Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 42, 1-9; http://www.math.u-szeged.hu/ejqtde/

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

Aurel Diamandescu University of Craiova

Abstract

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. Key Words: Ψ− bounded solution, Lyapunov matrix differential equation. 2000 Mathematics Subject Classification: 34D05, 34C11.

1. 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]).

2. Preliminaries. In this section we present some basic definitions and results which are useful

later on.

Let ℝ^{n} be the Euclidean n - space. For x = (x_{1}, x_{2}, x_{3}, ...,x_{n})^{T} ∈ ℝ^{n}, let ∥x∥ = max{|x_{1}|, |x_{2}|, |x_{3}|, ..., |x_{n}|} be the norm of x ( ^{T} denotes transpose).

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_{∥x∥≤1} ∥Ax∥ .

It is well-known that |A| = max_{1≤i≤n}{∑_{j=1}^{n} |a_{ij}|}.

Def inition 1. ([1]) Let A = (a_{ij}) ∈ 𝕄_{m×n} and B = (b_{ij}) ∈ 𝕄_{p×q}. The Kronecker product of A and B, written A⊗B, is defined to be the partitioned matrix

A ⊗ B = ( \begin{matrix}a_{11}B & a_{12}B & ⋯ & a_{1n}B \\ a_{21}B & a_{22}B & ⋯ & a_{2n}B \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m1}B & a_{m2}B & ⋯ & a_{mn}B \end{matrix} )

Obviously, A⊗B ∈ 𝕄_{mp×nq}.

Lemma 1. The Kronecker product has the following properties and rules, provided that the dimension of the matrices are such that the various expressions are defined:

1). A⊗(B⊗C) = (A⊗B)⊗C; 2). (A⊗B)^{T} = A^{T}⊗B^{T}; 3). (A⊗B)^{−1} = A^{−1}⊗B^{−1}; 4). (A⊗B)·(C⊗D) = AC⊗BD; 5). A⊗(B + C) = A⊗B + A⊗C; 6). (A + B)⊗C = A⊗C + B⊗C;

7). I_{p}⊗A = ( \begin{matrix}A & O & ⋯ & O \\ O & A & ⋯ & O \\ ⋮ & ⋮ & ⋮ & ⋮ \\ O & O & ⋯ & A \end{matrix} );

8). (A(t)⊗B(t))′ = A′(t)⊗B(t) + A(t)⊗B′(t); (here, ′ denotes derivative \frac{d}{dt} ). Proof. See in [1].

Def inition 2. The application Vec : 𝕄_{m×n} ⟶ ℝ^{mn}, defined by

Vec(A) = (a_{11},a_{21}, ⋯, a_{m1},a_{12},a_{22}, ⋯, a_{m2}, ⋯, a_{1n},a_{2n}, ⋯, a_{mn})^{T},

where A = (a_{ij}) ∈ 𝕄_{m×n}, is called the vectorization operator.

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]):

Lemma 3. If A, B, M ∈ 𝕄_{n×n}, then 1). Vec(AMB) = (B^{T}⊗A)·Vec(M); 2). Vec(MB) = (B^{T}⊗I_{n})·Vec(M); 3). Vec(AM) = (I_{n}⊗A)·Vec(M); 4). Vec(AM) = (M^{T}⊗A)·Vec(I_{n}). Proof. It is a simple exercise.

Let Ψ_{i} : ℝ ⟶ (0,∞), i = 1,2,...,n, be continuous functions and

Ψ = diag [Ψ_{1}, Ψ_{2}, ⋯ Ψ_{n}].

Def inition 3. ([3]). A function f : ℝ ⟶ ℝ^{n} is said to be Ψ− bounded on ℝ if Ψf is bounded on ℝ (i.e. sup_{t∈ℝ} ∥ Ψ(t)f(t)∥< +∞).

Def inition 4. ([3]). A matrix function M : ℝ ⟶ 𝕄_{n×n} is said to be Ψ− bounded on ℝ if the matrix function ΨM is bounded on ℝ (i.e. sup_{t∈ℝ} | Ψ(t)M(t)|<

+∞).

