AN EXPONENTIAL STABILITY CRITERION FOR q-PERIODIC EVOLUTION FAMILIES ON HILBERT SPACES

Constantin Buşe

West University of Timişoara, Bd. V. Parvan 4, 1900 Timişoara, România E-mail buse@hilbert.math.uvt.ro

Abstract

We consider a mild solution u_{μx} of a well-posed inhomogeneous, Cauchy problem

u̇(t) = A(t)u(t) + e^{iμt}x (t ≥ 0), μ ∈ R, u(0) = 0

on a separable Hilbert space H, where A(·) is periodic. We prove that if

sup_{μ∈R} sup_{t>0} ||u_{μx}(t)|| < ∞, ∀x ∈ H

then the solution of the Cauchy problem

u̇(t) = A(t)u(t) (t ≥ 0), u(0) = x ∈ H

is uniformly exponentially stable for every x ∈ H. The basic idea for the proof is to apply the spectral mapping theorem for an evolution semigroup associated with a q-periodic evolution family U on a Hvalued almost periodic functions space.

1991 Mathematics Subject Classification. 47D06, 34G10 Keywords and phrases. evolution semigroup, exponential stability, periodic evolution families, almost periodic function.

1. INTRODUCTION

Let X be a Banach space and L(X) the space of all bounded linear operators on X. We denote by || · || the norms of vectors and operators on X. Let A be a linear and bounded operator acting on X. We consider the system

ẋ(t) = Ax(t) (t ≥ 0)  (1.1)

and the Cauchy problem

u̇(t) = Au(t) + e^{iμt}x (t ≥ 0), x(0) = 0  (1.2)

where μ ∈ R and x ∈ X. It is well-known that the system (1.1) is uniformly exponentially stable, that is there exist the constants N > 0 and ν > 0 such that ||e^{tA}|| ≤ Ne^{−νt} for all t ≥ 0, if and only if the solution of the Cauchy problem (1.2) is bounded for every μ ∈ R and any x ∈ X, i.e.

sup_{t>0} || ∫_{0}^{t} e^{−iμξ}e^{ξA}xdξ|| < ∞, ∀μ ∈ R, ∀x ∈ X.

If A is an unbounded linear operator, the above result is false, see e.g. [12, Example 3.1], but some weakly results hold. For example, a strongly continuous semigroup T = {T (t)}_{t≥0} of bounded linear operators on X is uniformly exponentially stable, i.e. its growth bound

ω_{0}(T) := lim_{t→∞} \frac{ln ||T (t)||}{t}

is negative, if and only if for all μ ∈ R and every f ∈ P(q) the mild solution of the Cauchy problem

u̇(t) = A_{T} u(t) + e^{iμt}f(t) (t ≥ 0), u(0) = 0  (1.3)

is bounded, i.e.

sup_{t>0} || ∫_{0}^{t} e^{−iμξ}T (t − ξ)f(ξ)dξ|| < ∞, ∀μ ∈ R, ∀f ∈ P(q).  (1.4)

Here A_{T} is the infinitesimal generator of T, P(q) is the space of all X-valued continuous and q-periodic functions defined on R_{+} = [0, ∞) and q is some positive real number. More exactly in [13, Corollary 4.5 and its Reformulation], Vu and Schüler was shown that if (1.4) holds for μ = 0 then 1 ∈ ρ(T (q)). With different metods in [4] it has been shown that if (1.4) holds with P(q) replaced by P_{0}(q) := {f ∈ P(q) : f(0) = 0} and some μ ∈ R, then T (q) is power bounded and e^{iμ} ∈ ρ(T (q)).

