Pub Date : 2024-09-11DOI: 10.1134/s0012266124050112
A. N. Naimov, M. V. Bystretsky
Abstract
In the present paper, we study an a priori estimate and the existence of periodic solutions of a given period for a system of second-order ordinary differential equations with the main quasihomogeneous nonlinearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have nonzero bounded solutions. Under the conditions of the a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions is stated and proved for any perturbation in a given class. The results obtained differ from earlier results in that the set of zeros of the main nonlinearity is not taken into account.
{"title":"On the Existence of Periodic Solutions of a System of Second-Order Ordinary Differential Equations with a Quasihomogeneous Nonlinearity","authors":"A. N. Naimov, M. V. Bystretsky","doi":"10.1134/s0012266124050112","DOIUrl":"https://doi.org/10.1134/s0012266124050112","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In the present paper, we study an a priori estimate and the existence of periodic solutions\u0000of a given period for a system of second-order ordinary differential equations with the main\u0000quasihomogeneous nonlinearity. It is proved that an a priori estimate of periodic solutions takes\u0000place if the corresponding unperturbed system does not have nonzero bounded solutions. Under\u0000the conditions of the a priori estimate, using methods for calculating the mapping degree of vector\u0000fields, a criterion for the existence of periodic solutions is stated and proved for any perturbation\u0000in a given class. The results obtained differ from earlier results in that the set of zeros of the main\u0000nonlinearity is not taken into account.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175370","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 : 2024-09-11DOI: 10.1134/s0012266124050094
V. E. Khartovskii
Abstract
For a linear autonomous differential-difference system of the neutral type, the existence criterion is proved and a method is proposed for designing an observed output feedback controller providing the closed-loop system with finite stabilization (solution of the problem of complete (0 )-controllability) and a finite predetermined spectrum. This makes the closed-loop system exponentially stable. The constructiveness of the presented results is illustrated by an example.
{"title":"Finite Stabilization and Finite Spectrum Assignment by a Single Controller Based on Incomplete Measurements for Linear Systems of the Neutral Type","authors":"V. E. Khartovskii","doi":"10.1134/s0012266124050094","DOIUrl":"https://doi.org/10.1134/s0012266124050094","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For a linear autonomous differential-difference system of the neutral type, the existence\u0000criterion is proved and a method is proposed for designing an observed output feedback controller\u0000providing the closed-loop system with finite stabilization (solution of the problem of complete\u0000<span>(0 )</span>-controllability) and a finite predetermined\u0000spectrum. This makes the closed-loop system exponentially stable. The constructiveness of the\u0000presented results is illustrated by an example.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175368","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 : 2024-09-11DOI: 10.1134/s0012266124050045
S. N. Melikhov, B. I. Suleimanov, A. M. Shavlukov
Abstract
Formal asymptotics are substantiated that describe a typical dropping cusp singularity in the semiclassical approximations to solutions of two cases of the integrable nonlinear Schrödinger equation (-ivarepsilon Psi ^{prime }_{t} = varepsilon ^2Psi ^{prime prime }_{xx}pm 2|Psi | ^2Psi ), where (varepsilon ) is a small parameter. The substantiation uses the ideas and facts of the mathematical catastrophe theory and the part of Yu.F. Korobeinik’s theorem concerning analytical, as (hto 0), solutions (G(h,u) ) of the mixed type linear equation (hG^{prime prime }_{hh}=G^{prime prime }_{uu}) to which the hodograph images of both cases of the systems of equations of these semiclassical approximations are equivalent.
Abstract Formal asymptics are substantiated that describe a typical dropping cusp singularity in the semiclassical approximations to solutions of two cases of the integrable nonlinearSchrödinger equation (-ivarepsilon Psi ^{prime }_{t} = varepsilon ^2Psi ^{prime prime }_{xx}pm 2|Psi | ^2Psi )、其中 (varepsilon )是一个小参数。证明使用了数学灾难理论的概念和事实，以及 Yu.F.Korobeinik's storem concerning analytical, as (hto 0), solutions(G(h,u) ) of the mixed type linear equation(hG^{prime prime }_{hh}=G^{prime prime }_{uu}) to which the hodograph images of the both cases of the systems of equations of these semiclassicalapproximations are equivalent.
