On regularity of linear systems of differential equations on the whole real axis (Q2740275)

The linear homogeneous system (LHS) \(\dot x=A(t)x, A(t)\in C(\mathbb R \mapsto \mathbb R^{n\times n})\), with bounded coefficients is called regular (weakly regular) if the nonhomogeneous system \(\dot x=A(t)x+f(t)\) has a unique (at least one) solution bounded on \(\mathbb R\).NEWLINENEWLINENEWLINEThe author obtains a number of sufficient conditions for the regularity of LHS with the matrix \(A(t)\) of special structure. E.g., in the case where \(n=3m\), NEWLINE\[NEWLINEA(t)=\begin{pmatrix} B_1(t)&-A_{12}^*(t)&-A_{13}^*(t)\\ -A_{12}(t)&-B_2(t)&-A_{23}(t)\\ A_{13}(t)&-A_{23}(t)&-B_3(t) \end{pmatrix},NEWLINE\]NEWLINE \(B_i(t), A_{ij}(t)\in C(\mathbb R \mapsto \mathbb R^m)\) these sufficient conditions have the following form: \(B_3(t)\) is symmetric and uniformly positive, \(B_1(t), B_2(t)\) are symmetric and nonnegative, and the systems \(\dot x=-\frac{1}{2}B_i(t)\), \(i=1,2,\) are weakly regular.NEWLINENEWLINENEWLINEProofs are based on the construction of a nondegenerate quadratic form \(\langle S(t)x,x\rangle \) (Lyapunov function) with positive derivative along solutions to the LHS.