Let x ∈ X and ω(x) the infimum of all ω ∈ R for which there is M_{ω} > 0 such that ||T (t)x|| ≤ M_{ω}e^{ωt} for every t ≥ 0. Let ω_{1}(T) the supremum of all ω(x) with x ∈ D(A_{T}). In [11] J. van Neerven was shown that if there is a constant M > 0, independent of μ, such that

sup_{t>0} || ∫_{0}^{t} e^{−iμξ}T (ξ)xdξ|| ≤ M||x||, ∀μ ∈ R, ∀x ∈ X.  (1.5)

then ω_{1}(T) < 0, that is, that every solution of the system ẋ(t) = A_{T}x(t) starting in D(A_{T}) is exponentially stable. However may be exist solutions of the system ẋ(t) = A_{T}x(t) starting in X \D(A_{T}) which are not exponentially stable, even if (1.5) holds, see eg [2, Example 2]. Moreover, in [11, Corollary 5 and the Proof of Theorem 4] it is shown that if (1.5) holds then the operator resolvent R(z, A_{T}) exists and is uniformly bounded on {ℜ(z) > 0}. Combining this with a result of L. Gearhart [7], see also F. Huang [8] or G. Weiss [14], it results in the case when X is a complex Hilbert space that ω_{0}(T) is negative, i.e. every solution of the system ẋ(t) = A_{T}x(t) (t ≥ 0), is exponentially stable.

For a well-posed Cauchy problem

u̇(t) = A(t)u(t) (t ≥ 0), u(0) = x ∈ X,  (1.6)

on a complex Banach space X with (unbounded) linear operators A(t) the solutions of (1.6) lead to an evolution family U = {U(t, s) : t ≥ s ≥ 0} in the space L(X), that is

• (e_{1}) U(t, t) = I, U(t, τ)U(τ, s) = U(t, s) for t ≥ τ ≥ s ≥ 0;

• (e_{2}) the map (t, s) ↦ U(t, s)x is continuous for every x ∈ X.

When the Cauchy problem (1.6) is periodic, i.e. there exists q > 0 such that A(t + q) = A(t) for t ∈ R_{+}, the corresponding evolution family U is periodic, that is

• (e_{3}) U(t + q, s + q) = U(t, s), ∀t ≥ s ≥ 0.

An evolution family which satisfies (e_{1}), (e_{2}) and (e_{3}) will be called qperiodic evolution family on X. Every q-periodic evolution family U on X has exponential growth, i.e. there exist ω ∈ R and M > 0 such that

||U(t, s)|| ≤ Me^{ω(t−s)} ∀t ≥ s ≥ 0,  (1.7)

see [3, Lemma 1] or [6, Theorem 6.6]. We recall that a evolution family U as above is called uniformly expnentially stable if there are ω < 0 and M > 0 such that (1.7) holds. For a locally integrable function f : R_{+} → X a mild solution of the well-posed inhomogeneous Cauchy problem

u̇(t) = Au(t) + f(t), (t ≥ 0), u(0) = 0

is defined by

u_{f}(t) = ∫_{0}^{t} U(t, τ)f(τ)dτ (t ≥ 0)

In particular if f(t) = e^{iμt}x with μ ∈ R and x ∈ X then

u_{f}(t) = u_{μx}(t) = ∫_{0}^{t} e^{iμτ}U(t, τ)xdτ.  (1.8)

The main result of this paper is formulated as follows:

THEOREM 1.1 Let U be a q-periodic evolution family on a separable Hilbert space H. If

sup_{μ∈R} sup_{t>0} ||u_{μx}(t)|| < ∞, ∀x ∈ H  (1.9)

then U is uniformly exponentially stable.

2. PRELIMINARY RESULTS

For the remainder of this section U will be a q-periodic evolution family on a Banach space X, i.e. (e_{1}), (e_{2}), (e_{3}) and (1.7) hold. Let BUC(R_{+}, X) the Banach space of all X-valued bounded and uniformly continuous functions on R_{+} endowed with sup-norm and AP (R_{+}, X) the space of all X-valued almost periodic functions in the Bohr’s sense, i.e. the linear closed hull in BUC(R_{+}, X) of the set of all functions

{e^{iμ·} : μ ∈ R, x ∈ X}

Let AP_{0}(R_{+}, X) the set of all f ∈ AP (R_{+}, X) such that f(0) = 0. It is clear that AP_{0}(R_{+}, X) is a closed subspace of AP (R_{+}, X). Let P_{q}^{0}(R_{+}, X) the set of all X-valued continuous functions f defined on R_{+} such that f(t+q) = f(t) for every t ≥ 0 and f(0) = 0.

