STABILITY OF A CLASS OF MULTIDIMENSIONAL CONTINUOUS-DISCRETE LINEAR SYSTEMS

on the occasion of his 70th birthday

STABILITY OF A CLASS OF MULTIDIMENSIONAL CONTINUOUS-DISCRETE LINEAR SYSTEMS

VALERIU PREPELIT¸ ˘A

A class of multidimensional linear systems is considered. The states, controls and outputs are functions ofqcontinuous-time andrdiscrete-time variables. The solution of a multiple differential-difference equation is obtained and used to derive a formula for the state of the system and for its input-output map. The stability of these systems is studied and a necessary and sufficient condition of asymptotic stability is given, as well as a suitable Lyapunov equation. The stability property is characterized by using some Lyapunov functions.

AMS 2000 Subject Classification: 93C35, 93C05, 93D20, 93D30.

Key words: multidimensional continuous-discrete linear system, input-output map, stability, Lyapunov equation, Lyapunov function.

1. INTRODUCTION

The past three decades knew a continually growing interest in multi- dimensional (nD) systems, which become a distinct and important branch of system theory. Its development is determined by the wide variety of applica- tions in various domains, including control and image processing, computer tomography, geophysics, seismology, etc.

The history of this approach has begun with the study of 2D systems, the commonly used models being those of Roesser [14], Fornasini-Marchesini [3] and the subclass of the Attasi model [1]. Recently, a number of researchers have focussed on models with mixed structure, their behaviour depending on two variables, one continuous and the other one discrete [5], [7], [8], [10], [11], [12], [13]. They were used as models in the study of linear repetitive processes [2], [4], [15] with practical applications in long-wall coal cutting and metal rolling, or in iterative learning control synthesis [9].

In the present paper a class of time-variable multidimensional continuous- discrete systems of Attasi type is studied. In Section 2 the state space rep- resentation is given and formulas for the state and general response of the system are derived.

MATH. REPORTS9(59),1 (2007), 87–98

Section 3 is devoted to the study of stability of time-invariant multi- dimensional continuous-discrete systems. Necessary and sufficient conditions for asymptotic stability are obtained, which extend the conditions for 1D continuous-time and 1D discrete-time systems, including a suitable Lyapunov function. A necessary condition is expressed by using a generalized Lyapunov equation.

We shall use the following notation: q∈N and r ∈Ndenote the num- ber of continuous and discrete variables, respectively; a function x(t1, . . . , tq; k₁, . . . , kr), ti ∈ R, ki ∈ Z will be sometimes denoted by x(t;k), where t= (t₁, . . . , tq), k = (k₁, . . . , kr); m form ∈N^∗ denotes the set {1,2, . . . , m}

and P(m) the family of all subsets of m. If τ ={i₁, . . . , il} is a subset ofm, then|τ|:=land ˜τ :=m\τ; fori∈m, ˜i:=m\{i}and ˜i:={i+1, . . . , m}. The notation (τ, δ) ⊂(q, r) means thatτ andδ are subsets ofq andr, respectively, and (τ, δ) = (q, r). For τ ={i₁, . . . , il} and δ ={j₁, . . . , jh} the operators _∂τ^∂ andσδ are defined by

∂

∂τx(t;k) = ∂^l

∂ti₁. . . ∂ti_lx(t;k), σδx(t;k) =x(t;k+eδ), where eδ = ej₁ +· · ·+ej_h, ej = (0, . . . ,0

j−1

,1,0, . . . ,0) ∈ R^r; when τ = q and δ=r we write∂/∂τ =∂/∂tand σδ=σ.

If Ai, i= m is a family of matrices, then we agree that

i∈∅Ai = 0 and

i∈∅Ai=I. IfP is a positive definite matrix, one writes P >0.

2. STATE SPACE REPRESENTATION

The time set of the Attasi-type multidimensional system isT =R^q×Z^r, q, r∈N.

