Proceedings of the Steklov Institute of Mathematics最新文献

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The Method of Comparison with a Model Equation in the Study of Inclusions in Vector Metric Spaces 矢量公设空间夹杂物研究中的模型方程比较法
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030180
E. S. Zhukovskiy, E. A. Panasenko
{"title":"The Method of Comparison with a Model Equation in the Study of Inclusions in Vector Metric Spaces","authors":"E. S. Zhukovskiy, E. A. Panasenko","doi":"10.1134/s0081543824030180","DOIUrl":"https://doi.org/10.1134/s0081543824030180","url":null,"abstract":"<p>For a given multivalued mapping <span>(F:Xrightrightarrows Y)</span> and a given element <span>(tilde{y}in Y)</span>, the existence of a solution <span>(xin X)</span> to the inclusion <span>(F(x)nitilde{y})</span> and its estimates are studied. The sets <span>(X)</span> and <span>(Y)</span> are endowed with vector-valued metrics <span>(mathcal{P}_{X}^{E_{+}})</span> and <span>(mathcal{P}_{Y}^{M_{+}})</span>, whose values belong to cones <span>(E_{+})</span> and <span>(M_{+})</span> of a Banach space <span>(E)</span> and a linear topological space <span>(M)</span>, respectively. The inclusion is compared with a “model” equation <span>(f(t)=0)</span>, where <span>(f:E_{+}to M)</span>. It is assumed that <span>(f)</span> can be written as <span>(f(t)equiv g(t,t))</span>, where the mapping <span>(g:{E}_{+}times{E}_{+}to M)</span> orderly covers the set <span>({0}subset M)</span> with respect to the first argument and is antitone with respect to the second argument and <span>(-g(0,0)in M_{+})</span>. It is shown that, in this case, the equation <span>(f(t)=0)</span> has a solution <span>(t^{*}in E_{+})</span>. Further, conditions on the connection between <span>(f(0))</span> and <span>(F(x_{0}))</span> and between the increments of <span>(f(t))</span> for <span>(tin[0,t^{*}])</span> and the increments of <span>(F(x))</span> for all <span>(x)</span> in the ball of radius <span>(t^{*})</span> centered at <span>(x_{0})</span>\u0000for some <span>(x_{0})</span> are formulated, and it is shown that the inclusion has a solution in the ball under these conditions. The results on the operator inclusion obtained in the paper are applied to studying an integral inclusion.\u0000</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Perturbation Method and Regularization of the Lagrange Multiplier Rule in Convex Constrained Optimization Problems 凸约束优化问题中的扰动法和拉格朗日乘数规则的正规化
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030155
M. I. Sumin
{"title":"The Perturbation Method and Regularization of the Lagrange Multiplier Rule in Convex Constrained Optimization Problems","authors":"M. I. Sumin","doi":"10.1134/s0081543824030155","DOIUrl":"https://doi.org/10.1134/s0081543824030155","url":null,"abstract":"<p>We consider a regularization of the Lagrange multiplier rule (LMR) in the nondifferential form in a convex constrained optimization problem with an operator equality constraint in a Hilbert space and a finite number of functional inequality constraints. The objective functional of the problem is assumed to be strongly convex, and the convex closed set of its admissible elements also belongs to a Hilbert space. The constraints of the problem contain additively included parameters, which enables using the so-called perturbation method to study it. The main purpose of the regularized LMR is the stable generation of generalized minimizing sequences (GMSs), which approximate the exact solution of the problem using extremals of the regular Lagrange functional. The regularized LMR itself can be interpreted as a GMS-generating (regularizing) operator, which assigns to each set of input data of the constrained optimization problem the extremal of its corresponding regular Lagrange functional, in which the dual variable is generated following one or another procedure for stabilizing the dual problem. The main attention is paid to (1) studying the connection between the dual regularization procedure and the subdifferential properties of the value function of the original problem; (2) proving the convergence of this procedure in the case of solvability of the dual problem; (3) an appropriate update of the regularized LMR; (4) obtaining the classical LMR as a limiting version of its regularized analog. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sufficient Optimality Conditions for Hybrid Systems of Variable Dimension with Intermediate Constraints 具有中间约束条件的可变维度混合系统的充分最优条件
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030052
A. S. Bortakovskii
