Pub Date : 2024-04-09DOI: 10.1134/s0012266124010038
M. G. Yumagulov, L. S. Ibragimova
Abstract
New approaches are proposed in the problem of constructing equivalent scalar differential equations for multidimensional nonlinear systems of control theory, as well as in the problem of constructing equivalent Hamiltonian systems for nonlinear Lurie equations (scalar differential equations containing derivatives of only even orders). The conditions for the solvability of the corresponding problems are studied, and new formulas for the transition to equivalent equations and systems are proposed. For the Lurie equations, the proposed approaches are based on the transition from the linear part to the normal forms of the corresponding Hamiltonian systems with a subsequent transformation of the resulting system. Calculation formulas and algorithms are obtained, and their efficiency is illustrated by examples.
{"title":"Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems","authors":"M. G. Yumagulov, L. S. Ibragimova","doi":"10.1134/s0012266124010038","DOIUrl":"https://doi.org/10.1134/s0012266124010038","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> New approaches are proposed in the problem of constructing equivalent scalar differential\u0000equations for multidimensional nonlinear systems of control theory, as well as in the problem of\u0000constructing equivalent Hamiltonian systems for nonlinear Lurie equations (scalar differential\u0000equations containing derivatives of only even orders). The conditions for the solvability of the\u0000corresponding problems are studied, and new formulas for the transition to equivalent equations\u0000and systems are proposed. For the Lurie equations, the proposed approaches are based on the\u0000transition from the linear part to the normal forms of the corresponding Hamiltonian systems with\u0000a subsequent transformation of the resulting system. Calculation formulas and algorithms are\u0000obtained, and their efficiency is illustrated by examples.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574963","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010026
V. I. Mironenko, V. V. Mironenko
Abstract
The relationships between the notion of generalized integral and the notions of reflecting function and Poincaré map (period map) for periodic differential systems are traced. The notion of generalized first integral is used to study questions of the existence and stability of periodic solutions of periodic differential systems and analyze the center–focus problem.
{"title":"Reflecting Function and a Generalization of the Notion of First Integral","authors":"V. I. Mironenko, V. V. Mironenko","doi":"10.1134/s0012266124010026","DOIUrl":"https://doi.org/10.1134/s0012266124010026","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The relationships between the notion of generalized integral and the notions of reflecting\u0000function and Poincaré map (period map) for periodic differential systems are traced.\u0000The notion of generalized first integral is used to study questions of the existence and stability of\u0000periodic solutions of periodic differential systems and analyze the center–focus problem.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574625","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010051
V. I. Elkin
Abstract
We consider the symmetries of partial differential equations based on the use of differential-geometric and algebraic methods of the theory of dynamical control systems.
摘要 我们在使用动态控制系统理论的微分几何和代数方法的基础上，对偏微分方程的对称性进行了研究。
{"title":"Applying Differential-Geometric Control Theory Methods in the Theory of Partial Differential Equations. III","authors":"V. I. Elkin","doi":"10.1134/s0012266124010051","DOIUrl":"https://doi.org/10.1134/s0012266124010051","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the symmetries of partial differential equations based on the use of\u0000differential-geometric and algebraic methods of the theory of dynamical control systems.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574637","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010099
A. V. Metel’skii
Abstract
For a linear autonomous system of neutral type with commensurable delays, an algorithm is given for solving the modal controllability problem (in particular, the finite spectrum assignment problem), which provides a closed-loop system with a given characteristic quasipolynomial. A procedure for editing the finite part of the spectrum is proposed. A criterion for exponential stabilization of the system under study is constructively justified. When the criterion is met, the closed-loop system can be made exponentially stable according to the proposed spectral reduction algorithm. The obtained statements and spectrum assignment algorithms are illustrated with examples.
{"title":"Spectrum Assignment for a System of Neutral Type","authors":"A. V. Metel’skii","doi":"10.1134/s0012266124010099","DOIUrl":"https://doi.org/10.1134/s0012266124010099","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For a linear autonomous system of neutral type with commensurable delays, an algorithm\u0000is given for solving the modal controllability problem (in particular, the finite spectrum\u0000assignment problem), which provides a closed-loop system with a given characteristic\u0000quasipolynomial. A procedure for editing the finite part of the spectrum is proposed. A criterion\u0000for exponential stabilization of the system under study is constructively justified. When the\u0000criterion is met, the closed-loop system can be made exponentially stable according to the\u0000proposed spectral reduction algorithm. The obtained statements and spectrum assignment\u0000algorithms are illustrated with examples.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574516","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010105
K. A. Shchelchkov
Abstract
We consider the problem of stabilization to zero under disturbance in terms of a differential pursuit game. The dynamics is described by a nonlinear autonomous system of differential equations. The set of control values of the pursuer is finite, and that of the evader (disturbance) is a compact set. The control objective, i.e., the pursuer’s goal, is to bring the trajectory to any predetermined neighborhood of zero in finite time regardless of the disturbance. To construct the control, the pursuer knows only the state coordinates at some discrete times, and the choice of the disturbance’s control is unknown. In the paper, we obtain conditions for the existence of a neighborhood of zero from each point of which a capture occurs in the indicated sense. A winning control is constructed constructively and has an additional property specified in a theorem. In addition, an estimate of the capture time sharp in some sense is produced.
