A new proof for a Barbaşin’s theorem in the periodic case

by Constantin BUŞE and Marcela GIURGIULESCU

Abstract

It is proved that a q-periodic evolutionary process U = {U(t, s)}_{t≥s} of bounded linear operators on a Banach space X is uniformly exponentially stable if and only if the function t ↦ ||U(0, −t)|| belongs to L^{1}(R_{+}). We would use this result for a new proof of a Barbaşin’s theorem in this particular case.

1991 Mathematics Subject Classification: 34D05

Key words and phrases: evolutionary processes, uniformly exponentially stable.

1. Introduction

Let X be a Banach space (real or complex) and L(X) the Banach algebra of all bounded linear operators acting on X. We denote by || · || the norms of vectors and operators. A strongly continuous evolutionary process on X is a family of bounded linear operators U = {U(t, s)}_{t≥s} on X which satisfy the following three conditions:

(i) U(t, s) = U(t, r)U(r, s) for all t ≥ r ≥ s;

(ii) U(t, t) = Id, Id is the identity operator in L(X), for all t ∈ R; (iii) for each x ∈ X the maps (t, s) ↦ U(t, s)x are continous on the set

{(t, s) : t ≥ s}.

If, in addition, the family U satisfies the condition

(iv) there are M ≥ 1 and ω ∈ R such that

||U(t, s)|| ≤ Me^{ω(t−s)}, ∀t ≥ s,

then it is called strongly continuous evolutionary process with exponential growth on X.

Let U be a strongly continuous evolutionary process on X. If U(t, s) = U(t − s, 0) for all t ≥ s then the family G = {U(t, 0)}_{t∈R} is a C_{0}-group of bounded linear operators on X.

Let q > 0, fixed. A strongly continuous evolutionary process U on X is called q-periodic evolutionary process if

(v) U(t + q, s + q) = U(t, s) for all t ≥ s.

It is easy to see that a C_{0}-group or a q-periodic evolutionary process are strongly continuous evolutionary process with exponential growth, see e.g. [5, Lemma 1].

Examples. 1. Let A ∈ L(X) and G(t) := e^{tA} for all t ∈ R. The family G := {G(t)}_{t∈R} is a C_{0}-group of bounded linear operators on X. Moreover, the function t ↦ e^{tA} : R → L(X) is continuous in the uniform topology of L(X), so G is an uniformly continuous group of operators on X.

2. Let A(t) ∈ L(X) for t ∈ R such that the function t ↦ A(t) is continuos and bounded. Let P (·) the solution of the operatorial Cauchy problem Ẋ(t) = A(t)X(t), X(0) = Id. The family U = {U(t, s) : t ≥ s}, where U(t, s) := P (t)P^{−1}(s) for t ≥ s, is a strongly continuous evolutionary process with exponential growth.

3 Let A(·) and U as in Example 2. If A(t + q) = A(t) for all t ∈ R and some q > 0, then U is a q-periodic evolutionary process.

Let U = {U(t, s) : t ≥ s} be a strongly continuous evolutionary process. We say that U is uniformly stable if there is K > 0 such that ||U(t, s)|| ≤ K for all t ≥ s. We also say that U is uniformly exponentially stable if there exist N > 0 and ν > 0 such that

||U(t, s)|| ≤ Ne^{−ν(t−s)}, ∀t ≥ s.

It is not hard to see that the maps ξ ↦ ||U(t, ξ)|| are measurable on (−∞, t] for every t > 0. In fact this follows by the strongly continuous property of U and definition of the operatorial norm.

Theorem 1. A strongly continuous evolutionary process with exponential growth {U(t, s) : t ≥ s} on X is uniformly exponentially stable if and only if

sup_{t>0} ∫_{−∞}^{t} ||U(t, ξ)||dξ = M < ∞.  (1)

For reasons of self-containedness we give the proof of Theorem 1 in some details. In fact this proof is an easy modification of the proof from [6, Theorem 3.1.2]. The following lemma would be used latter.

Lemma 2. Let U be a strongly continuous evolutionary process with exponential growth on X. If there exists a function g : R_{+} → (0, ∞) with inf{g(t) : t > 0} < 1 and ||U(t, s)|| ≤ g(t − s), then U is uniformly exponentially stable.

