卷 60, 编号 2 (2024)

We study the local dynamics of a logistic equation with delay and with additional feedback containing a large delay. Critical cases in the problem of stability of the zero equilibrium state are identified and it is shown that they have infinite dimension. Well-known methods for studying local dynamics, based on the application of the theory of invariant integral manifolds and normal forms, are not applicable here. Methods of infinite-dimensional normalization proposed by the author are used and developed. As the main results, special nonlinear boundary value problems of parabolic type are constructed, which play the role of normal forms. They determine the main terms of the asymptotic expansions of solutions to the original equation. They are called quasinormal forms. 

In this paper, for an inhomogeneous string vibration equation, we consider a periodic in spatial variable and a mixed half-strip problems, the solutions of which are written in quadratures in the form of finite sums. When solving these problems we use the characteristic rectangle identity, Riemann invariants and the method of characteristics.

The paper considers two weakly singular Fredholm boundary integral equations of the first kind, to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on spectra, where they are ill-posed. For the first equation, it is shown that if its solution exists on the spectrum, it allows us to find a solution to the transmission problem. The second equation in this case always has infinitely many solutions, only one of which gives a solution to the transmission problem. The interpolation method for finding approximate solutions of the considered integral equations and the transmission problem is discussed.

The paper considers the problem of controlling processes, the mathematical model of which is an initial-boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation, which is typical in vibration theory. As examples, models of vibrations of moving elastic materials were considered. For model problems, an energy identity is established, and conditions for the uniqueness of a solution are formulated. As an optimization problem, we considered the problem of controlling the right side in order to minimize the quadratic integral functional, which evaluates the proximity of the solution to the objective function. From the original functional a transition was made to the majorant functional, for which the corresponding upper bound was established. An explicit expression for the gradient of this functional is obtained, and conjugate initial-boundary value problems are derived.

The paper addresses the consensus problem (i.e., the agreement of phase vectors) for a multi-agent system consisting of identical linear agents. The study focuses on the case where there is no communication between agents, meaning there is no exchange of information, and agent control is achieved through the agents’ own sensors, providing incomplete information about the phase vector of the agent and its neighbors, with the information possibly being noisy. To solve this problem, a linear protocol based on observer data for systems under uncertainty is proposed. Cascade observers based on the “super-twisting” method are suggested as such observers. Sufficient conditions for the existence of a controller are obtained, where the observation error converges to zero under limited disturbances. An example illustrating the proposed approach is provided. 

The paper discusses the development of a method for constructing asymptotic formulas for $x\to +\infty$ of a fundamental system of solutions of two-term singular symmetric differential equations of odd order with coefficients from a wide class of functions that allow oscillation (with weakened regularity conditions that do not satisfy the classical Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation $\left(i2\right)\left[\right(p{\left(x\right)y^{\text{'}}{)}^{\text{'}\text{'}}}^{}+\left(p\right(x\left){y^{\text{'}\text{'}}}^{}{)}^{\text{'}}\right]+q\left(x\right)y=\lambda y$ the asymptotics of solutions in the case of different behavior of the coefficients $qx$ is studied, $hx-1+1\sqrt{px}$. New asymptotic formulas are obtained for the case when $hx\notin L_1\infty$.

Ниже публикуются краткие аннотации докладов, состоявшихся в осеннем семестре 2023 г. (предыдущее сообщение о работе семинара дано в журнале “Дифференц. уравнения”. 2023. Т. 59. № 8; за дополнительной информацией обращаться по адресу: deq@cs.msu.ru)[7]