2D ACAUSAL LINEAR SYSTEMS DESCRIBED BY DIFFERENTIAL-DIFFERENCE EQUATIONS

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BY DIFFERENTIAL-DIFFERENCE EQUATIONS

VALERIU PREPELIT¸ ˘A

A class of 2D continuous-discrete acausal linear systems is considered. Well-posed boundary conditions are introduced for such a system and the formula of the state is derived by using a canonical boundary value operator as well as the input-output operator of the system. Some properties of the 1D acausal systems studied by I. Gohberg and M.A. Kaashoek are extended to this framework, and necessary and sufficient conditions of minimality of 2D acausal systems are obtained. The adjoint of a 2D acausal system is defined, and the relationship between the input- output operator of the system and its adjoint is emphasized.

AMS 2000 Subject Classification: 93C35, 93B20, 93B05, 93B07, 93C23, 45E10.

Key words: acausal system, continuous-discrete system, multidimensional system, controllability Gramian, observability Gramian, minimality, adjoint system.

1. INTRODUCTION

The papers of A.J. Krener ([13], [14]), have introduced the continuous- time linear systems with boundary conditions in state space representation, using them to model the problem of boundary value regulation. Motivated by the analysis of Wiener-Hopf integral equation, Gohberg and Kaashoek ([7], [8], [9], [10]) have developed the study of these systems. In the linear estimation theory of stochastic processes governed by acausal systems, some important results have been obtained by Adams, Willsky and Levy ([1], [2]).

In the last three decades the theory of two-dimensional (2D) systems (or multidimensional (nD) systems) knew a powerful development, due to the richness in its potential applications in various areas as image processing, seismology, geophysics or computer tomography. Different state space models for 2D systems have been proposed by Roesser [20], Fornasini and Marchesini [5], Attasi [4] and others.

An important subclass of 2D systems consists of the systems which are continuous with respect to one variable and discrete with respect to an- other one. This subclass was studied for instance in [11], [12], [17], [18]. The

MATH. REPORTS10(60),3 (2008), 265–276

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continuous-discrete variables have many applications in problems such as linear repetitive processes ([6], [21]), long-wall coal cutting and metal rolling [22]

and iterative learning control synthesis ([3], [15]).

In this paper we study the possible connections of the two approaches, namely the 2D continuous-discrete (2Dcd) systems with boundary conditions.

This class is the continuous-discrete counterpart of Attasi’s discrete-time 2D model. Its state space representation contains an equation which is differential with respect to one variable and of difference type with respect to the second one.

In Section 2 the state-space representation of 2Dcd systems with well- posed boundary conditions is given. Formulas for the states and the input- output maps of these systems are provided. Section 3 is devoted to semiseparable kernels and the existence of a realization of a semiseparable kernel is proved.

The minimality of 2Dcd acausal systems is discussed in Section 4 and a necessary and sufficient condition of minimality is expressed by means of suitable controllability and observability Gramians.

The adjoint of a 2D acausal system is defined in Section 5 and the relationship between the input-output operator of a 2Dcd system and its adjoint is emphasized.

2. 2D ACAUSAL CONTINUOUS-DISCRETE SYSTEMS WITH WELL-POSED BOUNDARY CONDITIONS

Consider the time set

T = [a1, b1]× {a₂, a2+ 1, . . . , b2}, where [a1, b1]⊂Rand a2, b2 ∈Z.

Definition 2.1. A two-dimensional acausal continuous-discrete (2Dcd) system is an ensemble

Σ = (A₁(t, k), A₂(t, k), B(t, k), C(t, k), D(t, k), N₁, N₂, M₁, M₂)

where, for any (t, k) ∈ T, A₁(t, k), A₂(t, k) are n×n real commutative matrices while B(t, k), C(t, k), D(t, k) are real n×m, p×n, p ×m matrices;

N1, N2, M1, M2aren×nreal matrices such that the matrixQ=

N1 N2

M₁ M₂

is nonsingular; A1(·, k) and A2(·, k) are assumed to be integrable on [a₁, b1], B(·, k) and C(·, k) are square integrable and D(·, k) is measurable and essen- tially bounded for any k∈ {a₂, a2+ 1, . . . , b2}.

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The state space representation of Σ is given by the equations

˙

x(t, k+ 1) =A1(t, k+ 1)x(t, k+ 1) +A2(t, k) ˙x(t, k)−

(2.1)

−A₁(t, k)A₂(t, k)x(t, k) +B(t, k)u(t, k), (2.2) y(t, k) =C(t, k)x(t, k) +D(t, k)u(t, k), where ˙x(t, k) = ^∂x_∂t(t, k),

(2.3) N1x(a1, a2) +N2x(b1, b2) =v, (2.4) z=M₁x(a₁, a₂) +M₂x(b₁, b₂).

