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

Nonlinear boundary value problems for degenerate differential-algebraic systems in the noncritical case

Sergey Chuiko — 2024-11-01

We have obtained the conditions of existence and a scheme for constructing solutions of a weakly nonlinear boundary value problem for a degenerate differential-algebraic system in the noncritical case. The boundary condition is determined by a weakly nonlinear vector functional. The linear part of the problem is a linear boundary value problem for a degenerate differential-algebraic system.
Linear differential-algebraic boundary value problems have been studied in monographs by S. Campbell, J.R. Magnus, A.M. Samoilenko and V.P. Yakovets. In the works of A.M. Samoilenko and O.A. Boichuk, using the central canonical form, the necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic boundary value problems were obtained.
We have obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems without using the central canonical form, which allows us to study the solvability of differential-algebraic boundary value problems that depend on arbitrary continuous functions. This approach significantly varies the classification of nonlinear differential-algebraic boundary value problems in critical and noncritical cases.

Our formulation of the weakly nonlinear differential-algebraic boundary value problem generalises the boundary value problems studied in the works of Yu.O. Mitropolsky, A.M. Samoilenko, and O.A. Boichuk. The case when a differential-algebraic system is not solvable with respect to the derivative is considered, and substitutions of the unknown are proposed. It leads the original system to a nonlinear differential-algebraic system solvable with respect to the derivative.
Finally, we present an example of a nonlinear differential-algebraic antiperiodic boundary value problem for a Riccati-type equation, which demonstrates the constructiveness of the obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems.
The obtained results can be transferred to the problems of finding conditions for the existence and schemes for constructing solutions of nonlinear degenerate differential-algebraic boundary value problems in critical cases, as well as to the problems of finding conditions for the stability of such solutions.

Mathematical modelling and virtual design of metamaterials for reducing noise and vibration in built-up structures

Emmanuel Akaligwo — 2024-11-25

Noise and vibration are pervasive challenges in built-up structures, impacting structural integrity, operational efficiency, and occupant well-being. These issues are particularly pronounced in urban and industrial settings, where traditional materials often struggle to deliver effective mitigation across the broad range of relevant frequencies. This paper introduces an integrated mathematical modeling and virtual design framework for the development of advanced metamaterials aimed at reducing noise and vibration in such complex structures. The approach combines finite element analysis, dynamic energy analysis, and optimization algorithms to design metamaterials with frequency-selective properties that create targeted barriers to acoustic and vibrational disturbances. The study not only develops a systematic methodology for designing these metamaterials but also validates their efficacy through comprehensive simulations and benchmarking against established solutions. The results highlight the advantages of the proposed metamaterials in terms of adaptability, efficiency, and performance robustness across various operating conditions. Sensitivity analyses and comparative evaluations further underscore the superiority of the framework in addressing frequency-dependent challenges, offering significant improvements over conventional materials. A unique aspect of this research is the inclusion of natural metamaterials (NMs) as a sustainable alternative for mitigating ground vibrations. The study reviews the potential of NMs for diverse functionalities, particularly in attenuating ground vibrations in urban environments. These findings emphasize the versatility and eco-friendliness of natural materials, providing a roadmap for their development and application in achieving clean and quiet environments. The proposed framework, therefore, bridges theoretical advancements with practical applications, paving the way for resilient and sustainable solutions to noise and vibration challenges in built-up structures.

Time-optimal controlonasubspaceforthetwoand three-dimensional system

Oleh Vozniak — 2024-12-23

This article is devoted to the problem of the optimal synthesis on a subspace for the 2 and 3-dimensional case for the linear control system $\dot{x_1} = u, \dot{x}_i=x_{i-1}, i=\overline{2,n}$ with $|u|\le1$. This problem is related to the problem of optimal synthesis into the point, which solution was presented by V. I. Korobov and G. M. Sklyar and is based on the moment min-problem, but its difference with respect to the original problem is that the number of unknown functions is greater than the number of variables, which requires using the methods for the parametric optimization. As in the problem of optimal synthesis into the point, we search for the optimal solution in the form of piecewise function with $u=\pm 1$ and $n-1$ switching points, which is optimal according to Pontryagyn's maximum principle and the theorem on $n$ intervals. In this article we consider approaches and problems related to the finding of the general solution of the optimization problem and solve the cases of $n=2$ and $n=3$. The interest of the obtain results is the fact that unlike the solution for the single endpoint $x_T$ the general solution for the subspace may have less than $n-1$ switching points, or not have switching points at all, depending on the parameters of the subspace. In our work, we give the solution for the problem of synthesis of the two-dimensional system onto a subspace $G: {x_2 = k x_1}$ for all values of $k$ using the moment min-problem and the optimization methods. For the three-dimensional system we consider the problem of synthesis on a plane $x_3 = k_1 x_1 + k_2 x_2$ and obtain the number of the switching points depending on the values of $k_1$ and $k_2$, construct trajectories and present the equations the for optimal time $\Theta$ for different cases.

Korobov’s controllability function method via orthogonal polynomials on [0,∞)

Abdon Choque — 2024-12-23

Given a controllable system described by ordinary or partial differential equations and an initial state, the problem of finding a set of bounded positional controls that transfer the initial state to another state, not necessarily an equilibrium point, in finite time is called the synthesis problem. In the present work, we consider a family of Brunovsky systems of dimension n. A family of bounded positional controls un(x) is developed to stabilize a given Brunovsky system in finite time. We employ orthogonal polynomials associated with a function distribution σ(τ, θ) defined for τ ∈ [0, +∞) and parameter θ > 0. The parameter θ is interpreted as a Korobov’s controllability function, θ = θ(x), which serves as a Lyapunovtype function. Utilizing θ(x), we construct the positional control un(x) = un(x, θ(x)).
Our analysis is based on the foundational work “A general approach to the solution of the bounded control synthesis problem in a controllability problem”. Matematicheskii Sbornik, 151(4), 582–606 (1979) by Korobov, V. I, in which the controllability function method was proposed. This method has been applied to solve bounded finite-time stabilization problems in various control scenarios, such as the control of the wave equation, optimal control with mixed cost functions, and other applications.
For the construction of the mentioned positional controls, we employ a member of a family of orthogonal polynomials on [0,∞). For orthogonal polynomials, we refer to “Orthogonal Polynomials”. American Mathematical Society, Providence, (1975) by G. Szego.
We also rely on the work “On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials”. Linear Algebra and its Applications, 476, 56–84 (2015) by Choque Rivero, A. E.
The results in the present work extend and develop the findings presented in the conference paper “Bounded finite-time stabilizing controls via orthogonal polynomials”. 2018 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC), Ixtapa, Mexico. –2018 by Choque-Rivero A. E., Orozco B. d. J. G.