Overdetermined 2D systems invariant in one direction …andreymath/Seminars/SLScatVesls.pdf ·...

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OutlineOverdetermined 2D systems and their transfer functions

Class SI and its properties

Overdetermined 2D systems invariant in onedirection and their transfer functions

Andrey Melnikov

Department of MathematicsDrexel University

Notes will be available at http://www.math.drexel.edu/∼andreymath

January 2010

Andrey Melnikov Overdetermined 2D systems invariant in one direction and their transfer functions

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Plan of the lecture:

1. Overdetermined 2D systems and their transfer functions

2. Definition of the class of functions SI3. Main theorems for the elements of SI4. Scattering theory of SL equation using the class SI

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2. Definition of the class of functions SI

3. Main theorems for the elements of SI4. Scattering theory of SL equation using the class SI

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2. Definition of the class of functions SI3. Main theorems for the elements of SI

4. Scattering theory of SL equation using the class SI

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2. Definition of the class of functions SI3. Main theorems for the elements of SI4. Scattering theory of SL equation using the class SI

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2D systems, invariant in one directionOverdeterminednesst1-invariant conservative vesselFrequency domain analysis

Overdetermined 2D systems and their transfer functions2D systems, invariant in one direction

Overdetermined t1-invariant 2D system is

∂∂t1

x(t1, t2) = A1(t2)x(t1, t2) + B̃(t2)σ1u(t1, t2)

x(t1, t2) = F (t2, t02 )x(t1, t

02 ) +

t2∫t02

F (t2, s)B̃(s)σ2u(t1, s)ds

y(t1, t2) = u(t1, t2)− B̃∗(t2)x(t1, t2),(1)

where for some Hilbert spaces H, E

A1(t2) : H → H, B̃(t2) : E → H,

are bounded operator-functions and F (x , y) is an evolutionsemi-group. u(t1, t2) ∈ E and y(t1, t2) ∈ E are called the input andthe output, x(t1, t2) ∈ H is called the state.

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Overdetermined 2D systems and their transfer functions2D systems, invariant in one direction

Overdetermined t1-invariant 2D system is

∂∂t1

x(t1, t2) = A1(t2)x(t1, t2) + B̃(t2)σ1u(t1, t2)

x(t1, t2) = F (t2, t02 )x(t1, t

02 ) +

t2∫t02

y(t1, t2) = u(t1, t2)− B̃∗(t2)x(t1, t2),(1)

where for some Hilbert spaces H, E

A1(t2) : H → H, B̃(t2) : E → H,

are bounded operator-functions and F (x , y) is an evolutionsemi-group. u(t1, t2) ∈ E and y(t1, t2) ∈ E are called the input andthe output, x(t1, t2) ∈ H is called the state.

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Overdetermined 2D systems and their transfer functionsOverdeterminedness

To ensure that the overdetermined systems equations (1) arecompatible, we shall demand the equality of the two transitions forour system:

1. (t01 , t0

2 ) −→ (t01 , t2) −→ (t1, t2),

2. (t01 , t0

2 ) −→ (t1, t02 ) −→ (t1, t2).

for arbitrary (t01 , t0

2 ), (t1, t2).

Substituting the expressions forx(t1, t2) and demanding compatibility conditions for the freeevolution (u(t1, t2) = 0), we shall obtain:

A1(t2)F (t2, t02 ) = F (t2, t

02 )A1(t

02 ). (2)

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Overdetermined 2D systems and their transfer functionsOverdeterminedness

To ensure that the overdetermined systems equations (1) arecompatible, we shall demand the equality of the two transitions forour system:

1. (t01 , t0

2 ) −→ (t01 , t2) −→ (t1, t2),

2. (t01 , t0

2 ) −→ (t1, t02 ) −→ (t1, t2).

for arbitrary (t01 , t0

2 ), (t1, t2). Substituting the expressions forx(t1, t2) and demanding compatibility conditions for the freeevolution (u(t1, t2) = 0), we shall obtain:

A1(t2)F (t2, t02 ) = F (t2, t

02 )A1(t

02 ). (2)

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Using (2) we will obtain that the input u(t1, t2) has to satisfy thefollowing PDE

B̃(t2)σ2∂

∂t1u(t1, t2)− B̃(t2)σ1

∂∂t2

u(t1, t2)−(A1(t2)B̃(t2)σ2 + F (t2, t

02 ) ∂

∂t2[F (t0

2 , t2)B̃(t2)σ1])u(t1, t2) = 0.