{"title":"Sufficient Optimality Conditions for Hybrid Systems of Variable Dimension with Intermediate Constraints","authors":"A. S. Bortakovskii","doi":"10.1134/s0081543824030052","DOIUrl":"https://doi.org/10.1134/s0081543824030052","url":null,"abstract":"<p>An optimal control problem is considered for a hybrid system in which continuous motion alternates with discrete changes (switchings) of the state space and control space. The switching times are determined as a result of minimizing a functional that takes into account the costs of each switching. Sufficient conditions for the optimality of such systems under additional constraints at the switching times are obtained. The application of the optimality conditions is demonstrated using academic examples.\u0000</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a Control Reconstruction Problem with Nonconvex Constraints 关于具有非凸约束条件的控制重构问题
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030143
N. N. Subbotina, E. A. Krupennikov
{"title":"On a Control Reconstruction Problem with Nonconvex Constraints","authors":"N. N. Subbotina, E. A. Krupennikov","doi":"10.1134/s0081543824030143","DOIUrl":"https://doi.org/10.1134/s0081543824030143","url":null,"abstract":"<p>A control reconstruction problem for dynamic deterministic affine-control systems is considered. This problem consists of constructing piecewise constant approximations of an unknown control generating an observed trajectory from discrete inaccurate measurements of this trajectory. It is assumed that the controls are constrained by known nonconvex geometric constraints. In this case, sliding modes may appear. To describe the impact of sliding modes on the dynamics of the system, the theory of generalized controls is used. The notion of normal control is introduced. It is a control that generates an observed trajectory and is defined uniquely. The aim of reconstruction is to find piecewise constant approximations of the normal control that satisfy given nonconvex geometric constraints.\u0000The convergence of approximations is understood in the sense of weak convergence in the space <span>(L^{2})</span>. A solution to the control reconstruction problem is proposed.\u0000</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Package Guidance Problem for a Fractional-Order System 分数阶系统的包引导问题
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030167
P. G. Surkov
{"title":"Package Guidance Problem for a Fractional-Order System","authors":"P. G. Surkov","doi":"10.1134/s0081543824030167","DOIUrl":"https://doi.org/10.1134/s0081543824030167","url":null,"abstract":"<p>The problem of guaranteed closed-loop guidance to a given set at a given time is studied for a linear dynamic control system described by differential equations with a fractional derivative of the Caputo type. The initial state is a priori unknown, but belongs to a given finite set. The information on the position of the system is received online in the form of an observation signal. The solvability of the guidance problem for the control system is analyzed using the method of Osipov–Kryazhimskii program packages. The paper provides a brief overview of the results that develop the program package method and use it in guidance problems for various classes of systems. This method allows us to connect the solvability condition of the guaranteed closed-loop guidance problem for an original system with the solvability condition of the open-loop guidance problem for a special extended system. Following the technique of the program package method, a criterion for the solvability of the considered guidance problem is derived for a fractional-order system. In the case where the problem is solvable, a special procedure for constructing a guiding program package is given. The developed technique for analyzing the guaranteed closed-loop guidance problem and constructing a guiding control for an unknown initial state is illustrated by the example of a specific linear mechanical control system with a Caputo fractional derivative.\u0000</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Finite Groups with $$mathbb{P}$$ -Subnormal Schmidt Subgroups 具有 $$mathbb{P}$ -Subnormal Schmidt 子群的有限群
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030179
Xiaolan Yi, Zhuyan Xu, S. F. Kamornikov
{"title":"Finite Groups with $$mathbb{P}$$ -Subnormal Schmidt Subgroups","authors":"Xiaolan Yi, Zhuyan Xu, S. F. Kamornikov","doi":"10.1134/s0081543824030179","DOIUrl":"https://doi.org/10.1134/s0081543824030179","url":null,"abstract":"<p>A subgroup <span>(H)</span> of a group <span>(G)</span> is called <span>(mathbb{P})</span>-subnormal in <span>(G)</span> whenever either <span>(H=G)</span> or there is a chain of subgroups\u0000<span>(H=H_{0}subset H_{1}subsetmathinner{ldotpldotpldotp}subset H_{n}=G)</span>\u0000such that <span>(|H_{i}:H_{i-1}|)</span> is a prime for every <span>(i=1,2,mathinner{ldotpldotpldotp},n)</span>. We study the structure of a finite group <span>(G)</span> all of whose Schmidt subgroups are <span>(mathbb{P})</span>-subnormal. The obtained results complement the answer to Problem 18.30 in the <i>Kourovka Notebook</i>..</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Modeling a Solution of Systems with Constant Delay Using Controlled Models 论利用受控模型为具有恒定延迟的系统解决方案建模
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030040
M. S. Blizorukova, V. I. Maksimov
