)

Multipliers Carleman estimates

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method)

Multipliers Carleman estimates Microlocal Analysis

(Many) methods to study the controllability

Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method)

Multipliers Carleman estimates Microlocal Analysis Fattorini H. O. and Russell D. L., Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 4 (1971), 272-292. J.-L. Lions, Controlabilit´ e exacte, stabilisation et perturbations des syst` emes distribu´ es, Vol. 1, Masson, Paris, 1988.

Optimization method

Lemma Let T > 0 and (u0, u1) ∈ H. The function v ∈ L2(0, T) is a control which drives to zero the solution of (1) in time T if and

• nly if, for any (ϕ0, ϕ1) ∈ H,

T v(t)ϕx(t, 1) dt = −

• u1(x), ϕ(0, x)
• −1,1+
• u0(x), ϕ′(0, x)
• 1,−1 ,

where (ϕ, ϕ′) ∈ H is the solution of the backward equation            ϕ′′(t, x) + ϕxxxx(t, x) = 0 (t, x) ∈ (0, T) × (0, 1) ϕ(t, 0) = ϕ(t, 1) = 0 t ∈ (0, T) ϕxx(t, 0) = ϕxx(t, 1) = 0 t ∈ (0, T) ϕ(T, x) = ϕ0(x) x ∈ (0, 1) ϕ′(T, x) = ϕ1(x) x ∈ (0, 1). (3)

Optimization method

For each (u0, u1) ∈ H, define the functional J : H → R,

J(ϕ0, ϕ1) = 1 2 T |ϕx(t, 1)|2 dt+

• u1(x), ϕ(0, x)
• −1,1−
• u0(x), ϕ′(0, x)
• 1,−1 ,

where (ϕ, ϕ′) is the solution of (3) with initial data (ϕ0, ϕ1).

Optimization method

For each (u0, u1) ∈ H, define the functional J : H → R,

J(ϕ0, ϕ1) = 1 2 T |ϕx(t, 1)|2 dt+

• u1(x), ϕ(0, x)
• −1,1−
• u0(x), ϕ′(0, x)
• 1,−1 ,

where (ϕ, ϕ′) is the solution of (3) with initial data (ϕ0, ϕ1). If J has a minimum at ( ϕ0, ϕ1) ∈ H then v(t) = ϕx(1, t) is a control for (1).

Optimization method

For each (u0, u1) ∈ H, define the functional J : H → R,

J(ϕ0, ϕ1) = 1 2 T |ϕx(t, 1)|2 dt+

• u1(x), ϕ(0, x)
• −1,1−
• u0(x), ϕ′(0, x)
• 1,−1 ,

where (ϕ, ϕ′) is the solution of (3) with initial data (ϕ0, ϕ1). If J has a minimum at ( ϕ0, ϕ1) ∈ H then v(t) = ϕx(1, t) is a control for (1). J has a minimum if it is coercive and it is coercive if the following observability inequality holds for any (ϕ0, ϕ1) ∈ H:

(ϕ(0), ϕ′(0))2

H ≤ C

T |ϕx(t, π)|2dt. (4)

Optimization method

For each (u0, u1) ∈ H, define the functional J : H → R,

J(ϕ0, ϕ1) = 1 2 T |ϕx(t, 1)|2 dt+

• u1(x), ϕ(0, x)
• −1,1−
• u0(x), ϕ′(0, x)
• 1,−1 ,

where (ϕ, ϕ′) is the solution of (3) with initial data (ϕ0, ϕ1). If J has a minimum at ( ϕ0, ϕ1) ∈ H then v(t) = ϕx(1, t) is a control for (1). J has a minimum if it is coercive and it is coercive if the following observability inequality holds for any (ϕ0, ϕ1) ∈ H:

(ϕ(0), ϕ′(0))2

H ≤ C

T |ϕx(t, π)|2dt. (4)

Hence, if (4) holds, for any initial data (u0, u1) ∈ H, there exists a control v ∈ L2(0, T) with the property vL2 ≤ √ C(u0, u1)H. (5)

Ingham’s inequality

Observability inequality (4) is equivalent to inequality of the form

• n∈Z∗

|αn|2 ≤ C(T)

• T

2

− T

2

• n∈Z∗

αneνn t

• 2

dt, (αn)n∈Z∗ ∈ ℓ2. (6) Ingham’s inequality For any T > 2π

γ∞ , γ∞ = lim inf n→∞ |νn+1 − νn|, inequality (6) holds.