{"title":"Typical Dropping Asymptotics in the Semiclassical Approximations to Solutions of the Nonlinear Schrödinger Equation","authors":"S. N. Melikhov, B. I. Suleimanov, A. M. Shavlukov","doi":"10.1134/s0012266124050045","DOIUrl":"https://doi.org/10.1134/s0012266124050045","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Formal asymptotics are substantiated that describe a typical dropping cusp singularity in\u0000the semiclassical approximations to solutions of two cases of the integrable nonlinear\u0000Schrödinger equation <span>(-ivarepsilon Psi ^{prime }_{t} = varepsilon ^2Psi ^{prime prime }_{xx}pm 2|Psi | ^2Psi )</span>,\u0000where <span>(varepsilon )</span> is a small parameter. The substantiation uses the\u0000ideas and facts of the mathematical catastrophe theory and the part of Yu.F. Korobeinik’s\u0000theorem concerning analytical, as <span>(hto 0)</span>, solutions\u0000<span>(G(h,u) )</span> of the mixed type linear equation\u0000<span>(hG^{prime prime }_{hh}=G^{prime prime }_{uu})</span> to\u0000which the hodograph images of both cases of the systems of equations of these semiclassical\u0000approximations are equivalent.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142223186","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 : 2024-09-11DOI: 10.1134/s0012266124050021
V. M. Budanov
Abstract
We justify an analytical method for constructing periodic solutions of nonlinear systems of ordinary differential equations of polynomial type. Periodic solutions are constructed in the form of Fourier series in which the coefficients are polynomials depending on a parameter, which is not assumed to be small. Two examples are considered: the van der Pol equation and the Lorenz system.
{"title":"Method for Constructing Periodic Solutions of Nonlinear Differential Equations","authors":"V. M. Budanov","doi":"10.1134/s0012266124050021","DOIUrl":"https://doi.org/10.1134/s0012266124050021","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We justify an analytical method for constructing periodic solutions of nonlinear systems of\u0000ordinary differential equations of polynomial type. Periodic solutions are constructed in the form\u0000of Fourier series in which the coefficients are polynomials depending on a parameter, which is not\u0000assumed to be small. Two examples are considered: the van der Pol equation and the Lorenz\u0000system.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175336","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 : 2024-09-11DOI: 10.1134/s0012266124050057
V. G. Nikolaev
Abstract
The Schwarz problem for (J)-analytic functions in an arbitrary ellipse is considered. The matrix (J) is assumed to be two-dimensional with distinct eigenvalues lying above the real axis. An example of a nonconstant solution of the homogeneous Schwarz problem in the form of a vector polynomial of degree three is given. A numerical parameter (l) of the matrix (J ), expressed via its eigenvectors, is introduced. After that, one relation derived earlier by the present author is analyzed. Based on this analysis, a method for computing the dimension and structure of the kernel of the Schwarz problem in an arbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed via the ellipse parameters, the eigenvalues of the matrix (J), and the parameter (l ) are obtained. Examples of one-dimensional and trivial kernels are given.