{"title":"On Modeling a Solution of Systems with Constant Delay Using Controlled Models","authors":"M. S. Blizorukova, V. I. Maksimov","doi":"10.1134/s0081543824030040","DOIUrl":"https://doi.org/10.1134/s0081543824030040","url":null,"abstract":"<p>The problem of modeling a solution is studied for a nonlinear system of differential equations with constant delay, inexactly known right-hand side,\u0000and inaccurately given initial state. The case is considered when the right-hand side of the system is a nonsmooth (it is only known that it is Lebesgue measurable)\u0000unbounded function (belonging to the space of square integrable functions in the Euclidean norm). An algorithm for solving this system that is stable to\u0000information noises and calculation errors is constructed. The algorithm is based on the concepts of feedback control theory.\u0000An estimate of the convergence rate of the algorithm is established. The possibility of using the algorithm to find an approximate solution to\u0000a system of ordinary differential equations is mentioned.\u0000</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Continuous Dependence of Sets in a Space of Measures and a Program Minimax Problem 度量空间中集合的连续依赖性和程序最小问题
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030064
A. G. Chentsov, D. A. Serkov
{"title":"Continuous Dependence of Sets in a Space of Measures and a Program Minimax Problem","authors":"A. G. Chentsov, D. A. Serkov","doi":"10.1134/s0081543824030064","DOIUrl":"https://doi.org/10.1134/s0081543824030064","url":null,"abstract":"<p>For conflict-controlled dynamical systems satisfying the conditions of generalized uniqueness and uniform boundedness, the solvability of the minimax problem in the class of relaxed controls is studied. The issues of properness of such a relaxation are considered; i.e., the possibility of approximating relaxed controls in the space of strategic measures by embeddings of ordinary controls is analyzed. For this purpose, the dependence of the set of measures on the general marginal distribution specified on one of the factors of the base space is studied. The continuity of this dependence in the Hausdorff metric defined by the metric corresponding to the <span>(ast)</span>-weak topology in the space of measures is established. The density of embeddings of ordinary controls and control–disturbance pairs in sets of corresponding relaxed controls in the <span>(ast)</span>-weak topologies is also shown. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Control of Acceleration of a Dynamic Object by the Modified Linear Tangent Law in the Presence of a State Constraint 在状态约束条件下利用修正线性切线法控制动态物体的加速度
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030131
S. A. Reshmin, M. T. Bektybaeva
{"title":"Control of Acceleration of a Dynamic Object by the Modified Linear Tangent Law in the Presence of a State Constraint","authors":"S. A. Reshmin, M. T. Bektybaeva","doi":"10.1134/s0081543824030131","DOIUrl":"https://doi.org/10.1134/s0081543824030131","url":null,"abstract":"<p>The paper is devoted to trajectory optimization for an inertial object moving in a plane with thrust bounded in magnitude in the presence of external forces. The aim is to maximize the longitudinal terminal velocity with the state constraint satisfied at each time to avoid a lateral collision with an obstacle. The linear tangent law is used as the basis for an algorithm that controls the direction of the thrust. Conditions for the existence of a solution are studied. Constraints on the initial lateral velocity and the time of the motion of the object are obtained. Since the linear tangent law violates the constraint for some motion times, a modified control law is proposed. A transcendental equation is obtained to find a critical value of time above which an undesired collision occurs. The corresponding conjecture is formulated, which allows us to eliminate the ambiguity that arises during the solution process. A method for solving the problem is presented and confirmed by numerical calculations.\u0000</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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 Problem of Optimal Stimulation of Demand 论最佳刺激需求问题
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-08-20 DOI: 10.1134/s0081543824030039
A. S. Aseev, S. P. Samsonov
{"title":"On the Problem of Optimal Stimulation of Demand","authors":"A. S. Aseev, S. P. Samsonov","doi":"10.1134/s0081543824030039","DOIUrl":"https://doi.org/10.1134/s0081543824030039","url":null,"abstract":"<p>We study the problem of optimal stimulation of demand based on a controlled version of Kaldor’s business cycle model. Using the approximation method, we prove a version of Pontryagin’s maximum principle in normal form containing an additional pointwise condition on the adjoint variable. The results obtained develop and strengthen the previous results in this direction.\u0000</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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