Definition2.1. A (q, r)-Dcontinuous-discrete systemis a set Σ = (Aci, Adj, B, C, D) with Aci(t;k), i∈q, andAdj(t;k), j∈r, commuting n×nmatrices

∀t∈ R^q and ∀k∈ Z^r; B(t;k), C(t;k), D(t;k) are respectively n×m, p×n andp×m real matrices, all of them being continuous with respect tot∈R^q for any k∈Z^r; the state equation is

∂

∂tσx(t;k) =

(τ,δ)⊂(q,r)

(−1)^{q+r−|τ|−|δ|−1}× (2.1)

×

i∈˜τ

Aci(t;k)

j∈δ˜

Adj(t;k) ∂

∂τσδx(t;k) +B(t;k)u(t;k)

while the output equation is

(2.2) y(t;k) =C(t;k)x(t;k) +D(t;k)u(t;k),

wherex(t;k) =x(t₁, . . . , tq;k₁, . . . , kr) ∈Rⁿ is the state, u(t;k) ∈R^m is the inputand y(t;k)∈R^p is the output.

Forτ ={i₁, . . . , il} ⊂q,δ ={j₁, . . . , jh} ⊂r and ti ∈R,i∈τ, t⁰_i ∈R, i∈τ,kj ∈Z,j∈δ,k⁰_j ∈Z,j∈˜δ, we use the notation

x(tτ, t⁰_τ˜;kδ, k_δ⁰_˜) :=x(t⁰₁, . . . , t⁰_i1−1, ti₁, t⁰_i1+1, . . . , t⁰_il−1, ti_l, t⁰_il+1, . . . , t⁰q; k₁⁰, . . . , k_j⁰₁₋₁, kj₁, k_j⁰₁₊₁, . . . , k_j⁰_h−1, kj_h, k_j⁰_h+1, . . . , k⁰_jr).

Let Φ_i(ti, t⁰_i;t˜i;k) be the (continuous) fundamental matrix of Aci(t;k) with respect to the variables ti, t⁰_i, i ∈ q, i.e., the unique matrix solution of the system ^∂Y_∂t

i(t;k) =Aci(t;k)Y(t;k), Y(t_˜i, t⁰_i;k) =I for anytl∈R,l∈˜iand k∈R^r. IfAci is a constant matrix, then Φ_i(ti, t⁰_i;t˜i, k) = e^{Aci(ti−t0i)}.

The discrete fundamental matrix Fj(t;kj, k_j⁰;k_˜j) of the matrix Adj(t;k) is defined by

Fj(t;kj, k⁰j;k˜j) =

=

Adj(t;kj−1, k_˜j)Adj(t;kj −2, k_˜j). . . Adj(t;k_j⁰, k_˜j) forkj > k_j⁰

In fork=k_j⁰,

for any kh ∈Z,h ∈˜j and t∈R^q.

Fj(t;kj, k_j⁰;k˜j) is the unique matrix solution of the difference system Y(t;kj+ 1, k_˜j) =Adj(t;k)Y(t;kj, k_˜j), Y(t;k⁰j;k_˜j) =I.

IfAdj is a constant matrix, thenFj(t;kj, k⁰j;k˜j) =A^k_dj^j−kj0.

Definition2.2. The vectorx₀ ∈Rⁿ is called aninitial stateof the system Σ if

(2.3) x(tτ, t⁰_˜τ;kδ, k⁰_˜δ) =

i∈δ

Φ_i(ti, t⁰_i;t_˜i;k)

j∈δ

Fj(t;kj, k_j⁰;k_˜j)

x₀ for any (τ, δ)⊂(q, r). Equations (2.3) are called initial conditions of Σ.

Proposition2.3. The solution of the initial value problem

∂

∂tσx(t;k) =

(τ,δ)⊂(q,r)

(−1)^{q+r−|τ|−|δ|−1}× (2.4)

×

i∈˜τ

(σδAci(t;k))

j∈δ˜

Adj(t, k) ∂

∂τσδx(t;k) +f(t;k)

with initial conditions(2.3) is x(t;k) =

q i=1

Φi(ti, t⁰i;t⁰_i−1, t_˜¯i;k) r

j=1

Fj(t⁰;kj, k⁰j;k_j⁰₋₁, k_˜¯j)

x0+ (2.5)

+ _t1

t⁰₁

. . . _tq

t⁰_q k₁−1 l₁=k⁰₁

. . .

kr−1 l_r=k_r⁰

q i=1

Φ_i(ti, si;s_i−1, t_˜¯i;k)

×

× r

j=1

Fj(s;kj, lj+ 1;l_j−1, k_˜¯j)

f(s;l)ds₁. . .dsq.

Here, s = (s₁, . . . , sq), l = (l₁, . . . , lr) and if, for instance, i = 1 then the corresponding variable t_i−1 = t⁰_∅ lacks; f : R^q×Z^r → Rⁿ is a continuous function.

