Universal perturbations of linear differential equations (Q2759017)

Let \(\mathbb{R}^n\) be endowed with a norm \(\|\cdot\|\), and let \(L(\mathbb{R}^n)\) be the space of all linear endomorphisms endowed with the corresponding operator norm \(\|\cdot\|\).NEWLINENEWLINENEWLINEFor a given function \(A\in \mathbb{C}^1 (0,\infty, L(\mathbb{R}^n))\), the author considers a fundamental system \(X: [0,\infty)\to L(\mathbb{R}^n)\) for \(x'(t)= A(t) x(t)\). He assumes that \(A\) is such that \(X\) and \(X^{-1}\) are bounded on \([0,\infty)\). The author proves the theorem:NEWLINENEWLINENEWLINELet \(n\geq 2\) and let \(A\) be as above. Let \(y_0\in \mathbb{R}^n\), \(y_0\neq 0\), and let \((t_k)\), \(k\in (1,\infty)\), be a strictly increasing sequence in \([0,\infty)\) with \(t_k\to \infty\) \((k\to\infty)\). Let \(U\) denote the set of all \(B\in F\) with the following property.: The solution \(y: [0,\infty)\to \mathbb{R}^n\) to NEWLINE\[NEWLINEy'(t)= (A(t)+ B(t)) y(t),\quad y(0)= y_0,NEWLINE\]NEWLINE satisfies \(\overline{(y(t_k):k\in\mathbb{N})}= \mathbb{R}^n\). Then \(U\) is a dense \(G_0\) subset of \(F\).