We denote by Φ(t, t0;k) the (continuous) fundamental matrix ofA1(t, k) with respect to t ∈ [a1, b1], for any fixed k ∈ {a₂, a2 + 1, . . . , b2} and by F(t;k, k₀) the discrete fundamental matrix ofA₂(t, k), i.e.,

F(t;k, k₀) =

( [A2(t, k−1)A2(t, k−2)· · ·A2(t, k0)]⁻¹ fork > k0

In fork=k0

for any fixedt∈[a1, b1]. SinceA1(t, k) andA2(t, k) are commutative matrices, Φ(t, t₀;k) and F(s;l, l₀) are commutative matrices too.

Definition 2.2. A vectorx₀ ∈Rⁿ is said to be theinitial stateof Σ if (2.5) x(t, a2) = Φ(t, a1;a2)x0, x(a1, k) =F(a1;k, a2)x0

for any (t, k)∈T.

Obviously equations (2.5) yieldx(a1, a2) =x0. From [16, Proposition 2.3]

we have

Proposition2.3. The solution of(2.1)under conditions (2.5)is x(t, k) = Φ(t, a₁;k)F(a₁;k, a₂)x(a₁, a₂)+

(2.6)

+ Z t

a1

k−1

X

l=a2

Φ(t, s;k)F(s;k, l+ 1)B(s, l)u(s, l)ds.

Definition 2.4. The boundary condition (2.3) is said to be well-posed if the homogeneous problem corresponding to (2.1) and (2.3) (i.e. with u ≡ 0 and v= 0) has the unique solution x= 0.

Proposition2.5. The boundary condition(2.3)is well-posed if and only if the matrix R=N1+N2Φ(b1, a1;b2)F(a1;b2, a2) is nonsingular.

Proof. By (2.6) with u≡0 we get

x(b1, b2) = Φ(b1, a1;b2)F(a1;b2, a2)x(a1, a2).

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Replace it in (2.3) with v= 0. It follows that (2.3) is well-posed if and only if the equation [N₁+N₂Φ(b₁, a₁;b₂)F(a₁;b₂, a₂)]x(a₁, a₂) = 0 has the unique so- lutionx(a₁, a₂) = 0, condition which is equivalent toRbeing nonsingular.

In the sequel we shall consider systems with well-posed boundary condition (2.3) and which verify (2.5). Moreover, the discrete-time character of Σ with respect to the variable kimposes that the matricesA2 depend only onk and A2(k) is nonsingular for any k∈ {a₂, a2+ 1, . . . , b2}.

In this case, fork < l, the discrete fundamental matrix of A₂ becomes F(k, l) = [A2(l−1)A2(l−2)· · ·A2(k+ 1)A2(k)]⁻¹.

Then the semigroup property F(k, l)F(l, i) = F(k, i) holds for any k, l, i ∈ {a₂, a₂+ 1, . . . , b₂}.

Definition 2.6. The matrix P = P_Σ = R⁻¹N₂Φ(b₁, a₁;b₂)F(b₂, a₂) is called the canonical boundary value operator of the system Σ with well-posed boundary condition.

Theorem2.7. The state of the system Σ determined by the control u : T →Rⁿ and by the input vector v∈Rⁿ is

x(t, k) = Φ(t, a1;k)F(k, a2)R⁻¹v−

(2.7)

− Z b1

a1

b2−1

X

l=a2

Φ(t, a₁;k)F(k, a₂)PΦ(a₁, s;b₂)F(a₂, l+ 1)B(s, l)u(s, l)ds+

+ Z t

a1

k−1

X

l=a2

Φ(t, s;k)F(k, l+ 1)B(s, l)u(s, l)ds.

Proof. We replace x(b₁, b₂) expressed by (2.6) in the boundary condition (2.3). We get

[N₁+N₂Φ(b₁, a₁;b₂)F(b₂, a₂)]x₀+ +N2

Z b1

a1

b2−1

X

l=a2

Φ(b1, s;b2)F(b2, l+ 1)B(s, l)u(s, l)ds=v, hence

(2.8) x0 =R⁻¹v−R⁻¹N2

Z b1

a1

b2−1

X

l=a2

Φ(b1, s;b2)F(b2, l+ 1)B(s, l)u(s, l)ds.

