Stability problems in non autonomous linear differential equations in infinite dimensions

One goal of this paper is to study robustness of stability of nonautonomous linear ordinary differential equations under integrally small perturbations in an infinite dimensional Banach space. Some applications are obtained to the case of rapid oscillatory perturbations, with arbitrary small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (1970). Based in Rodrigues (1970) and in Kloeden and Rodrigues (2011) we introduce a class of functions that we call Generalized Almost Periodic Functions that extend the usual almost periodic functions and are suitable to deal with oscillatory perturbations. We also present an infinite dimensional example of the previous results. We show in another example that it is possible to stabilize an unstable system using a perturbation with large period and small mean value. Finally, we give an example where we stabilize an unstable linear ODE with small perturbation in infinite dimensions using some ideas developed in Rodrigues andSol\`a-Morales (2019)} and in an example of Kakutani, see Rickard (1960).

Following this philosophy, in this paper we consider the following linear system of ordinary differential equations in an infinite dimensional Banach space X,ẋ = A(t)x and a perturbed systemẏ = A(t)y + B(t)y, where A(t) and B(t) are continuous in R. We suppose first that for each t ∈ R, A(t) and B(t) are bounded operators, the first system is asymptotically stable and that B(t) is integrally small in an arbitrary interval of length bounded by h > 0.
We establish conditions on the smallness of B(t) in such a way that the perturbed system will also be asymptotically stable. This is stablished inTheorem 1. Then we extend to the case such that A : D → X is ubbounded and generates an C 0 -semigoup T (t), t ≥ 0. This is stablished inTheorem 8.
In Daleckii & Krein [4] page 178 and in Carvalho et al. [1] similar results are presented about robustness of stability but with the stronger assumption 1 τ 0 t+τ 0 t B(τ ) dτ ≤ δ, for some τ 0 > 0, for every t ∈ R for sufficiently small δ. One observes that the smallness condition is imposed with the norm inside the integral and in our case the norm appears outside the integral and this makes a significant difference, as it is shown in Theorem (1).
Then we introduce a class of functions that we call Generalised Almost Periodic Functions that contains the usual almost periodic functions. In fact part, of it was introduced in Kloeden & Rodrigues [10], where the authors studied perturbations of an hyperbolic equilibrium.
In the present paper we use also the concept of mean value to define the class of Generalized Almost Periodic Functions (GAP).
This new class of functions has some important advantages compared with the almost periodic functions, namely, if we perturb an almost periodic function with a local perturbation in time it will not be almost periodic. Therefore it is not robust with respect to this kind of perturbations. It is also not also robust with respect to some more general perturbations, like chaotic functions.
As a consequence we study a system of the formẏ = A(t)y + B(ωt)y and prove that if ω > 0 is sufficiently large the the stability is preserved. When B(t) is periodic the result says that for sufficiently small periods and large oscillations the stability is preserved. The function B(t) does not need to be small and so if we have a linear perturbation with large oscillations the stability is preserved. This is shown in Theorem 7. In the periodic case the perturbation will have very small period. We present an example in the infinite dimensions case, in the space 2 where we show that the stability is preserved. These results extend to infinite dimensions some results of Coppel [3].
Then in Theorem (8) we extend the above results to the case where we have an unbounded infinitesimal generator. Henry [8] proves similar results with different applications, but using a different method where he passes from the continuous case to a discrete case and then recover the results for the continuous problem. Our method follows more the method of Coppel [3] (finite dimension).
In Section 7 we present a two dimensional example where we show that it is possible to stabilise an unstable system with a periodic perturbation with large period and small mean value.
Finally in Section 8 using some ideas developed in Rodrigues &Solà-Morales [21] and in an example of Kakutani [13], we give an example in infinite dimensions where we estabilize an unstable linear system using a small linear perturbation.
These two last examples seem to be new in the literature, to our knowledge.

