Pub Date : 2024-07-08DOI: 10.1134/s0012266124030054
A. V. Glushak
Abstract
In a Banach space, we consider the Cauchy problem and the Dirichlet and Neumann boundary value problems for a functional-differential equation generalizing the Euler–Poisson–Darboux equation. A sufficient condition for the solvability of the Cauchy problem is proved, and an explicit form of the resolving operator is indicated, which is written using the Bessel and Struve operator functions introduced by the author. For boundary value problems in the hyperbolic case, we establish conditions imposed on the operator coefficient of the equation and the boundary elements that are sufficient for the unique solvability of these problems.
{"title":"On the Solvability of Initial and Boundary Value Problems for Abstract Functional-Differential Euler–Poisson–Darboux Equations","authors":"A. V. Glushak","doi":"10.1134/s0012266124030054","DOIUrl":"https://doi.org/10.1134/s0012266124030054","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In a Banach space, we consider the Cauchy problem and the Dirichlet and Neumann\u0000boundary value problems for a functional-differential equation generalizing the\u0000Euler–Poisson–Darboux equation. A sufficient condition for the solvability of the Cauchy problem\u0000is proved, and an explicit form of the resolving operator is indicated, which is written using the\u0000Bessel and Struve operator functions introduced by the author. For boundary value problems in\u0000the hyperbolic case, we establish conditions imposed on the operator coefficient of the equation\u0000and the boundary elements that are sufficient for the unique solvability of these problems.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"20 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576712","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 : 2024-07-08DOI: 10.1134/s0012266124030091
A. V. Chernov
Abstract
For the Cauchy problem associated with a controlled semilinear evolution equation with an operator (not necessarily bounded) in a Hilbert space, we obtain sufficient conditions for exact controllability into a given terminal state (and also into given intermediate states at interim time moments) on an arbitrarily fixed (without additional constraints) time interval. Here we use the Browder—Minty theorem and also a chain technology of successive continuation of the solution of the controlled system to intermediate states. As examples, we consider a semilinear pseudoparabolic equation and a semilinear wave equation.
{"title":"On the Exact Global Controllability of a Semilinear Evolution Equation","authors":"A. V. Chernov","doi":"10.1134/s0012266124030091","DOIUrl":"https://doi.org/10.1134/s0012266124030091","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For the Cauchy problem associated with a controlled semilinear evolution equation with\u0000an operator (not necessarily bounded) in a Hilbert space, we obtain sufficient conditions for exact\u0000controllability into a given terminal state (and also into given intermediate states at interim time\u0000moments) on an arbitrarily fixed (without additional constraints) time interval. Here we use the\u0000Browder—Minty theorem and also a chain technology of successive continuation of the solution of\u0000the controlled system to intermediate states. As examples, we consider a semilinear\u0000pseudoparabolic equation and a semilinear wave equation.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"144 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576831","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 : 2024-06-05DOI: 10.1134/s0012266124020095
Ya. T. Sultanaev, N. F. Valeev, E. A. Nazirova
Abstract
The paper discusses the development of a method for constructing asymptotic formulas as (xto infty ) for the fundamental solution system of two-term singular symmetric differential equations of odd order with coefficients in a broad class of functions that allow oscillation (with relaxed regularity conditions that do not satisfy the classical Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation (({i}/{2})bigl [(p(x)y^{prime })^{prime prime }+(p(x)y^{prime prime })^{prime }bigr ] +q(x)y =lambda y), the asymptotics of solutions in the case of various behavior of the coefficients (q(x)) and (h(x)=-1+{1}big /{sqrt {p(x)}}) is studied. New asymptotic formulas are obtained for the case in which (h(x) notin L_1[1,infty ) ).
