On a method for studying the norm and the stability of solutions (Q2473728)

Several sufficient conditions for stability (respectively, asymptotic stability, instability) for certain classes of systems of time-varying ordinary differential equations are presented. The first theorem concerns linear systems of the form \(\dot x= A(t)x\), where \(x\in\mathbb{C}^n\), \(n\geq 2\) and \(A(t)\) is normal, that is \(AA^*= A^*A\) (\(A^*\) is the conjugate matrix of \(A\)). This result is extended to quasi-linear systems (i.e., systems of the form \(\dot x= A(x,t)x\)), systems with quasi-normal matrices (i.e., of the form \(A(t)+ B(t)\), \(A(t)\) being normal), and linearizable systems (i.e., systems of the form \(\dot x= A(x, t)x+ f(t, x)\)). A further extension of these results concerns singular systems.