Proof. We shall first prove (2.5) for the case q =r = 1, that is, for the equation

(2.6) ∂

∂tx(t;k+ 1) =

=Ac(t;k+1)x(t;k+1)+Ad(t;k) ∂

∂tx(t;k)−Ac(t;k)Ad(t;k)x(t;k)+f(t;k) with initial conditions

(2.7) x(t;k₀) = Φ(t, t₀;k₀)x₀, x(t₀;k) =F(t₀;k, k₀)x₀,

where Φ(t, t₀;k₀) is the fundamental matrix of Ac(t;k) and F(t;k, k₀) is the discrete fundamental matrix ofAd(t;k).

Let us consider the vector (2.8) z(t;k) = ∂

∂tx(t;k)−Ac(t;k)x(t;k).

From (2.7) and the first property of the fundamental matrix we have z(t;k₀) = ∂

∂tx(t;k₀)−Ac(t;k₀)x(t;k₀) =

= ∂

∂tΦ(t;t₀, k₀)x₀−Ac(t;k₀)Φ(t;t₀, k₀)x₀ = 0,

hencez(t;k₀) = 0,∀t≥t₀. Equation (2.6) can be written as the 1D difference equation z(t;k+ 1) = Aj(t;k)z(t;k) + f(t;k) whose solution, given by the discrete-time variation of parameter formula, is z(t;k) = F(t;k, k0)z(t, k0) +

k−1 l=k₀

F(t;k, l+ 1)f(t, l). Hence

(2.9) z(t, k) =

k−1

l=k0

F(t;k, l+ 1)f(t, l)

sincez(t, k0) = 0. But equation (2.8) can be written as the differential equation

∂

∂tx(t;k) =Ac(t;k)x(t;k) +z(t;k) and by the variation of parameter formula its solution is

x(t;k) = Φ(t, t₀;k)x(t₀;k) + t

t₀

Φ(t, s;k)z(s;k)ds.

By replacingx(t₀;k) and z(s;k) by their expressions (2.7) and (2.9) we get (2.10) x(t;k) = Φ(t, t₀;k)F(t₀;k, k₀)x₀+

_t

t₀ k−1

l=k₀

Φ(t, s;k)F(s;k, l+1)f(s, l)ds hence formula (2.5) holds forq= 1, r= 1.

Assume that (2.5) holds for some q, r ∈ N and consider the equation (2.4) withq+ 1 instead ofq. Introduce the function

z(t₁;t₂, . . . , tq+1;k₁, . . . , kr) = ∂x

∂t₁(t₁, t₂, . . . , tq+1;k₁, . . . , kr)− (2.11)

−Ac1(t₁, t₂, . . . , tq+1;k₁, . . . , kr)x(t₁, t₂, . . . , tq+1;k₁, . . . , kr). Then equation (2.4) of order (q+ 1, r) can be written as a (q, r) equation of the same type withzinstead ofxand initial conditions (2.3) that give zero initial conditions forz. By the induction assumption the corresponding solution is

z(t₁;t₂, . . . , tq+1;k₁, . . . , kr) = (2.12)

= _t2

t⁰₂

. . . _tq+1

t⁰_q+1 k₁−1 l₁=k₁⁰

. . .

k_r−1 lr=k⁰_r

q+1 i=2

Φi(ti, si;t₁;s_i−1\{1}, t_˜¯i;k

×

× r

j=1

Fj(t₁;s₂, . . . , sq+1;kj, lj+ 1;l_j−1;k_˜¯j

f(s;l)ds₂. . .dsq+1.

But the variation of parameter formula yields the solution of (2.11), that is, x(t1, t2, . . . , tq+1;k1, . . . , kr) =

= Φ₁(t₁, t⁰₁;t₂, . . . , tq+1;k₁, . . . , kr)x(t⁰₁;t₂, . . . , tq+1;k₁, . . . , kr)+

+ _t1

t⁰₁

Φ₁(t₁, s₁;t₂, . . . , tq+1;k₁, . . . , kr)z(s₁;t₂, . . . , tq+1;k₁, . . . , kr)ds₁ and, by replacingzfrom (2.12), we get formula (2.5) for the case (q+ 1, r).

Similarly, one can derive the case (q, r+ 1) from the case (q, r), hence we proved (2.5) by induction for anyq, r ∈N.