• A. E. Ingham, Some trigonometric inequalities with applications to the

theory of series, Math. Zeits., 41 (1936), 367-379.

• J. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization
• f distributed semilinear control systems, Comm. Pure Appl. Math., 32

(1979), 555-587.

• J. P. Kahane: Pseudo-P´

eriodicit´ e et S´ eries de Fourier Lacunaires, Ann.

• Sci. Ecole Norm. Super. 37, 93-95 (1962).

Observability inequality

In our particular case νn = i sgn(n) n2, γ∞ = lim inf

n→∞ |νn+1 − νn| = ∞.

Ingham’s inequality implies that the observability inequality (4) is verified for any T > 0. Consequently, given any T > 0, there exists a control v ∈ L2(0, T) for each (u0, u1) ∈ H. The control function v is not unique.

Moment problem for the beam equation

The null-controllability of the beam equation is equivalent to solve a moment problem. Lemma Let T > 0 and (u0, u1) = ∞

n=1 a0 n sin(nx), ∞ n=1 a1 n sin(nx)

• ∈ H. The

function v ∈ L2(0, T) is a control which drives to zero the solution

• f (1) in time T if and only if
• T

2

− T

2

v

• t + T

2

• etνndt = (−1)ne− T

2 νn

√ 2nπ

• νna0

n − a1 n

• (n ∈ Z∗),

(7) where νn = i sgn(n) n2 are the eigenvalues of the unbounded skew-adjoint differential operator corresponding to (1). A solution v of the moment problem may be constructed by means

• f a biorthogonal family to the sequence (eνn t)n∈Z∗.

Moment problem for the beam equation

Definition A family of functions (φm)m∈Z∗ ⊂ L2 − T

2 , T 2

• with the property
• T

2

− T

2

φm(t)eνn tdt = δmn ∀ m, n ∈ Z∗, (8) is called a biorthogonal sequence to (eνn t)n∈Z∗ in L2 − T

2 , T 2

• .

Moment problem for the beam equation

Definition A family of functions (φm)m∈Z∗ ⊂ L2 − T

2 , T 2

• with the property
• T

2

− T

2

φm(t)eνn tdt = δmn ∀ m, n ∈ Z∗, (8) is called a biorthogonal sequence to (eνn t)n∈Z∗ in L2 − T

2 , T 2

• .

Once we have a biorthogonal sequence to (eνn t)n∈Z∗, a “formal” solution of the moment problem is given by v(t) =

• n∈Z∗

(−1)ne− T

2 νn

√ 2nπ

• νna0

n − a1 n

• φn
• t − T

2

• .

(9)

Ingham’s inequality and the existence of a biorthogonal

Consider a Hilbert space H and a family (fn)n∈Z∗ ⊂ H such that

• n∈Z∗

|an|2 ≤ C1

• n∈Z∗

anfn

• 2

, (an)n∈Z∗ ∈ ℓ2. (10)

Then there exists a biorthogonal sequence to the family (fn)n∈Z∗. (fn)n∈Z∗ is minimal i. e. fm / ∈ Span

• (fn)n∈Z∗\{m}
• (m ∈ Z∗).

Apply Hahn-Banach Theorem to {fm} and Span

• (fn)n∈Z∗\{m}
• . There exists φm ∈ H such that

(φm, fm) = 1 and (φm, fn) = 0 for any n = m. The biorthogonal sequence which is bounded:

• n∈Z∗

bnφn

• 2

≤ 1 C1

• n∈Z∗

|bn|2.