Replacing x0 = x(a1, a2) in (2.6), equation (2.7) follows by using the semigroup property, i.e. Φ(b1, s;b2) = Φ(b1, a1;b2)Φ(a1, s;b2) and F(b2, l+ 1) = F(b2, a2)F(a2, l+ 1).

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Theorem 2.8. The input-output map of the 2Dcd system Σ is H : L²(T,R^m)×Rⁿ→L²(T,R^p)×Rⁿ, H(u, v) = (y, z), where

y(t, k) =C(t, k)Φ(t, a1;k)F(k, a2)R⁻¹v−

(2.9)

− Z b1

a1

b2−1

X

l=a2

C(t, k)Φ(t, a₁;k)F(k, a₂)PΦ(a₁, s;b₂)F(a₂, l+ 1)B(s, l)u(s, l)ds+

+ Z t

a1

k−1

X

l=a2

C(t, k)Φ(t, s;k)F(k, l+ 1)B(s, l)u(s, l)ds+D(t, k)u(t, k) and

z= [M₁+M₂Φ(b₁, a₁;b₂)F(b₂, a₂)]R⁻¹v−

(2.10)

−[M₁+M2Φ(b1, a1;b2)F(b2, a2)]P Z b1

a1

b2−1

X

l=a2

Φ(a1, s;b2)F(a2, l+1)B(s, l)u(s, l)ds+

+M2

Z b1

a1

b2−1

X

l=a2

Φ(b1, s;b2)F(b2, l+ 1)B(s, l)u(s, l)ds.

Proof. Formula (2.9) follows by replacing x(t, k) given by (2.7) in the output equation (2.2). Then, by replacing x(a₁, a₂) = x₀ (2.8) and x(b₁, b₂) given by (2.7) in (2.4) and using the semigroup property, we obtain

z=M1x(a1, a2) +M2x(b1, b2) =

=M1R⁻¹v−M1R⁻¹N2

Z b1

a1

b2−1

X

l=a2

Φ(b1, s;b2)F(b2, l+ 1)B(s, l)u(s, l)ds+

+M₂Φ(b₁, a₁;b₂)F(b₂, a₂)R⁻¹v−

−M₂ Z _b₁

a1

b2−1

X

l=a2

Φ(b₁, a₁;b₂)F(b₂, a₂)PΦ(a₁, s;b₂)F(a₂, l+ 1)B(s, l)u(s, l)ds+

+M₂ Z b1

a1

b2−1

X

l=a2

Φ(b₁, s;b₂)F(b₂, l+ 1)B(s, l)u(s, l)ds=

= [M1+M2Φ(b1, a1;b2)F(b2, a2)]R⁻¹v−

−M₁R⁻¹N₂Φ(b₁, a₁;b₂)F(b₂, a₂) Z b1

a1

b2−1

X

l=a2

Φ(a₁, s;b₂)F(a₂, l+1)B(s, l)u(s, l)ds−

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−M₂Φ(b₁, a₁;b₂)F(b₂, a₂)P Z b1

a1

b2−1

X

l=a2

Φ(a₁, s;b₂)F(a₂, l+ 1)B(s, l)u(s, l)ds+

+M₂ Z b1

a1

b2−1

X

l=a2

Φ(b₁, s;b₂)F(b₂, l+ 1)B(s, l)u(s, l)ds.

Now, (2.10) follows by Definition 2.6.

3. SEMISEPARABLE KERNELS

Let us consider the case A1 : [a1, b1] → R^n×n and A2 : {a₂, a2 + 1, . . . , b2} → R^n×n. Therefore the fundamental matrices of A1 and A2 are Φ(t, s) and F(k, l), respectively.

Assume thatv= 0. Then, by the semigroup property, Φ(t, s) = Φ(t, a₁)·

·Φ(a₁, s),F(k, l+ 1) =F(k, a₂)F(a₂, l+ 1), and (2.9) can be written as

y(t, k) =− Z t

a1

k−1

X

l=a2

C(t, k)Φ(t, a₁)F(k, a₂)PΦ(a₁, s)F(a₂, l+1)B(s, l)u(s, l)ds−

(3.1)

− Z b1

t b2−1

X

l=a2

C(t, k)Φ(t, a₁)F(k, a₂)PΦ(a₁, s)F(a₂, l+ 1)B(s, l)u(s, l)ds−

− Z t

a1

b2−1

X

l=k

C(t, k)Φ(t, a1)F(k, a2)PΦ(a1, s)F(a2, l+ 1)B(s, l)u(s, l)ds+

+ Z t

a1

k−1

X

l=a2

C(t, k)Φ(t, a₁)F(k, a₂)Φ(a₁, s)F(a₂, l+ 1)B(s, l)u(s, l)ds+

+D(t, k)u(t, k).