Proposition 2.4. The state of the system Σ determined by the initial state x₀ ∈Rⁿ and the control u is

x(t;k) = ^q

i=1

Φi(ti, t⁰i;t⁰_i−1, t_˜¯i;k) r

j=1

Fj(t⁰;kj, k⁰j;k_j⁰₋₁, k_˜¯j)

x0+ (2.13)

+ _t1

t⁰₁

. . . _tq

t⁰_q k₁−1 l₁=k⁰₁

. . .

kr−1 l_r=k_r⁰

q i=1

Φ_i(ti, si;s_i−1, t_˜¯i;k)

×

× r

j=1

Fj(s;kj, lj+ 1;l_j−1, k_˜¯j)

B(s;l)u(s;l)ds1. . .dsq.

Proof. Equation (2.1) has the form (2.4) with f(t;k) = B(u;k)u(t;k) and (2.13) follows from (2.5) by replacingf(t;k).

Corollary2.5. If Aci, Adj and B are constant matrices, then the state of Σdetermined by the control u and the initial state x0 with initial moments t⁰_i = 0, i∈q, k_j⁰ = 0, j∈r, is

(2.14) x(t;k) = q

i=1

e^Aciti r

j=1

A^k_dj^j

x₀+

+ _t1

0 . . . _tq

0

q i=1

e^Aci(ti−si) k₁−1

l₁=0

. . .

k_r−1 lr=0

r j=1

A^k_dj^j−lj−1

B(s;l)u(s, l)ds1. . .dsq. By replacing the statex(t;k) given by (2.13) in the output equation (2.2), we obtain

Proposition2.6. The input-output map of the (q, r)-D continuous-dis- crete systemΣis

y(t;k) =C(t;k) q

i=1

Φi(ti, t⁰_i;t⁰_i−1, t_˜¯i, k) r

j=1

Fj(t⁰;kj;k_j⁰₋₁, k_˜¯j)

x₀+ (2.15)

+ _t1

t⁰₁

. . . _tq

t⁰_q k₁−1 l₁=k₁⁰

. . .

kr−1 lr=k⁰_r

C(t;k) q

i=1

Φ_i(ti, si;s_i−1, t_˜¯i;k)

×

× r

j=1

Fj(s;kj, lj;l_j−1, k_˜¯j)

B(s;l)u(s;l)ds₁. . .dsq+D(t;k)u(t;k).

Similarly to Corollary 2.5, we can derive from (2.15) the input-output map of a time-invariant system Σ.

3. STABILITY OF(q, r)-D CONTINUOUS-DISCRETE SYSTEMS

We consider the time-invariant (q, r)-D continuous-discrete system Σ with state equation (2.1). For the control u(t;k) ≡0, from (2.14) we deduce the “free” state of Σ determined by the initial statex₀∈Rⁿ as

(3.1) x(t;k) = q

i=1

e^Aciti r

j=1

A^k_dj^j

x0.

We shall extend the results concerning the stability of 1D continuous-time and discrete-time systems to (q, r)-D systems. Ifti ≥0,∀i∈q, we write t≥0.

Definition 3.1. i) The system Σ = ((Aci)i∈q,(Adj)j∈r) is stableif∀ε >0,

∃δ >0 such that for anyx₀ ∈Rⁿwith x₀ < δ we have x(t;k) < ε,∀t≥0 and∀k≥0.

ii) The system Σ isunstableif it is not stable.

iii) Σ is asymptotically stable if it is stable and for any x₀ ∈ Rⁿ the corresponding solution x(t;k) verifies

t_ilim→∞x(t₁, . . . , ti, . . . , tq;k₁, . . . , kj, . . . , kr) = 0 and

klimj→∞x(t₁, . . . , ti, . . . , tq;k₁, . . . , kj, . . . , kr) = 0

∀i∈q and∀j∈r.

We denote byσ(A) the spectrum of a matrixA ∈R^n×n, byma(λ) and mg(λ), respectively, the algebraic and geometric multiplicities of an eigenvalue λ∈σ(A), byC⁻,C⁰andC⁺the set of the complex numbers with respectively negative, zero and positive real parts and by D⁻₁,D⁰₁ and D⁺₁ the set of the complex numbers with modulus less than, equal to or greater than 1.

