We consider the problem of the boundary control of one-dimensional oscillations of an effective (averaged) medium corresponding to a two-phase medium consisting of periodically alternating layers of elastic and viscoelastic materials with long-term memory or various viscoelastic materials with Kelvin–Voigt friction and long-term memory. The averaged model is described by a boundary value problem for an integrodifferential equation. It is shown that for this model, it is impossible to bring oscillations to a state of rest in finite time (in contrast to the equation of string oscillations) by a force acting at one end of the band. A hypothesis is formulated on the possibility of bringing the specified object to a state of rest with the help of force effects distributed along the entire length of the object.

We consider the problem of the boundary control of one-dimensional oscillations of an effective (averaged) medium corresponding to a two-phase medium consisting of periodically alternating layers of elastic and viscoelastic materials with long-term memory or various viscoelastic materials with Kelvin–Voigt friction and long-term memory. The averaged model is described by a boundary value problem for an integrodifferential equation. It is shown that for this model, it is impossible to bring oscillations to a state of rest in finite time (in contrast to the equation of string oscillations) by a force acting at one end of the band. A hypothesis is formulated on the possibility of bringing the specified object to a state of rest with the help of force effects distributed along the entire length of the object.

We consider a nonlinear problem of equilibrium of an elastic body with periodically located cracks. On the edges of these cracks, non-penetration conditions are given. The nonlinear problem is formulated in the form of variational inequality. The period of distribution of the cracks, as well as their sizes, depends on a small parameter. The behavior of the solution to the problem with periodically located cracks is determined by the first two terms $u^0(x)$ and $u^1(x, y)$ of the asymptotic expansion. In this paper, we study the solution of the variational inequality on a periodicity cell (a local problem). For the first corrector $u^1(x, y)$, we construct a penalty equation and a linear iterative equation in integral form. We prove that the sequence of solutions of the problem with penalty converges to the solution of the problem on the cell when the small regularization parameter tends to zero. We show that the approximate solution of the iteration equation converges strongly to the solution of the penalty equation. Рассматривается нелинейная задача о равновесии упругого тела с периодически расположенными трещинами. При этом на краях трещин ставятся односторонние ограничения, что приводит к вариационному неравенству. Период распределения трещин, а также их размеры зависят от малого параметра. Поведение решения задачи с периодически расположенными трещинами определяется двумя первыми членами $u^0(x)$, $u^1(x, y)$ асимптотического разложения. В статье изучается решение вариационного неравенства на ячейке периодичности (локальная задача). Для первого корректора $u^1(x, y)$ строится уравнение со штрафом и линейное итерационное уравнение в интегральной форме. Доказано, что последовательность решений задачи со штрафом при стремлении малого параметра регуляризации к нулю сходится к решению задачи на ячейке. Показано, что приближенное решение итерационного уравнения сходится сильно к решению уравнения со штрафом.