# On Uniform Exponential Stability of Exponentially Bounded Evolution Families

## Abstract.

A result of Barbashin (, ) states that an exponentially bounded evolution family $$\{U(t, s)\}_{t \geq s \geq 0}$$ defined on a Banach space and satisfying some measurability conditions is uniformly exponentially stable if and only if for some 1 ≤ p < ∞, we have that:

$$\mathop {\sup}\limits_{t \geq 0} \int^{t}_{0} ||U(t, s)||^{p} ds < \infty .$$

Actually the Barbashin result was formulated for non-autonomous differential equations in the framework of finite dimensional spaces. Here we replace the above ”uniform” condition be a ”strong” one.

Among others we shall prove that the evolution family $$\{U(t, s)\}_{t \geq s \geq 0}$$ is uniformly exponentially stable if there exists a non-decreasing function $$\phi : {\bf R}_+ \rightarrow {\bf R}_+$$ with $$\phi(r) > 0$$ for all r > 0 such that for each $$x^{*} \in X^{*}$$, one has:

$$\mathop {\sup}\limits_{t \geq 0} \int^{t}_{0} \phi (||U(t, s)^{*}x^{*}||) ds < \infty .$$

In particular, the family U is uniformly exponentially stable if and only if for some 0 < p < ∞ and each $$x^{*} \in X^{*}$$, the inequality

$$\mathop {\sup}\limits_{t \geq 0} \int^t_0 ||U(t, s)^{*}x^{*}||^p ds < \infty$$

is fulfilled. The latter result extends a similar one from the recent paper . Related results for periodic evolution families are also obtained.

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Correspondence to C. Buşe.

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Buşe, C., Choudary, A.D.R., Dragomir, S.S. et al. On Uniform Exponential Stability of Exponentially Bounded Evolution Families. Integr. equ. oper. theory 61, 325–340 (2008). https://doi.org/10.1007/s00020-008-1592-7

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### Mathematics Subject Classification (2000).

• Primary 47D06
• Secondary 35B35

### Keywords.

• Operator semigroups
• rearrangement function space
• evolution families of bounded linear operators
• uniform exponential stability