No Ingham?

If we are in a context in which no Ingham’s type inequality is available? We can take the inverse way: Construction of the biorthogonal Paley-Wiener Theorem: Let F : C → C be an entire function

• f exponential type (|F(z)| ≤ MeT|z|) which belongs to

L2(R) on the real axis. Then

• R F(t)eixtdt is a function from

L2(−T, T).

• R. E. A. C. Paley and N. Wiener, Fourier Transforms in Complex

Domains, AMS Colloq. Publ., Vol. 19, Amer. Math. Soc., 1934. f(x) = 1 2π

• R

F(t)eixtdt ⇒        F(t) = T

−T

f(x)e−ixtdx; fL2 = √ 2πFL2(R).

Evaluation of its norm Construction of the control

Finite differences for the beam equation

N ∈ N∗, h =

π N+1, xj = jh, 0 ≤ j ≤ N + 1,

x−1 = −h, xN+2 = π + h.          u′′

j (t) = − uj+2(t)−4uj+1+6uj(t)−4uj−1(t)+uj−2(t) h4

, t > 0 u0(t) = uN+1(t) = 0, u−1(t) = −u1(t), t > 0 uN+2 = −uN + h2vh(t), t > 0 uj(0) = u0

j, u′ j(0) = u1 j, 1 ≤ j ≤ N.

(11)

Finite differences for the beam equation

N ∈ N∗, h =

π N+1, xj = jh, 0 ≤ j ≤ N + 1,

x−1 = −h, xN+2 = π + h.          u′′

j (t) = − uj+2(t)−4uj+1+6uj(t)−4uj−1(t)+uj−2(t) h4

, t > 0 u0(t) = uN+1(t) = 0, u−1(t) = −u1(t), t > 0 uN+2 = −uN + h2vh(t), t > 0 uj(0) = u0

j, u′ j(0) = u1 j, 1 ≤ j ≤ N.

(11) Discrete controllability problem: given T > 0 and (U 0

h, U1 h) = (u0 j, u1 j)1≤j≤N ∈ C2N, there exists a control function

vh ∈ L2(0, T) such that the solution u of (11) satisfies uj(T) = u′

j(T) = 0, ∀j = 1, 2, ..., N.

(12) System (11) consists of N linear differential equations with N unknowns u1, u2, ..., uN. uj(t) ≈ u(t, xj) if (U 0

h, U1 h) ≈ (u0, u1).

Discrete controls

Existence of the discrete control vh. Boundedness of the sequence (vh)h>0 in L2(0, T). Convergence of the sequence (vh)h>0 to a control v of the beam equation (1).

Discrete controls

Existence of the discrete control vh. Boundedness of the sequence (vh)h>0 in L2(0, T). Convergence of the sequence (vh)h>0 to a control v of the beam equation (1).

• L. LEON and E. ZUAZUA: Boundary controllability of the

finite-difference space semi-discretizations of the beam equation. ESAIM:COCV, A Tribute to Jacques- Louis Lions, Tome 2, 2002, pp. 827-862.

Equivalent vectorial form

System (11) is equivalent to    U ′′

h(t) + (Ah)2Uh(t) = Fh(t)

t ∈ (0, T) Uh(0) = U 0

h

U ′

h(0) = U 1 h,

(13)

Ah = 1 h2         2 −1 . . . −1 2 −1 . . . −1 2 −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 −1 . . . −1 2         , Uh(t) =      u1(t) u2(t) . . . uN(t)      Fh(t) = 1 h2        . . . −vh(t)        , U 0

h =

     u0

1

u0

2

. . . u0

N

     , U 1

h =

     u1

1

u1

2

. . . u1

N

     .