For the proof of Lemma 2, see e.g. [3, Lemma 4].

Proof of Theorem 1. Let ω > 0 and K ≥ 1 such that

||U(t, s)|| ≤ Ke^{ω(t−s)} ∀t ≥ s.

Firstly we prove that U is uniformly stable. Indeed, if it is not true, then for every n = 1, 2, ⋯, there are t_{n} and s_{n} with ||U(t_{n}, s_{n})|| > n. It follows that n < Ke^{ω(t_{n}−s_{n})} so, lim_{n→∞}(t_{n} − s_{n}) = ∞. On the other hand for every ξ ∈ [s_{n}, t_{n}], we have

n ≤ ||U(t_{n}, s_{n})|| ≤ Ke^{ω(ξ−s_{n})}||U(t_{n}, ξ)||.

By integration on [s_{n}, t_{n}] we obtain

n[1 − e^{−ω(t_{n}−s_{n})}] ≤ KMω,

which is a contradiction. It results that there exists K_{1} > 0 such that ||U(t, s)|| ≤ K_{1} for all t ≥ s. Moreover,

(t − s)||U(t, s)|| = ∫_{s}^{t} ||U(t, s)|| ≤ K_{1} ∫_{s}^{t} ||U(t, ξ)||dξ ≤ K_{1}M,

so (t − s + 1)||U(t, s)|| ≤ K_{1}(M + 1). Using Lemma 2 it follows that U is uniformly exponentially stable.

Remark 3. Seems to be an open question if the condition

sup_{t>0} ∫_{−∞}^{t} ||U(t, ξ)x||dξ < ∞, ∀x ∈ X

implies the uniform exponentiall stability property of U.

2. The periodic case.

Let q > 0, fixed, and U be a q-periodic evolutionary process on X. We would prove that if the function ξ ↦ ||U(0, −ξ)|| belongs to L^{1}(R_{+}) then U is uniformly exponentially stable. More exactly, the following theorem holds.

Theorem 4. A q-periodic evolutionary process U = {U(t, s) : t ≥ s} on X is uniformly exponentially stable if and only if^{∫_{0}^{∞}||U(0, −ξ)||dξ < ∞. (2)} Proof: A simple calculus give: ∫_{0}^{∞}||U(0, −ξ)||dξ = ∑^{∞}_{k=0} ∫_{kq}^{(k+1)q}||U(0,−ξ)||dξ = ∑^{∞}_{k=0} ∫_{0}^{q} ||U(k + 1)q,r)||dr.

From Beppo-Levi’s theorem if follows that there exists a null set A_{0} ⊂ [0, q] such that lim_{k→∞} ||U((k + 1)q, t)|| = 0 for all t ∈ [0, q] and t ∉ A_{0}. Moreover, let t_{1} ∈ [0, q) and t > t_{1}, t ∈ [0, q], t ∉ A_{0}. Then

||U((k + 1)q, t_{1})|| ≤ Ke^{ωq}||U((k + 1)q, t)|| → 0 as k → ∞

so, lim_{k→∞} ||U((k + 1)q, s)|| = 0 for all s ∈ [0, q]. Now, is easy to see that U is uniformly exponentially stable, see e.g. [5, Lemma 2].

Remark 5. A q-periodic evolutionary process U = {U(t, s) : t ≥ s} on X is uniformly exponentially stable if and only if

∫_{0}^{∞}||U(ξ, 0)x||dξ < ∞, ∀x ∈ X.

Proof. Let U_{0} the restriction of U to the set Δ := {(t, s) : t ≥ s ≥ 0}. From [4, Theorem 2.3] it follows that U_{0} is uniformly exponentially stable. Let N > 0 and ν > 0 such that ||U(t, s)|| ≤ Ne^{−ν(t−s)} for all t ≥ s ≥ 0. Then for each real numbers ξ and η with ξ ≥ η, one has

||U(ξ, η)|| = ||U(ξ + nq, η + nq)|| ≤ Ne^{−ν(ξ−η)},

where n ∈ N is such that η +nq ≥ 0, i.e. U is uniformly exponentially stable. Corollary 6. Let U be a q-periodic evolutionary process on X. If

sup_{t>0} ∫_{0}^{t} ||U(t, ξ||dξ < ∞,