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

x′ = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n})x + f(t),  (2)

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)’.

Lemma 5. ([3]). The matrix function M(t) is Ψ− bounded on ℝ if and only if the vector function VecM(t) is I_{n} ⊗ Ψ − bounded on ℝ.

Proof. From the proof of Lemma 2, it results that

\frac{1}{n} | A | ≤ ∥ VecA ∥_{ℝ^{^{n}^{2}}} ≤ | A |,

for every A ∈ 𝕄_{n×n}.

Setting A = Ψ(t)M(t), t ∈ ℝ and using Lemma 3, we have the inequality

\frac{1}{n} | Ψ(t)M(t) | ≤ ∥ (I_{n} ⊗ Ψ(t)) · VecM(t) ∥_{ℝ^{^{n}^{2}}} ≤ | Ψ(t)M(t) | , t ∈ ℝ (3)

for all matrix function M(t).

Now, the Lemma follows immediately.

Lemma 6. ([3]). Let X(t) and Y(t) be fundamental matrices for the systems

x′(t) = A(t)x(t)  (4)

and

y′(t) = y(t)B(t)  (5)

respectively.

Then, the matrix Z(t) = Y^{T}(t)⊗X(t) is a fundamental matrix for the system

z′(t) = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n})z(t).  (6)

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 X̃ 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 X̃ supplementary to X_{0} : X̃ = 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.

3. The main results.

The main results of this paper are the following.

Theorem 1. If A and B are continuous n×n real matrices on ℝ then, the equation (1) has at least one Ψ− bounded solution on ℝ for every continuous and Ψ− bounded matrix function F : ℝ ⟶ 𝕄_{n×n} if and only if there exists a positive constant K such that

∫_{−∞}^{t} | (Y^{T}(t) ⊗ (Ψ(t)X(t)))P_{−} ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds+

+ ∫_{t}^{0} | (Y^{T}(t) ⊗ (Ψ(t)X(t))) (P_{0} + P_{+}) ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds+

+ ∫_{0}^{∞} | (Y^{T}(t) ⊗ (Ψ(t)X(t)))P_{+} ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds ≤ K, t < 0; ∫_{−∞}^{0} | (Y^{T}(t) ⊗ (Ψ(t)X(t)))P_{−} ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds+

+ ∫_{0}^{t} | (Y^{T}(t) ⊗ (Ψ(t)X(t))) (P_{0} + P_{−}) ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds+

+ ∫_{t}^{∞} | (Y^{T}(t) ⊗ (Ψ(t)X(t)))P_{+} ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds ≤ K, t ≥ 0.

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) = Vec^{−1}z(t) is a Ψ− bounded solution of the equation (1) on ℝ ( because F(t) = Vec^{−1}f(t) ).

Thus, the linear nonhomogeneous Lyapunov matrix differential equation (1) has at least one Ψ − bounded solution on ℝ for every continuous and Ψ− bounded matrix function F on ℝ.

Now, we prove the ‘only if’ part.

Suppose that the equation (1) has at least one Ψ − bounded solution on ℝ for every continuous and Ψ− bounded matrix function F : ℝ ⟶ 𝕄_{n×n} on ℝ.

Let f : ℝ ⟶ ℝ^{n^{2}} be a continuous and I_{n} ⊗ Ψ− bounded function on ℝ. From Lemma 5, it follows that the matrix function F(t) = Vec^{−1}f(t) is continuous and Ψ− bounded on ℝ. From the hypothesis, the equation

\frac{dZ}{dt} = A(t)Z + ZB(t) + Vec^{−1}f(t)

has at least one Ψ − bounded solution Z(t) on ℝ.

From Lemma 4 and Lemma 5, it follows that the vector valued function z(t) = VecZ(t) is a I_{n} ⊗ Ψ− bounded solution on ℝ of the differential system

\frac{dz}{dt} = (I_{n}⊗A(t) + B^{T}(t)⊗I_{n})z + f(t).