Assuming the existence of a function γ(t2) satisfying

A1(t2)B̃(t2)σ2+F (t2, t02 )

∂s[F (t0

2 , t2)B̃(t2)σ1] = −B̃(t2)γ(t2) (3)

we obtain that it is enough that u(t1, t2) satisfies the PDE

σ2∂

∂t1u(t1, t2)− σ1

∂t2u(t1, t2) + γ(t2)u(t1, t2) = 0. (4)

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Using (2) we will obtain that the input u(t1, t2) has to satisfy thefollowing PDE

B̃(t2)σ2∂

∂t1u(t1, t2)− B̃(t2)σ1

∂∂t2

u(t1, t2)−(A1(t2)B̃(t2)σ2 + F (t2, t

02 ) ∂

∂t2[F (t0

2 , t2)B̃(t2)σ1])u(t1, t2) = 0.

Assuming the existence of a function γ(t2) satisfying

A1(t2)B̃(t2)σ2+F (t2, t02 )

∂s[F (t0

2 , t2)B̃(t2)σ1] = −B̃(t2)γ(t2) (3)

we obtain that it is enough that u(t1, t2) satisfies the PDE

σ2∂

∂t1u(t1, t2)− σ1

∂t2u(t1, t2) + γ(t2)u(t1, t2) = 0. (4)

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For the output signal we shall look for the output compatibilityconditions of the same type:

σ2∂

∂t1y(t1, t2)− σ1

∂t2y(t1, t2) + γ∗(t2)y(t1, t2) = 0 (5)

for an operator (t2 dependent) γ∗(t2) : E → E . We recall thaty(t1, t2) = u(t1, t2)− B∗(t2)x(t1, t2) and we insert it into the lastequation. Using the system equations (1) and the compatibilitycondition for the input, we are led to require

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σ2∂

∂t1y(t1, t2)− σ1

∂t2y(t1, t2) + γ∗(t2)y(t1, t2) = 0 (5)

for an operator (t2 dependent) γ∗(t2) : E → E .

We recall thaty(t1, t2) = u(t1, t2)− B∗(t2)x(t1, t2) and we insert it into the lastequation. Using the system equations (1) and the compatibilitycondition for the input, we are led to require

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σ2∂

∂t1y(t1, t2)− σ1

∂t2y(t1, t2) + γ∗(t2)y(t1, t2) = 0 (5)

for an operator (t2 dependent) γ∗(t2) : E → E . We recall thaty(t1, t2) = u(t1, t2)− B∗(t2)x(t1, t2) and we insert it into the lastequation. Using the system equations (1) and the compatibilitycondition for the input, we are led to require

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0 = σ2B̃(t2)∗A1(t2)F (t2, t

02 )− (6)

−σ1∂

∂t2[B̃(t2)

∗F (t2, t02 )] + γ∗(t2)B̃(t2)

∗F (t2, t02 )

γ(t2) = σ1B̃(t2)∗B̃(t2)σ2 − (7)

−σ2B̃(t2)∗B̃(t2)σ1 + γ∗(t2).

