# Observability of a class of linear dynamic infinite systems on time scales/lineaarsete dunaamiliste lopmatumootmeliste susteemide klassi jalgitavus ajaskaaladel.

1. INTRODUCTIONThe theory of dynamical systems on time scales unifies theories of continuous-time and discrete-time systems. It also allows considering a dynamical system with time, which is partly continuous and partly discrete. Observability of finite-dimensional, stationary, linear systems on time scales was studied in [1]. It was shown that the standard Kalman criterion of observability holds for systems on an arbitrary time scale. We prove here a similar result for a class of infinite systems of linear differential equations with output. The systems are described by infinite matrices whose rows have only finitely many nonzero elements and the state space consists of all infinite sequences. In [2] we studied observability of discrete-time systems of this type and proved an extension of the Kalman criterion to infinite systems. As its consequence the following characterization of observability was derived: the system is observable if and only if every state variable can be expressed as a finite linear combination of the output and its time-shifts. In [3] we showed that the same holds for continuous-time infinite systems, with time-shifts replaced by time-derivatives. However, in that case, to have existence and uniqueness of solutions, we were forced to consider formal solutions defined by formal power series. As we consider here a more general situation of arbitrary time case, the difficulties that appeared in the continuous-time case must inevitably show up also in this paper. For this purpose we introduce formal polynomial series on time scales, formal solutions described by such series, and formal exponential matrices. Once we obtain the Kalman criterion of observability, we can restate all the results of [2] and [3], as the time scale becomes no longer essential.

The systems we study here are infinite-dimensional, but they are far from the mainstream of the infinite-dimensional system theory, which consists mainly of two areas: systems described by partial differential equations and systems on Banach or Hilbert spaces (see, e.g., [4]). The space of all real sequences is neither a Banach space nor a Hilbert space. It has the structure of a Frechet space, but the theory of systems on Frechet spaces is not developed yet. Most of the examples of infinite-dimensional systems lead to systems on Banach spaces, but the systems that we study here can also serve as models of certain dynamics. For example, the infinite extension of a finite-dimensional control system leads to a system whose state space consists of all infinite sequences. Such a system appears also as an effect of dicretization of the heat equation. Moreover, during this process we can obtain systems with time belonging to different time scales. If we discretize all the variables, we obtain a system with the time scale Z, while discretizing only the space variables, we end up with the time scale R. The results concerning observability obtained in this paper coincide in principal with the results obtained for systems on Banach spaces, but cannot be deduced from them. Moreover, since we study systems on concrete spaces, we can make the most of the structure of such systems. For example, the row-finiteness of the matrices that describe the systems allows for a nice characterization of observability as a possibility of expressing the state variables as linear combinations of finitely many [DELTA]-derivatives of the output functions.

2. INFINITE MATRICES

Let K be a nonempty countable set. Consider the countable product [R.sup.K] = [[PI].sub.k[member of]K] R as the set of all functions K [right arrow] R: If K = N, then [R.sup.N] is the linear space of all infinite sequences of real numbers represented by infinite columns x = [([x.sub.1], ..., [x.sub.i], ...).sup.T], [x.sub.i] [member of] R, I [member of] N. The space [R.sup.N] with the product topology (the Tikhonov topology) is metrizable. However, there is no norm for this topology, so [R.sup.N] is not a Banach space.

Recall that a linear topological space is called a Frechet space if it is metrizable, complete and locally convex. Note that [R.sup.N] is a Frechet space [5].

We have the following:

Proposition 2.1. [5] A function f : [R.sup.N] [right arrow] R is linear and continuous iff there is a finite set S [subset] N and a set of real numbers [{[a.sub.i]}i[member of]S] such that for all x [member of] [R.sup.N]: f(x) = [[SIGMA].sub.i[member of]S]] [a.sub.i][x.sub.i], where [x.sub.i] = x(i):

Let K = NxN: Then each element A [member of] [R.sup.NxN]; A : NxN [member of] (i; j)[??][a.sub.ij] [member of] R; is called an infinite matrix. We will denote it in the standard way [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] By I = [([[delta].sub.i,j]ij[member of]N], where [[delta].sub.ij] = 0 for i [not equal to] j, [[delta].sub.ii] = 1, we denote the identity matrix in [R.sup.NxN]: The set [R.sup.NxN] of all infinite matrices is a linear space over R with the standard operations.