Discrete observability inequality

   W ′′

h (t) + A2 hWh(t) = 0

t ∈ (0, T) Wh(T) = W 0

h ∈ CN

W ′

h(T) = W 1 h ∈ CN.

(14) The energy of (14) is defined by Eh(t) = 1 2

• AhWh(t), Wh(t) + A−1

h W ′ h(t), W ′ h(t)

• ,

(15) and the following relation holds: d dtEh(t) = 0. (16) The exact controllability in time T of (11) holds if the following discrete observability inequality is true Eh(t) ≤ C(T, h) T

• WhN(t)

h

• 2

dt, (W 0

h, W 1 h) ∈ C2N.

(17)

One or two problems

Eigenvalues: νn = i sgn (n) µn, µn =

4 h2 sin2 nπh 2

• ,

1 ≤ |n| ≤ N. Eigenvectors form an orthogonal basis in C2N: φn = 1 √2µn   ϕn −νn ϕn   , ϕn = √ 2      sin(nhπ) sin(2nhπ) . . . sin(Nnhπ)      , 1 ≤ |n| ≤ N. The observability constant is not uniform in h: (W 0

h, W 1 h) = φN ⇒ C(T, h) =

1 T cos2 Nπh

2

≈ 1 Th2 . There are initial data (u0, u1) ∈ H such that the sequence of discrete minimal L2−norm controls ( vh)h>0 diverges!!!

Cures (L. Leon and E. Zuazua, COCV 2002)

Problems from the bad numerical approximation of high eigenmodes (spurious numerical eigenmodes). Control the projection of the solution over the space Span{φn : 1 ≤ |n| ≤ γN}, with γ ∈ (0, 1).

• 1≤|n|≤γN

|αn|2 ≤ C

• T

2

− T

2

• 1≤|n|≤γN

αneνn t

• 2

dt. (18)

Introduce a new control which vanishes in the limit

Eh(t) ≤ C T

• WhN(t)

h

• 2

dt + h2 T

• W ′

hN(t)

h

• 2

dt

• .

(19)

C = C(T) ⇒ uniform controllability ⇒ convergence of the discrete controls.

Regularity and filtration of the initial data

We consider the controlled system    U ′′

h(t) + (Ah)2Uh(t) = Fh(t)

t ∈ (0, T) Uh(0) = U 0

h

U ′

h(0) = U 1 h,

(20) We suppose that one of the following properties holds: Initial data (u0, u1) are sufficiently smooth (for instance, in H3(0, 1) × H1

0(0, 1)) and discretized by points

U 0 = (u0(jh))1≤j≤N, U1 = (u1(jh))1≤j≤N; Initial data (u0, u1) are in the energy space H and the high frequencies of their discretization are filtered out, (U 0, U1) =

• 1≤|n|≤δN

anhΦn (δ ∈ (0, 1)); Can we obtain the uniform controllability in any T > 0?

Discrete moments problem

Lemma Let T > 0 and ε > 0. System (20) is null-controllable in time T if and only if, for any initial datum (U 0

h, U1 h) ∈ C2N of form

(U 0

h, U 1 h) =

 

N

• j=1

a0

jhϕj, N

• j=1

a1

jhϕj

  , (21)

there exists a control vh ∈ L2(0, T) such that

T vh(t)eνntdt = (−1)nh √ 2 sin(|n|πh)

• −a1

|n|h + νna0 |n|h

• ,

(22)

for any n ∈ Z∗ such that |n| ≤ N.

Biorthogonal family

If (θm)1≤|m|≤N ⊂ L2 − T

2 , T 2

• is a biorthogonal sequence to the

family of exponential functions

• eνnt

1≤|n|≤N in L2

− T

2 , T 2

• then

a control of (13) will be given by

vh(t) =

• 1≤|n|≤N

(−1)nhe−νn T

2

√ 2 sin(|n|πh)

• −a1

|n|h + νna0 |n|h

• θn
• t − T

2

• .