Robustness of Stability.
The next theorem extends to infinite dimensional Banach spaces a result of W. A. Coppel [3], Proposition 6, p.6. Consider the equations:ẋ Let T (t, s) = X(t)X −1 (s) the evolution operator of (1). Suppose that T (t, s) ≤ Ke α(t−s) for t ≥ s, t, s ∈ R, where α ∈ R and K ≥ 1.
Let δ, h be two positive numbers. If If α is negative, h is sufficiently large and δ sufficiently small in such a way that β < 0 then it follows that system (2) is asymptotically stable.
Proof: By the variation of constants formula Integrating the above equation, we obtain And so, Therefore, We first suppose that s ≤ t ≤ s + h and estimate Therefore, and so, using gronwall's inequlity it follows that in an arbitrary interval of length h, say for For t ≥ s there exists n ∈ N, n = n(t, s) such that s + nh ≤ t ≤ s + (n + 1)h and so We are going to prove by induction that for s + nh ≤ t ≤ s + (n + 1)h The case n = 0 has already been proved.

The space of generalised almost periodic functions
Let (X, | · |) be a Banach space and recall the definition of an almost periodic function [5].

Definition 2.
A continuous function f : R → X is said to be almost periodic if for every sequence (α n ) there exists a subsequence (α n ) such that the lim n→∞ f (t+α n ) exists uniformly in R. Proof: The proof is trivial.
Theorem 3. F is a closed subspace of BU C(R, L(X)) and hence a Banach space.
Proof: This proof can be found in Kloeden-Rodrigues [10].
then it is independent of a.
Proof: Let a ∈ R.
Then we define: Definition 4. We say that A ∈ F is a generalized almost periodic function if there exists the Definition 5. We define the class of generalized almost periodic functions as Since there exists the lim T →∞ Using Cauchy Criterion we conclude that there exists Definition 6. For A ∈ GAP we define the mean value of A as:

Corollary 2. Any function A ∈ GAP can be written as
and B ∈ O.
The next theorem shows that stability is preserved if the linear perturbation has sufficiently large frequency: is a generalized almost periodic function with mean value zero (GAP). Consider the equations: where α > 0 and K > 1. Then there existsK and ω 0 > 0 such that for ω > ω 0 where S ω (t, s) indicates the evolution operator of (4).

Proof:
We are going to show that for any h > 0, δ > 0 there exists To complete the proof we consider now the case h ≥ |t By a change of variables, The result follows from Theorem 7 for δ sufficiently small.
= e A 0 t be the semigroup generated by (5) and S ω (t, s) be the evolution operator of (6).

An infinite dimensional example
In this section we will construct a true infinite dimension example to apply the results of the previous section. We are going to use some results of the paper Rodrigues and Solà-Morales [19]. Consider the space X = 2 . We consider the operator J ∈ L(X) given by the infinite dimensional Jordan matrix: As it is proved in Rodrigues and Solà-Morales [19] the spectrum of J is the closed unity circle of the complex plane. Now we take 0 < a < 1 and we define the operator: From the same paper above it follows that the spectrum of L is the closed disc B ν (a) with center in a and radius ν. Then we take 0 < ν < min{a, 1 − a} Then we let A := log L = (log a)I + log(I − D). But In the space X = 2 . We consider the operator A ∈ L(X) given above.  Then there existsK,α and ω 0 > 0 such that for ω > ω 0 Proof: Follows from Theorem 7 .
Next we will present a simple example where the perturbation B(t) belongs to GAP but it is not almost periodic.
Then B ∈ GAP and has mean value zero. Let d(t) .
In the special case that we take b(t) .
= d(t) + cos t, B(t) is not almost periodic.