Now let A = ([a.sub.ij]), B = ([b.sub.ij]) be infinite matrices. Then the product AB = [([c.sub.ik)i;k[member of]N] is well defined if the series [c.sub.ik] = [[SIGMA].sub.j[member of]N [a.sub.ij][b.sub.jk] is convergent for each (i; k) [member of ] N x N:

Definition 2.2. We say that A = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

a) row-finite if for each i [member of] N there is [alpha](i) [member of] N : [a.sub.ij] = 0 for j > [alpha](i);

b) column-finite if [A.sup.T] is row-finite;

c) lower-diagonal if [a.sub.ij] = 0 for j > i;

d) upper-diagonal if [a.sub.ij] = 0 for j < i.

Of course, a lower-diagonal matrix is a particular case of a row-finite matrix. The identity matrix I is row-finite, column-finite, and lower(upper)-diagonal.

Proposition 2.3.

1. The set of all infinite row-finite matrices forms an algebra over R with a unit--the identity matrix I.

2. The set of all infinite column-finite matrices forms an algebra over R with a unit.

Remark 2.4. The associativity of multiplication is the most essential property for row-finite (column-finite) matrices. It does not hold for all infinite matrices, however.

From Proposition 2.3 we get that if A is a row-finite matrix, then all powers [A.sup.k]; k [member of] [union] {0}; have the same property. Let us denote by [([A.sup.k]).sub.ij] the element at the ith row and jth column of matrix [A.sup.k]. Now we recall the definition of the matrix exponential of an infinite matrix. It was discussed in [6].

Definition 2.5. [7] Let A = [([a.sub.ij])i[member of]N;j[member of]N] be an infinite matrix and suppose that there is r > 0 such that for each i; j [member of] N the power series [[SIGMA].sup.[infinity].sub.k=0] [t.sup.k]/k! ([A.sup.k]).sub.ij] has the radius of convergence greater than or equal to r > 0: Then we define the matrix [e.sup.tA] by [([e.sup.tA]).sub.ij] = [[SIGMA].sup.[infinity].sub.k=0][t.sup.k]/k! [([A.sup.k].sub.ij].

Example 2.6. Let A be the upper-diagonal matrix of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the exponential matrix exists and is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 2.7. If an infinite matrix A is a lower-diagonal matrix, then the exponential matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is well defined [8].

In Section 4 we shall give the definition of formal exponential matrix on an arbitrary time scale.

3. CALCULUS ON TIME SCALES

In this section we give a short collection of the most important facts from the calculus on time scales. For further reading we refer to [9-11].

By a time scale, denoted here by T, we mean a nonempty closed subset of R: As the theory of time scale gives the way to unify continuous and discrete analysis, the standard cases of time scales are the following: T = R; T = Z or T = hZ; h > 0:

For t [member of] T we define the forward jump operator [sigma] and the graininess [mu] by:

a) [sigma](t) = inf {s [member of] T : s > t} and [sigma](sup T) = sup T if sup T < + [infinity];

b) [mu](t) = [sigma](t) - t.

Moreover, we have the backward operator [rho] defined by: [rho](t) = sup{s [member of] T: s < t} and [rho](inf T) = inf T if inf T > -[infity]. In the continuous-time case, when T = R, we have that for all t [member of] R : [sigma](t) = [sigma](t) = t and [mu](t) = 0: In the discrete-time case, for each t [member of] T = Z : [sigma](t) = t + 1; [sigma](t) = t + 1; [mu](t) = 1:

Additionally we define the set [T.sup.k] as: [T.sup.k] := T\([rho](sup T); sup T] if sup T < [infinity] and [T.sup.k] = T if sup T = [infinity]

For a function f : T [right arrow] R we define the delta derivative of f at t [member of] [T.sup.k], denoted by [f.sup.[DELTA](t), to be the number, if it exists, with the property that for all [epsilon] > 0 there is a neighbourhood U [subset] T of t such that for all s [member of] U it holds that

|f([sigma](t)) - f(s) - [f.sup.[DELTA]](t)([sigma](t)- s)|[less than or equal to] [epsilon]|[sigma](t)-s|.

Moreover, we say that the function f is delta differentiable on [T.sup.k] provided [f.sup.[DELTA]](t) exists for all t [member of] [T.sup.k]:

Example 3.1.

1. Let T = R; then [f.sup.[DELTA]](t) = f'(t) and f is delta differentiable iff it is differentiable in the ordinary sense.