Theorem3.2. i)The system Σ is asymptotically stable if and only if

i∈q

σ(Aci)⊂C⁻ and

j∈r

σ(Adj)⊂D⁻₁.

ii) If

i∈q

σ(Aci)

∩C⁺

∪

j∈r

σ(Adj)

∩D⁺₁

=∅,

then Σ is unstable.

iii)If

i∈q

σ(Aci)⊂C⁻∪C⁰,

j∈r

σ(Adj)⊂D⁻₁ ∪D⁰₁ and ma(λ) =mg(λ) for any

λ∈

i∈q

σ(Aci) ∩C⁰

∪

j∈r

σ(Adj) ∩D⁰₁

, then Σ is stable (but not asymptotically stable).

iv)If∃λ∈ (

i∈q

σ(Aci))∩C⁰

∪ (

j∈r

σ(Adj))∩D⁰₁

withma(λ)> mg(λ), then Σ is unstable.

Proof. Using the Jordan canonical form of the matrices Aci and Adj

we deduce that the elements of the matrix q

i=1e^Aciti r

j=1A^kj_dj

, hence by (3.1) those of the vector solution x(t;k), are linear combinations of product functions of the form t^α_iⁱe^βiti and k^γ_j^jµ^kj, where λi = αi + iβi ∈ σ(Aci) and µ∈σ(Adj). If Reλ <0 and |µ|<1, both functions tend to 0 as ti → ∞and kj → ∞, hence lim

ti→∞

kj→∞

x(t;k) = 0, i.e., Σ is asymptotically stable.

If for some eigenvaluesλwe have Reλ >0 or|µ|>1, then these functions converge to ∞, hence Σ is unstable. In case iii), ma(λ) =mg(λ) implies that these functions have the form e^iβt or µ^kj, |µ|= 1, hence 1 = |e^iβt| →/ 0 and

|µ^kj|= 1. It follows that all these functions are bounded, hence Σ is stable.

In case (iv), the elements ofx(t;k) contain functions of the formt^αle^iβltl or k_j^γjµ^kj with αl ≥ 1, γj ≥ 1. Therefore these functions have the limit ∞, i.e., Σ is unstable.

Remark 3.3. It follows from Theorem 3.2i that it is possible to check the asymptotic stability of Σ by applying the Routh-Hurwitz criterion to each matrixAci and the same criterion, via a linear fractional map, to the matri- cesAdj.

A necessary condition of asymptotic stability can be obtained by intro- ducing a generalized Lyapunov equation.

Theorem 3.4. If the system Σ = ((Aci)i∈q,(Adj)j∈r) is asymptotically stable, then for any positive definite matrix Qthe equation

(3.2)

τ∈P(q)

δ∈P(r)

(−1)^q−|δ|−1

j∈δ

A^Tdj

i∈τ

A^Tci

P+

+P

i∈τ

Aci

j∈δ

Adj

=−Q has a unique positive solution P.

Proof. Equation (3.2) can be written as (3.3) A^TdrP1Adr−P1 =−Q, where

(3.4) −P1 =

τ∈P(q)

δ∈P(r−1)

(−1)^q−|δ|−1

j∈δ

A^Tdj

i∈τ

A^Tci

P+

+P

i∈τ

Aci

j∈δ

Adj

.

Since σ(Adr)⊂ D⁻₁, for anyQ > 0 the 1D discrete Lyapunov equation (3.3) has the unique solution

(3.5) P₁ =

∞ kr=0

(A^T_dr)^krQ(Adr)^kr >0.

By repeating this procedure for equation (3.4) (which is similar to (3.2)) we obtain a chain of 1D discrete Lyapunov equations

A^T_d,r−jPj+1Ad,r−j−Pj+1=−Pj, Pj >0, j∈r−1,

and since σ(Ad,r−j) ⊂ D⁻₁, their positive definite solutions Pj are similar to (3.5).

Finally, (3.2) becomes an equation of the typeq >0,r= 0, namely,

τ∈P(q)

(−1)^q−1

i∈τ

A^Tci

P +P

i∈τ

Aci

=−Pr

which can be written as an 1D Lyapunov equation (3.6) A^TcqW₁+W₁Acq =−Pr, where

(3.7) −W1 =

τ∈P(q−1)

(−1)^q−2

i∈τ

A^Tci

P +P

i∈τ

Aci

.

Sinceσ(A^T_cq)⊂C⁻, (3.6) has the unique solution W1 =

_∞

0 e^ATcqtqPre^Acqtqdtq >0.