A q-periodic evolution family U defines an evolution semigroup E = {E^{t}}_{t≥0} of bounded linear operators on AP_{0}(R_{+}, X) by setting

[E^{t}f](s) = \{ \begin{matrix}U(s, s − t)f(s − t), & s ≥ t \\ 0, & 0 ≤ s ≤ t \end{matrix}  (1.10)

for f ∈ AP_{0}(R_{+}, X). It can easily seen that E^{t} acts on AP_{0}(R_{+}, X) for all t ≥ 0 and E is strongly continuous, see [10, Lemma 2]. Let G be the infinitesimal generator of E and

s(G) := sup{ℜ(z) : z ∈ σ(G)}

It is well-known that s(G) ≤ ω_{0}(G).

LEMMA 2.1 Let u, f ∈ AP_{0}(R_{+}, X). The following statements are equivalent:

• (i) u ∈ D(G) and Gu = −f;

• (ii) u(t) = u_{f}(t) for all t ≥ 0.

Proof. It follows as in [9, Lemma 1.1]. We omit the details.

LEMMA 2.2 A q-periodic evolution family U is uniformly exponentially stable if and only if the growth bound of the associated evolution semigroup E on AP_{0}(R_{+}, X) is negative.

Proof. It follows as in [5, Theorem 2.2].

LEMMA 2.3 Let U be a q-periodic evolution family on X. The following statements are equivalent:

• (i) U is uniformly exponentially stable;

• (ii) G is invertible;

• (iii) u_{f}(·) belongs to AP_{0}(R_{+}, X) for all f ∈ AP_{0}(R_{+}, X);

• (iv) sup_{t>0} ||u_{f}(t)|| < ∞ for all f ∈ AP_{0}(R_{+}, X);

• (v) for every g ∈ P_{q}^{0}(R_{+}, X) and any μ ∈ R the solution of the Cauchy problem (1.8) with f(t) = e^{iμt}g(t) for t ≥ 0 is bounded on R_{+}. Proof. The equivalence of (i) anf (v) has been shown in [3]. Another way for the proof of (iv) ⇒ (i) is indicated in [5, Theorem 2.5]. A such proof uses the fact the growth bound of arbitrary strongly continuous semigroup T = {T (t)}_{t≥0} on X is negative if and only if the map t ↦ ∫_{0}^{t} T (ξ)f(t − ξ)dξ belongs to AP_{0}(R_{+}, X) for all f ∈ AP_{0}(R_{+}, X). LEMMA 2.4 Let U be a q-periodic evolution family on the Banach space X. Then the evolution semigroup E on AP_{0}(R_{+}, X) satisfies the spectral mapping theorem, i.e., σ(G) = {z ∈ C : ℜ(z) ≤ s(G)} and e^{tσ(G)} = σ(E^{t}) \ {0}, t ≥ 0. Proof. It follows reasoning as in [9, Corollary 2.4] using Lemma 2.1 and Lemma 2.3 above. 3. PROOF OF STABILITY CRITERION Proof of Theorem 1.1 From (1.9) using the uniformly boundedness prin- ciple it follows that there exist a constant K > 0 such that^{sup_{ν∈R} sup_{s≥0} || ∫_{0}^{s} e^{iνξ}U(s, ξ)xdξ|| ≤ K||x||, ∀x ∈ H. (3.1)} We denote by P T_{0}(R_{+}, H) the set of all functions t ↦ h_{N}(t) := ∑^{N}_{k=1} a^{k}e^{iμ_{k}t}x^{k} : R^{+} →H, where N = 1, 2, ⋯, a_{k} ∈ C, μ_{k} ∈ R, {x_{k} : k = 1, 2, ⋯} is a complete ortogonal set in H such that ||x_{k}|| = 1 for all k and ∑_{k=1}^{N} a_{k}x_{k} = 0. It is not hard to see that P T_{0}(R_{+}, H) is dense in AP_{0}(R_{+}, H). Let E the evolution 6