On the mathematical theory of vibration (Q1267414)

The authors consider the system \[ \dot x= f(x, u_n, t),\quad x\in \mathbb{R}^{d(x)},\quad u\in \mathbb{R}^n,\quad t\in [t_0,t_1],\tag{1} \] where \(f\) is a continuous function with respect to \(x\), \(u_n\). The problem to obtain the limit system for (1) in an explicit form as \(u_n\rightharpoonup 0\) is solved by regarding the weak convergence of the argument as a weak convergence measure in another conjugate space. This approach has advantages with respect to the one considered by the authors in [Russ. Math. Surv. 42, No. 3, 211-212 (1987); translation from Usp. Mat. Nauk 42, No. 3(255), 179-180 (1987; Zbl 0677.34051)] and allows one to describe the limit system in explicit form in the case, in which the right-hand side of (1) is discontinuous. An application of the results to model vibration problems in mechanics is indicated.