By repeatedly applying this procedure to (3.7), we finally get the equation A^Tc1P+P Ac1=−Wq−1

whose solution is

P = _∞

0 e^ATc1t1Wq−1e^Ac1t1dt1,

hence the unique solution (3.8) P =

_∞

0 . . . _∞

0

∞ k₁=0

. . . ^∞

k_r=0

e

q i=1A^T_cit_i

r j=1

(A^Tdj)^kj

Q r

j=1

A^k_dj^j

×

×e

q

i=1A_cit_idt₁. . .dtq>0 of (3.2).

Remark 3.5. The above necessary condition is not sufficient, as the fol- lowing counterexample shows. Letn= 1,q = 1,r = 1,Ac >0,Ad>1, hence Σ is unstable. Equation (3.2) has the form

A^TcA^TdP Ad+A^TdP AdAc−A^TcP −P Ac =Q,

i.e. (sincen= 1), 2Ac(A²_d−1)P =Q. ForQ >0 this equation has the solution P = _2A ^Q

c(A²_d−1) >0 but Σ is unstable.

If for someQ >0 the generalized Lyapunov equation (3.2) has a solution P > 0, we can only show relation (3.9) below concerning the eigenvalues of the matrices of Σ. Since Σ = ((Aci)_i∈q,(Adj)_j∈r) is a commuting family, there is a vector v ∈Cⁿ which is an eigenvector of every matrix in the family (see [6, Lemma 1.3.17]). Letλi,i∈q and µj,j∈r, be the eigenvalues (associated withv) of Aci and Adj,respectively.

Proposition 3.6. Assume that equation (3.2) has a solution P >0 for some Q > 0. If λi, i∈ q, and µj, j ∈r, are the eigenvalues of Aci and Adj

respectively, associated with a common eigenvector v, then

(3.9)

i∈q

Reλi

j∈r

(|µj|²−1)

>0.

Proof. We premultiply and postmultiply (3.2) respectively by v^∗ and v. Since Aciv = λiv, Adjv = µjv, v^∗A^T_ci = λiv^∗, v^∗A^T_dj = µ_jv^∗, λiλi = 2Reλi, µjµj =|µj|², from (3.2) we obtain

i∈q

(λi+λi)

j∈r

(µjµj−1)

v^∗P v=v^∗Qv, which implies (3.9) sincev^∗P v >0 andv^∗Qv >0.

Definition 3.7. Let P be a positive definite matrix of order n. The quadratic form V(x) = x^TP x is a Lyapunov function for the system Σ if for any initial state x₀ ∈ Rⁿ the corresponding solution x(t;k) of (2.1) (with u(t, k)≡0) verifies for anyt≥0 and k≥0 the conditions

i)V(x(t;k_j, k_˜j))< V(x(t;k_j, k_˜j)), ∀k_j> k_j, ∀j∈r;

ii) _∂t^∂

iV(x(ti, t_˜i;k))<0,∀i∈q.

Theorem 3.8. The system Σ is asymptotically stable iff there exists a Lyapunov function for Σ.

Proof. Assume that Σ is asymptotically stable. Let Q be a positive definite matrix and letP be the solution (3.8) of the Lyapunov equation (3.2).

Consider the quadratic formV(x) =x^TP x. It follows from (3.2), that (3.10) x^T(A^TciP+P Aci)x <0

for any x∈Rⁿ,x= 0, hence, by (3.1),

∂

∂tiV(x(t;k)) =x(t;k)^Te^ATciti(A^TciP+P Aci)e^Acitix(t;k)<0. From

(3.11) x^T(A^T_djP Adj−P)x <0, ∀x= 0, we get

V(x(t;kj+ 1))−V(x(t;kj, k˜j)) =x(t;k)^T(A^TjP Aj−P)x(t;k)<0, henceV(x) =x^TP x is a Lyapunov function for Σ.

Conversely, if there exists a Lyapunov function V(x) = x^TP x for Σ, it follows from (3.10) and (3.11) that σ(Aci) ⊂ C⁻, ∀i ∈ q and σ(Adj) ⊂ D⁻₁,

∀j∈r, hence Σ is asymptotically stable.

CONCLUSION

The state space representation of a class of time-varying (q, r)-D conti- nuous-discrete system was studied and some results concerning stability of these systems were obtained. This study can be continued by analyzing for this class other important concepts such as controllability, observability, reali- zations etc.

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Received 15 September 2006 “POLITECHNICA” University of Bucharest Department of Mathematics I

Splaiul Independent¸ei 313 060032 Bucharest, Romania

vprepelita@mathem.pub.ro