{"title":"On the Structure of the Kernel of the Schwarz Problem for First-Order Elliptic Systems on the Plane","authors":"V. G. Nikolaev","doi":"10.1134/s0012266124050057","DOIUrl":"https://doi.org/10.1134/s0012266124050057","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The Schwarz problem for <span>(J)</span>-analytic functions in\u0000an arbitrary ellipse is considered. The matrix <span>(J)</span> is assumed to be\u0000two-dimensional with distinct eigenvalues lying above the real axis. An example of a nonconstant\u0000solution of the homogeneous Schwarz problem in the form of a vector polynomial of degree three is\u0000given. A numerical parameter <span>(l)</span> of the matrix\u0000<span>(J )</span>, expressed via its eigenvectors, is introduced. After\u0000that, one relation derived earlier by the present author is analyzed. Based on this analysis, a\u0000method for computing the dimension and structure of the kernel of the Schwarz problem in an\u0000arbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed via the\u0000ellipse parameters, the eigenvalues of the matrix <span>(J)</span>, and the parameter\u0000<span>(l )</span> are obtained. Examples of one-dimensional and\u0000trivial kernels are given.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175364","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 : 2024-09-11DOI: 10.1134/s0012266124050033
A. A. Levakov, D. A. Dolzhenkova
Abstract
A stochastic differential-difference hybrid system is a system of coupled variables whose dynamics is described by stochastic differential equations for some of them and difference equations for the others. Systems with two types of difference equations are examined: first, a difference equation in the form of a process involving a multiplicative Wiener process, and second, a difference equation with delay. The existence and uniqueness theorems for both systems are proved. The basic conditions on the system’s parameters are local Lipschitz conditions and linear growth order.
{"title":"Existence and Uniqueness Theorems for Stochastic Differential-Difference Hybrid Systems","authors":"A. A. Levakov, D. A. Dolzhenkova","doi":"10.1134/s0012266124050033","DOIUrl":"https://doi.org/10.1134/s0012266124050033","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A stochastic differential-difference hybrid system is a system of coupled variables whose\u0000dynamics is described by stochastic differential equations for some of them and difference\u0000equations for the others. Systems with two types of difference equations are examined: first,\u0000a difference equation in the form of a process involving a multiplicative Wiener process, and\u0000second, a difference equation with delay. The existence and uniqueness theorems for both systems\u0000are proved. The basic conditions on the system’s parameters are local Lipschitz conditions and\u0000linear growth order.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175363","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 : 2024-09-11DOI: 10.1134/s0012266124050070
A. E. Golubev
Abstract
The problem of stabilizing the zero value of the state vector of constrained nonlinear dynamical systems written in a special form is solved. The proposed control design accounts for magnitude constraints on the values of state variables and is based on the integrator backstepping approach using logarithmic Lyapunov barrier functions. The obtained stabilizing feedbacks, in contrast to similar known results, are based on the use of linear virtual stabilizing functions that do not grow unboundedly as the state variables approach boundary values. As an example, we consider a state constraints aware solution of the control problem of positioning an autonomous underwater vehicle at a given point in space.
{"title":"Backstepping Stabilization of Nonlinear Dynamical Systems under State Constraints","authors":"A. E. Golubev","doi":"10.1134/s0012266124050070","DOIUrl":"https://doi.org/10.1134/s0012266124050070","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The problem of stabilizing the zero value of the state vector of constrained nonlinear\u0000dynamical systems written in a special form is solved. The proposed control design accounts for\u0000magnitude constraints on the values of state variables and is based on the integrator backstepping\u0000approach using logarithmic Lyapunov barrier functions. The obtained stabilizing feedbacks, in\u0000contrast to similar known results, are based on the use of linear virtual stabilizing functions that\u0000do not grow unboundedly as the state variables approach boundary values. As an example, we\u0000consider a state constraints aware solution of the control problem of positioning an autonomous\u0000underwater vehicle at a given point in space.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175367","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 : 2024-09-11DOI: 10.1134/s0012266124050082
P. A. Tochilin, I. A. Chistyakov
Abstract
A nonlinear system of ordinary differential equations with control parameters is considered. Pointwise restrictions are imposed on the possible values of these parameters. It is required to solve the problem of transferring the trajectory of the system from an arbitrary initial position to the smallest possible neighborhood of a given target set on a fixed time interval by selecting an appropriate feedback control. To solve this problem, it is proposed to construct a continuous piecewise cubic function of a special kind. The level sets of this function correspond to internal estimates of the solvability sets of the system. Using this function, it is also possible to construct a feedback control function that solves the target control problem on a fixed time interval. The paper proposes formulas for calculating the values of the piecewise cubic function, examines its properties, and considers an algorithm for searching for parameters defining this function.
