CONNECTIONS BETWEEN EXPONENTIAL STABILITY AND BOUNDEDNESS OF SOLUTIONS OF A COUPLE OF DIFFERENTIAL TIME DEPENDING AND PERIODIC SYSTEMS

Among others, we prove that the vectorial time de- pendent q-periodic differential system u x(t) = A(t)x(t), t 2 R, x(t) 2 C n (A(t)) is uniformly exponentially stable (i.e. all its solutions decay ex- ponentially at infinity) if and only if for each vector b 2 C n , the solution of the Cauchy Problem u y(t) = A(t)y(t) + e iµt b, t � 0, b 2 C n , y(0) = 0 is bounded on R+, uniformly in respect with the parameterµ on the entire real axis. As a consequence, we get that the system (A(t)) is uniformly exponentially stable if and only if for each vector x 2 C n , the map t 㜡 t Z

By F Φ (R + , C n ) will denote the set of all q-periodic and continuous functions f b : R + → C n , whose restrictions to the interval [0, q] are given by f b (s) = h(s)Φ(s)b, with b ∈ C n .Here Φ(•) is the fundamental matrix associated to the vectorial homogeneous system (A(t)).See the next section for further details.The map h(•) belongs to the set {h 1 (•), h 2 (•)} where h 1 and h 2 are scalar valued functions, defined on the interval [0, q], by: and h 2 (s) = s(q − s).
As is known, see [7], the system (A(t)) is uniformly exponentially stable if and only if for each real number µ and each function f b ∈ F Φ (R + , C n ), the solution y µ,f b of the Cauchy Problem (A(t), µ, f b , 0), is bounded on R + .In the present note we improve this result showing that it may be preserved if consider only C n -valued constant functions instead of functions in F Φ (R + , C n ).However, our boundedness assumption is stronger than the given one in [7].More exactly, we require that the solution of the Cauchy Problem is bounded on R + uniform in respect with the parameter µ on the real axis.Our interest to the present result is stimulated by the discrete analogous one proved recently in [3] and [1].
The main ingredient of the proof in [3] is that to use a development in a Fourier series of a smooth, q-periodic, C n -valued function whose restriction to the set of all positive integer numbers is a given q-periodic sequence (z n ) decaying at zero.Then the assertion is a consequence of a result in [6].In the discrete case, q is an integer number greater than one.The infinite dimensional version of the described result is also proved in [3].This was possible because the linear span of the range of the sequence (z n ) is a finite dimensional one.The Fourier method was successfully applied by Lan Thanh Nguyen in his research on higher order differential equations in Hilbert spaces.See [11].In this paper we use other trick.Namely, we prove that each function in F Φ (R + , C n ) satisfies a Lipschitz condition on R + and then, by an well-known classical Fourier theorem ( [16], pp.93), it belongs to the space AP 1 (R + , C n )) of all almost periodic functions in the sense of Bohr, whose associated series of the Fourier-Bohr coefficients is absolutely convergent in C n .Further details about the space AP 1 (R + , C n )) may be found, for example, in the recent monograph [8]  Our interest is also stimulated by the possibility to find new real integral characterizations for the exponential stability of the system (A(t)).It is widely known that such characterizations are very useful in the control theory and in the stability theory, especial when we want to build Lyapunov functions associated to the system (A(t)).We refer only to the theorems of the Datko type and to the theorems of Barbashin type which are briefly presented in as follows.The theorem of Datko states that a (non necessarily periodic) system (A(t)) is uniformly exponentially stable if and only if each tra- for some (and then for all) p ≥ 1.Further details refereing to this type of theorems in the general framework of strongly continuous evolution families of bounded linear operators acting on a Banach space may be found in [9].For further proofs and generalizations of such theorems and for different approaches of this theory we refer to [13], [10], [14], [15], and the references therein.The uniform variant of the theorem of Barbashin states that the system (A(t)) is uniformly exponentially stable if and only if for some (and then for all) p ≥ 1.Further details and proofs may be found in [2], where the finite dimensional case is treated, and in [12] where the general framework of strongly continuous evolution families of bounded linear operators acting on a Banach space is analyzed.The strong variant of the theorem of Barbashin asserts that the evolution family {U(t, s)} of bounded linear operators acting on a Banach space X is uniformly exponentially stable if for each x ∈ X and some p ≥ 1, the following estimation holds true.
Surprisingly, the proof of the strong variant of the theorem of Barbashin seems to be more difficult and as long we can see, it is still an open problem for strongly continuous evolution families acting on an arbitrary Banach space.In this direction, some progress has made in [5], where the dual family of U was involved, and the estimation like EJQTDE, 2011 No. 90, p. 3 (SBA), is related to the strong operator topology in L(X * ).In the discrete and periodic case the strong variant of the Barbashin problem is completely solved in [1] and [3].In this paper we clarify such result in the continuous case of the finite dimensional and time dependent and periodic systems.
The paper is organized as follows.Second section contains the necessary definitions and preliminary results for that the paper to be selfcontained.In the third section we state and prove the announced results and establish its natural consequences.