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The fact that the system is lossless comes from the requirement ofthe so called energy balance equations:

∂ti〈x(t1, t2), x(t1, t2)〉Ht2

+ 〈σiy(t1, t2), y(t1, t2)〉E =

= 〈σiu(t1, t2), u(t1, t2)〉E , i = 1, 2,

which means that the energy of the output is distributed betweenthe energy of the input and the change of the energy of the stateof the system. Immediate consequences of this requirement are

0 = A1(t2) + A∗1(t2) + B̃(t2)σ1B̃(t2)∗, (8)

dt2[F ∗(t2, t

02 )F (t2, t

02 )] = F ∗(t2, t

02 )B̃(t2)

∗σ2B̃(t2)F (t2, t02 ). (9)

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The fact that the system is lossless comes from the requirement ofthe so called energy balance equations:

∂ti〈x(t1, t2), x(t1, t2)〉Ht2

+ 〈σiy(t1, t2), y(t1, t2)〉E =

= 〈σiu(t1, t2), u(t1, t2)〉E , i = 1, 2,

which means that the energy of the output is distributed betweenthe energy of the input and the change of the energy of the stateof the system. Immediate consequences of this requirement are

0 = A1(t2) + A∗1(t2) + B̃(t2)σ1B̃(t2)∗, (8)

dt2[F ∗(t2, t

02 )F (t2, t

02 )] = F ∗(t2, t

02 )B̃(t2)

∗σ2B̃(t2)F (t2, t02 ). (9)

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Overdetermined 2D systems and their transfer functionst1-invariant vessel

In this manner we obtain the notion of conservative vessel in theintegral form, which is a collection of operators and spaces:

V = (A1(t2),F (t2, t02 ), B̃(t2);σ1, σ2, γ(t2), γ∗(t2);Ht2 , E)

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where the operators satisfy certain regularity assumptions and

0 = A1(t2) + A∗1(t2) + B̃(t2)∗σ1B̃(t2) (8)

‖F (t2, t02 )x(t1, t

02 )‖2 − ‖x(t1, t

02 )‖2 =

=∫ t2t02〈σ2B̃(s)x(t1, s), B̃(s)x(t1, s)〉ds (9)

F (t2, t02 )A1(t

02 ) = A1(t2)F (t2, t

02 ) (2)

0 = ddt2

(F (t02 , t2)B̃(t2)σ1)+

+F (t02 , t2)A1(t2)B̃(t2)σ2 + F (t0

2 , t2)B̃(t2)γ(t2) (3)

0 = σ1∂

∂t2[B̃(t2)

∗F (t2, t02 )]−

−σ2B̃(t2)∗A1(t2)F (t2, t

02 )− γ∗(t2)B̃(t2)

∗F (t2, t02 ) (6)

γ(t2) = −σ2B̃(t2)∗B̃(t2)σ1 + σ1B̃(t2)

∗B̃(t2)σ2 + γ∗(t2) (7)

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Then the vessel V is naturally associated to the system (1)

∂t1x(t1, t2) = A1(t2) x(t1, t2) + B̃(t2)σ1 u(t1, t2)

x(t1, t2) = F (t2, t02 )x(t1, t

02 ) +

t2∫t02

y(t1, t2) = u(t1, t2)− B̃(t2)∗ x(t1, t2).

together with the compatibility conditions for the input/ outputsignals:

σ2∂

∂t1u(t1, t2)− σ1

∂∂t2

u(t1, t2) + γ(t2)u(t1, t2) = 0

σ2∂

∂t1y(t1, t2)− σ1

∂∂t2

y(t1, t2) + γ∗(t2)y(t1, t2) = 0

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Then the vessel V is naturally associated to the system (1)

∂t1x(t1, t2) = A1(t2) x(t1, t2) + B̃(t2)σ1 u(t1, t2)

x(t1, t2) = F (t2, t02 )x(t1, t

02 ) +

t2∫t02

y(t1, t2) = u(t1, t2)− B̃(t2)∗ x(t1, t2).

together with the compatibility conditions for the input/ outputsignals:

σ2∂

∂t1u(t1, t2)− σ1

∂∂t2

u(t1, t2) + γ(t2)u(t1, t2) = 0

σ2∂

∂t1y(t1, t2)− σ1

∂∂t2

y(t1, t2) + γ∗(t2)y(t1, t2) = 0

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Remarks: 1. The first equation is the Lax equation, which playsan important role in completely integrable non-linear PDEs. Itfollows from the Lax equation that the spectrum of A1(t2) isindependent of t2.