A case where the infinitesimal generator is unbounded.
Consider the equations:ẋ = Ax (12) We suppose that D is dense in X and A : D → X is the infinitesimal generator of a C 0 semigroup T (t), such that |T (t)| ≤ Ke αt , t ≥ 0, K ≥ 1, α ∈ R. Now we will analyse some smallness conditions on the perturbation B(t), such that the equation 13 is also asymptotically stable in the case α < 0. The case when B(t) is uniformly small is studied in Kloeden-Rodrigues [10] without leaving the continuous case. Similar results are obtained by Carvalho et all [1], but they first find the result for the discrete case.
Similar results to the next theorem are treated by Carvalho et all [1] and Dalekii-Krein [4] but they use the stronger assumption that t τ |B(t)| is small, with the norm inside the integral and in the first one they prove via a discretiztion method. Similar results are obtained by Henry [8] in Thorem7.6.11, pag. 238, where he also consider first the discrete case, and requires that B(t) is uniformly small and integrally small.
Our result is an extension of a classical result of Coppel [3] for the infinite dimensional case, and A being an unbounded operator.
We will follow the steps of Theorem 1 where we imposed that |B(t)| ≤ M for every t ∈ R and that | u t B(τ )dτ | ≤ δ for t ≤ u ≤ t + h. We also assume that the range of B(t) is contained in the domain of A.
Theorem 8. We assume besides the above assumptions on A and T (t), B : R → L(X) is a continuous function and such that for each t ∈ R AB(t) is a bounded operator and B(t)A can be extended to the whole space as a bounded operator. We suppose also that AB(t) and where h is a positive real number. We suppose that there are positive numbers M and δ such that Let S(t, s) be the evolution operator associated to system 13. Then If α is negative, h is sufficiently large and δ sufficiently small in such a way that β < 0 then it follows that system (13) is asymptotically stable.
Proof: The proof follows the ideas of (1). By the variation of constants formula Integrating the above equation, we obtain

u)B(u)S(u, s)du
And so, Therefore, We first suppose that s ≤ t ≤ s + h and estimate |S(t, s)|. Therefore, and so using Gronwall's inequality it follows that in an arbitrary interval of length h, say for For t ≥ s there exists n ∈ N, n = n(t, s) such that s + nh ≤ t ≤ s + (n + 1)h and so We are going to prove by induction that for s + nh ≤ t ≤ s + (n + 1)h The case n = 0 has already been proved.

Applications
Consider the following result from Henry [8] pg. 30.
Theorem 9. Suppose A is a closed operator in the Banach space X and suppose that σ 1 is a bounded spectral set of A, and σ 2 = σ(A) − σ 1 so σ 2 ∪ {∞} is another spectral set. Let E 1 , E 2 be the projections associated with these spectral sets, and X j = E j (X), j = 1, 2. Then X = X 1 ⊕ X 2 , the X j are invariant under A, and if A j is the restriction of A to X j , then Theorem 10. Let h and δ be positive real numbers.

Suppose that
and |B(t)| ≤ M for every t ∈ R. Suppose we can decompose σ(A) . = σ 1 ∪ σ 2 , where σ 1 is a bounded spectral set and σ 2 = σ(A) − σ 1 so σ 2 ∪ {∞} is another spectral set. Suppose there is a smooth curve Γ, oriented positively, that contains σ 1 in its interior and σ 2 is in the exterior of Γ. Consider the projection P 1 .
for t ≥ 0, where µ > α. Then AP 1 is a bounded operator and P 1 A = AP 1 and so P 1 A is also a bounded operator.
The above decomposition is chosen in such a way that |P 2 B(t)| ≤ M δ for every t ∈ R .
In analogy with the bounded case if C t (u) . = u t B(τ )dτ , we suppose that Consider the equations:ẋ = Ax (15) y = Ay + B(t)y (16) If the above assumptions are satisfied if δ is sufficiently small, h is sufficiently large and (15) is asymptotically stable then system (16) is also asymptotically stable.
Proof: The proof follows the ideas of Theorem 8. The next example is in R 2 and it shows that it is possible stabilise an unstable system under a small (in mean value) periodic perturbation.
Let 0 < α < β and δ < T . Let the T -periodic operator given by Consider the systems:ẋ = Ax (18) First we observe that lim T →∞ 1 T T 0 D(s)ds = 0, that is B(t) has zero mean value, but has large period.
Nest we are going to prove, using Floquet Theorem that system (19) is uniformly asymptotically stable.
For the sistemẋ = A(t)x, where A(t) is continuous and T -periodic, will use Floquet's Theorem even if , A(t) is not continuous, according to the comment in [7] page 118.
Consider the matrix solution X(t) of (18) such that X(0) = I the identity matrix. Then it is given by 0   then we have the rotation matrix:   , the fundamental matrix Y (t) ofẏ = (A + D(t))y, such that Y (0) = I will be given by Then the monodromy matrix will be Now we can find the eigenvalues of the monodromy Y (T ) and they will be the caracteristic multipliers of (19) The caracteristic polynomial is given by p(λ) Thereforeẏ = (A + D(t))y is uniformly asymptotic stable.

