Stabilization of solutions to differential equations in a Hilbert space (Q1363943)

The following differential equation is studied \[ y''(t)+ Ay'(t)+ By(t)= 0\quad (t\geq 0), \tag{1} \] where \(A\) and \(B\) are commuting selfadjoint operators in a Hilbert space \(H\). Solutions of (1) are considered in the space \(C^2(\mathbb{R}_+,H)\). A few properties of solutions of the equation (1) at infinity are established. For example, the limit \(\lim_{t\to \infty} y(t)/t\) exists if and only if the operators \(A\) and \(B\) are nonnegative. Another result asserts that any solution of (1) is bounded iff operators \(A\) and \(B\) are nonnegative and \((0,0)\in \sigma (A,B)\), where \(\sigma (A,B)\) denotes the joint spectrum of \(A\) and \(B\). Some applications to the problems of mathematical physics are derived.