The sum of the first and the fourth integrals in (3.1) is equal to Z t

a1

k−1

X

l=a2

C(t, k)Φ(t, a1)F(k, a2)(I−P)Φ(a1, s)F(a2, l+ 1)B(s, l)u(s, l)ds, hence (3.1) can be written as

(3.2) y(t, k) = Z _b₁

a1

b2−1

X

l=a2

K(t, s;k, l)u(s, l)ds+D(t, k)u(t, k).

We denote by T(t, k) and T^∗(t, k) the setsT(t, k) = [a1, t)× {a₂, a2+ 1, . . . , k−1}and T^∗(t, k) =T\T(t, k), respectively.

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Definition 3.1. The function K(t, s;k, l) is called the kernel of equation (3.2). A kernel is said to be 2D semiseparable if it has the form

(3.3) K(t, s;k, l) =

( E₁(t, k)G₁(s, l) if (s, l)∈T(t, k),

−E₂(t, k)G2(s, l) if (s, l)∈T^∗(t, k).

If equation (3.2) is obtained from the input-output equation (3.1) of a system Σ, the kernel K is denoted by KΣ and is called the kernel of the system Σ.

Proposition3.2.The kernelk_Σof any2Dcdacausal system is2Dsemi- separable.

Proof. From (3.1) and (3.2) we get K_Σ(t, s;k, l) =

=

(C(t, k)Φ(t, a₁)F(k, a₂)(I−P)Φ(a₁, s)F(a₂, l+1)B(s, l) if (s, l)∈T(t, k)

−C(t, k)Φ(t, a₁)F(k, a₂)PΦ(a₁, s)F(a₂, l+ 1)B(s, l) if (s, l)∈T^∗(t, k) hence KΣ is 2D semiseparable.

Definition 3.3. Given a 2D semiseparable kernelK, a 2Dcd acausal system Σ is said to be a realizationof K ifK=K_Σ.

Proposition 3.4. For any 2D semiseparable kernel K there exists a realization of K.

Proof. If K has the form (3.3), a realization of K is the system Σ for whichA1=On,A2 =In,B(t, k) =

"

G₁(t, k) G2(t, k)

#

,C(t, k) = [E1(t, k) E2(t, k)], D = O_p^m ∈ R^p×m, N₁ =

"

I_n₁ 0

0 0

#

, N₂ =

"

0 0

0 I_n₂

#

with n₁, n₂ > 0, n1+n2 =nwhileM1andM2 are arbitrary. Obviously, Φ(t, s) =In,F(k, l) = I_n, hence R = N₁ +N₂ = I_n. It follows that this system has well-posed boundary condition and its canonical operator is P =R⁻¹N₂ =N₂.

Example 3.5. Let us consider the 2D continuous-discrete Wiener-Hopf equation

(3.4) y(t, k)− Z b1

a1

b2−1

X

l=a2

K(t−s, k−l)u(s, l)ds=Du(t, s), (t, s)∈T, where K(t, k) ∈ R^p×m for any (t, k) ∈ T. Assume that K(t, k) can be extended to a function ¯K(t, k) defined onR×Zwhich admits a proper rational 2D continuous-discrete Laplace transform (see [19]) of the form T(s, z) =

θ(s,z)

π1(s)π2(z) with θ(s, z) ∈ R^p×n[s, z], π1(s) ∈ R[s], π2(z) ∈ R[z]. Then, using

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the algorithm of minimal realization described in [17], we can determine the constant matrices A₁, A₂, B, C, D with A₁A₂ = A₂A₁ such that T_Σ(s, z) = C(s−A1)⁻¹(z−A2)⁻¹B+D. By considering the matrices N1 =I −P and N2=P, whereP is a suitable spectral projection andM1, M2 arbitrary matrices, we obtain a system Σ whose input-output map (3.1) coincides with the 2Dcd Wiener-Hopf equation (3.4).

4. MINIMALITY OF 2D CONTINUOUS-DISCRETE ACAUSAL SYSTEMS

We shall now consider 2Dcd systems Σ having the state space representation (2.1)–(2.4) with A₁(t), A₂(k) and v= 0. The dimensiondim Σ of Σ is n (see Definition 2.1).

