Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

Sergey Zagorodnyuk — Tue, 10 Oct 2023 00:00:00 +0000

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1,
\dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$

$$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1,
\dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1,
\dots,\kappa_\rho+\delta_\rho+1;x), $$

and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots,
\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$

$$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1;
\alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x),
\qquad n\in\mathbb{Z}_+, $$

where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters.

The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures.

The importance of the orthogonality property was our main reason to concentrate our attention on polynomials

$\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces

of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$.

The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties.

We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

On integration with respect to filter

Dmytro Seliutin — Wed, 20 Dec 2023 16:55:35 +0000

This article is devoted to the study of one generalization of the Riemann integral. Namely, in the paper, it was observed that the classical definition of the Riemann integral over a finite segment as a limit of integral sums, when the diameter of the division of the segment tends to zero, can be replaced by a limit of integral sums over a filter of sets, which can be described in a certain "good way". This idea was continued, and in the work we propose a new concept - the integral of a function over a filter on the set of all tagged partitions of a segment. Using of filters is a very good method in questions related to convergence or some of its analogues in general topological vector spaces. Namely, if the space is non-metrizable, then the concept of convergence is introduced precisely with the help of filters. Also, using filters, you can formulate the concept of completeness and its analogues. The completeness of spaces is one of the central concepts of the theory of topological vector spaces, since Banach spaces are complete. That is, using a generalization of the completeness of spaces constructed using filters, we can explore various generalizations of Banach spaces. We study standard issues related to integration. For example, does the integrability of the filter function imply its boundedness? The answer to this question is affirmative. Namely: the concept of filter boundedness of a function is introduced, and it is shown that if a function is integrable over filter, then its integral sums are bounded over the filter, and this function itself is bounded in the classical sense. Next, we showed that the filter integral satisfies the linearity property, namely, the integral over filter of the sum of two functions is the sum of the filter integrals of these functions. In addition, we can to subtract the constant factor from the sign of the integral over filter. We introduce the concept of an exactly tagged filter, and with the help of such filters we study the filter integrability of unbounded functions on a segment. We give an example of a specific unbounded function and a specific filter under which this function is integrable. Next, we prove a theorem that describes unbounded filter-integrable functions on a segment. The last section of the article is devoted to the integration of functions relative to the filter on a subsegment of this segment.

On linear stabilization of a class of nonlinear systems in a critical case

Maxim Bebiya, Vladyslava Maistruk — Wed, 20 Dec 2023 16:56:35 +0000

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise

stabilization. This method requires a sequential investigation of lower-dimensional subsystems.

Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such

controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient

conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the

corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the

sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic

stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from

their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with

higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing

powers can be applied to systems of higher dimensions.

Stability of minimal surfaces in the sub-Riemannian manifold $\widetilde{E(2)}$

Eugene Petrov, Ihor Havrylenko — Wed, 20 Dec 2023 16:57:31 +0000

In the paper we study smooth oriented surfaces in the universal covering space of the group of orientation-preserving Euclidean plane isometries, which has a three-dimensional sub-Riemannian manifold structure. This structure is constructed as a restriction of the Euclidean metric on the group to some completely non-integrable left invariant distribution. The sub-Riemannian area of a surface is then defined as the integral of the length of its unit normal field projected orthogonally onto this distribution. We calculate the first variation formula of the sub-Riemannian surface area and derive the minimality criterion from it. Here we call a surface minimal if it is a critical point of the sub-Riemannian area functional under normal variations with compact support. We show that the minimality in this case is not equivalent to the vanishing of the sub-Riemannian mean curvature. We then prove that a Euclidean plane is minimal if and only if it is parallel or orthogonal to the $z$-axis (where the $z$-coordinate corresponds to the rotation angle of an isometry). Also we obtain the minimality condition for a graph and give examples of minimal graphs. The examples considered in the paper demonstrate, in particular, that the minimality of a surface in the Riemannian (in this case Euclidean) sense does not imply its sub-Riemannian minimality, and vice versa.

Next, we consider the stability of minimal surfaces. For this purpose, we derive the second variation formula of the sub-Riemannian area and show with it that minimal Euclidean planes are stable. We introduce a class of surfaces for which the tangent planes are perpendicular to the planes of the sub-Riemannian structure, and call them vertical surfaces. In particular, for such surfaces the second variation formula is simplified significantly. Then we prove that complete connected vertical minimal surfaces are either Euclidean planes or helicoids and that helicoids are unstable. This implies a following Bernstein type result: a complete connected vertical minimal surface is stable if and only if it is a Euclidean plane orthogonal to the $z$-axis.