Notations and preliminary results
Let X be a Banach space and let L(X) be the space of all bounded linear operators acting on X.The norm in X and in L(X) is denoted by the same symbol, namely ) for all t ≥ s ≥ r ≥ 0, where I denote the identity operator on L(X).An evolution family U is called strongly continuous if for each x ∈ X the map is continuous.Such a family is called q-periodic (with some q > 0) if U(t + q, s + q) = U(t, s), for all pairs (t, s) with t ≥ s ≥ 0.
Clearly, a q-periodic evolution family also satisfies The family U is called uniformly exponential stable if there exist two positive constants N and ν such that In the next proposition we collect some equivalent characterizations for the exponential stability of a q-periodic evolution family.EJQTDE, 2011 No. 90, p. 4 Proposition 2.1.Let U = {U(t, s) : t ≥ s ≥ 0} be a strongly continuous and q-periodic evolution family acting on the Banach space X.
The following four statements are equivalent: (1) The family U is uniformly exponentially stable.
(2) There exist two positive constants N and ν such that (3) The spectral radius of U(q, 0) is less than one, i.e.
The result of this section is based on the next technical lemma.
Lemma 2.2.Let us consider the functions h 1 , h 2 : [0, q] → C, defined by: Denote H 1 (µ) := q 0 h 1 (s)e −iµs ds and H 2 (µ) := q 0 h 2 (s)e −iµs ds.Then, The first assertion is now clear.Let µ k := 4kπ q .Then q 0 h 2 (s)e −iµ k s ds = q q 0 se −iµ k s ds − q 0 s 2 e −iµ k s ds EJQTDE, 2011 No. 90, p. 5 ) be the set of all continuous X-valued functions f defined on R + , with f (0) = 0 and f (t + q) = f (t) for t ∈ R + .Next theorem is essentially contained in article [4] of the second named author of this paper, but we include here by sake of completeness and because there the proof is not complete.
Theorem 3.1.The following two statements hold true: (1) If the system (A(t)) is uniformly exponentially stable (i.e if the spectral radius of U(q, 0) is less then one) then for each real number µ and each vector b ∈ C n , the solution of the Cauchy Problem (A(t), µ, b, 0) is bounded on R + .(2) Conversely, if for each real number µ and each vector b ∈ C n the solution of the Cauchy Problem (A(t), µ, b, 0) is bounded on R + and if the matrix Φ µ (q) is invertible, then the system (A(t)) is uniformly exponentially stable.
Proof.(1) Let ν be the integer part of t q and let r := (t−qν) ∈ [0, q).Then  The boundedness of I 1 follows because the spectral radius of U(q, 0) is less than 1.The family U has exponential growth and 0 ≤ t−s ≤ r ≤ q hence we have: (2) Using the invertibility of the matrices U(r, 0) and Φ µ (q) and the identity The assertion follows from Proposition 2.1.
The second result of this section may be read as follows: Theorem 3.2.The system (A(t)) is uniformly exponentially stable if and only if for each b ∈ C n , the solution of (A(t), µ, b, 0) is bounded on R + uniformly in respect to the parameter µ on R, i.e.Before giving the proof, we state a new lemma.Recall that F Φ (R + , C n ) is the set of all C n -valued continuous and q-periodic functions defined on R + , given on [0, q] by f b (s) = h i (s)Φ(s)b, for i ∈ {1, 2} and b ∈ C n , where h 1 and h 2 was introduced in the previous section.
Integrating the equality ∂ ∂t Φ(t) = A(t)Φ(t) between the positive numbers t 1 and t 2 , with t 2 ≥ t 1 , we get: The last inequality holds true also for t 2 ≤ t 1 .Moreover, one has In the previous estimation was used the following inequality which is an easy consequence of the Lagrange's Mean Value Theorem.
Proof of Theorem 3.2.We establish that (2.1) is a consequence of (3.1).It is known (see [16], pp.93) that each q-periodic function f satisfying a Lipschitz condition on R + belongs to AP 1 (R + , C n ), i.e. there exists a sequence (b ν ) ν∈Z , with b ν ∈ C n , such that where b ν are the Fourier-Bohr coefficients of f given by b ν = 1 q q 0 e −2πiν t q f (t)dt, for ν ∈ Z.
The last inequality holds true because the map f belongs to AP 1 (R + , C n ).The assertion follows from Theorem 2.3.Case 2. We assume only the fact that the family U has an exponential growth.There exist M ≥ 1 and ω > 0 such that U(t, s) ≤ Me ω(t−s) , for all t ≥ s.Let V (t, s) := e −ω(t−s) U(t, s).Integrating by parts, we get As is stated above may find two positive constants N and ν such that U(t, s) = e ω(t−s) V (t, s) ≤ Ne (ω−ν)(t−s) , ∀t ≥ s.
The assertion is obtained by repeating this reasoning for a certain number of times.Theorem 3.2 yields the following weak version of the Barbashin theorem.This result seems to be a new one and it opens the problem if similar result could be preserved in infinite dimensional spaces.