Definition4.1. A realization Σ of a 2D semiseparable kernelK is said to be minimalif dim Σ≤dim ˆΣ for any realization ˆΣ of K.

We introduce the controllability and the observability Gramians of the system Σ as

C(Σ) = Z b1

a1

b2−1

X

l=a2

Φ(a₁, s)F(a₂, l)B(s, l)B(s, l)^TF(a₂, l)^TΦ(a₁, s)^Tds,

O(Σ) = Z b1

a1

b2−1

X

l=a2

Φ(s, a₁)^TF(l, a₂)^TC(s, l)^TC(s, l)F(l, a₂)Φ(s, a₁)ds.

The canonical boundary value operator of Σ is

P = [N₁+N₂Φ(b₁, a₁)F(b₂, a₂)]⁻¹N₂Φ(b₁, a₁)F(b₂, a₂).

We shall extend [9, Theorem 3.1] to the case of 2Dcd acausal systems.

Theorem4.2.A realizationΣof the2Dsemiseparable kernelK is minimal if and only if

(4.1) Im[C(Σ) PC(Σ)] =Rⁿ,

(4.2) Ker

"

O(Σ) O(Σ)P

#

={0},

(4.3) KerO(Σ)⊂ImC(Σ).

Proof. Necessity. Let us consider an arbitrary direct sum decomposition Rⁿ = X1 ⊕X2 with n1 = dimX1, 0 < n1 < n and the corresponding partitions Φ(a₁, t)F(a₂, k)B(t, k) =

"

B1(t, k) B₂(t, k)

#

,C(t, k)Φ(t, a₁)F(k, a₂) =

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[C₁(t, k) C₂(t, k)], D(t, k) =

"

D₁(t, k) D₂(t, k)

# , P =

"

P₁₁ P₁₂ P₂₁ P₂₂

#

. Let us denote by Σ¹ the 2Dcd system with well-posed boundary conditions determined by the matrices A¹₁ =On1, A¹₂ =In1, B1(t, k), C1(t, k), D1(t, k), N₁¹ =I −P11, N₂¹ =P₁₁.

If condition (4.1) is not fulfilled, we take X₁ = Im [C(Σ) PC(Σ)] and X₂ =X₁^⊥. If (4.2) is not fulfilled, we take X₂ = Ker

"

O(Σ) O(Σ)P

#

and X₁ = X₂^⊥. If (4.3) is not fulfilled,X2 is a subspace of KerC(Σ) such that Im C(Σ) + KerO(Σ) = Im C(Σ)⊕X₂ and X₁ is the complement of X₂ in Rⁿ which includes ImC(Σ). As in [9, Lemmas 3.2–3.4] we can prove that in all these casesK_Σ₁ =K, hence Σ₁ is a realization ofK and dim Σ₁ =n₁ < n= dim Σ, i.e., Σ is not minimal.

Sufficiency. If conditions (4.1)–(4.3) hold for some realization Σ of K, we consider the direct sum decomposition of the state space Rⁿ given by X₂ = KerO(Σ), X₁⊕X₂ = ImC(Σ) and X₁⊕X₂⊕X₃ =Rⁿ. Following the lines of [9, Theorem 3.1], we can prove that dim Σ≤dim ˆΣ for any realization Σ ofˆ K.

5. ADJOINT 2D CONTINUOUS-DISCRETE ACAUSAL SYSTEMS

Consider the system Σ = (A1(t), A2(k), B(t, k), C(t, k), D(t, k), N1, N2) with well-posed boundary conditions, given by (2.1)–(2.3), where the matrix A2(k) is nonsingular ∀k ∈ {a₂, a2+ 1, . . . , b2}. In order to cover the case of systems over C, we shall denote by A^∗ the adjoint of a matrix A. Obviously, ifA is a real matrix, A^∗ =A^T. By A^−∗ we denote (A^∗)⁻¹.

Definition5.1. The 2Dcd system Σ having the state space representation x(t, k˙˜ + 1) =−A₁(t)^∗x(t, k˜ + 1) +A2(k)^−∗x(t, k)+˙˜

(5.1)

+A1(t)^∗A2(k)^−∗x(t, k)˜ −C(t, k)^∗u(t, k),˜

˜

y(t, k) =B(t, k)^∗x(t, k) +˜ D(t, k)^∗u(t, k),˜ (5.2)

˜

x(a₁, a₂) =N₁^∗λ, x(b˜ ₁, b₂) =−N₂^∗λ (5.3)

is called theadjoint of Σ.