Stabilizing Unstable Linear ODE in Infinite Dimensions.
There is a classical example in Operator Theory due to S. Kakutani of a bounded operator in an infinite-dimensional Hilbert space whose spectrum shrinks drastically from a disk to a single point under an arbitrarily small bounded perturbation. The example can be found in [13] (p. 282) and [6] when the systemẋ(t) = Ax(t) is unstable and the perturbation B(t) is small in some senses.
Roughly speaking, we could say that the examples of this section show that while stability is a robust feature, instability does not need to be so.
Let us describe briefly the example of Kakutani with the notations and choices of [21]. In a real separable Hilbert space H with a Hilbert orthonormal basis (e n ) n≥1 a weighted shift operator W ∈ L(H) is a bounded linear operator defined by the relations W e i = α i e i+1 for a bounded sequence of real numbers (α n ) n≥1 . One readily sees that W = sup{|α n |} and W k = sup{|α n α n+1 · · · α n+k−1 |}.
We choose first the sequence ε m = M/K m−1 for some M > 0 and some K > 1, and define a weighted shift W ε by α n = ε m if n = 2 (m−1) (2 + 1), where is a non-negative integer.
This sophisticated way of distributing the numbers ε m into a sequence α n makes a number ε m to appear for the first time in the α n sequence at the position n = 2 (m−1) and from that position onwards to appear periodically, infinitely many times, with a period of 2 m .
Then, one also defines the weighted shifts L m by a sequence of weights α n that are all of them equal to zero, except at the positions n = 2 (m−1) (2 + 1), where is a non-negative integer, where α n = ε m . With this choice, the operator W ε − L m is also a weighted shift, and it has zeroes along its sequence of weights, distributed each 2 m places, and starting at the 2 (m−1) position. This means, according to 21, that W ε − L m is nilpotent of index 2 m , (W ε − L m ) 2 m = 0. Consequently, its spectral radius ρ(W ε − L m ) = 0. One can also obtain, after some work, that ρ(W ε ) = M/K and that the spectrum σ(W ε ) is the whole disk of radius M/K centered at zero. Concerning the norms, by using (21) one gets that W ε = M and In this way, Kakutani's example shows the existence of a bounded linear operator W ε with positive spectral radius that is approximated, in the operator norm, by a sequence W ε − L m of operators whose spectrum reduces to the single point 0.
Our fist example of translation of these ideas to (20) is very simple. Let us choose a number which is unstable since R + M/K > 1.
We construct now the sequence of operators S m = R I + W ε − L m . All of these operators have their spectra reduced to the single point z = R, and these operators converge in the operator norm to T = R I + W ε , which spectrum is the disk of radius M/K centered at z = R. If we take now A m = log(R I + W ε − L m ), we again have that the sequence A m tends to A = log(T ) as m → ∞ in the operator norm, by the continuity of the logarithm. Also, by the properties of the exponential, perhaps by using adapted norms, for all δ > 0 and all m, there exists a number D m,δ such that which implies stability since log(R) < 0, and δ can be chosen small enough.
In this way we have perturbed an autonomous unstable systemẋ(t) = Ax(t) to a new autonomous systemẋ(t) = Ax(t) + (A m − A)x(t), with a perturbation that can be taken as small as we wish in the operator norm, and the new system is asymptotically stable.
This example deserves to be commented in relation of Theorem 4 of [10] (p. 2704).