2. This object is interesting, because it is time varying on the onehand, but has all the advantages of the time-invariant case on theother hand: transfer function, functional model.3. We shall always assume that σ1 is invertible.

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Remarks: 1. The first equation is the Lax equation, which playsan important role in completely integrable non-linear PDEs. Itfollows from the Lax equation that the spectrum of A1(t2) isindependent of t2.2. This object is interesting, because it is time varying on the onehand, but has all the advantages of the time-invariant case on theother hand: transfer function, functional model.

3. We shall always assume that σ1 is invertible.

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Remarks: 1. The first equation is the Lax equation, which playsan important role in completely integrable non-linear PDEs. Itfollows from the Lax equation that the spectrum of A1(t2) isindependent of t2.2. This object is interesting, because it is time varying on the onehand, but has all the advantages of the time-invariant case on theother hand: transfer function, functional model.3. We shall always assume that σ1 is invertible.

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Overdetermined 2D systems and their transfer functionsFrequency domain analysis

Performing a partial separation of variables for the system (1),

u(t1, t2) = uλ(t2)eλt1 ,

x(t1, t2) = xλ(t2)eλt1 ,

y(t1, t2) = yλ(t2)eλt1 ,

we arrive at the notion of the transfer function.

Compatibility PDEs for u(t1, t2), y(t1, t2) become ODEs foruλ(t2), yλ(t2) with the spectral parameter λ,

λσ2uλ(t2)− σ1∂

∂t2uλ(t2) + γ(t2)uλ(t2) = 0, (10)

λσ2yλ(t2)− σ1∂

∂t2yλ(t2) + γ∗(t2)yλ(t2) = 0. (11)

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Overdetermined 2D systems and their transfer functionsFrequency domain analysis

Performing a partial separation of variables for the system (1),

u(t1, t2) = uλ(t2)eλt1 ,

x(t1, t2) = xλ(t2)eλt1 ,

y(t1, t2) = yλ(t2)eλt1 ,

we arrive at the notion of the transfer function.Compatibility PDEs for u(t1, t2), y(t1, t2) become ODEs foruλ(t2), yλ(t2) with the spectral parameter λ,

λσ2uλ(t2)− σ1∂

∂t2uλ(t2) + γ(t2)uλ(t2) = 0, (10)

λσ2yλ(t2)− σ1∂

∂t2yλ(t2) + γ∗(t2)yλ(t2) = 0. (11)

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The corresponding i/s/o system becomesλxλ(t2) = A1(t2)xλ(t2) + B̃(t2)σ1uλ(t2)

∂∂t2

xλ(t2) = F (t2, t02 )xλ(t0

2 ) +t2∫t02

F (t2, s)B̃(s)σ2uλ(s)ds

yλ(t2) = uλ(t2)− B̃(t2)∗xλ(t2)

The output yλ(t2) = uλ(t2)− B̃(t2)∗xλ(t2) may be found from the

first i/s/o equation:

yλ(t2) = S(λ, t2)uλ(t2), (12)

using the transfer function

S(λ, t2) = I − B̃(t2)∗(λI − A1(t2))

−1B̃(t2)σ1

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∂∂t2

xλ(t2) = F (t2, t02 )xλ(t0

2 ) +t2∫t02

yλ(t2) = uλ(t2)− B̃(t2)∗xλ(t2)

yλ(t2) = S(λ, t2)uλ(t2), (12)

S(λ, t2) = I − B̃(t2)∗(λI − A1(t2))

