Anna Sukretna

This paper investigates the linear–quadratic optimal stabilization problem in the so-called critical case, that is, the situation in which the spectrum of the system matrix contains purely imaginary eigenvalues or the standard positive-definiteness conditions on the weight matrices of the objective functional are violated. To address these challenges, new regularization methods for critical problems via perturbation of the system matrices and the functional are studied, and novel algorithms for decomposing multidimensional problems into a collection of one-dimensional canonical systems are developed. The main contribution of this work lies in providing a systematic framework for critical cases where standard methods fail. The results are of practical significance for the construction of optimal synthesis in various engineering and applied systems; in particular, they are applicable to the stabilization of unmanned aerial vehicles, robotic complexes, and intelligent power grids.

Mathematical models of the infection spread dynamics have remained a relevant area of research for the past few decades. Moreover, these models have also proven to be suitable for modeling DDoS attacks in computer networks. The qualitative behaviour of solutions in such models is usually arranged as follows: there is some threshold value at which the global asymptotic stability of the solution with a zero component, which is responsible for the number of infected individuals, changes to the stability of the non-zero equilibrium position. From a meaningful point of view, this means that the self-exhaustion and disappearance of the infection changes to a constant circulation of the infection in the population. However, the presence of an impulsive perturbation can make the infection-free solution asymptotically stable regardless of the threshold value, and such impulsive perturbations allow a natural interpretation within the framework of the epidemiological model as the process of vaccination of a certain part of individuals. Such problems have already been considered for the SIR- and SEIR-models in the case of a simple structure of impulsive points, due to which the impulsive problem was actually reduced to a problem without impulsive perturbation. However, a more complex structure of the set and magnitudes of impulsive effects requires the application of the theory of impulsive differential equations, which is done in this work. Namely, the article considers the SEIR-system of differential equations with impulsive perturbations that have an almost periodic character. The existence and properties of an infection-free impulsive almost periodic solution are investigated, and its asymptotic stability is proven.

The article analyzes the linear-quadratic optimal stabilization problem in the so-called "critical case", namely, the situation is considered when the spectrum of the system matrix contains purely imaginary eigenvalues or when the standard conditions of positive definiteness of the weight matrices of the quality functional are violated. Methods for regularizing critical problems by perturbing the system matrices and the functional are investigated, and algorithms for decomposing multidimensional problems into a set of one-dimensional canonical systems are proposed. The results are of practical importance for constructing optimal synthesis in various engineering and economic systems, in particular, the results can be used for stabilizing unmanned aerial vehicles, robotic complexes and intelligent power grids.

This article is dedication to the life and scientific path of the famous Ukrainian mathematician M.O. Perestyuk. The problems of the differential equations with impulse action theory, which was one of the leading M.O. Perestyuk’s research directions, are illustrated by examples accessible to a wide audience.

In this paper we analyze the problem of the optimal bounded control with semi-definite quality criterion for parabolic process with rapidly oscillating coefficients on finite time interval. The case when optimal control goes out both on lower, and on upper restriction, that is has two switch-points is considered. Our aim is to build a synthesized control which is practically preferable to program one. But after Fourier method application such a control will contain coefficients which are expressed by series and, in addition, it irregularly depends on small parameter ε. Thus, for practical application it is natural to restrict infinite series by finite sums and to get rid of dependence on a small parameter using average procedure. So, besides the law of optimal synthesis also approximate averaged feedback control, which provides control system behavior that is close to optimal one and thus has a series of advantages from the practical application point of view, is proposed and proved. The efficiency of approximate averaged control construction procedure consisted of cutting to finite sums of series in optimal control and replacing of Fourier coefficients nonregular depended on small parameter by corresponding average values is illustrated by an example of controlled system for parabolic process. We compare the properties of the averaged control and the sequence of optimal controls calculated at the different values of the small parameter. For comparison, we use the switching points of optimal control and averaged control, deviation between optimal control and averaged control, the difference in the values of the quality criterion on optimal control and averaged control. In considered example the precision of quality criteria value on approximate control has ε-order and for sufficiently small ε the precision is one order better then ε.

The article is devoted to the life and scientific path of academician of the National Academy of Sciences of Ukraine, Нonored professor of Taras Shevchenko Kyiv National University, Mykola Oleksiyovych Perestyuk. In particular, the material contains a complete list of Mykola Oleksiyovych's students and the topics of their dissertations, a list of monographs, textbooks, study guides and major scientific publications, a list of major state awards and titles.

For the problem of optimal control of a parabolic-hyperbolic process with nonlocal point boundary conditions, an explicit form of the solution is obtained in the form of formal series according to the system of eigenfunctions, which are generated by the spatial differential operator and boundary conditions. At the same time, the unequivocal solvability of the intermediate problems is established for each iteration. In addition, sufficient conditions for the convergence of the series are established, which determine the obtained formal solution of the optimal control problem, which justifies its correctness system of differential equations; Lyapunov exponents; fractal dimension

Processes that involve jump-like changes are observed in mechanics (the movement of a spring under an impact; clockwork), in radio engineering (pulse generation), in biology (heart function, cell division). Therefore, high-quality research of pulse systems is a relevant task in the modern theory of mathematical modeling. This paper considers the issue related to the existence of bounded solutions along the entire real axis (semi-axis) of the weakly nonlinear systems of differential equations with pulse perturbation at fixed time moments. A concept of the regular and weakly regular system of equations for the class of the weakly nonlinear pulse systems of differential equations has been introduced. Sufficient conditions for the existence of a bounded solution to the heterogeneous system of differential equations have been established for the case of poorly regularity of the corresponding homogeneous system of equations. The conditions for the existence of singleness of the bounded solution along the entire axis have been defined for the weakly nonlinear pulse systems. The results were applied to study bounded solutions to the systems with pulse action of a more general form. The established conditions make it possible to use the classical methods of differential equations to obtain statements about solvability and the continuous dependence of solutions on the parameters of a pulse system. It has been shown that classical qualitative methods for studying differential equations are mainly naturally transferred to dynamic systems with discontinuous trajectories. However, the presence of a pulse action gives rise to a series of new specific problems. The theory of systems with pulse influence has a wide range of applications. Such systems arise when studying pulsed automatic control systems, in the mathematical modeling of various mechanical, physical, biological, and other processes.