Vol 224 (2023)

Necessary and sufficient criteria of Lyapunov stability for systems of ordinary differential equations are obtained. The criteria are obtained in the multiplicative form based on the transformation of difference schemes for numerical integration and are converted to the additive and integral forms. The formal restrictions for these criteria are constructed and their applicability conditions are indicated.

Using the Fourier transform, we examine the first boundary-value problem for two elliptic systems in a half-space. We prove that for both systems, the homogeneous problem has infinitely many solutions depending on one arbitrary function. At the same time, one of the systems is strongly connected under certain conditions for the coefficients of the system, whereas the second system is always strongly connected.

We consider the optimal control problem for a linear discrete system with unknown limited disturbances, which must be transferred to a terminal set in a finite time, while providing a minimum guaranteed value of the terminal quality criterion. We discuss two approaches to constructing optimal feedbacks: the disconnectable feedback, which is determined through optimal guarantee programs, and the closed feedback based on optimal control strategies with closures. We discuss disadvantages of the first approach and propose an effective method of constructing optimal closed real-time feedback.

In this paper, we consider the Goursat problem for a partial differential equation containing a small parameter $\epsilon$ in the coefficient of the highest derivative. For $\epsilon =0$, the order of the equation does not decrease, but a singularity appears, which has the nature of a power boundary layer. A solution of the singularly perturbed Gaussian problem is constructed in the form of a formal series in powers of the small parameter. The asymptotic nature of the constructed series is proved.

We present an existence and uniqueness theorem for a nontrivial analytical zero-front solution of a problem for a nonlinear evolution parabolic predator-prey system. In special cases, we construct exact solutions by reduction to the Cauchy problem for a system of ordinary differential equations, which inherits all features of the original problem. We propose an algorithm for the numerical solution of the problem based on the method of specific solutions and present the result of computational experiments.

We examine combinatorial objects of pyramidal structure. We consider one of the ways of representing rules in hierarchical, sequential structures: the method of decision trees, where each object corresponds to a single node that provides a solution. An algorithm for constructing a decision tree based on the generalized Pascal pyramid is suggested. Also, we propose a method for constructing a search index, which displays the proportion of relevant material and allows one to perform comparisons in the variety of terms based on the weight coefficients of terms and paths.

In this paper, we consider problems of stabilization of stationary motions (equilibrium positions and regular precessions) of a satellite near the center of mass in gravitational and magnetic fields under the assumption that the center of mass moves in a circular orbit. Solutions for a number of problems of stabilizing stationary motions of a satellite with the help of magnetic systems are proposed. We present the results of mathematical modeling of the algorithms, which confirm the effectiveness of the developed methodology. This paper is the final part of the work. The first part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. — 220. — P. 71–85. The second part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. — 221. — P. 71–92. The third part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. — 222. — P. 42–63. The fourth part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. — 223. — P. 84–106.

In this paper, we consider an inverse control-type problem of determining the minor coefficient of a parabolic equation with an integral boundary-value condition and an additional integral condition. The well-posedness of the problem is examined. The Fréchet differentiability of the target functional based on the additional integral condition is proved and an expression for its gradient is found. A necessary condition for the optimality of control is established.

Degenerate linear systems of differential equations of a special form in Banach spaces are considered. The structure of the solution of the Cauchy problem for such systems is completely determined by the properties of the matrix and operator pencils of the system. Solutions are constructed in the space of distributions with support bounded on the left and are restored using the matrix fundamental operator function of the system. Based on the analysis of the generalized solution constructed in this way, one can obtain theorems on the solvability in the space of functions of finite smoothness of the original Cauchy problem.