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Vol 60, No 5 (2024)
ORDINARY DIFFERENTIAL EQUATIONS
CONSTRUCTING DIAGONAL LYAPUNOV–KRASOVSKII FUNCTIONALS FOR A CLASS OF POSITIVE DIFFERENTIAL-ALGEBRAIC SYSTEMS
Abstract
A coupled system describing the interaction of a differential subsystem with nonlinearities of a sector type and a linear difference subsystem is considered. It is assumed that the system is positive. A diagonal Lyapunov–Krasovskii functional is constructed and conditions are determined under which the absolute stability of the investigated system can be proved with the aid of such a functional. In the case of nolinearities of the power form, estimates for the convergence rate of solution to the origin are derived. The stability analysis of the corresponding system with parameter switching is fulfiled. Sufficient conditions guaranteeing the asymptotic stability of the zero solution for any admissible switching law are obtained.
Differencial'nye uravneniya. 2024;60(5):579–589
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METHOD FOR CONSTRUCTING PERIODIC SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS
Abstract
A justification is given for the analytical method for constructing periodic solutions for nonlinear systems of ordinary differential equations of polynomial type. Periodic solutions are constructed in the form of Fourier series, in which the coefficients are polynomials depending on the parameter without the assumption of its smallness. Two examples are considered — the van der Pol equation and the Lorentz system.
Differencial'nye uravneniya. 2024;60(5):590-603
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THE EXISTENCE AND UNIQUENESS THEOREMS FOR STOCHASTIC DIFFERENTIAL-DIFFERENCE HYBRID SYSTEMS
Abstract
A stochastic differential-difference hybrid system is a system of interacting variables whose dynamics are described by stochastic differential equations for some of them and difference equations for others. Systems with two types of difference equations are examined: first, a difference equation in the form of a process involving the multiplicative Wiener process, and second, a difference equation with delay. The existence and uniqueness theorems for both systems have been proven. The basic conditions on the system’s parameters are local Lipschitz conditions and linear growth order.
Differencial'nye uravneniya. 2024;60(5):604–617
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PARTIAL DERIVATIVE EQUATIONS
TYPICAL DROPPING ASYMPTOTICS OF QUASICLASSICAL APPROXIMATIONS TO SOLUTIONS OF THE NONLINEAR SCHRÖDINGER EQUATION
Abstract
Formal asymptotics are substantiated that describe typical dropping cusp singularity of quasiclassical approximations to solutions of two cases of the integrable nonlinear Schr¨odinger equation −𝑖𝜀Ψ′𝑡=𝜀2Ψ′′𝑥𝑥±2|Ψ|2Ψ, where 𝜀 is a small parameter. The substantiation uses the ideology and facts of the mathematical catastrophe theory and the part of the theorem of Yu.F. Korobeinik, concerning analytical as ℎ → 0 solutions 𝐺(ℎ, 𝑢) of the mixed type linear equation ℎ𝐺′′ℎℎ = 𝐺′′𝑢𝑢 to which the hodograph images of both cases of the systems of equations of these quasiclassical approximations are equivalent.
Differencial'nye uravneniya. 2024;60(5):618–631
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ABOUT THE CORE STRUCTURE OF THE SCHWARTZ PROBLEM FOR FIRST-ORDER ELLIPTIC SYSTEMS ON A PLANE
Abstract
The Schwarz problem for 𝐽-analytic functions in an arbitrary ellipse is considered. The matrix 𝐽 is assumed to be two-dimensional, with different eigenvalues lying above the real axis. An example of a nonconstant solution of the homogeneous Schwarz problem in the form of a vector-polynomial of degree three is given. The numerical parameter 𝑙 of the matrix 𝐽, which is expressed through its eigenvectors, is introduced. After that, one relation derived earlier by the author is analyzed. Based on this analysis, a method for computing the dimension and structure of the kernel of the Schwarz problem in an arbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed through the ellipse parameters, the eigenvalues of the matrix 𝐽 and the parameter 𝑙 are obtained. Examples of one-dimensional and trivial kernels are given.
