Pub Date : 2023-01-01DOI: 10.21638/11701/spbu10.2023.301
Alexander Yu. Aleksandrov
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.
研究了一类非线性微分方程组在离散化条件下的稳定性保持问题。考虑了Persidskii系统、Lurie间接控制系统和其右侧具有正则结构的系统。假设这些系统的零解是全局渐近稳定的。确定了相应差分系统零解渐近稳定的保证条件。以前，这样的条件是建立在使用显式欧拉方法进行离散化的情况下。本文在隐式欧拉方法的基础上构造了差分格式。对于得到的离散系统，证明了局部和全局渐近稳定的定理，导出了暂态过程时间的估计。对于具有正则右侧结构的系统，基于V. I. Zubov的方法，提出了一种改进的隐式计算格式，保证了微分系统和相应的差分系统解的收敛速率与原点的匹配。结果表明，与采用显式方法得到的约束相比，隐式计算格式在对系统的离散步长和右侧约束较宽松的条件下，能够保证系统保持渐近稳定性。最后给出了一个算例来说明所得的理论结论。
{"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}
Pub Date : 2023-01-01DOI: 10.21638/11701/spbu10.2023.308
Aleksei P. Zhabko, Vladimir V. Karelin, Vyacheslav V. Provotorov, Sergey M. Sergeev
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.
{"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}
Pub Date : 2023-01-01DOI: 10.21638/11701/spbu10.2023.306
Pavel A. Shamanaev
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.
{"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}
Pub Date : 2023-01-01DOI: 10.21638/spbu05.2023.304
Yuri Zaytsev
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.
{"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}
Pub Date : 2023-01-01DOI: 10.21638/spbu01.2023.301
Nikolai А. Vavilov
The present survey describes the contribution of St Petersburg mathematicians to the theory of linear, classical and algebraic groups. The first part is dedicated to the prehistorical period, the historical genesis of Tartakowski and Faddeev algebra schools, and to the general outline of the works by Borewicz and Suslin of the mid-1970s that initiated systematical research in the fields of classical groups and algebraic K-theory in St Petersburg.
{"title":"St Petersburg school of linear groups. I. Prehistorical period","authors":"Nikolai А. Vavilov","doi":"10.21638/spbu01.2023.301","DOIUrl":"https://doi.org/10.21638/spbu01.2023.301","url":null,"abstract":"The present survey describes the contribution of St Petersburg mathematicians to the theory of linear, classical and algebraic groups. The first part is dedicated to the prehistorical period, the historical genesis of Tartakowski and Faddeev algebra schools, and to the general outline of the works by Borewicz and Suslin of the mid-1970s that initiated systematical research in the fields of classical groups and algebraic K-theory in St Petersburg.","PeriodicalId":477285,"journal":{"name":"Вестник Санкт-Петербургского университета","volume":"43 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":"135913415","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}
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