Therefore, the system ˜Σ is characterized by the matrices ˜A₁(t) =−A₁(t)^∗, A˜2(k) =A2(k)^−∗ (= (A2(k)^∗)⁻¹), ˜B(t, k) =−C(t, k)^∗, ˜C(t, k) =B(t, k)^∗. We shall consider for ˜Σ boundary conditions of the form (2.5), corresponding to

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the matrices ˜A1(t) and ˜A2(k). We shall use the notation

♦ Z t

a1

k−1

X

l=a2

def= Z b1

a1

b2−1

X

l=a2

− Z t

a1

k−1

X

l=a2

.

Theorem5.2. The input-output map of the adjoint systemΣ˜ is the operator T˜:L²(T,R^p)→L²(T,R^m)×Rⁿ given by T˜(˜u) = (˜y,λ), where˜

˜

y(t, k)^∗ = (5.4)

=− Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, a1)F(l, a2)PΦ(a1, t)F(a2, k)B(t, k)ds+

+♦ Z _t

a1

k−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, t)F(l, k)B(t, k)ds+ ˜u(t, k)^∗D(t, k) and

(5.5) λ^∗= Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, a1)F(l+ 1, a2)R⁻¹ds, where R=N1+N2Φ(b1, a1)F(b2, a2) andP =R⁻¹N2Φ(b1, a1).

Proof. The fundamental matrices of A₁ and ˜A₁ = −A^∗₁ are related by Φ_A_˜

1(t, s) = Φ_A₁(s, t)^∗. For k > l, the (discrete) fundamental matrix of ˜A₂ = (A^∗₂)⁻¹ is

FA˜2(k, l) = [ ˜A2(k−1) ˜A2(k−2)· · ·A˜2(l)] =

= [(A2(k−1)^∗)⁻¹(A2(k−2)^∗)⁻¹· · ·(A2(l)^∗)⁻¹] =

= ([A₂(k−1)A₂(k−2)· · ·A₂(l)]⁻¹)^∗ =F_A₂(l, k)^∗ and we can similarly prove that F_A_˜

2(k, l) =F_A₂(l, k)^∗ fork < l.

By applying (2.6) to the system ˜Σ we obtain

˜

x(t, k)^∗= ˜x(a₁, a₂)^∗Φ(a₁, t)F(a₂, k)−

(5.6)

− Z t

a1

k−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, t)F(l+ 1, k)ds.

By (5.3) and (5.6) we have

−λ^∗N2 = ˜x(b1, b2)^∗ = ˜x(a1, a2)^∗Φ(a1, b1)F(a2, b2)−

(5.7)

− Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, b₁)F(l+ 1, b₂)ds.

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We replace ˜x(a1, a2)^∗ by λ^∗N1 and postmultiply the equality obtained by Φ(b₁, a₁)F(b₂, a₂). By the semigroup property we get

λ^∗[N₂+N₁Φ(b₁, a₁)F(b₂, a₂)] = Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, a₁)F(l+ 1, a₂)ds, which yields (5.5). Now, we postmultiply again (5.7) by Φ(b₁, a₁)F(b₂, a₂) and replace λ^∗ by (5.5). Since P =R⁻¹N2Φ(b1, a1)F(b2, a2), we obtain

˜

x(a1, a2)^∗ = Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, a1)F(l+ 1, a2)ds−

− Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, a1)F(l+ 1, a2)Pds.

Replacing ˜x(a1, a2)^∗ in (5.6), we obtain the state of ˜Σ, namely,

˜

x(t, k)^∗ = Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, t)F(l+ 1, k)−

(5.8)

− Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, a1)F(l+ 1, a2)PΦ(a1, t)F(a2, k)ds−

− Z b1

a1

b2−1

X

l=a2

˜

u(s, l)^∗C(s, l)Φ(s, t)F(l+ 1, k)ds.

Now, (5.4) follows by replacing ˜x(t, k)^∗ given by (5.8) in the output equation (5.2).

Conclusion. A class of 2D continuous-discrete acausal systems with well- posed boundary conditions has been considered. The existence of realizations for semiseparable kernels, as well as minimality conditions and properties of the adjoint systems have been studied. The study can be continued to inves- tigate related topics such as similarity, reduction and irreducibility, controllability, observability, generalized systems etc.

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Received 20 December 2007 “Politehnica” University of Bucharest Department of Mathematics I

Splaiul Independent¸ei 313 060032 Bucharest, Romania

vprepelita@mathem.pub.ro