{"title":"On Piecewise Cubic Estimates of the Value Function in a Target Control Problem for a Nonlinear System","authors":"P. A. Tochilin, I. A. Chistyakov","doi":"10.1134/s0012266124050082","DOIUrl":"https://doi.org/10.1134/s0012266124050082","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A nonlinear system of ordinary differential equations with control parameters is considered.\u0000Pointwise restrictions are imposed on the possible values of these parameters. It is required to\u0000solve the problem of transferring the trajectory of the system from an arbitrary initial position to\u0000the smallest possible neighborhood of a given target set on a fixed time interval by selecting an\u0000appropriate feedback control. To solve this problem, it is proposed to construct a continuous\u0000piecewise cubic function of a special kind. The level sets of this function correspond to internal\u0000estimates of the solvability sets of the system. Using this function, it is also possible to construct a\u0000feedback control function that solves the target control problem on a fixed time interval. The\u0000paper proposes formulas for calculating the values of the piecewise cubic function, examines its\u0000properties, and considers an algorithm for searching for parameters defining this function.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175366","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 : 2024-09-11DOI: 10.1134/s0012266124050100
D. M. Korosteleva
Abstract
We propose a new symmetric variational functional-algebraic statement of the eigenvalue problem in a Hilbert space with a linear dependence on the spectral parameter for a class of mathematical models of thin-walled structures with an attached oscillator. The existence of eigenvalues and eigenvectors is established. A new symmetric approximation of the problem in a finite-dimensional subspace with a linear dependence on the spectral parameter is constructed. Error estimates are obtained for the approximate eigenvalues and eigenvectors. The theoretical results are illustrated with an example of a structural mechanics problem.
{"title":"Approximation of Functional-Algebraic Eigenvalue Problems","authors":"D. M. Korosteleva","doi":"10.1134/s0012266124050100","DOIUrl":"https://doi.org/10.1134/s0012266124050100","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We propose a new symmetric variational functional-algebraic statement of the eigenvalue\u0000problem in a Hilbert space with a linear dependence on the spectral parameter for a class of\u0000mathematical models of thin-walled structures with an attached oscillator. The existence of\u0000eigenvalues and eigenvectors is established. A new symmetric approximation of the problem in\u0000a finite-dimensional subspace with a linear dependence on the spectral parameter is constructed.\u0000Error estimates are obtained for the approximate eigenvalues and eigenvectors. The theoretical\u0000results are illustrated with an example of a structural mechanics problem.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175369","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 : 2024-07-30DOI: 10.1134/s0012266124040098
D. V. Georgievskii, N. A. Rautian
Abstract
We discuss the well-posed solvability and exponential stability of solutions of abstract integro-differential equations where the kernels of integral operators are of general type and lie in the space of functions integrable on the positive half-line. The abstract integro-differential equations studied in the present paper are operator models of viscoelasticity theory problems. The proposed approach to the study of these integro-differential equations is related to an application of semigroup theory and can also be used to study other integro-differential equations containing integral terms of the Volterra convolution type.
{"title":"Well-Posed Solvability of Volterra Integro-Differential Equations Arising in Viscoelasticity Theory","authors":"D. V. Georgievskii, N. A. Rautian","doi":"10.1134/s0012266124040098","DOIUrl":"https://doi.org/10.1134/s0012266124040098","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We discuss the well-posed solvability and exponential stability of solutions of abstract\u0000integro-differential equations where the kernels of integral operators are of general type and lie in\u0000the space of functions integrable on the positive half-line. The abstract integro-differential\u0000equations studied in the present paper are operator models of viscoelasticity theory problems. The\u0000proposed approach to the study of these integro-differential equations is related to an application\u0000of semigroup theory and can also be used to study other integro-differential equations containing\u0000integral terms of the Volterra convolution type.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867280","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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