We look for a biorthogonal sequence (θm)1≤|m|≤N to

• eiνnt

1≤|n|≤N and we try to estimate the right hand side sum.

The exponents are real: νn = sgn(n) 4 h2 sin nπh 2

• (1 ≤ |n| ≤ N).

Biorthogonal sequence

Taking into account that

νn+1−νn = 4 h2 sin nπh 2

• sin

(2n + 1)πh 2

• >

n if δ < |n| < δN 4

• therwise,

we can use Ingham’s inequality and a Kahane’s argument to show that, for any T > 0, there exists a biorthogonal (θm)1≤|m|≤N to the family

• eiνnt

1≤|n|≤N with the property that

• 1≤|n|≤N

bnθn

• 2

≤ C exp C T

• 1≤|n|≤N

|bn|2.

It follows that

vh(t)2 =

• 1≤|n|≤N

(−1)nhe−νn T

2

√ 2 sin(|n|πh)

• −a1

|n|h + νna0 |n|h

• θn
• t − T

2

• 2

≤ C exp C T

• 1≤|n|≤N

h2 sin2(nπh)

• |a1

|n|h|2 + |νn|2|a0 |n|h|2

.

Regularity or filtration

vh(t)2 ≤ C exp C T

• 1≤|n|≤N

h2 sin2(nπh)

• |a1

|n|h|2 + |νn|2|a0 |n|h|2

.

The initial data to be controlled are in H3(0, 1) × H1

0(0, 1)

• 1≤|n|≤N

n2 |a1

|n|h|2 + |νn|2|a0 |n|h|2

≤ C(u0, u1)2

3,1

⇒ vh2 ≤ C exp C T

• (u0, u1)2

3,1.

The high frequencies of the discrete initial data are filtered out

vh2 ≤ C(δ) exp C T

• 1≤|n|≤δN

1 n2

• |a1

|n|h|2 + |νn|2|a0 |n|h|2

≤ C′(δ) exp C T

• (u0, u1)2

1,−1.

Numerical results

Figure: Initial data to be controlled.

N = 100; T = .3; A conjugate gradient method for the corresponding discrete

• ptimization approach.

Numerical results

Figure: Example 2 - The first four iterations of the conjugate gradient method for the approximation of vh with N = 100 without filtration.

Numerical results

Figure: The approximation of the control vh with N = 100, 200, 500 and 1000 by using filtration of the initial data with δ =

1 40.

Figure: Controlled solution and the approximation of the control with N = 100 by using filtration of the initial data δ =

1 40.

Numerical vanishing viscosity

Instead of (13) we consider the system    U ′′

h(t) + (Ah)2Uh(t) + εAhU ′ h(t) = Fh(t)

t ∈ (0, T) Uh(0) = U 0

h

U ′

h(0) = U 1 h,

(23) ε = ε(h), limh→0 ε = 0 If Fh = 0, dEh dt (t) = −εAhU ′

h(t), U′ h(t) ≤ 0

The term εAhU ′

h(t) represents a numerical vanishing viscosity.

Numerical vanishing viscosity

Instead of (13) we consider the system    U ′′

h(t) + (Ah)2Uh(t) + εAhU ′ h(t) = Fh(t)

t ∈ (0, T) Uh(0) = U 0

h

U ′

h(0) = U 1 h,

(23) ε = ε(h), limh→0 ε = 0 If Fh = 0, dEh dt (t) = −εAhU ′

h(t), U′ h(t) ≤ 0

The term εAhU ′

h(t) represents a numerical vanishing viscosity.

Can we obtain the uniform controllability in any T > 0 (without projection or additional controls) using this new discrete scheme?

Bibliography I

• R. J. DiPerna : Convergence of approximate solutions to

conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70.

• L. R. Tcheugou´

e T´ ebou and E. Zuazua: Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numer. Math. 95 (2003), 563-598.

