Singular exponents and properness criteria for linear differential systems (Q2567800)

The author studies the system of linear differential equations \[ \dot x= A(t)x, \quad x\in\mathbb{R}^n, \quad t\geq 0, \] with coefficient matrix \(A(\cdot):[0,+\infty)\to \text{End}(\mathbb{R}^n)\) piecewise continuous and uniformly bounded on \(t\geq 0\). If \(\sigma_1(t)\geq\cdots\geq \sigma_n(t)\) are the singular numbers corresponding to the principal solution matrix \(X(t)\) of this system, then the numbers \[ \underline\sigma_k(A)= \varliminf_{t\to\infty}{1\over t}\ln\sigma_{n-k+1}(t) \text{ and }\overline\sigma_k(A)= \varlimsup_{t\to\infty}{1\over t}\ln\sigma_{n-k+1}(t), \quad k= 1,\dots, n, \] are called the \(k\)th lower and \(k\)th upper singular exponents of this system, respectively. As his main result, the author proves that the above system is proper if and only if \(\underline\sigma_k(A)= \overline\sigma_k(A)\) for \(k= 1,\dots, n\).