2. Let T = Z, then [f.sup.[DELTA]](t) = f(t + 1) - f(t) and it always exists.

3. Let T be any time scale, then the delta derivative of [t.sup.2] is t+[sigma](t) and [(t+[sigma](t)).[DELTA] may not exist.

A function f : T [right arrow] R is called regulated if its right-side limits exist (finite) at all points t [member of] T with [sigma](t) = t and its left-side limits exist (finite) at all points t [member of] T with [rho](t) = t: A regulated function f has a pre-antiderivative, i.e. a continuous F with [F.sup.[DELTA]] = f on a set D where T\D is countable and does not contain any points t with [sigma](t) > t: Then the Cauchy integral can be defined as: [[integral].sup.b.sub.a]f(t)[DELTA].sup.t] = F(b) - F(a).

Example 3.2

1. Let T = R; then [[integral].sup.b.sub.a] f(t)[DELTA]t is equal to the usual Riemann integral.

2. Let T = Z, then [[integral].sup.b.sub.a] f(t)[DELTA]t = [[SIGMA].sup.b-1].sub.t=a] f(t).

Let [N.sub.0] mean the set of natural numbers with zero. For t [member of] R and k [member of] N we define

[t.sup.(0)] := 1 and [t.sup.(k)] := t(t - 1) ... (t - k + 1): (3.1)

If t [member of] Z and k [greater than or equal to] t + 1; k [member of] N, then [t.sup.(k)] = 0:

Let T be a time scale. We use the functions [h.sub.k] : T x T [right arrow] R; k [member of] N [union] (0); defined recursively as follows [9;10]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Let [h.sup.[DELTA].sub.k] (t; [t.sub.0]) denote, for each fixed [t.sub.0], the derivative of [h.sub.k](t; [t.sub.0]) with respect to t.

Then, for t [member of] [T.sup.[kappa]],

[h.sup.[DELTA].sub.0](t, [t.sub.0]) [equivalent to] 0 and [h.sup.[DELTA].sub.k] (t, [t.sub.0] = [h.sub.k-1](t, [t.sub.0]), k [greater than or equal to] 1. (3.3)

In the case T = Z we have that [h.sub.k](t; [t.sub.0]) = [(t-[t.sub.0]).sup.(k)/k! for all t; [t.sub.0] [member of] Z and k [member of] N: Moreover, if T = R, then [h.sub.k](t; [t.sub.0]) = [(t.[t.sub.0]).sup.k], and it gives a universal upper bound in the following way:

Theorem 3.3. [11] Let k [member of] N [union] {0}: Then for all t [greater than or equal to] [t.sub.0] it holds that

0 [less than or equal to] [h.sub.k](t; [t.sub.0]) [less than or equal to][(t - [t.sub.0].sup.k]/k! (3.4)

4. FORMAL POLYNOMIAL SERIES ON TIME SCALES

Let X be a linear space (over R). Then by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we denote the set of all infinite sequences of elements from X: Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] We define the addition of two sequences by ([a.sub.k]) + ([b.sub.k]) = ([a.sub.k] + [b.sub.k]) and the multiplication by scalars by [alpha]([a.sub.k]) = ([alpha][a.sub.k]); [alpha] [member of] R. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with these operations on sequences becomes a linear space over R.

Remark 4.1. If in X we have a metric d, then in the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we can introduce a metric defined by [rho](a; b) = [[SIGMA].sup.[infinity].sub.k=0] d([a.sub.k],[b.sub.k]) / 1+d([a.sub.k],[b.sub.k]), where a; b [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we can define the shift operator by D([a.sub.0]; [a.sub.1], ...) = ([a.sub.1]; [a.sub.2], ...). This operation is R-linear, i.e. D([alpha]a + [beta]b) = [alpha]D(a) + [beta]D(b), for [alpha], beta] [member of] R and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A formal power series in a real variable t with coefficients from a linear space X over R has the form of an infinite sum of monomials: [[SIGMA].sup.[infinity].sub.k=0] [a.sub.k][t.sup.k], where for every k [member of] [N.sub.0], [a.sub.k] [member of] X. Operations like sum, and multiplying by scalars are defined on series in a similar way as on sequences. Shifting of sequences from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is better represented as an operation on polynomial series given by [[SIGMA].sup.[infinity].sub.k=0] [a.sub.k] [t.sup.k]/k!, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the operation of formal differentiation of the series corresponds to the action of the shift operator on the sequence of coefficients of the series.

