Вестник Санкт-Петербургского университета最新文献_第3页

Application of the implicit Euler method for the discretization of some classes of nonlinear systems 隐式欧拉方法在若干非线性系统离散化中的应用

Вестник Санкт-Петербургского университета Pub Date : 2023-01-01 DOI: 10.21638/11701/spbu10.2023.301

Alexander Yu. Aleksandrov

{"title":"Application of the implicit Euler method for the discretization of some classes of nonlinear systems","authors":"Alexander Yu. Aleksandrov","doi":"10.21638/11701/spbu10.2023.301","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2023.301","url":null,"abstract":"The problem of stability preservation under discretization of some classes of nonlinear differential equations systems is studied. Persidskii systems, Lurie systems of indirect control, and systems whose right-hand sides have a canonical structure are considered. It is assumed that the zero solutions of these systems are globally asymptotically stable. Conditions are determined that guarantee the asymptotic stability of the zero solutions for the corresponding difference systems. Previously, such conditions were established for the case where discretization was carried out using the explicit Euler method. In this paper, difference schemes are constructed on the basis of the implicit Euler method. For the obtained discrete systems, theorems on local and global asymptotic stability are proved, estimates of the time of transient processes are derived. For systems with a canonical structure of right-hand sides, based on the approach of V. I. Zubov, a modified implicit computational scheme is proposed that ensures the matching of the convergence rate of solutions to the origin for the differential and corresponding difference systems. It is shown that implicit computational schemes can guarantee the preservation of asymptotic stability under less stringent constraints on the discretization step and right-hand sides of the systems under consideration compared to the constraints obtained using the explicit method. An example is presented illustrating the obtained theoretical conclusions.","PeriodicalId":477285,"journal":{"name":"Вестник Санкт-Петербургского университета","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135151175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Optimal control of thermal and wave processes in composite materials 复合材料热波过程的最优控制

Вестник Санкт-Петербургского университета Pub Date : 2023-01-01 DOI: 10.21638/11701/spbu10.2023.308

Aleksei P. Zhabko, Vladimir V. Karelin, Vyacheslav V. Provotorov, Sergey M. Sergeev

{"title":"Optimal control of thermal and wave processes in composite materials","authors":"Aleksei P. Zhabko, Vladimir V. Karelin, Vyacheslav V. Provotorov, Sergey M. Sergeev","doi":"10.21638/11701/spbu10.2023.308","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2023.308","url":null,"abstract":"The paper indicates the approach and the corresponding to it method of penalty functions for analyzing the problems of optimal control of thermal and wave processes in structural elements made of composite materials (composites). An object that is quite common in the industrial sphere, the structure of which is a set of layers (phases) of unidirectional composites — layered composites, is considered. When solving problems related to the analysis and description of the states of composites, quantitative characteristics of layers that are not functions of the coordinates of the points of the medium are usually used in order not to solve the corresponding problems for an inhomogeneous medium. Such functions are elements of Sobolev spaces, first of all, functions summable with a square. The convenience lies in the fact that when finding the conditions for solvability of initial-boundary value problems of various types (in most cases, such problems are the basis of mathematical models of many physical processes), it is possible to reduce to operator-difference systems, for which it is easy to construct a priori estimates of weak solutions. The next step after establishing the weak solvability of the initial-boundary value problem of the thermal or wave process in composites is the formulation and solution of the problem of optimal control of these processes. The proposed method of penalty functions on the example of solving such problems is a general method. It is applicable with slight modifications also not only in the case of elliptic, parabolic and other problems (including nonlinear) for scalar functions, but also for vector functions. An example of the latter is the Navier–Stokes system, widely used in the description of network-like hydrodynamic processes, considered in Sobolev spaces, the elements of which are functions with carriers on n-dimensional network-like domains, n greater or equal to 2.","PeriodicalId":477285,"journal":{"name":"Вестник Санкт-Петербургского университета","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135151850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the stability of the zero solution with respect to a part of variables in linear approximation 线性逼近中零解对部分变量的稳定性