{"title":"On the Problem of Controlling a Nonlinear System by a Discrete Control under Disturbance","authors":"K. A. Shchelchkov","doi":"10.1134/s0012266124010105","DOIUrl":"https://doi.org/10.1134/s0012266124010105","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the problem of stabilization to zero under disturbance in terms of\u0000a differential pursuit game. The dynamics is described by a nonlinear autonomous system of\u0000differential equations. The set of control values of the pursuer is finite, and that of the evader\u0000(disturbance) is a compact set. The control objective, i.e., the pursuer’s goal, is to bring the\u0000trajectory to any predetermined neighborhood of zero in finite time regardless of the disturbance.\u0000To construct the control, the pursuer knows only the state coordinates at some discrete times, and\u0000the choice of the disturbance’s control is unknown. In the paper, we obtain conditions for the\u0000existence of a neighborhood of zero from each point of which a capture occurs in the indicated\u0000sense. A winning control is constructed constructively and has an additional property specified in\u0000a theorem. In addition, an estimate of the capture time sharp in some sense is produced.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574504","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010087
S. Iskandarov, A. T. Khalilov
Abstract
Sufficient conditions are established for the boundedness of all solutions and their first two derivatives of a third-order linear integro-differential equation of the Volterra type on the half-line. To this end, using a method proposed by the first author in 2006, first, we reduce the equation under consideration to an equivalent system consisting of one first-order differential equation and one second-order Volterra integro-differential equation. Then a new generalized Lyapunov functional is proposed for this system, the nonnegativity of this functional on solutions of this system is proved, and an upper bound is given for the derivative of this functional via the original functional. The resulting estimate is an integro-differential inequality whose solution gives an estimate of the functional.
{"title":"The Method of Lyapunov Functionals and the Boundedness of Solutions and Their First and Second Derivatives for a Third-Order Linear Equation of the Volterra Type on the Half-Line","authors":"S. Iskandarov, A. T. Khalilov","doi":"10.1134/s0012266124010087","DOIUrl":"https://doi.org/10.1134/s0012266124010087","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Sufficient conditions are established for the boundedness of all solutions and their first two\u0000derivatives of a third-order linear integro-differential equation of the Volterra type on the half-line.\u0000To this end, using a method proposed by the first author in 2006, first, we reduce the equation\u0000under consideration to an equivalent system consisting of one first-order differential equation and\u0000one second-order Volterra integro-differential equation. Then a new generalized Lyapunov\u0000functional is proposed for this system, the nonnegativity of this functional on solutions of this\u0000system is proved, and an upper bound is given for the derivative of this functional via the original\u0000functional. The resulting estimate is an integro-differential inequality whose solution gives an\u0000estimate of the functional.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574502","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}
Pub Date : 2024-04-09DOI: 10.1134/s001226612401004x
D. K. Durdiev
Abstract
Direct and inverse problems for a model equation of mixed parabolic-hyperbolic type are studied. In the direct problem, we consider a Tricomi-type problem for this equation with a noncharacteristic line of type change. The unknowns of the inverse problem are the variable coefficients of the lower-order terms in the equation. To determine these coefficients, an integral overdetermination condition is specified relative to the solution defined in the parabolic part of the domain, and in the hyperbolic part, conditions are specified on the characteristics: on one characteristic it is the value of the normal derivative and on the other, the value of the function itself. Theorems for the unique solvability of the posed problems in the sense of classical solution are proved.
{"title":"Inverse Problem of Determining Two Coefficients of Lower-Order Terms in a Mixed Parabolic-Hyperbolic Equation","authors":"D. K. Durdiev","doi":"10.1134/s001226612401004x","DOIUrl":"https://doi.org/10.1134/s001226612401004x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Direct and inverse problems for a model equation of mixed parabolic-hyperbolic type are\u0000studied. In the direct problem, we consider a Tricomi-type problem for this equation with a\u0000noncharacteristic line of type change. The unknowns of the inverse problem are the variable\u0000coefficients of the lower-order terms in the equation. To determine these coefficients, an integral\u0000overdetermination condition is specified relative to the solution defined in the parabolic part of the\u0000domain, and in the hyperbolic part, conditions are specified on the characteristics: on one\u0000characteristic it is the value of the normal derivative and on the other, the value of the function\u0000itself. Theorems for the unique solvability of the posed problems in the sense of classical solution\u0000are proved.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574623","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010075
V. A. Pavlenko
Abstract
This paper continues a series of papers in which simultaneous (2times 2 ) matrix solutions of two scalar evolution equations, which are analogs of time-dependent Schrödinger equations, were constructed. In the constructions in the present paper, these equations correspond to the Hamiltonian system (H^{2+2+1} )—one of the representatives of the hierarchy of degenerations of the isomonodromic Garnier system. The mentioned hierarchy was described by H. Kimura in 1986. In terms of solutions of linear systems of differential equations in the method of isomonodromic deformations, the consistency condition for which is the Hamiltonian equations of the (H^{2+2+1} ) system, the constructed simultaneous matrix solutions of analogs of time-dependent Schrödinger equations are written out explicitly in this paper.