Thus, this system has at least one I_{n} ⊗ Ψ− bounded solution on ℝ for every continuous and I_{n} ⊗ Ψ− bounded function f : ℝ ⟶ ℝ^{n^{2}}.

Theorem 1.1 ([4]) tell us that there exists a positive constant K such that the fundamental matrix U(t) of the differential system

\frac{dz}{dt} = (I_{n}⊗A(t) + B^{T}(t)⊗I_{n})z

satisfies the condition (1.3) of Theorem 1.1 ([4]).

Lemma 6 tell us that U(t) = Y^{T}(t)⊗X(t). After computation, it follows that (7) holds.

The proof is now complete.

Remark. Theorem 1 generalizes Theorem 1.1 ([4]).

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

Z′ = A(t)Z + ZB(t)  (8)

has no nontrivial Ψ− bounded solution on ℝ, then, the equation (1) has a unique Ψ− bounded solution on ℝ for every continuous and Ψ− bounded matrix function F : ℝ ⟶ 𝕄_{n×n} if and only if there exists a positive constant K such that for t ∈ ℝ,

∫_{−∞}^{t} | (Y^{T}(t) ⊗ (Ψ(t)X(t)))P_{−} ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds+

(9)

+ ∫_{t}^{∞} | (Y^{T}(t) ⊗ (Ψ(t)X(t)))P_{+} ((Y^{T}(s))^{−1} ⊗ (X^{−1}(s)Ψ^{−1}(s))) | ds ≤ K.

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) for some K > 0; (b). the condition lim_{t→±∞} |Y^{T}(t)⊗(Ψ(t)X(t))P_{0} | = 0;

(2). The continuous and Ψ− bounded matrix function F : ℝ ⟶ M_{n×n} is such that

lim_{t→±∞} Ψ(t)F(t) = O_{n}.

Then, every Ψ− bounded solution Z on ℝ of the equation (1) satisfies the condition

lim_{t→±∞} Ψ(t)Z(t) = O_{n}.

Proof. Let Z(t) be a Ψ− bounded solution on ℝ of the equation (1). From Lemma 4 and Lemma 5, it follows that the vector valued function z(t) = VecZ(t) is a I_{n} ⊗ Ψ− bounded solution on ℝ of the differential system

\frac{dz}{dt} = (I_{n}⊗A(t) + B^{T}(t)⊗I_{n})z + f(t),

where f(t) = VecF(t).

Also, from Lemma 5, the function f is continuous and I_{n} ⊗ Ψ− bounded on ℝ. From Theorem 1.3 ([4]), it follows that

lim_{t→±∞} ∥(I_{n} ⊗ Ψ(t))z(t)∥_{ℝ^{n^{2}}} = 0.

Now, from the inequality (3), we have

| Ψ(t)Z(t)| ≤ n∥(I_{n} ⊗ Ψ(t))z(t)∥_{ℝ^{n^{2}}}, t ∈ ℝ

and then,

lim_{t→±∞} Ψ(t)Z(t) = O_{n}.

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;

(3). The continuous and Ψ− bounded matrix function F : ℝ ⟶ M_{n×n} is such that

lim_{t→±∞} Ψ(t)F(t) = O_{n}.

Then, the equation (1) has a unique solution Z on ℝ such that

lim_{t→±∞} Ψ(t)Z(t) = O_{n}.

Proof. Indeed, in this case, we have P_{0} = O_{n} in Theorem 3.

Remark. If the function F does not fulfill the condition 2 of theorem 3, then, the Ψ− bounded solution Z(t) of equation (1) may be such that lim_{t→±∞} Ψ(t)Z(t) ≠ O_{n}. This is shown in the next simple example.

Example. Consider the linear equation (1) with

A(t) = ( \begin{matrix}−1 & 0 \\ 0 & 4 \end{matrix} ) , B(t) = ( \begin{matrix}−2 & 0 \\ 0 & −2 \end{matrix} ) and F(t) = ( \begin{matrix}e^{3t} & 0 \\ 0 & e^{−2t} \end{matrix} ) .

