The importance of Sylvester matrix and Lyapunov matrix differential equations and their occurrence in a number of areas of applied mathematics such as control systems, dynamic programming, optimal filters, quantum mechanics and systems engineering etc., are well known. The two main objectives of this paper are therefore (1) to develop the theory and methods to solve dynamical system on time scales (2) to explore the techniques of controllability and observability. In this paper we mainly focus our attention to first order Sylvester matrix dynamical systems of the form

where X(t) is an n×n matrix, U(t) is m×n input piecewise rd-continuous matrix called control and Y (t) is r×n output rd-continuous matrix. Here A(t), B(t), C(t) and K(t) are n × n, n × n, n × m and r × n rd-continuous matrices respectively. D(t), L(t) are rd-continuous matrices of order n × n. X^Δ is the generalized delta derivative of X,(Definition 1.10 of [2]) and t is from a time scale T, which is a nonempty closed subset of ℝ and μ is a graininess function. When B = A^* (* denotes the transpose of matrix) equation (1.1) is called matrix Lyapunov dynamical system on time scales. Many authors [4, 11], were obtained complete controllability and complete observability criteria for similar systems of the type (1.1) and (1.2) with B(t) = 0,D(t) and L(t) are identity matrices and X(t) is a vector.

If the time scale T = ℝ, then μ = 0, the system (1.1) becomes Sylvester matrix dynamical system of the form

If the time scale T = Z, then μ = 1, the system (1.1) become Sylvester matrix delta-difference system of the form

where ΔX(t) = X(t + 1) − X(t). In the above system, if we put A(t) = A₁(t) − I_n and B(t) = B₁(t)–I_n then system (1.4) becomes Sylvester matrix difference system of the form

Therefore, study of behavior of controllability of Sylvester matrix system (1.1) unify the study of (1.3), (1.4), (1.5) and extended to matrix dynamic systems on time scales. The analytical, numerical solutions and control aspects of Sylvester matrix differential system (1.3) was studied by Fausett [5]. The existence and uniqueness, controllability and observability of matrix delta-difference system (1.4) were studied by Murty [10]. The calculus of time scales was initiated by Stefan Hilger [6] in order to create a theory that can unify discrete and continuous analysis. The study of dynamic equations on time scales, is an area of mathematics that has recently received a lot of attention and sheds new light on the discrepancies between continuous differential equations and discrete difference equations. It also prevents one from proving a result twice, once for differential equations and once for difference equations. The general idea, which is the main goal of Bohner and Peterson’s excellent introductory text [2, 3] is to prove a result for a dynamic equation where the domain of the unknown function is so-called time scale. If T = ℝ, the general result obtained yields the same result concerning an ordinary differential equation. If T = Z, the general result is the same result one would obtain concerning a difference equation. This paper is well organized as follows:

In section 2 we study some basic properties of time scales, Kronecker product of matrices and develop preliminary results by converting the given problem into a Kronecker product problem. The solution to the corresponding initial value problem obtained in terms of two transition matrices of the systems X^Δ(t) = A(t)X(t) and X^Δ(t) = B^* (t)X(t) by using the standard technique of variation of parameters [8].

In Section 3 we address the necessary and sufficient conditions for complete controllability and complete observability under certain smoothness conditions.

A great deal of work has been done since 1988, unifying the theory of differential equations and the theory of difference equations by establishing the corresponding results in time scale setting. For more detailed information refer the books [2] and [3].

Definition 2.1

A nonempty closed subset of ℝ is called a time scale. It is denoted by T. By an interval we mean the intersection of the given interval with a time scale. For t < sup T and t > inf T, define the forward jump operator σ, and backward jump operator ρ, respectively by

For all t, r ∈ T if σ(t) = t, t is said to be right dense, (otherwise t is said to be right scattered) and ρ(r) = r, r is said to be left dense, (otherwise r is said to be left scattered). The graininess function μ(t) : T[0,∞) is defined by μ(t) = σ(t) – t.

Definition 2.2

A function x : T → ℝ is right dense continuous (rd-continuous) if it is continuous at every right dense point t ∈ ℝ and its left hand limit exists at each left dense point t ∈ ℝ.

Definition 2.3

A mapping f : T → X, where X is a banach space, is called rd-continuous if

• (i) It is continuous at each right dense t ∈ T.

• (ii) At each left dense point the left side limit f(t⁻) exists.

Definition 2.4

F : T^k → ℝ is called an antiderivative of f : T^k → ℝ provided F^Δ(t) = f(t) holds for all t ∈ T^k. We then define the integral by

Theorem 2.5

Assume f : T → ℝ is a function and let t ∈ T^k. Then we have the following:

• (i) If f is differentiable at t, then f is continuous at t.

• (ii) If f is continuous at t and t is right-scattered, then f(t) is differentiable at t withfΔ(t)=f(σ(t))-f(t)μ(t)

• (iii) If t is right-dense, then f is differentiable at t if and only if the limitlimt→sf(t)-f(s)t-s

  exists as a finite number. In this casefΔ(t)=lims→tf(t)-f(s)t-s

• (iv) If f is differentiable at t, thenf(σ(t))=f(t)+μ(t)fΔ(t).

