Existence of Ψ−bounded solutions for nonhomogeneous Lyapunov matrix differential equations on R

In this paper, we give a necessary and sufficient condition for the existence of at least one Ψ− bounded solution of a linear nonhomogeneous Lyapunov matrix differential equation on R. In addition, we give a result in connection with the asymptotic behavior of the Ψ− bounded solution of this equation.


Introduction.
This paper deals with the linear nonhomogeneous Lyapunov matrix differential equation X ′ = A(t)X + XB(t) + F(t) (1) where A, B and F are continuous n×n matrix-valued functions on R.
Recently, the existence of at least one Ψ− bounded solution on R of equation (1) for every Lebesgue Ψ− integrable matrix function F on R has been studied in [8].
Our aim is to determine necessary and sufficient condition for the existence of at least one Ψ− bounded solution on R of equation (1), for every continuous and Ψ− bounded matrix function F on R.
Here, Ψ is a matrix function. The introduction of the matrix function Ψ permits to obtain a mixed asymptotic behavior of the components of the solutions.
In order to be able to solve our problem, we use a bounded input -bounded output approach which has been used in the past few years (see [2], [10], [11] and [12]).
The approach used in our paper is essentially based on a trichotomic type decomposition of the space R n at the initial moment (which has been used in the past few years both in the finite-dimensional spaces (see [4], [5] and [8]) and in general case of Banach spaces (see [6], [7] and [13])) and the technique of Kronecker product of matrices (which has been successfully applied in various fields of matrix theory).
Thus, we obtain results which extend the recent results regarding the boundedness of solutions of the equation (1) (according to [4]).

Preliminaries.
In this section we present some basic definitions and results which are useful later on.
Let M m×n be the linear space of all m×n real valued matrices. For a n×n real matrix A = (a ij ), we define the norm |A| by |A| = sup Ax .
It is well-known that |A| = max The Kronecker product of A and B, written A⊗B, is defined to be the partitioned matrix .
Lemma 2. The vectorization operator Vec : M n×n −→ R n 2 , is a linear and one-to-one operator. In addition, Vec and Vec −1 are continuous operators.
Proof. See in [3]. Remark. Obviously, if F is a continuous matrix function on R, then f = Vec(F) is a continuous vector function on R and vice-versa.
We recall that the vectorization operator Vec has the following properties as concerns the calculations (see [9]): ..,n, be continuous functions and We shall assume that A, B and F are continuous n×n -matrices on R. By a solution of (1), we mean a continuous differentiable n×n -matrix function X satisfying the equation (1) for all t ∈ R.
The following lemmas play a vital role in the proofs of the main results.
Lemma 4. ( [3]). The matrix function X(t) is a solution of (1) on the interval J ⊂ R if and only if the vector valued function x(t) = VecX(t) is a solution of the differential system where f(t) = VecF(t), on the same interval J.
Def inition 5. The above system (2) is called 'corresponding Kronecker product system associated with (1)'. Proof. From the proof of Lemma 2, it results that for every A ∈ M n×n . Setting A = Ψ(t)M(t), t ∈ R and using Lemma 3, we have the inequality for all matrix function M(t). Now, the Lemma follows immediately.
. Let X(t) and Y(t) be fundamental matrices for the systems and respectively. Then, the matrix Z(t) = Y T (t)⊗X(t) is a fundamental matrix for the system If, in addition, X(0) = I n and Y(0) = I n , then Z(0) = I n 2 .
Now, let Z(t) be the above fundamental matrix for the system (6) with Z(0) = I n 2 .
Let the vector space R n 2 represented as a direct sum of three subspaces X − , X 0 and X + defined as follows: a solution z of the sistem (6) is I n ⊗ Ψ− bounded on R if and only if z(0) ∈ X 0 ; let X denote the subspace of R n 2 consisting of all vectors which are values of I n ⊗ Ψ− bounded solutions of (6) on R + for t = 0; let X − denote an arbitrary fixed subspace of X supplementary to X 0 : X = X − ⊕ X 0 ; finally, the subspace X + is an arbitrary fixed subspace of R n 2 , supplementary to X − ⊕ X 0 . Let P − , P 0 and P + denote the corresponding projections of R n 2 onto X − , X 0 and X + respectively.

The main results.
The main results of this paper are the following.
Proof. First, we prove the 'if' part. Suppose that (7) holds for some K > 0. Let F : R −→ M n×n be a continuous and Ψ− bounded matrix function on R. From Lemma 5, it follows that the vector function f(t) = VecF(t) is I n ⊗ Ψ − bounded on R. From Theorem 1.1 ([4]), it follows that the differential system (2) has at least one I n ⊗ Ψ − bounded solution on R ( because condition (7) for the system (1) becomes condition (1.3) in Theorem 1.1 ([4]) for system (6)).
Let z(t) be this solution.
From Lemma 4 and Lemma 5, it follows that the matrix function Z(t) = Thus, the linear nonhomogeneous Lyapunov matrix differential equation (1)  Lemma 6 tell us that U(t) = Y T (t)⊗X(t). After computation, it follows that (7) holds.
The proof is now complete.
As a particular case, we have the following result: Corollary 1. If A and B are continuous n×n real matrices on R and the equation has no nontrivial Ψ− bounded solution on R, then, the equation (1) Proof. Indeed, in this case, P 0 = O n .
The next result shows us that the asymptotic behavior of Ψ− bounded solutions of (1) is determined completely by the asymptotic behavior of F(t) as t −→ ±∞.
Theorem 3. Suppose that (1). The fundamental matrices X and Y for the systems (4) and (5) respectively satisfy: (a). the condition (7)   The proof is now complete.

Remark. Theorem generalizes Theorem 1.3 ([4]).
As a particular case, we have Corollary 2. Suppose that (1). The homogeneous equation (8) has no nontrivial Ψ− bounded solution on R; (2). The fundamental matrices X and Y for the systems (4) and (5) respectively satisfy the condition (9) for some K > 0; ( In addition, the matrix function F is Ψ− bounded on R. On the other hand, the solutions of the equation (1)  . Note that the asymptotic properties of the components of the solutions are not the same. On the other hand, we see that the asymptotic properties of the components of the solutions are the same, via matrix function Ψ. This is obtained by using a matrix function Ψ rather than a scalar function.
This example shows that the hypothesis (2) of theorem 3 is an essential condition for the validity of the theorem.