卷 59, 编号 3 (2023)
Articles
On the Preservation of a Quadratic Lyapunov Function of a Linear Differential Autonomous System under Constant Perturbations of the Coefficients
摘要
For an autonomous linear homogeneous asymptotically stable differential system, we obtain sufficient conditions for the smallness of perturbations in the class of autonomous linear homogeneous systems under which a quadratic form that is a Lyapunov function of the original system remains a Lyapunov function of the perturbed system.
On a Nonclassical Eigenvalue Problem with Nonlinearizable Solutions
摘要
We study a nonlinear eigenvalue problem of a special form arising in electrodynamics. The problem is posed for a system of two equations with boundary conditions of the first kind and two additional local conditions. There is one spectral parameter in the problem, and two more parameters, which are subject to an additional constraint, occur in the above-mentioned local conditions. Thus, there are two unknown parameters in the problem: one is the spectral parameter, and the other is some additional parameter, chosen so as to ensure that there exists a nontrivial solution of the problem in question. The existence of nonlinearizable solutions of the problem is proved.
On the Spectral Properties of High-Order Differential Operators with Periodic Boundary Conditions
摘要
We study the spectral properties of the differential operator $L_0,$
generated by the differential expression $l_0(y)=(-1)^{m}y^{2m}+q(x)y,$ $0< x< 1,$
,
, and the boundary conditions
$y^{(s)}(1)-y^{(s)}(0)=0$ $(s=\overline{0,2m-1}),$
, where $m\in\mathbb{N},$ $q(x) $
and
is an arbitrary complex-valued function in the class $L_1^{+}(0,1)=\{q(x)\in L_1(0,1):\int_0^1q(t)e^{-2\pi ikt} dt=0,$ $k\le0\}.$
,
.
Asymptotics of the Solution of a Singularly Perturbed System of Equations with a Single-Scale Internal Layer
摘要
We consider a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations with different powers of the small parameter multiplying the second derivatives. A specific feature of the problem is that one of the two equations of the degenerate system has a double root and the other has three nonintersecting simple roots. It is proved that for sufficiently small values of the small parameter the problem has a solution that has a fast transition in a neighborhood of some interior point of the interval. A complete asymptotic expansion of this solution is constructed and justified. It qualitatively differs from the well-known expansion in the case where all the roots of the degenerate equations are simple but also does not coincide with the expansions in the previously studied problems with double roots; in particular, the inner transition layer turns out to be single-scale.
A Nonlocal Problem with an Integral Matching Condition for a Loaded Parabolic-Hyperbolic Equation with a Fractional Caputo Derivative
摘要
A nonlocal problem with an integral matching condition is studied for a parabolic-hyperbolic equation with two lines of type change containing a nonlinear loaded term. The uniqueness of the solution of the problem is proved by the method of energy integrals and the existence, using the theory of integral equations. Classes and sufficient conditions are determined for given functions that ensure the unique solvability of the problem under study.
A Time-Nonlocal Inverse Problem for the Beam Vibration Equation with an Integral Condition
摘要
We study the direct problem for transverse vibrations of a homogeneous beam of finite length with time-nonlocal conditions and obtain necessary and sufficient conditions for the existence of its solution. For the direct problem, the inverse problem of determining the time-dependent coefficients of the lower derivative and the right-hand side in the equation is studied. The existence and uniqueness of the solution of the inverse problem are proved. The solution is based on separation of variables, which is used to reduce the problems to an integral equation and a system of integral equations.
Smooth Solutions of Hyperbolic Equations with Translation by an Arbitrary Vector in the Free Term
摘要
We construct three-parameter families of solutions of hyperbolic differential-difference equations in a half-space with a general shift operator in the free term (or in a nonlocal operator potential). It is proved that the solutions obtained are classical if the real part of the symbol of the corresponding differential-difference operators is positive. Classes of equations for which the indicated condition is satisfied are given.
Pullback Attractors of the Bingham Model
摘要
Based on the theory of trajectory pullback attractors, we study the qualitative behavior of weak solutions for the Bingham model with periodic conditions in the space variables. For the model under consideration, a family of trajectory spaces is introduced and the existence of pullback attractors is proved.
On the Solvability of a System of Nonlinear Integral Equations with the Hammerstein–Stieltjes Operator on the Half-Line
摘要
We study a system of nonlinear integral equations of the Hammerstein–Stieltjes type whose pre-kernels are continuous distribution functions. A constructive existence theorem for a nontrivial, nonnegative, and bounded solution of the system is proved. The integral asymptotics of the constructed solution is studied. Examples of systems for which all the conditions of the theorem are satisfied are given.
Multistep Methods for Second-Order Differential-Algebraic Equations
摘要
We consider the initial value problem for linear systems of second-order ordinary differential equations with an identically singular matrix multiplying the principal part. Sufficient conditions for the existence of a unique solution are given in terms of matrix polynomials. For such problems, multistep difference schemes are proposed. An analysis of their stability and calculations of a model example are carried out.
A Numerical Method for the Optimization of the Diffraction Efficiency of Thin-Layer Coatings with Diffraction Gratings
摘要
A method is proposed for optimizing the diffraction efficiency of multilayer dielectric gratings in the problem of spectral addition of signals in a wide range of wavelengths. From a physical point of view, we pose a direct problem of electromagnetic wave diffraction on multilayer dielectric gratings for the solution of which a modified method of separation of variables is applied. To optimize the diffraction efficiency, a gradient method with a constant step is used, while the gradient is calculated analytically. Numerical results are presented
On the Stability of Solutions to Control Problems for a Nonlinear Reaction–Diffusion–Convection Model
摘要
The problems of multiplicative control for the reaction–diffusion–convection model with coefficients that are nonlinearly dependent on the solution and also dependent on spatial variables are studied. In the case of a power-law dependence of the model coefficients on the solution, optimality systems are derived for extremum problems. With their help, estimates of the local stability of solutions of specific control problems with respect to small perturbations of both the cost functionals and one of the given functions of the boundary value problem are obtained.
On Some Extremal Problems Associated with Motion in a Velocity Field
摘要
The extremals of the Pontryagin maximum principle for problems related to motion in the velocity field are studied. Controls are continuous functions. It is shown that in the state space there exists a neighborhood of the final point through each point of which there passes a single extremal trajectory leading to the final point. It is also shown that if the trajectory of an extremal contains a point that another extremal with the same value of the functional passes through, then this point cuts off the nonoptimal part from the trajectory. It is proved that the remaining part leading to the final point is optimal.
Retraction Note: Proof of the Jacobian Conjecture in the Two-Dimensional Case and Global Isochronous Centers of Polynomial Hamiltonian Differential Systems
摘要
The author has retracted this article. The main results of the work, namely Theorem 1 and the corollaries arising from it, have not been strictly proven: formulas (8) are valid only under the condition
. Yet, even when the indicated condition is met, formulas (9) need to be proven. Because of this, this work in its current form should be recognized as incorrect. The author is grateful to Professor L.G. Makar-Limanov, who drew attention to the lack of correct proof of formula (8) and Professor V.V. Bakhtin, who pointed to other inaccurate statements in this article.