Definition 2.6

A function f : T → ℝ is called rd-continuous provided it is continuous at right dense points in T and its left sided limits exists (finite) at left dense points in T. The set rd-continuous functions f : T → ℝ will be denoted by

The set of functions f : T → ℝ that are differentiable and whose derivative is rd-continuous is denoted by

Results 2.7

A,B ∈ ℝ are matrix-valued functions on T, then

• φ₀(t, s) ≡ I and φ_A(t, t) ≡ I

• φ_A(σ(t), s) ≡ (I + μ(t)A(t) φ_A(t, s);

• φA-1(t,s)≡φΘA**(t,s);

• φA(t,s)=φA-1(s,t)=φΘA**(s,t)

• φ_A(t, s)φ_A(s, r) = φ_A(t, r)

• φ_A(t, s)φ_B(t, s) = φ_A⊗B(t, s) if φ_A(t, s) and B(t) commute

Theorem 2.8.([2])

Let A ∈ ℝ be an n×n matrix-valued function on T and suppose that f : T → ℝⁿis rd-continuous. Let t_{0 ∈}T and y_{0 ∈} ℝⁿ. Then the initial value problem

has a unique solution y : T → ℝ. Moreover, this solution is given by

Now we present some properties and results for Kronecker products which are useful for studying Kronecker product matrix dynamical systems on time scales. Kronecker product is also known as a direct product or a tensor product is a concept having its origin in group theory and has important applications in particle physics. This technique has been successfully applied in various fields of matrix theory.

Definition 2.9

Let A ∈ ℂ^m×n(ℝ^m×n) and B ∈ ℂ^p×q(ℝ^p×q), then the Kronecker product of A and B written as A ⊗ B, is defined to be the partitioned matrix

is an mp × nq matrix and is in ℂ^mp×nq(ℝ^mp×nq).

The Kronecker product has the following properties and rules [1].

• (A ⊗ B)^* = A^* ⊗ B^*

• (A ⊗ B)⁻¹ = A⁻¹ ⊗ B⁻¹ (provided A and B are invertible)

• (A ⊗ B)⁻ = A⁻ ⊗ B⁻ (A⁻ is the generalized inverse of A)

• The mixed product rule: (A⊗B)(C ⊗D) = (AC ⊗BD) provided the dimensions of the matrices are such that the various expressions exist.

• ||A ⊗ B|| = ||A|| ||B||, where norm of A is defined by ||A|| = max_i;j |a_ij |.

• There exists a zero element O_mn = O_m ⊗ O_n

• There exists a unit element I_mn = I_m ⊗ I_n

• V ec(Y AB) = (B^* ⊗ A)V ecY

• If A and B are matrices both of order n × n, then

  • V ec(AX) = (I_n ⊗ A)V ecX

  • V ec(XA) = (A^* ⊗ I_n)V ecX

Definition 2.10

Let A = [a_ij ] ∈ ℝ^m×n, we denote

Definition 2.11

Let A and B are rd-continuous matrices on time scale T, then

A^Δ is the generalized delta derivative of A, t is from a time scale T, which is a known non-empty closed subset of ℝ.

Now by applying the Vec operator to the Δ-differentiable matrix dynamical system (1.1) also the output equation (1.2) and using Kronecker product properties, we have

where Z(t) = V ecX(t), Û (t) = V ecU(t), Ŷ (t) = V ecY (t) and

is a n² × n² matrix. Let A(t) and B(t) be regressive and rd-continuous.

From the definition of Kronecker product G : T^k → ℝ^n2×n2is regressive and rd-continuous. System (2.1) and (2.2) is called the Kronecker product system associated with (1.1) and (1.2).

Remark 2.12

It is easily seen that, if X(t) is the solution of (1.1) then V ecX(t) = Z(t) is the solution of (2.1) and vice-versa.

Now we confine our attention to corresponding homogeneous matrix dynamical system on time scales (2.1) given by

Lemma 2.13

Let φ₁(t, s) and φ₂(t, s) are denote state transition matrices of the systems X^Δ(t) = A(t)X(t) and X^Δ(t) = B^*(t)X(t) respectively. Then the matrix φ(t, s) defined by

is the state transition matrix of (2.3) and every solution of (2.3) is of the form Z(t) = φ(t, s)Ψ (where Ψ is any constant matrix of order n²).

Theorem 2.14. ([7])

Let φ(t, s) = φ₂(t, s)⊗φ₁(t, s) be a transition matrix of (2.3), then the unique solution of (2.1), subject to the initial condition Z(t₀) = Z₀is

In this section, we prove necessary and sufficient conditions for complete controllability and complete observability of the matrix dynamical systems on time scales (2.1) and (2.2).

Definition 3.1

The Δ-differential systems S₁ given by (2.1) and (2.2) is said to be completely controllable if for t₀, any initial state Z(t₀) = Z₀ and any given final state Z_f there exists a finite time t₁ > t₀ and a control Û (t), t₀ ≤ t₁ such that Z(t₁) = Z_f.

Theorem 3.2

The time scale dynamical system S₁is completely controllable on the closed interval J = [t₀, t₁] if and only if the n² × n²symmetric controllability matrix

where φ(t, s) is defined in (2.4), is non-singular. In this case the control

defined on t₀ ≤ t₁, transfers Z(t₀) = Z₀to Z(t₁) = Z_f.

Definition 3.3

The timescale dynamical system (2.1), (2.2) is completely observable on J = [t₀, t₁] if for any time t₀ and any initial state Z(t₀) = Z₀ there exists a finite time t₁ > t₀ such that the knowledge of Û (t) and Ŷ (t) for t₀ ≤ t₁ suffices to determine Z₀ uniquely.

Now we present a necessary and sufficient condition for the system (2.1), (2.2) to be completely observable.

Theorem 3.4

The system S₁is completely observable on J if and only if the n² × n²symmetric observability matrix

is non singular.

The authors are grateful to Prof. M.S.N. Murty for his valuable suggestions in preparing the manuscript.