−1B̃(t2)σ1

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∂∂t2

xλ(t2) = F (t2, t02 )xλ(t0

2 ) +t2∫t02

yλ(t2) = uλ(t2)− B̃(t2)∗xλ(t2)

yλ(t2) = S(λ, t2)uλ(t2), (12)

S(λ, t2) = I − B̃(t2)∗(λI − A1(t2))

−1B̃(t2)σ1

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Multiplication by S(λ, t2) maps solutions of the input ODE withthe spectral parameter λ to solutions of the output ODE with thesame spectral parameter. This can be written by means offundamental matrices Φ(λ, t2, t

02 ) and Φ∗(λ, t2, t

02 ) of the input

and of the output ODE’s. Namely:

S(λ, t2)Φ(λ, t2, t02 ) = Φ∗(λ, t2, t

02 )S(λ, t0

2 ) (13)

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The theory of such vessels, developed in [M, MV1] enables to finda more convenient form of the vessel.

Denoting

H = Ht02

the same for all t2,

A1 = A1(t02 ), F ∗(t2, t

02 )F (t2, t

02 ) = X−1(t2),

B(t2) = F (t02 , t2)B̃(t2),

we shall obtain the following notion, first introduced in [M2].

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The theory of such vessels, developed in [M, MV1] enables to finda more convenient form of the vessel. Denoting

H = Ht02

the same for all t2,

A1 = A1(t02 ), F ∗(t2, t

02 )F (t2, t

02 ) = X−1(t2),

B(t2) = F (t02 , t2)B̃(t2),

we shall obtain the following notion, first introduced in [M2].

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DefinitionA (differential) conservative vessel is:

V = (A1,B(t2), X(t2);σ1, σ2, γ(t2), γ∗(t2);H, E), (14)

where operators satisfy the following vessel conditions:

dt2(B(t2)σ1) + A1B(t2)σ2 + B(t2)γ(t2), (15)

A1X(t2) + X(t2)A∗1 = B(t2)σ1B(t2)

∗, (16)

dt2X(t2) = B(t2)σ2B(t2)

∗, (17)

γ∗(t2) = γ(t2) + σ1B(t2)∗X−1(t2)B(t2)σ2− (18)

−σ2B(t2)∗X−1(t2)B(t2)σ1

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DefinitionA (differential) conservative vessel is:

V = (A1,B(t2), X(t2);σ1, σ2, γ(t2), γ∗(t2);H, E), (14)

where operators satisfy the following vessel conditions:

dt2(B(t2)σ1) + A1B(t2)σ2 + B(t2)γ(t2), (15)

A1X(t2) + X(t2)A∗1 = B(t2)σ1B(t2)

∗, (16)

dt2X(t2) = B(t2)σ2B(t2)

∗, (17)

γ∗(t2) = γ(t2) + σ1B(t2)∗X−1(t2)B(t2)σ2− (18)

−σ2B(t2)∗X−1(t2)B(t2)σ1

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Lyapunov equation (16) is redundant

LemmaSuppose that B(t2) satisfies (15) and X (t2) satisfies (17), then ifthe Lyapunov equation (16)

A1X(t2) + X(t2)A∗1 + B(t2)σ1B(t2)

∗ = 0

holds for a fixed t02 , then it holds for all t2.

Proof: By differentiating we will obtain that LHS is constant.

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Transfer function of such a vessel is

S(λ, t2) = I − B(t2)∗X−1(t2)(λI − A1)

−1B(t2)σ1

and has the following properties:

1. For all t2, S(λ, t2) is an analytic function of λ in theneighborhood of ∞, where it satisfies:

S(∞, t2) = Ip

2. For all λ, S(λ, t2) is a continuous function of t2.

3. For λ in the domain of analyticity of S(λ, t2):

S(λ, t2)∗σ1S(λ, t2) ≤ σ1, <λ > 0, (19)

andS(λ, t2)

∗σ1S(λ, t2) = σ1, <λ = 0. (20)

4. Maps solutions of the input LDE (10) with spectral parameterλ to the output LDE (11) with the same spectral parameter.