The fundamental matrices for the systems (4) and (5) are

X(t) = ( \begin{matrix}e^{−t} & 0 \\ 0 & e^{4t} \end{matrix} ), Y(t) = ( \begin{matrix}e^{−2t} & 0 \\ 0 & e^{−2t} \end{matrix} )

respectively.

Consider

Ψ(t) = ( \begin{matrix}e^{−3t} & 0 \\ 0 & e^{2t} \end{matrix} ) .

It is easy to see that the conditions of theorem 3 are satisfied with

P_{0} = O_{4}, P_{−} = diag [1,0,1,0], P_{+} = diag [0,1,0,1] and K = \frac{5}{12}.

In addition, the matrix function F is Ψ− bounded on ℝ.

On the other hand, the solutions of the equation (1) are

Z(t) = ( \begin{matrix}c_{1}e^{−3t} + \frac{1}{6}e^{3t} & c_{2}e^{−3t} \\ c_{3}e^{2t} & c_{4}e^{2t} − \frac{1}{4}e^{−2t} \end{matrix} ) ,

where c_{1}, c_{2}, c_{3}, c_{4} ∈ ℝ.

There exists a unique Ψ− bounded solution on ℝ, namely

Z(t) = ( \begin{matrix}\frac{1}{6}e^{3t} & 0 \\ 0 & −\frac{1}{4}e^{−2t} \end{matrix} ) ,

but lim_{t→±∞} | Ψ(t)Z(t)| = \frac{1}{4}.

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.

References

[1] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill Book Company, Inc. New York 1960 (translated in Romanian).

[2] L. Berezansky, E. Braverman, On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations, Journal of Mathematical Analysis and Applications, 304(2) (2005): 511-530.

[3] A. Diamandescu, Ψ − bounded solutions for a Lyapunov matrix differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 2009, No. 17, 1-11.

[4] A. Diamandescu, A note on the existence of Ψ− bounded solutions for a system of differential equations on ℝ, Electronic Journal of Differential Equations, Vol. 2008(2008), No. 128, pp. 1-11.

[5] A. Diamandescu, A note on existence of Ψ− bounded solutions for linear difference equations on ℤ, Kuwait Journal of Science and Engineering, Vol. 36, No. 2A, 2009, pp. 35-49.

[6] S. Elaydi, K. Janglajew, Dichotomy and trichotomy of difference equations, Journal of Difference Equations and Applications, 3(5&6) (1998): 417-448.

[7] M. Megan, C. Stoica, On uniform exponential trichotomy of evolution operators in Banach spaces, Integral Equations and Operator Theory, 60 (2008): 499-506.

[8] M. S. N. Murty, G. Suresh Kumar, On Ψ− bounded solutions for nonhomogeneous matrix Lyapunov systems on ℝ, Electronic Journal of Qualitative Theory of Differential Equations, 2009, No. 62, 1-12.

[9] M. S. N. Murty, G. Suresh Kumar, On Ψ− boundedness and Ψ− stability of matrix Lyapunov systems, J. Appl. Math. Comput(2008) 26: 67-84.

[10] M. Pituk, A Criterion for the exponential stability of linear difference equations, Applied Mathematics Letters, 17(7) (2004): 779-783.

[11] B. Sasu, A. L. Sasu, Stability and stabilizability for linear systems of difference equations, Journal of Difference Equations and Applications, 10(12) (2004): 1085-1105.

[12] B. Sasu, A. L. Sasu, Exponential dichotomy and (l^{p},l^{q}) - admissibility on the half-line, Journal of Mathematical Analysis and Applications, 316(2) (2006): 397-408.

[13] B. Sasu, A. L. Sasu, Exponential trichotomy and p - admissibility for evolution families on the real line, Mathematische Zeitschrift, 253 (2006): 515-536.

Aurel Diamandescu, University of Craiova, Department of Applied Mathematics, 13, ”Al. I. Cuza” st., 200585 Craiova, Romania E - mail address: adiamandescu@central.ucv.ro

(Received April 2, 2010)