• A. M¨

unch and A. F. Pazoto: Uniform stabilization of a viscous numerical approximation for a locally damped wave equation, ESAIM Control Optim. Calc. Var. 13 (2007), 265-293.

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Exponentially Stable Approximations for a Class of Second Order Evolution Equations, ESAIM: COCV 13 (2007), 503-527.

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approximations for a class of damped systems, J. Math. Pures Appl. 91 (2009), 20-48.

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nonlinear Schr¨

• dinger equation, SIAM J. Numer. Anal. 47 (2009),

1366-1390.

Bibliography II

At the interface between parabolic and hyperbolic equations: singular limit control problem.

• A. L´
• pez, X. Zhang and E. Zuazua, Null controllability of the

heat equation as singular limit of the exact controllability of dissipative wave equations, J. Math. Pures Appl. 79 (2000), 741-808. J.-M. Coron and S. Guerrero, Singular optimal control: a linear 1-D parabolic-hyperbolic example, Asymptot. Anal. 44 (2005), 237-257.

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uniform controllability of a transport equation in the vanishing viscosity limit, Journal of Functional Analysis 258 (2010), 852-868.

• M. L´

eautaud, Uniform controllability of scalar conservation laws in the vanishing viscosity limit, SIAM J. Control Optim. 50 (2012), 1661-1699.

Spectral analysis. Good news but no Ingham.

Eigenvalues: λn = 1

2

• ε + i sgn (n)

√ 4 − ε2

• µ|n|, 1 ≤ |n| ≤ N.

Eigenvectors: φn = 1 √2µn   ϕn −λn ϕn   , ϕn = √ 2      sin(nhπ) sin(2nhπ) . . . sin(Nnhπ)      , 1 ≤ |n| ≤ N. If (W 0

h, W 1 h) = φN we obtain that

C(T, h) = T

• WhN(t)

h

• 2

dt (Wh(0), W ′

h(0))2 ≈

1 cos2 Nπh

2

• ℜ(λN)

e2Tℜ(λN) − 1. To ensure the uniform observability of these initial data we need ε > C ln 1 h

• h2

Spectral analysis. Good news but no Ingham.

Eigenvalues: λn = 1

2

• ε + i sgn (n)

√ 4 − ε2

• µ|n|, 1 ≤ |n| ≤ N.

Eigenvectors: φn = 1 √2µn   ϕn −λn ϕn   , ϕn = √ 2      sin(nhπ) sin(2nhπ) . . . sin(Nnhπ)      , 1 ≤ |n| ≤ N. If (W 0

h, W 1 h) = φN we obtain that

C(T, h) = T

• WhN(t)

h

• 2

dt (Wh(0), W ′

h(0))2 ≈

1 cos2 Nπh

2

• ℜ(λN)

e2Tℜ(λN) − 1. To ensure the uniform observability of these initial data we need ε > C ln 1 h

• h2 ⇒ ℜ(λN) > C ln

1 h

• .

Discrete moments problem

Lemma Let T > 0 and ε > 0. System (13) is null-controllable in time T if and only if, for any initial datum (U 0

h, U1 h) ∈ C2N of form

(U 0

h, U 1 h) =

 

N

• j=1

a0

jhϕj, N

• j=1

a1

jhϕj

  , (24)

the exists a control vh ∈ L2(0, T) such that

T vh(t)eλntdt = (−1)nh √ 2 sin(|n|πh)

• −a1

|n|h + (λn − εµ|n|)a0 |n|h

• ,

(25)

for any n ∈ Z∗ such that |n| ≤ N.

Biorthogonal family

If (θm)1≤|m|≤N ⊂ L2 − T

2 , T 2

• is a biorthogonal sequence to the

family of exponential functions

• eλnt

1≤|n|≤N in L2

− T

2 , T 2

• then

a control of (13) will be given by

vh(t) =

• 1≤|n|≤N

(−1)nhe−λn T

2

√ 2 sin(|n|πh)

• −a1

|n|h + (λn − εµ|n|)a0 |n|h

• θn
• t − T

2

• .