To do the same on an arbitrary time scale T, we use the generalized polynomials, defined by (3.2), in the definition of formal polynomial series on time scales.

Definition 4.2. Let T be a time scale and t; [t.sub.0] [member of] T: Let X be a linear space over R. By a formal polynomial series (on T, centred at [t.sub.0]) over X we mean a formal expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The sequence a is called the corresponding sequence of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Two formal polynomial series on the time scale T, at the point [t.sub.0] [member of] T, are equal if their corresponding sequences are equal, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The set of all formal polynomial series (on T, centred at [t.sub.0]) over X are denoted by X[[t; [t.sub.0]]].

Similarly as sequences from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the series like (4.1) can be added and multiplied by scalars in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

Proposition 4.3. The set of all formal polynomial series (on T, centred at [t.sub.0]) over the linear (metrizable) space X is a linear (metrizable) space (over R) with the addition and the multiplication by scalars defined by (4.2) and (4.3).

Given a formal polynomial series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we define its formal [DELTA]-derivative by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

Hence the formal c-derivative of the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a is the new formal polynomial series (on T, centred at [t.sub.0]) with the shifted corresponding sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 4.4. If we fix the value of t [member of] T, then the expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] becomes the series of elements from the space X. As we do not assume that X is a metric space, hence, in general, we cannot consider convergence of the series of elements. But if X is metrizable or if the sum is finite, we can investigate the convergence.

By Theorem 3.3, we have the following:

Corollary 4.5. Let T be any time scale and let [t.sub.0] [member of] T. Then the convergence of the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies the convergence of the series with the same corresponding sequence [[SIGMA].sup.[infinity].sub.k=0][a.sub.k][h.sub.k](t, [t.sub.0] on T.

In particular, when T = Z, for every t [member of] Z, the series [[SIGMA].sup.[infinity].sub.k=0] [a.sub.k][h.sub.k](t, [t.sub.0] reduces to the finite sum [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If, additionally, for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let P be the ring of infinite row-finite matrices. Then P[[t; [t.sub.0]]] is the set of all formal polynomial series (on T, centred at [t.sub.0]) with corresponding matrix sequences. Let us take X = [R.sub.N]. Then for a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the formal polynomial series (on T, centred at [t.sub.0]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has a vector form. So we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or as the family of scalar series: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 4.6. Let P be the ring of infinite row-finite matrices and X = [R.sup.N]: Then for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [x.sub.0] [member of] X we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the formal scale-power series from X[[t; [t.sub.0]]].

Definition 4.7. Let P be the ring of infinite row-finite matrices. Let A [member of] P: Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then we define the formal matrix exponent of A (at [t.sub.0]) as the formal polynomial series on T:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

and for [x.sub.0] [member of] [R.sup.N] we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.6)

As in the expression (4.5) we have values of the functions [h.sub.k]; [x, t.sub.0]), the formal exponent depends on the time scale. If T = R and A is lower-diagonal, then the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the matrix of scalar series and this coincides with Definition 2.5.

If T = Z, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for fixed t [member of] T: Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and has the same form as in the finite-dimensional case.

Proposition 4.8. Let t, [t.sub.0][member of] T. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corollary 4.9. Let t; [t.sub.0] [member of] T. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Let us observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, as [A.sub.i] has only a finite number of elements different from zero, we have the following: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, for each i [member of] N it holds that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

5. INFINITE SYSTEMS OF [DELTA] DIFFERENTIAL EQUATIONS

We consider here linear systems of infinitely many [DELTA]-differential equations of the form

[[x.sup.[DELTA](t) = Ax(t); (5.1)

where x(t) [member of] [R.sup.N] and A is a row-finite infinite real matrix (Definition 2.2).

Such systems may appear as infinite extensions of finite-dimensional control systems on time scales (in particular, continuous-time or discrete-time systems) of the form [y.sup.[DELTA]](t) = [F.sub.y](t) + Gu(t), when we extend the state space adding new variables: u and all its [DELTA]-derivatives [u.sup.[k]], for k [member of] N.

Such systems are also related to infinite-dimensional systems described by partial differential equations. Let us consider a problem of discretization for such a system.

Example 5.1. The parabolic equation [partial derivative]u / [partial derivative]t = [[partial derivative].sup.2]u / [[partial derivative][x.sup.2] may be discretized with respect to the space variable x. Let [z.sub.k](t) = u(t; k), k [member of] Z. If we replace [[partial derivative].sup.2]u(t,x)/[partial derivative][x.sup.2] by u(t, x + 2) - 2x(t, x + 1) + x(t; x), then the discretized system is given by the infinite number of differential equations:

[[??].sub.k](t)=[z.sub.k](t)-2[z.sub.k+1](t) + [z.sub.k+2](t), k [member of] Z.

