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Differential Equations最新文献

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Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems 控制论和哈密顿系统理论问题中的等价微分方程
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010038
M. G. Yumagulov, L. S. Ibragimova
{"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}
引用次数: 0
Reflecting Function and a Generalization of the Notion of First Integral 反射函数与第一积分概念的广义化
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010026
V. I. Mironenko, V. V. Mironenko
{"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}
引用次数: 0
Applying Differential-Geometric Control Theory Methods in the Theory of Partial Differential Equations. III 在偏微分方程理论中应用微分几何控制论方法。三
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010051
V. I. Elkin
{"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}
引用次数: 0
Spectrum Assignment for a System of Neutral Type 中性系统的频谱分配
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010099
A. V. Metel’skii
{"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}
引用次数: 0
On the Problem of Controlling a Nonlinear System by a Discrete Control under Disturbance 论扰动下离散控制非线性系统的控制问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010105
K. A. Shchelchkov
{"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}
引用次数: 0
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 李雅普诺夫函数法与半直线上的 Volterra 型三阶线性方程的解及其一阶和二阶衍生物的有界性
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010087
S. Iskandarov, A. T. Khalilov
{"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}
引用次数: 0
Inverse Problem of Determining Two Coefficients of Lower-Order Terms in a Mixed Parabolic-Hyperbolic Equation 确定抛物线-双曲混合方程中两个低阶项系数的逆问题
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s001226612401004x
D. K. Durdiev
{"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}
引用次数: 0
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 时变薛定谔方程对应于等单调伽尼耶系统退化层次中一对 $$H^{2+2+1}$ 哈密顿系统的解的类似物
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010075
V. A. Pavlenko
{"title":"Solutions of Analogs of Time-Dependent Schrödinger Equations Corresponding to a Pair of $$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}
引用次数: 0
Holomorphic Regularization of Singularly Perturbed Integro-Differential Equations 奇异扰动积分微分方程的全态正则化
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010014
V. S. Besov, V. I. Kachalov
{"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}
引用次数: 0
On Error Estimates for Discretization Operators for the Solution of the Poisson Equation 论求解泊松方程的离散化算子的误差估计值
IF 0.6 4区 数学
Differential Equations Pub Date : 2024-04-09 DOI: 10.1134/s0012266124010117
A. B. Utesov
{"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}
引用次数: 0
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