Now the main task in to show that there exists a biorthogonal sequence (θm)1≤|m|≤N and to evaluate its L2−norm in order to estimate the right hand side sum.

S.M., Uniform boundary controllability of a semi–discrete 1–D wave equation with vanishing viscosity, SIAM J. Cont. Optim., 47 (2008), 2857-2885. Main differences: We have the optimal value of the viscosity parameter ε: ε ≥ Ch2 ln 1 h

• .

S.M., Uniform boundary controllability of a semi–discrete 1–D wave equation with vanishing viscosity, SIAM J. Cont. Optim., 47 (2008), 2857-2885. Main differences: We have the optimal value of the viscosity parameter ε: ε ≥ Ch2 ln 1 h

• .

The controllability time T should be arbitrarily small.

Construction of a biorthogonal (I) - The big picture

Suppose that (θm)1≤|m|≤N is a biorthogonal sequence to the family

• f exponential functions
• eλnt

1≤|n|≤N in L2

− T

2 , T 2

• and define

Ψm(z) =

• T

2

− T

2

θm(t)e−i tzdt. Ψm(iλn) = δnm Ψm is an entire function of exponential type T

2

Ψn ∈ L2(R) Paley-Wiener Theorem ensures that the reciprocal is true and gives a constructive way to obtain a biorthogonal sequence. Ψm(z) = Pm(z) × Mm(z) =

• n=m

iλn − z iλn − iλm × Mm(z). Pm (the product) and Mm (the multiplier) should have small exponential type and good behavior on the real axis.

Construction of a biorthogonal (II) - A small picture

Construction of a biorthogonal (II) - A small picture

(ξ1

l )l is a biorthogonal to family F1 which is finite.

(ξ2

k)k is a biorthogonal to family F2 with good gap properties.

A biorthogonal (θm)m to full family F1 ∪ F2 can be constructed by using the Fourier transforms θ1

k and

θ2

l .

Construction of a biorthogonal (III): The main result

Theorem Let T > 0. There exist two positive constants h0 and ε0 such that for any h ∈ (0, h0) and ε ∈

• c0h2 ln

1

h

• , c0h
• there exists a

biorthogonal (θm)m to (eλnt)n and two constants α < T and C = C(T) > 0 (independent of ε and h) such that

• T

2

− T

2

• m

αmθm(t)

• 2

dt ≤ C(T)

• m

|αm|2eα|ℜ(λm)|, (26)

for any finite sequence (αm)m.

Construction of a biorthogonal (III): The main result

Theorem Let T > 0. There exist two positive constants h0 and ε0 such that for any h ∈ (0, h0) and ε ∈

• c0h2 ln

1

h

• , c0h
• there exists a

biorthogonal (θm)m to (eλnt)n and two constants α < T and C = C(T) > 0 (independent of ε and h) such that

• T

2

− T

2

• m

αmθm(t)

• 2

dt ≤ C(T)

• m

|αm|2eα|ℜ(λm)|, (26)

for any finite sequence (αm)m. Since

vh(t) =

• 1≤|n|≤N

(−1)nhe− T λn

2

√ 2 sin(|n|πh)

• −a1

|n|h + (λn − εµ|n|)a0 |n|h

• θn
• t − T

2

• .

we obtain immediately from (26) the uniform boundedness (in h)

• f the family of controls (vh)h>0.

Numerical results

Figure: Initial data to be controlled.

N = 100; T = 2.3; ε = h A conjugate gradient method for the corresponding discrete

• ptimization approach.

Numerical results

Figure: The first four iterations with ε = 0.

Numerical results

Figure: The first four iterations with ε = h.

Figure: Controlled solution and the control.

