On properties of the Izobov sigma-exponent (Q2761382)

The author deals with the system NEWLINE\[NEWLINE \frac{dx}{dt} = A(t)x,\qquad x\in \mathbb{R}^n,\quad t\in \mathbb{R}^+,\tag{1} NEWLINE\]NEWLINE where \(A: \mathbb{R}^+\to \text{End} \mathbb{R}^n \) is a continuous bounded operator-function. System (1) is studied in terms of the upper Izobov sigma-exponent [\textit{N. A. Izobov}, Differ. Uravn. 5, No. 7, 1186-1192 (1969; Zbl 0175.09902)] NEWLINE\[NEWLINE\nabla_\sigma(A) =\sup\limits_{Q\in K_\sigma} \lambda_n(A+Q), NEWLINE\]NEWLINE where \(K_\sigma = \{Q(\cdot):\|Q(t)\|\leq C(Q)e^{-\sigma t}\}\). The author studies properties of the functional \( A\mapsto \nabla_\sigma(A) \) in terms of the Baire classification of functions. An exact Baire class of this functional is established on the space \(M_n^{\sigma_0}\) with the metric \( \rho(A,B)=\sup_{t\in\mathbb{R}^+} e^{\sigma_0t}\|A(t)-B(t)\|\). It is proved that at a typical Baire point this functional is upper semi-continuous.