Abstract The paper discusses the development of a method for constructing asymptotic formulas as(xto infty ) for the fundamental solution system of two-termsingular symmetric differential equation of odd order with coefficients in a wide class offunctions that allow oscillation (with relaxed regularity conditions that not satisfy the classicalTitchmarsh-Levitan regularity conditions).以三阶二项式方程为例（({i}/{2})bigl [(p(x)y^{prime })^{prime }+(p(x)y^{prime })^{prime }bigr ] +q(x)y =lambda y ）、和（h(x)=-1+{1}big /sqrt {p(x)}）的各种行为情况下的解的渐近性进行了研究。对于 (h(x) notin L_1[1,infty ) 的情况，得到了新的渐近公式。).
{"title":"On the Asymptotic Behavior of Solutions of Third-Order Binomial Differential Equations","authors":"Ya. T. Sultanaev, N. F. Valeev, E. A. Nazirova","doi":"10.1134/s0012266124020095","DOIUrl":"https://doi.org/10.1134/s0012266124020095","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper discusses the development of a method for constructing asymptotic formulas as\u0000<span>(xto infty )</span> for the fundamental solution system of two-term\u0000singular symmetric differential equations of odd order with coefficients in a broad class of\u0000functions that allow oscillation (with relaxed regularity conditions that do not satisfy the classical\u0000Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation\u0000<span>(({i}/{2})bigl [(p(x)y^{prime })^{prime prime }+(p(x)y^{prime prime })^{prime }bigr ] +q(x)y =lambda y)</span>, the asymptotics of solutions in\u0000the case of various behavior of the coefficients <span>(q(x))</span> and\u0000<span>(h(x)=-1+{1}big /{sqrt {p(x)}})</span> is studied. New asymptotic\u0000formulas are obtained for the case in which <span>(h(x) notin L_1[1,infty ) )</span>.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"13 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512205","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 : 2024-06-05DOI: 10.1134/s0012266124020034
O. M. Jokhadze, S. S. Kharibegashvili
Abstract
For the inhomogeneous string vibration equation in a half-strip, we consider a problem periodic in the spatial variable and a mixed problem. The solutions of these problems in the form of finite sums are obtained by quadratures. When solving these problems, we use the characteristic rectangle identity, Riemann invariants, and the method of characteristics.
{"title":"Solution of Some Problems for the String Vibration Equation in a Half-Strip by Quadratures","authors":"O. M. Jokhadze, S. S. Kharibegashvili","doi":"10.1134/s0012266124020034","DOIUrl":"https://doi.org/10.1134/s0012266124020034","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For the inhomogeneous string vibration equation in a half-strip, we consider a problem\u0000periodic in the spatial variable and a mixed problem. The solutions of these problems in the form\u0000of finite sums are obtained by quadratures. When solving these problems, we use the\u0000characteristic rectangle identity, Riemann invariants, and the method of characteristics.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"45 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550809","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 : 2024-06-05DOI: 10.1134/s001226612402006x
A. M. Romanenkov
Abstract
The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.
{"title":"Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations","authors":"A. M. Romanenkov","doi":"10.1134/s001226612402006x","DOIUrl":"https://doi.org/10.1134/s001226612402006x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"97 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553245","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 : 2024-06-05DOI: 10.1134/s0012266124020083
V. V. Fomichev, A. I. Samarin
Abstract
The paper addresses the consensus problem (i.e., the agreement of state vectors) for a multiagent system consisting of identical linear agents. The study focuses on the case where there is no communication between agents, meaning there is no exchange of information, and agent control is achieved through the agents’ own sensors, providing incomplete information about the state vector of the agent and its neighbors, with the information possibly being noisy. To solve this problem, a linear protocol based on observer data for systems under uncertainty is proposed. Cascade observers based on the “super-twisting” method are suggested as such observers. Sufficient conditions are obtained for the existence of a controller where the observation error converges to zero under bounded disturbances. An example illustrating the proposed approach is provided.