Вестник Санкт-Петербургского университета Pub Date : 2023-01-01 DOI: 10.21638/11701/spbu10.2023.306

Pavel A. Shamanaev

{"title":"On the stability of the zero solution with respect to a part of variables in linear approximation","authors":"Pavel A. Shamanaev","doi":"10.21638/11701/spbu10.2023.306","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2023.306","url":null,"abstract":"The article presents the sufficient conditions for stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system in the linear approximation. the case is considered when the matrix of the linear approximation may contain eigenvalues with zero real parts and the algebraic and geometric multiplicities of these eigenvalues may not coincide. The approach is based on establishing some correspondence between the solutions of the investigated system and its linear approximation. The solutions of such systems starting in a sufficiently small zero neighborhood and the systems themselves possess the same componentwise asymptotic properties in this case. Such solutions’ properties are stability and asymptotic stability with respect to some variables, and for systems componentwise local asymptotic equivalence and componentwise local asymptotic equilibrium. Considering the correspondence between the solutions of systems as an operator defined in a Banach space, there is proved that it has at least one fixed point according to the Schauder’s principle. The operator allows to construct a mapping that establishes the relationship between the initial points of the investigated system and its linear approximation. Further, a conclusion about the componentwise asymptotic properties of solutions of the nonlinear system is made on the basis of estimates of the fundamental matrix of the linear approximation rows’ entries. There is given an example of the investigation of stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system is given, when the linear approximation matrix contains one negative and one zero eigenvalues, and the algebraic and geometric multiplicities of the zero eigenvalue do not coincide.","PeriodicalId":477285,"journal":{"name":"Вестник Санкт-Петербургского университета","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135152210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Economic aspects of state support of Russian steel industry enterprises in the conditions of external shocks 经济方面国家对俄罗斯钢铁工业企业在外部冲击条件下的支持

Вестник Санкт-Петербургского университета Pub Date : 2023-01-01 DOI: 10.21638/spbu05.2023.304

Yuri Zaytsev

{"title":"Economic aspects of state support of Russian steel industry enterprises in the conditions of external shocks","authors":"Yuri Zaytsev","doi":"10.21638/spbu05.2023.304","DOIUrl":"https://doi.org/10.21638/spbu05.2023.304","url":null,"abstract":"The metallurgical sector of Russia has been a recipient of state support at the federal and regional levels for many years. The sector receives subsidies from more than 90 programs and indirectly benefits from pricing policies for electricity, natural gas and rail transport. However, the analysis of measures to support the sector under the conditions of Western sanctions and the coronavirus pandemic is of particular relevance. The aim of the work is to determine the role of state support for steel producers in the face of external shocks — the coronavirus pandemic and Western sanctions. The working hypothesis of the study is the assertion that direct and indirect measures related to supporting the sector in the face of sanctions and the negative consequences of coronavirus contribute to reducing the adaptation costs of producers, as well as increasing their competitiveness. The stated goal and the indicated hypothesis determine the structure of the paper. Thus, the author examines the theoretical and practical aspects of the impact of sanctions on the national economy, the Russian steel market, state policy towards the steel industry, as well as direct and indirect measures of state assistance that affect the steel production sector. The research methodology is based on a comprehensive review of indirect and direct mechanisms of state support for the industry. The study resulted in the identification of channels of state influence on the steel market in the context of the COVID-19 pandemic and sanctions. The results of the study can be used in the formation of the Russian industrial policy.","PeriodicalId":477285,"journal":{"name":"Вестник Санкт-Петербургского университета","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135556987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

St Petersburg school of linear groups. I. Prehistorical period 圣彼得堡线性群学派。一、史前时期

Вестник Санкт-Петербургского университета Pub Date : 2023-01-01 DOI: 10.21638/spbu01.2023.301

Nikolai А. Vavilov

引用次数: 0