Abstract This paper continues a series of papers in which simultaneous (2times 2 ) matrix solutions of two scalar evolution equations, which are analogs of time-dependent Schrödinger equations, were constructed.在本文的构造中，这些方程对应于哈密顿系统（H^{2+2+1} ）--等单调伽尼耶系统退化层次的代表之一。上述层次结构由 H. Kimura 在 1986 年描述。木村（Kimura）于 1986 年描述了上述层次结构。在等单旋转变形方法中的线性微分方程系的解方面，其一致性条件是 (H^{2+2+1} )系统的哈密顿方程，本文明确写出了构建的时变薛定谔方程类似物的同步矩阵解。
{"title":"Solutions of Analogs of Time-Dependent Schrödinger Equations Corresponding to a Pair of $$H^{2+2+1}$$ Hamiltonian Systems in the Hierarchy of Degenerations of an Isomonodromic Garnier System","authors":"V. A. Pavlenko","doi":"10.1134/s0012266124010075","DOIUrl":"https://doi.org/10.1134/s0012266124010075","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> This paper continues a series of papers in which simultaneous <span>(2times 2 )</span> matrix solutions of two scalar evolution equations,\u0000which are analogs of time-dependent Schrödinger equations, were constructed. In the\u0000constructions in the present paper, these equations correspond to the Hamiltonian system\u0000<span>(H^{2+2+1} )</span>—one of the representatives of the hierarchy\u0000of degenerations of the isomonodromic Garnier system. The mentioned hierarchy was described by\u0000H. Kimura in 1986. In terms of solutions of linear systems of differential equations in the method\u0000of isomonodromic deformations, the consistency condition for which is the Hamiltonian equations\u0000of the <span>(H^{2+2+1} )</span> system, the constructed simultaneous matrix\u0000solutions of analogs of time-dependent Schrödinger equations are written out explicitly in\u0000this paper.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574635","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010014
V. S. Besov, V. I. Kachalov
Abstract
S.A. Lomov’s regularization method has long been used to solve integro-differential singularly perturbed equations, which are very important from the viewpoint of applications. In this method, the series in powers of a small parameter representing the solutions of these equations converge asymptotically. However, in accordance with the main concept of the method, to construct a general singular perturbation theory one must indicate conditions for the ordinary convergence of these series. This is the subject of the present paper.
{"title":"Holomorphic Regularization of Singularly Perturbed Integro-Differential Equations","authors":"V. S. Besov, V. I. Kachalov","doi":"10.1134/s0012266124010014","DOIUrl":"https://doi.org/10.1134/s0012266124010014","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> S.A. Lomov’s regularization method has long been used to solve integro-differential\u0000singularly perturbed equations, which are very important from the viewpoint of applications. In\u0000this method, the series in powers of a small parameter representing the solutions of these\u0000equations converge asymptotically. However, in accordance with the main concept of the method,\u0000to construct a general singular perturbation theory one must indicate conditions for the ordinary\u0000convergence of these series. This is the subject of the present paper.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574617","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}
Pub Date : 2024-04-09DOI: 10.1134/s0012266124010117
A. B. Utesov
Abstract
A discretization operator for the solution of the Poisson equation with the right-hand side from the Korobov class is constructed and its error is estimated in the (L^{p} )-metric, (2leq pleq infty ). It is proved that for (p=2 ) the resulting error estimate for the discretization operator is order sharp on the power scale. An error in calculating the trigonometric Fourier coefficients used when constructing the discretization operator is also found. It should be noted that the obtained estimate in one case is better than previously known estimates of the errors of discretization operators constructed from the values of the right-hand side of the equation at the nodes of the modified Korobov grid and the Smolyak grid, and in the other case it coincides with them up to constants.
{"title":"On Error Estimates for Discretization Operators for the Solution of the Poisson Equation","authors":"A. B. Utesov","doi":"10.1134/s0012266124010117","DOIUrl":"https://doi.org/10.1134/s0012266124010117","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A discretization operator for the solution of the Poisson equation with the right-hand side\u0000from the Korobov class is constructed and its error is estimated in the <span>(L^{p} )</span>-metric, <span>(2leq pleq infty )</span>. It is proved that for <span>(p=2 )</span> the resulting error estimate for the discretization\u0000operator is order sharp on the power scale. An error in calculating the trigonometric Fourier\u0000coefficients used when constructing the discretization operator is also found. It should be noted\u0000that the obtained estimate in one case is better than previously known estimates of the errors of\u0000discretization operators constructed from the values of the right-hand side of the equation at the\u0000nodes of the modified Korobov grid and the Smolyak grid, and in the other case it coincides with\u0000them up to constants.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574501","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}
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