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List of possible operations on SITau functionMoments of a function in SI

Definition of the class of functions SI

We consider a class SI = I(σ1, σ2, γ, γ∗) of matrix-valuedfunctions S(λ, t2)

that have the properties of the transfer functionof a vessel, defined in the previous slide.

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Definition of the class of functions SI

We consider a class SI = I(σ1, σ2, γ, γ∗) of matrix-valuedfunctions S(λ, t2) that have the properties of the transfer functionof a vessel, defined in the previous slide.

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System constructions

Looking at the definition of SI it is possible to notice that thisclass is closed under certain operations. Each one of the operationscorresponds to a system construction. Let S1(λ, t2), S2(λ, t2),S(λ, t2) ∈ SI

1. S−1(λ, t2) - Inversion of the system,

2. σ−11 S∗(λ, t2)σ1 - Adjoint system,

3. S1(λ, t2)S2(λ, t2) ∈ SI - Cascade connection of the systems,

4. Finding right divisor (in SI) of S(λ, t2) - Projection of thesystem,

5. Finding left divisor (in SI) of S(λ, t2) - Compression of thesystem,

6. Many more...

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6. Many more...

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6. Many more...

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6. Many more...

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6. Many more...

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6. Many more...

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Tau function

Tau function of a vessel is well defined up to a positive non zeroconstant

DefinitionLet S ∈ SI with a realization

S(λ, t2) = Ip − B∗(t2)X−1(t2)(λI − A1)−1B(t2)σ1.

The functionτ(t2) = det X(t2).

is called the τ -function associated to S.

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Moments of a function in SI

TheoremMoments of S(λ, t2) are Hi (t2) = B∗(t2)X−1(t2)A

i1B(t2):

S(λ, t2) = Ip−B∗(t2)X−1(t2)(λI−A1)−1B(t2)σ1 = Ip−

∞∑i=0

Hi (t2)

λi+1σ1,

and satisfy the following equations

σ−11 σ2Hi+1 − Hi+1σ2σ

−11 = d

dt2Hi − σ−1

1 γ∗Hi + Hiγσ−11 . (21)

Hi+1 + (−1)iH∗i+1 =i∑

(−1)j+1Hi−jσ1H∗j . (22)

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Moments of a function in SI

TheoremMoments of S(λ, t2) are Hi (t2) = B∗(t2)X−1(t2)A

i1B(t2):

S(λ, t2) = Ip−B∗(t2)X−1(t2)(λI−A1)−1B(t2)σ1 = Ip−

∞∑i=0

Hi (t2)

λi+1σ1,

and satisfy the following equations

σ−11 σ2Hi+1 − Hi+1σ2σ

−11 = d

dt2Hi − σ−1

1 γ∗Hi + Hiγσ−11 . (21)

Hi+1 + (−1)iH∗i+1 =i∑

(−1)j+1Hi−jσ1H∗j . (22)

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D. Alpay, A. Melnikov, V. Vinnikov, On the class RCI ofrational conservative functions intertwining solutions of lineardifferential equations, http://arxiv.org/abs/0912.2014.

A. Melnikov, Overdetermied 2D systems invariant in onedirection and their transfer functions, Phd Thesis, Ben Gurionuniversity, July 2009.

A. Melnikov, Finite dimensional Sturm Liouville vessels andtheir tau functions, submitted.

A. Melnikov, V. Vinnikov, Overdetermined 2D SystemsInvariant in One Direction and Their Transfer Functions,http://arXiv.org/abs/0812.3779.

A. Melnikov, V. Vinnikov, Overdetermined conservative 2DSystems, Invariant in One Direction and a Generalization ofPotapov’s theorem, http://arxiv.org/abs/0812.3970.

Overdetermined 2D systems invariant in one direction …andreymath/Seminars/SLScatVesls.pdf ·...

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