Observe that the equations are indexed by integers and not natural numbers as it was presented earlier. This leads to a row-finite matrix whose columns and rows are indexed by integers as well. One can also discretize the time variable replacing [[??].sub.k](t) by [z.sub.k](t + 1) - [z.sub.k](t), i.e. by [z.sup.[DELTA](t), where the [DELTA]-derivative is taken on the time scale Z. Other time scales can be used for time discretization as well (for example nonhomogeneous ones). Thus [z.sup.[DELTA]](t) may mean different things, expressing different ways of discretization.

More examples of systems of differential equations described by infinite matrices can be found in [7]. Although such systems do not belong to the mainstream of the infinite-dimensional system theory, they pose interesting mathematical problems and have many important applications.

Let us consider now the initial value problem on T

[x.sup.[DELTA](t) = Ax(t); x([t.sub.0]) = [x.sub.0] [member of] [R.sup.N], (5.2)

where A is a row-finite matrix, t [member of] [[t.sub.0], [infinity]) [intersection] T, and x [member of] [R.sup.N]: In the finite-dimensional case the initial value problem [x.sup.[DELTA]](t) = Ax(t); x([t.sub.0]) = [x.sub.0] has the unique forward solution (for t > [t.sub.0]; where t; [t.sub.0] [member of] T) even if A is not regressive [10].

If T = N, then (5.2) is a discrete-time and infinite-dimensional initial value problem. Because [A.sup.k]; k [member of] [N.sub.0] is row-finite in case A is row-finite, there is no difficulty with the existence and uniqueness of the forward solution. The solution, corresponding to the initial condition x([t.sub.0]) = [x.sub.0], is in the following form: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If T = R, the initial value problem (5.2) may have infinitely many smooth solutions and to get uniqueness we have considered, in [3], formal solutions in the form of formal power series. Hence, similarly as in the continuous-time case, we shall use formal expressions for solutions of (5.2).

Proposition 5.2. Let T be any time scale and for [t.sub.0] [member of] T : x([t.sub.0]) = [x.sub.0] [member of] [R.sup.N]: Then the vector of formal polynomial series (on T, centred at [t.sub.0]) defined by (4.6) is the unique (formal) solution of the initial value problem (5.[member of]).

Let [LAMBDA] be the system on T of the following form:

[x.sup.[DELTA]](t) = Ax(t); (5.3)

y(t) = Cx(t); (5.4)

where x(t) [member of] [R.sup.N] and y(t) [member of] [R.sup.J], where J = N or J = p < [infinity]. The matrices A and C are row-finite. In the case where J = p the matrix C has a finite number of rows but infinitely many columns. If T = N, then x is a discrete-time system and, as we notice in the above discussion, there exists a unique (forward) solution of (5.3), and the output trajectory corresponding to the initial condition x([t.sub.0]) = [x.sub.0] has just the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For any other time scale T we have formal output [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The initial condition x([t.sub.0]) = [x.sub.0] may be represented as a pair ([t.sub.0]; [x.sub.0]) [member of] T x[R.sup.N]: Such a pair is called an event.

Definition 5.3. We say that two events ([t.sub.0]; [x.sub.1]) and ([t.sub.0]; [x.sub.2]) are indistinguishable by the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Otherwise, the events ([t.sub.0]; [x.sub.1]) and ([t.sub.0]; [x.sub.2]) are distinguishable. We say that the system x is observable if for any two distinct points [x.sub.1], [x.sub.2] [member of] [R.sup.N] there is [t.sub.0] [member of] T such that the events ([t.sub.0]; [x.sub.1]); ([t.sub.0]; [x.sub.2]) are distinguishable.

Proposition 5.4. The events ([t.sub.0]; [x.sub.1]) and ([t.sub.0]; [x.sub.2]) are indistinguishable iff for all k [member of] [N.sub.0] : C [A.sup.k][x.sup.1] = C [A.sup.k][x.sup.2].

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 5.5. If there is [t.sub.0] [member of] T such that two events ([t.sub.0], [x.sub.1]), ([t.sub.0], [x.sub.2]) are distinguishable, then it means that for all t [member of] T the events (t; [x.sub.1]), (t; [x.sub.2]) are distinguishable.