According to that theorem, if an equationẋ(t) = A(t)x(t) exhibits an exponential dichotomy with nontrivial stable and an unstable part (which in particular means that it is unstable), then a new systemẋ(t) = A(t)x(t) + B(t)x(t) will exhibit a similar dichotomy (which means that it is also unstable) if sup{ B(t) ; t ∈ R} is sufficiently small, and if some compactness conditions are met, that are automatically satisfied in our case since B does not depend on t. This robustness of the instability is broken in our example, since the spectrum of A is a connected set that has points both in Re z < 0 and in Re z > 0, but it is not possible to divide it into two spectral sets by the vertical line Re z = 0. This is something very typical from infinite dimensional functional analysis, that cannot be expected in finite dimensions.
Our second example, also based on Kakutani's construction, starts with the same systeṁ x(t) = Ax(t) as above, with A = log(R I + W ε ) and W ε , with the relations 0 < R − M/K < R < 1 < R + M/K, whose instability is expressed by the inequality (22) above. We want to add to it now a time-dependent perturbation B(t), depending continuously on t ≥ 0 such that sup{ B(t) ; t ∈ [0, ∞)} can be taken as small as we wish, but with the novelty that lim t→∞ B(t = 0. Despite of this, we want to obtain a systemẋ(t) = Ax(t) + B(t)x(t) that will be stable.
for an increasing sequence t k with t 0 = 0 and t k + 1 < t k+1 , to be defined later. It is clear that B(t) is a continuous function from [0, ∞) to L(H). Since B m → 0 it is clear that Therefore, B(t) ≤ E m 0 for all t ≥ 0, and this can be made as small as we like by choosing m 0 sufficiently large.
In order to define the sequence (t k ) k≥0 let us now bound the solutions of     ẋ (t) = Ax(t) + B(t)x(t), For t between t k and t k+1 − 1 we will have A + B(t) = A m 0 +k and, because of (23), x(t) ≤ x(t k ) D m 0 +k e −ω(t−t k ) .
To fix ideas, let us start with k = 0. For 0 = t 0 ≤ t ≤ t 1 − 1 we can write x(t) ≤ D m 0 e −ωt x(0) . Then, for t 1 − 1 ≤ t ≤ t 1 we can broadly bound as and, putting the two parts together which obviously implies the weaker bound both for 0 ≤ t ≤ t 1 . Then, we continue with t 1 ≤ t ≤ t 2 − 1, and for this range of t we have A + B(t) = A m 0 +1 and and, as before, now for the whole t 1 ≤ t ≤ t 2 . Putting this together with (26) we get, again for t 1 ≤ t ≤ t 2 , x(t) ≤ e ( A +Em 0 ) D m 0 +1 e −ω(t−t 1 ) e ( A +Em 0 ) D m 0 e −ωt 1 x(0) , that we can write again as x(t) ≤ e ( A +Em 0 ) D m 0 +1 e −ω(t−t 1 ) e ( A +Em 0 ) D m 0 e − 1 2 ωt 1 e − 1 2 ωt 1 x(0) , and at this point we see that we can choose t 1 large enough in such a way that e ( A +Em 0 ) D m 0 +1 e − 1 2 ωt 1 ≤ 1.
With this choice we get for t 1 ≤ t ≤ t 2 , which will be needed in the next interval, and also deduce, together with (27) the weaker but more global bound x(t) ≤ e ( A +Em 0 ) D m 0 e − 1 2 ωt x(0) , now for all t such that 0 ≤ t ≤ t 2 .
Now we proceed inductively. Suppose that along the interval t k−1 ≤ t ≤ t k , where t k is still to be chosen, we have obtained, as in (28), the bound for t k−1 ≤ t ≤ t k , and the weaker inequality for 0 ≤ t ≤ t k . Then we analyze for t k ≤ t ≤ t k+1 and obtain that x(t) ≤ e ( A +Em 0 ) D m 0 +k e −w(t−t k ) e ( A +Em 0 ) D m 0 e −ω(t k −t k−1 ) e − 1 2 ωt k−1 x(0) .
Then we choose t k in such a way that and obtain for t k ≤ t ≤ t k+1 , and the weaker inequality for 0 ≤ t ≤ t k+1 .
With these choices of the t k one can make k → ∞ and obtain the final bound for all t ≥ 0, that proves the exponential asymptotic stability of the solutions of (25).