Controlled clamped beam equation

Given any time T > 0 and initial data (u0, u1) ∈ H := L2(0, π) × H−2(0, π), the exact controllability in time T of the linear clamped beam equation,            u′′(t, x) + uxxxx(t, x) = 0, x ∈ (0, π), t > 0 u(t, 0) = u(t, π) = ux(t, 0) = 0, t > 0 ux(t, π) = v(t), t > 0 u(0, x) = u0(x), u′(0, x) = u1(x), x ∈ (0, π) (27) consists of finding a scalar function v ∈ L2(0, T), called control, such that the corresponding solution (u, u′) of (27) verifies u(T, · ) = u′(T, · ) = 0. (28)

Finite differences for the clamped beam equation

N ∈ N∗, h =

π N+1, xj = jh, 0 ≤ j ≤ N + 1,

x−1 = −h, xN+2 = π + h.          u′′

j (t) = − uj+2(t)−4uj+1+6uj(t)−4uj−1(t)+uj−2(t) h4

, t > 0 u0(t) = uN+1(t) = 0, u−1(t) = u1(t), t > 0 uN+2 = uN + 2hvh(t), t > 0 uj(0) = u0

j, u′ j(0) = u1 j, 1 ≤ j ≤ N.

(29) Discrete controllability problem: given T > 0 and (U 0

h, U1 h) = (u0 j, u1 j)1≤j≤N ∈ C2N, there exists a control function

vh ∈ L2(0, T) such that the solution u of (11) satisfies uj(T) = u′

j(T) = 0, ∀j = 1, 2, ..., N.

(30)

Discrete observability inequality

   W ′′

h (t) +

BhWh(t) = 0 t ∈ (0, T) Wh(T) = W 0

h ∈ CN

W ′

h(T) = W 1 h ∈ CN.

(31) The energy of (31) is defined by Eh(t) = 1 2

• BhWh(t), Wh(t) + W ′

h(t), W ′ h(t)

• ,

(32) and the following relation holds: d dtEh(t) = 0. (33) The exact controllability in time T of (29) holds if the following discrete observability inequality is true Eh(t) ≤ C(T, h) T

• 2WhN(t)

h2

• 2

dt, (W 0

h, W 1 h) ∈ C2N.

(34)

Spectral analysis

Continuous spectrum: The eigenvalues of the corresponding differential operator are given by the positive roots of the equation cos(z) − cosh−1(z) = 0, which are asymptotically exponentially close to the zeros of the cos(z) function. Discrete spectrum: The eigenvalues of the corresponding discrete operator are given by the positive roots of the equation f(z) = 0, where

f(z) = cos z ± sin2 hz 2

• +

2

• 1 − sin4 hz

2

• rN+1(z)

r2(N+1)(z) − 2 sin2 hz

2

• rN+1(z) + 1,

r(z) = 1 + 2 sin2 zh 2

• +
• sin2

zh 2 1 + sin2 zh 2

• .

Function f has a sequence of well separated roots (zn)1≤n≤N ⊂ (0, (N + 1)π). We obtain that our problem has a sequence of eigenvalues λn =

1 h4 cos4 znh 2

• and a complete set of

eigenfunctions Φn, 1 ≤ n ≤ N.

Observability inequality for discrete clamped beam

The observability inequality is equivalent to

• 1≤|n|≤N

|an|2 ≤ C T

• 1≤|n|≤N

aneisgn(n)√

λ|n|t Φ|n| N

λ|n|

• 2

dt. (35)

Inequality (35) follows with C = C(T) = O κ

T

• since

1 For any T > 0 there exists nT = O(1/T) ∈ N, independent of

h, such that the following inequality holds

• λn+1 −
• λn ≥ 2π

T (nT ≤ n ≤ N − nT ) . (36)

2 There exists a constant C > 0, independent of h, such that

Φn

N ≥ C

• λn

(1 ≤ n ≤ N) . (37) We obtain that the discrete clamped beam equation is uniformly controllable in any time. As in the continuous case, the

• bservability constant explodes as exp(κ/T) as T tends to zero.

Thank you very much for your attention!