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Vol 61, No 11 (2025)
ORDINARY DIFFERENTIAL EQUATIONS
STABILITY OF FIXED POINTS IN ORDERED SPACES. APPLICATIONS TO BOUNDARY VALUE PROBLEMS FOR HOPFIELD-TYPE EQUATIONS OF A NEURAL NETWORK
Abstract
The ordinal structure of the set of fixed points of a monotone operator acting in a partially ordered space is investigated. The conditions for the stability of the set of fixed points to changes in the monotone operator are obtained. Based on these results, a boundary value problem and a control system for the differential equation of a neural network — a Hopfield-type model of the electrical activity of the brain with a continuous and discontinuous activation function — are investigated.
Differential Equations. 2025;61(11):1443-1459
1443-1459
ULTIMATE BOUNDS, EQUILIBRIUM POINTS AND BIFURCATIONS IN A THREE-DIMENSIONAL CANCER MODEL
Abstract
A nonlinear system describing the dynamics of cancer growth is investigated. For all values of the system parameters, the existence of an attractor is proved and positively invariant sets containing it are found. The estimates of ultimate bounds are calculated. All equilibrium points are found, the conditions of their existence and bifurcation are proved. In the parameter space of the system, sets are found where these conditions are fulfilled. Examples of constructing intersections of these sets with two-dimensional planes are given. Other characteristics associated with the appearance of periodic trajectories and chaotic dynamics are calculated.
Differential Equations. 2025;61(11):1460-1473
1460-1473
PARTIAL DERIVATIVE EQUATIONS
UPPER ESTIMATES OF THE GROWTH OF PERTURBATIONS BY TRIAXIAL SPREADING-DRAIN IN INFINITE VISCOUS SPACE
Abstract
The development over time of a system of small perturbations imposed on a triaxial homogeneous spreading-drain in an infinite three-dimensional space of a Newtonian incompressible fluid is investigated. In the first part of the work, it is assumed that the main motion is stationary and the velocity field is defined by only two constants. In this case, the linearized problem with respect to velocity and pressure perturbations is reduced to a spectral problem in which the real part of the spectral parameter is related to the nature of the exponential decay or growth of the initial perturbations. Based on the method of integral relations for quadratic functionals, an upper bound for this parameter is performed. In the second part of the paper, a more general case of unsteady triaxial spreading-drain is considered. An upper integral estimate of the growth of perturbations is derived, which includes a time function completely determined by the velocity field of the main fluid flow.
Differential Equations. 2025;61(11):1474-1481
1474-1481
CONTROL THEORY
ON RECONSTRUCTION OF DISTURBANCES OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH INACCURATE MEASUREMENT OF PART OF THE COORDINATES
Abstract
The article studies the problem of approximate online reconstruction of an unknown disturbance acting on a system described by ordinary differential equations. Under the assumption that some of the system coordinates are measured inaccurately, an algorithm for solving it is proposed, based on a combination of feedback control methods and methods of the theory of ill-posed problems. The convergence of the constructed approximations to the exact disturbance is established with appropriate matching of the measurement error and appropriately selected grids involved in the calculations.
Differential Equations. 2025;61(11):1482-1489
1482-1489
GENERALIZED SOLUTIONS OF HAMILTON-JACOBI EQUATIONS WITH FRACTIONAL COINVARIANT DERIVATIVES AND TIME-MEASURABLE HAMILTONIAN
Abstract
The paper is devoted to the study of generalized in the minimax sense solutions of a Cauchy problem for a (path-dependent) Hamilton-Jacobi equation with fractional coinvariant derivatives under a right-end boundary condition in the case where the Hamiltonian of the equation depends on the time variable in a measurable way. Theorems on the existence and uniqueness of the minimax solution and a theorem on the continuous dependence of this solution on variations of the Hamiltonian and boundary functional are proved. An application of the obtained results to the study of a differential game for a dynamical system described by a differential equation with a Caputo fractional derivative is given.
Differential Equations. 2025;61(11):1490-1509
1490-1509
ON AN ALGORITHM OF DISTURBANCE RECONSTRUCTION FOR A NONLINEAR DIFFERENTIAL-ALGEBRAIC SYSTEM
Abstract
The problem of reconstructing an unknown disturbance in a nonlinear system consisting of a combination of differential and algebraic equations is considered. Two cases are discussed. In the first case, a disturbance enters the system linearly; while in the second one, nonlinearly. In the case of linearity, the problem has two specific features. First, it is assumed that only a part of phase coordinates of the system (namely, the coordinates described by the differential equation) is inaccurately measured at discrete times. Second, it is only known about the disturbance acting on the system that it is an element of the space of square integrable functions; i.e., it can be unbounded. These assumptions imply the impossibility of exact reconstruction. Taking into account this peculiarity, we construct a solving algorithm, which is stable with respect to informational noises and computational errors. This algorithm is based on a combination of elements of the theory of ill-posed problems and the extremal shift method well-known in the theory of positional differential games. A similar algorithm is designed for a general case when a disturbance enters the system nonlinearly.
Differential Equations. 2025;61(11):1510-1526
1510-1526
ON THE PROBLEM OF TARGET OUTPUT FEEDBACK CONTROL FOR A MONOTONE SYSTEM
Abstract
The article is devoted to solving the problem of controlling a system defined by ordinary differential equations with a nonlinear right-hand side, which has the property of quasi-monotonicity with respect to off-diagonal elements. The equations also contain control parameters and uncertainties (errors), the possible values of which should satisfy some point-wise restrictions. The problem of control over a finite time interval is considered in order to transfer the state of the system to a given target set. The current state of the system is unknown, and for the formation of a control strategy, only a priori estimates of the initial state are available, as well as the results of incomplete and inaccurate measurement results received online. To solve the problem, a well-known general scheme is used, according to which it is necessary to consistently solve three subtasks: the approximate construction of information sets of the system, solvability sets, and, finally, the control synthesis problem. In this paper, this general scheme is successfully implemented for the special class of nonlinear systems under consideration. Theorems on external interval estimates of information sets, internal estimates of solvability sets, as well as on sufficient conditions for the solvability of the control problem are proved. Formulas for feedback control are obtained, depending on the so-called generalized position, formed on the basis of available information about the system and measurement results. The possibility of applying the theoretical results obtained to solve specific control problems is confirmed by the model example analyzed in the work.
Differential Equations. 2025;61(11):1527-1545
1527-1545
CASCADING OBSERVER DESIGN FOR UNCERTAIN UNCONTROLLABLE DYNAMICAL SYSTEMS
Abstract
Asymptotic observer design problem is considered for uncertain dynamical system. A method was developed, allowing the application of previously proposed cascading observer for a wider class of dynamical systems. More specifically, requirements of system's matrices controllability and zero dynamics stability are lifted.
Differential Equations. 2025;61(11):1546-1553
1546-1553
ORBITAL DECOMPOSITIONS OF CONTROL SYSTEMS AND TRAJECTORY PLANNING
Abstract
The paper studies the problem of transferring a nonlinear dynamic control system from a given initial state to a given final state. An approach is developed based on constructing such a decomposition of the system, in which the specified problem is transformed into two coupled Cauchy problems. One of these problems has boundary conditions at the initial moment, and the second — at the final moment. The case is considered when the boundary conditions of the problem are imposed only on a part of the state variables. To transform the system into a decomposable form, invertible transformations of the most general type are used — orbital equivalences. The given nontrivial example demonstrates the possibility of implementing the specified approach.
Differential Equations. 2025;61(11):1554-1565
1554-1565
CHRONICLE
1566-1584