Observe that the conditions on observability that were obtained for T = R, in [3], are the same for any time scale T. Hence we recall main propositions without repeating the proofs, which can be found in [2;3].

Let f : N[N.sub.0] [right arrow] N be an isomorphism. Then for each i [member of] N there is only one pair (k; n) [member of] Nx[N.sub.0] such that i = f(k; n): Now let D = ([d.sub.ij]) be an infinite matrix whose ith row [D.sub.i] is equal to [C.sub.k][A.sup.n]; where i = f(k; n): If C has only finitely many rows, then the matrix D can be written in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let D(x) = Dx;D : [R.sup.N] [right arrow] [R.sup.N]:

Let [e.sub.i]; i [member of] N be the infinite row with 1 at the ith position and 0 at other positions.

Proposition 5.6. System x is observable iff [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 5.7. Since the rows of D correspond to derivatives of the output, one can characterize observability as a possibility of computing every state variable as a linear combination of finitely many outputs and their derivatives.

If C has finitely many rows, then for all 0 [less than or equal to] k < [infinity] the rank of the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is finite.

Corollary 5.8. The system [LAMBDA] (with the matrix C of finitely many rows) is observable iff [for all]i [member of] N [there exists][k.sub.i] [member of] [N.sub.0]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 5.9. If x is observable, then rank D = [infinity]

Example 5.10. Let x be the system, on the time scale T = Z, of the form [x.sup.[DELTA](t) = Ax(t); y(t) = Cx(t); where A;C are infinite row-finite matrices and C has finitely many rows. Then the necessary and sufficient condition on observability is given by Corollary 5.8. The system x can be rewritten in the shift form of the discrete-time system: x(t + 1) = (A + I)x(t); y(t) = C(t): Then in the condition of observability we should write the matrix (A + I) instead of the matrix A. But it does not change this condition because for each k [member of] N

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Remark 5.11. For infinite-dimensional systems observability is usually dual to approximate controllability, which means that the reachable space is dense. Such duality was proved for continuous-time systems described by infinite matrices in [6]. One could try to extend this result to systems on time scales. This would require, however, developing controllability of such systems.

ACKNOWLEDGEMENTS

D. Mozyrska was supported by the Bialystok Technical University grant No. W/WI/7/07, Z. Bartosiewicz by the Biaystok Technical University grant No. W/WI/1/07.

Received 23 February 2007, in revised form 4 July 2007

REFERENCES

[1.] Bartosiewicz, Z. and Pawluszewicz, E. Realizations of linear control systems on time scales. Control Cybern., 2006, 35, 769-786.

[2.] Bartosiewicz, Z. and Mozyrska, D. Observability of infinite-dimensional finitely presented discrete-time linear systems. Zesz. Nauk. Politech. Biaostockiej. Mat.-Fiz-Chem., 2001, 20, 5-14.

[3.] Bartosiewicz, Z. and Mozyrska, D. Observability of row-finite countable systems of linear differential equations. In Proceedings of 16th IFAC Congress, 4-8 July 2005, Prague (Piztek, P., ed.). Elsevier, Oxford, 2006.

[4.] Curtain, R. F. and Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, 1995.

[5.] Banach, S. Theorie des operations lineaires. Warsaw, 1932.

[6.] Mozyrska, D. and Bartosiewicz, Z. Dualities for linear control differential systems with infinite matrices. Control Cybern., 2006, 36, 887-904.

[7.] Deimling, R. Ordinary Differential Equations in Banach Spaces. Lecture Notes Math., Vol. 596. Springer-Verlag, 1977.

[8.] Cooke, R. G. Infinite Matrices and Sequence Spaces. Macmillan, London, 1950.

[9.] Agarwal, R. P. and Bohner, M. Basic calculus on time scales and some of its applications. Results Math., 1999, 35, 3-22.

[10.] Bohner, M. and Peterson, A. Dynamic Equations on Time Scales. Birkhauser, Boston, 2001.

[11.] Bohner, M. and Lutz, A. Asymptotic expansions and analytic dynamic equations. Published online: http://web.umr.edu/bohner/papers, 2005.

Dorota Mozyrska and Zbigniew Bartosiewicz

Faculty of Computer Science, Bialystok Technical University, Wiejska 45A, 15-351 Bialystok, Poland; admoz@w.tkb.pl, bartos@pb.bialystok.pl