Preamble

In this section, we consider the linear differential equation in a Banach space $E$:
$$ \frac{dx(t)}{dt} = A(t)x(t), \quad t \in [0, +\infty) \quad \tag{0.1} $$
where $A(t)$ is a bounded linear operator such that $\sup_{t \ge 0} |A(t)| \le M$. Let $U(t, s)$ be the evolution operator (propagator) for equation (0.1), satisfying:
$$ \frac{dU(t, s)}{dt} = A(t)U(t, s), \quad U(s, s) = I \quad \tag{0.4} $$
As established in [1], the upper and lower Lyapunov exponents for the system (0.1) can be characterized by the growth rates of the norm of the evolution operator. Specifically, the upper exponent $\alpha^$ is defined as:
$$ \alpha^ = \lim_{t-s \to \infty} \frac{\ln |U(t, s)|}{t-s} \quad \tag{0.6} $$
where the limit is taken such that $0 \le s \le t < +\infty$. It is known that for any $\epsilon > 0$, there exists a constant $N_\epsilon$ such that $|U(t, s)| \le N_\epsilon \exp[(\alpha^* + \epsilon)(t-s)]$.

1. Stability and Growth Estimates

Let $\alpha_s$ denote the Bohl exponent. For any $\beta > 0$ and all $t > 0$, the following inequality holds:
$$ |U(t, 0)| \le N_\beta \exp[(\alpha_s + \beta)t] \quad \tag{1.1} $$
If we consider a solution $x(t)$ with initial condition $x_0 \in E$, we can analyze the behavior of the system in various domains $D$. Under the assumption that $\rho < 0$, the system exhibits stability. Furthermore, if there exists $A_0 > 0$ such that the conditions in (1.2) are satisfied for a non-empty set $D_1$, then according to the results in [1], we have the estimate:
$$ |U(t, s)| \le N \exp[(\alpha_s + \beta)(t-s)] \quad \tag{1.3} $$
where $N$ depends on the parameters of the operator $A(t)$. Conversely, for $s \in D_2$, the norm of the evolution operator satisfies the lower bound:
$$ |U(t, s)| \ge \exp[(\alpha_s + \rho)(t-s)] \quad \tag{1.4} $$
where $0 < \rho < \rho_0$.

2. Asymptotic Behavior and Perturbations

We assume that the operator $A(t)$ satisfies a condition of slow variation or regularity, such as:
$$ |T_{\chi_j} A(t) - A(t)| \le C \exp(-\gamma h_j) \quad \tag{2.1} $$
where $T_{\chi_j} A(t) = A(t + \chi_j)$. Under these conditions, we investigate the relationship between the spectral properties of $A(t)$ and the exponents of the system. If $\rho_0 = 0$ in (2.2), then for $t - s > T$, the evolution operator satisfies:
$$ |U(t, s)| \le \exp\left \alpha_s + \rho_0 + \epsilon \right \quad \tag{2.3} $$
As $j \to \infty$ and $t \to \infty$, we can show that for $t \in [l, l+L]$, the norm $|U(t, s)|$ is bounded from below by:
$$ |U(t, s)| \ge \exp\left \alpha_s + \beta_j - \epsilon \right \quad \tag{2.5} $$
By applying the translation operator $T_{\chi_j}$ and considering the identity (2.13):
$$ T_{\chi_j} U(t, s) - U(t, s) = \int_s^t U(t, \tau) [T_{\chi_j} A(\tau) - A(\tau)] T_{\chi_j} U(\tau, s) d\tau \quad \tag{2.13} $$
we can derive that the difference between the perturbed and original evolution operators vanishes asymptotically. Specifically, using the estimates (2.1) and (2.11), we obtain:
$$ \lim_{j \to \infty} |T_{\chi_j} U(t_j, \tau_j) - U(t_j, \tau_j)| \exp(-\alpha_s t_j) = 0 \quad \tag{2.15} $$
This implies that $\alpha_s = \alpha^*$, confirming that for operators satisfying the regularity condition (2.1), the Bohl and Lyapunov exponents coincide.

3. Numerical Ranges and Stability Criteria

Following the approach in [3], let us define the numerical range of the operator $A(t)$. Suppose there exist functions $\alpha(t)$ and $\rho(t)$ such that for all $\phi \in E$ with $|\phi| = 1$:
$$ \alpha(t) \le \text{Re}(A(t)\phi, \phi) \le \rho(t) \quad \tag{3.2} $$
Then for any solution $x(t)$ of (3.1), the growth is constrained by:
$$ \lim_{t \to \infty} \frac{1}{t} \int_0^t \alpha(s) ds \le \lim_{t \to \infty} \frac{\ln |x(t)|}{t} \le \lim_{t \to \infty} \frac{1}{t} \int_0^t \rho(s) ds \quad \tag{MATH_0003} $$
If the condition $\int_0^\infty \rho(s) ds < 0$ is satisfied, the system (3.1) is asymptotically stable. In the context of the class $Z(\nu, N)$ defined in [1], the system is exponentially stable if there exist constants $N$ and $\nu > 0$ such that:
$$ |x(t)| \le N \exp[-\nu(t-s)] |x(s)| $$
This stability is closely linked to the property $\alpha^* = \alpha_s$ under the perturbation conditions discussed above.

References

1. Krein, S. G., Linear Differential Equations in Banach Space, Nauka, Moscow, 1967.
2. Bylov, B. F., Vinograd, R. E., Grobman, D. M., Nemytskii, V. V., Theory of Lyapunov Exponents, Nauka, Moscow, 1966.
3. Daletskii, Yu. L., Krein, M. G., Stability of Solutions of Differential Equations in Banach Space, Nauka, Moscow, 1970.
4. Riesz, F., Sz.-Nagy, B., Functional Analysis, Ungar, New York, 1955.