Andrei Alekseevich Galyaev

Adequate mathematical modeling of various dynamical control processes leads to control problems with nonlocal conditions, which are described by both partial differential equations and ordinary differential equations. Control problems with nonlocal conditions arise, on the one hand, as mathematical models of real processes, and on the other hand, as a result of the inability of formulating problems correctly using local conditions. Nonlocal problems in the form of integral conditions, in particular, occur in the mathematical modeling of phenomena of various natures. Research into problems with nonlocal conditions, in various formulations, is important not only for theoretical purposes but also for practical necessity. This paper investigates the problem of controlling a linear dynamical system subject to a nonlocal integral-type condition. The control problem for a linear dynamical system is formulated with an integral condition imposed on the components of the phase vector over a certain portion of the time interval during the system’s operation. Conditions are obtained under which a solution to the system of ordinary linear differential equations with the given integral condition exists and this solution is constructed. Control functions are developed for the control problem, under the influence of which the phase trajectories are realized, satisfying the integral conditions of the problem. The continuity and nonuniqueness of the control functions are demonstrated. A criterion of complete controllability of a linear system with phase constraints of the integral type is obtained. An example is given as an illustration.

We propose an analytical representation of complexity diagrams based on information about the number of discretes in the signal spectrum and the signal-to-noise ratio. This approach provides additional criteria for solving the classification problem. We formulate and prove lemmas on the construction of analytical metric grids for complexity diagrams. The obtained results are verified by numerical experiments, which have confirmed the efficiency of these methods.

Развивается тематика предыдущих работ авторов, а именно исследуются порядковые статистики дискретного нормированного спектрального распределения аддитивного белого гауссовского шума для решения задачи обнаружения детерминированного сигнала в шумовой смеси с помощью информационных признаков. В данной работе не только устанавливается дополнительная связь между дискретным спектральным распределением статистики однооконной реализации белого шума, но и приводится новый результат, задающий формулы для точного вычисления математического ожидания и дисперсии нормированной порядковой статистики. На основе полученных аналитических результатов предложена новая формула вычисления спектральной сложности, а также уточнена уже известная. Теоретические результаты верифицированы статистическим численным моделированием.

Рассматривается задача оптимального управления группой, состоящей из произвольного числа несинхронных осцилляторов с общим скалярным управляющим воздействием, по критерию быстродействия. Проведено аналитическое исследование задачи. Доказано свойство сильной достижимости и глобальной управляемости, найдено программное управление, которое переводит систему из начала координат в фиксированную точку по критерию быстродействия. Для перевода группы осцилляторов в состояние покоя найдены траектории, удовлетворяющие как уравнениям движения системы, так и дополнительным уравнениям, полученным на основе матричных условий невырожденности релейного управления. Проведено сравнение классификаций траекторий по количеству переключений управления, найденных с использованием необходимых условий экстремума и численного алгоритма Нейштадта–Итона.

This paper further develops the topic of the authors' previous research. In particular, order statistics of the discrete normalized spectral distribution of the additive white Gaussian noise are investigated to detect a deterministic useful signal in a noise mixture using information features. An additional relationship is established between the discrete spectral distribution of the single-window realization statistics of the white noise. Another novel result consists in exact formulas for calculating the mean and variance of normalized order statistics. Based on the analytical expressions derived, a new formula for calculating the spectral complexity is proposed and the already known one is refined. The theoretical results are verified by statistical simulation.

Note-addition to the article “A new spectral measure of complexity and its capabilities for detecting signals in noise”.

This article is devoted to the improvement of signal recognition methods based on the information characteristics of the spectrum. A discrete function of the normalized ordered spectrum is established for a single window function included in the DFT. Lemmas on estimates of entropy, imbalance and statistical complexity in processing a time series of independent Gaussian quantities are proved. New concepts of one-dimensional and two-dimensional spectral complexities are proposed. The theoretical results obtained were verified by numerical experiments, which confirmed the effectiveness of the new information characteristic when detecting a signal mixed with white noise at low signal-to-noise ratios.

An addition to the article “A new spectral measure of complexity and its capabilities for detecting signals in noise” is presented.