We show the role which plays a recent theorem on the strong asymptotic stability in the analysis of the strong stabilizability problem in Hilbert spaces. We consider a control system with skew-adjoint operator and one-dimensional control. We examine in details the property for a linear feed-back control to stabilize such a system. A robustness analysis of stabilizing controls is also given.

We consider the strong stabilizability problem for delayed systems of neutral type. For simplicity the case of one delay in state is studied. We distinguish a special class of such systems for which we give a constructive solution, without using the derivative of the localized delayed state. Our results are based on an abstract theorem on the strong stabilizability of contractive systems in Hilbert space. An illustrative example is also given.

In this paper necessary and sufficient conditions of null-controllability and approximate null-controllability are obtained for the wave equation on a half-axis. Controls solving these problems are found explicitly. Moreover bang-bang controls solving the approximate null-controllability problem are constructed with the aid of solutions of the Markov trigonometric moment problem.

Розглянуто керованість балки, яку прикріплено до диска, що повільно обертається. Спочатку балка знаходиться у стані спокою; необхідно виконати обертання на заданий кут і досягти заданого стану. Рух відбувається за системою двох диференціальних рівнянь із неоднорідними коефіцієнтами: густина маси, інерція обертання, жорсткість на згинання та жорсткість на зсув. Задачу керованості зведено до проблеми моментів, яка, в свою чергу, розв'язується з використанням асимптотики спектра оператора, що зв'язаний із рухом.

Controllability of slowly rotating non-homogeneous beam clamped to a disc is considered. It is assumed that at the beginning the beam remains at the position of rest and it is supposed to rotate by the given angle and achieve desired position. The rotor of propelling engine is in the middle of the disk. The movement is governed by the system of two differential equations with non-constant coefficients: linear mass density, flexural rigidity, rotational inertia and shear stiffness. To solve the problem of controllability, the spectrum of the operator generating the dynamics of the model is studied. Then the problem of controllability is reduced to the moment problem that is, in turn, solved with the use of the asymptotics of the spectrum and Ullrich Theorem.

The paper is a survey of the main ideas and results on using of the Markov moment problem method in the optimal control theory. It contains a version of the presentation of the Markov moment approach to the time-optimal control theory, linear and nonlinear.