卷 59, 编号 8 (2023)

We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions.

We study singularly perturbed problems in the presence of spectral singularities of the limit operator using S.A. Lomov’s regularization method. In particular, a regularized asymptotic solution is constructed for a singularly perturbed inhomogeneous mixed problem on the half-line for a parabolic equation with a strong turning point of the limit operator. Based on the idea of asymptotic integration of problems with unstable spectrum, it is shown how regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for this type of singularity is described in detail, this algorithm is justified, and an asymptotic solution of any order in a small parameter is constructed.

We define the elementary solution of the polyharmonic equation, with the help of which an explicit representation of the Green’s function of the Dirichlet problem for the polyharmonic equation in the unit ball is given for all space dimensions except for some finite set. On the basis of the obtained Green’s function, the solution of the homogeneous Dirichlet problem in the unit ball is constructed. As an example, an explicit form of the solution of the homogeneous Dirichlet problem for the inhomogeneous polyharmonic equation with the simplest polynomial right-hand side is found.

In the space of convex compact sets with the Minkowski addition operation and the operation of multiplication of a matrix by a set, we consider linear recurrent equations of the first order. We give a complete description of such equations whose all solutions have a constant diameter. For equations of a special form, the Lyapunov exponents of the sequences of diameters of their solutions are calculated.

The dependence of the radius of a ball centered at zero inscribed in the values of the integral of a set-valued mapping on the upper integration limit is studied. For some types of integrals, exact asymptotics of the radius with respect to the upper limit are found when the upper limit tends to zero. Examples of finding this radius are considered. The results obtained are used to derive new sufficient conditions for the uniformly continuous dependence of the minimum time and solution-point in the linear minimum time control problem on the initial data. We also consider applications in some algorithms with a reachability set of a linear control system

For linear time-varying singularly perturbed systems (LTVSPS) with quasidifferentiable coefficients and a small parameter multiplying some derivatives, the problem of uniform observability is considered. Necessary and sufficient conditions for the quasidifferentiability of the set of output functions independent of the small parameter are proved, observability matrices independent of the small parameter for the slow subsystem and the family of fast subsystems associated with the LTVSPS are constructed, and a connection is established between them and the observability matrix of the original system. On the basis of the complete decomposition of the original LTVSPS with respect to the action of the group of linear nonsingular transformations, we prove rank sufficient conditions for the uniform observability of the LTVSPS that are independent of the small parameter and valid for all sufficiently small values of it. The conditions are expressed in terms of the observability matrices of the slow subsystem and the family of fast subsystems of smaller dimensions than the original LTVSPS.

Ниже публикуются краткие аннотации докладов, состоявшихся в весеннем семестре 2023 г. (предыдущее сообщение о работе семинара дано в журнале ``Дифференц. уравнения''. 2023. Т. 59. № 2; за дополнительной информацией обращаться по адресу: nds@cs.msu.su. Семинар основан академиками РАН С.В. Емельяновым и С.К. Коровиным. Составитель хроники А.В. Ильин.