Differencial'nye uravneniya. 2024;60(5):632-642
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CONTROL THEORY
MULTIPLICATIVE CONTROL PROBLEMS FOR THE DIFFUSION-DRIFT CHARGING MODEL OF AN INHOMOGENEOUS POLAR DIELECTRIC
Abstract
A two-parameter multiplicative control problem is studied for a model of electron-induced charging of an inhomogeneous polar dielectric. Exact estimates of the local stability of its optimal solutions with respect to small perturbations of both the cost functionals and the given function of the boundary value problem are derived. For one of the controls, the relay property or the bang-bang principle is established.
Differencial'nye uravneniya. 2024;60(5):643–659
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BACKSTEPPING STABILIZATION OF NONLINEAR DYNAMICAL SYSTEMS UNDER STATE CONSTRAINTS
Abstract
The problem of stabilizing the zero value of the state vector of constrained nonlinear dynamical systems written in a special form is solved. The proposed control design accounts for magnitude constraints on the values of state variables and is based on the integrator backstepping approach using logarithmic Lyapunov barrier functions. The obtained stabilizing feedbacks, in contrast to similar known results, are based on the use of linear virtual stabilizing functions that do not grow unbounded as state variables approach the boundary values. As an example, we consider a state constraints aware solution to the control problem of positioning an autonomous underwater vehicle at a given point in space.
Differencial'nye uravneniya. 2024;60(5):660–671
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ON PIECEWISE CUBIC ESTIMATES OF THE VALUE FUNCTION IN THE PROBLEM OF TARGET CONTROL FOR A NONLINEAR SYSTEM
Abstract
A nonlinear system of ordinary differential equations with control parameters is considered. Pointwise restrictions are imposed on the possible values of these parameters. It is required to solve the problem of transferring the trajectory of the system from an arbitrary initial position to the smallest possible neighborhood of a given target set at a fixed time interval by selecting the appropriate feedback control. To solve this problem, it is proposed to construct a continuous piecewise cubic function of a special kind. The level sets of this function correspond to internal estimates of the solvability sets of the system. Using this function, it is also possible to construct a feedback control function that solves the target control problem at a fixed time interval. The paper proposes formulas for calculating the values of a piecewise cubic function, examines its properties, and considers an algorithm for searching for parameters defining this function.
Differencial'nye uravneniya. 2024;60(5):672–685
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FINITE STABILIZATION AND ASSIGNMENT OF A FINITE SPECTRUM BY A SINGLE CONTROLLER FOR INCOMPLETE MEASUREMENTS FOR LINEAR SYSTEMS OF NEUTRAL TYPE
Abstract
For linear autonomous differential-difference of a neutral type system, the existence criterion was proved and a method was proposed for designing a controller with feedback on the observed output, providing the closed loop system finite stabilization (solution to the problem of complete 0-controllability) and a finite predetermined spectrum. This makes the closed loop system exponentially stable. The constructiveness of the presented results is illustrated by an example.
Differencial'nye uravneniya. 2024;60(5):686–706
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BRIEF MESSAGES
APPROXIMATION OF FUNCTIONAL-ALGEBRAIC EIGENVALUE PROBLEMS
Abstract
A new symmetric variational functional-algebraic formulation of the eigenvalue problem in the Hilbert space with linear dependence on the spectral parameter for a class of mathematical models of thin-walled structures with an attached oscillator is proposed. The existence of eigenvalues and eigenvectors is established. A new symmetric approximation of the problem in a finite-dimensional subspace with linear dependence on the spectral parameter is constructed. Error estimates of the approximate eigenvalues and eigenvectors are obtained. The theoretical results are illustrated on the example of the problem of structural mechanics.
Differencial'nye uravneniya. 2024;60(5):707–713
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ON THE EXISTENCE OF PERIODIC SOLUTIONS OF A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH QUASI-HOMOGENEOUS NON-LINEARITY
Abstract
In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.
Differencial'nye uravneniya. 2024;60(5):714–720
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