{"title":"Cascade Super-Twisting Observer for Linear Multiagent Systems without Communication","authors":"V. V. Fomichev, A. I. Samarin","doi":"10.1134/s0012266124020083","DOIUrl":"https://doi.org/10.1134/s0012266124020083","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper addresses the consensus problem (i.e., the agreement of state vectors) for a\u0000multiagent system consisting of identical linear agents. The study focuses on the case where there\u0000is no communication between agents, meaning there is no exchange of information, and agent\u0000control is achieved through the agents’ own sensors, providing incomplete information about the\u0000state vector of the agent and its neighbors, with the information possibly being noisy. To solve\u0000this problem, a linear protocol based on observer data for systems under uncertainty is proposed.\u0000Cascade observers based on the “super-twisting” method are suggested as such observers. Sufficient\u0000conditions are obtained for the existence of a controller where the observation error converges to\u0000zero under bounded disturbances. An example illustrating the proposed approach is provided.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"16 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550810","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 : 2024-06-05DOI: 10.1134/s0012266124020010
S. A. Kashchenko
Abstract
We study the local dynamics of the delay logistic equation with an additional feedback containing a large delay. Critical cases in the problem of stability of the zero equilibrium state are identified, and it is shown that they are infinite-dimensional. The well-known methods for studying local dynamics based on the theory of invariant integral manifolds and normal forms do not apply here. The methods of infinite-dimensional normalization proposed by the author are used and developed. As the main results, special nonlinear boundary value problems of parabolic type are constructed, which play the role of normal forms. They determine the leading terms of the asymptotic expansions of solutions of the original equation and are called quasinormal forms.
{"title":"Logistic Equation with Long Delay Feedback","authors":"S. A. Kashchenko","doi":"10.1134/s0012266124020010","DOIUrl":"https://doi.org/10.1134/s0012266124020010","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the local dynamics of the delay logistic equation with an additional feedback\u0000containing a large delay. Critical cases in the problem of stability of the zero equilibrium state are\u0000identified, and it is shown that they are infinite-dimensional. The well-known methods for\u0000studying local dynamics based on the theory of invariant integral manifolds and normal forms do\u0000not apply here. The methods of infinite-dimensional normalization proposed by the author are\u0000used and developed. As the main results, special nonlinear boundary value problems of parabolic\u0000type are constructed, which play the role of normal forms. They determine the leading terms of\u0000the asymptotic expansions of solutions of the original equation and are called quasinormal forms.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"24 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550811","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 : 2024-06-05DOI: 10.1134/s0012266124020046
M. V. Turbin, A. S. Ustiuzhaninova
Abstract
The paper deals with proving the weak solvability of an initial–boundary value problem for the modified Kelvin–Voigt model taking into account memory along the trajectories of motion of fluid particles. To this end, we consider an approximation problem whose solvability is established with the use of the Leray–Schauder fixed point theorem. Then, based on a priori estimates, we show that the sequence of solutions of the approximation problem has a subsequence that weakly converges to the solution of the original problem as the approximation parameter tends to zero.
{"title":"Solvability of an Initial–Boundary Value Problem for the Modified Kelvin–Voigt Model with Memory along Fluid Motion Trajectories","authors":"M. V. Turbin, A. S. Ustiuzhaninova","doi":"10.1134/s0012266124020046","DOIUrl":"https://doi.org/10.1134/s0012266124020046","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper deals with proving the weak solvability of an initial–boundary value problem for\u0000the modified Kelvin–Voigt model taking into account memory along the trajectories of motion of\u0000fluid particles. To this end, we consider an approximation problem whose solvability is established\u0000with the use of the Leray–Schauder fixed point theorem. Then, based on a priori estimates, we\u0000show that the sequence of solutions of the approximation problem has a subsequence that weakly\u0000converges to the solution of the original problem as the approximation parameter tends to zero.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"25 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552936","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 : 2024-06-05DOI: 10.1134/s0012266124020058
A. A. Kashirin, S. I. Smagin
Abstract
The paper considers two weakly singular Fredholm boundary integral equations of the first kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on the spectra, where they are ill posed. For the first equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the transmission problem. The second equation in this case always has infinitely many solutions, with only one of them giving a solution of the transmission problem. The interpolation method for finding approximate solutions of the integral equations and the transmission problem in question is discussed.
{"title":"On the Solvability of Fredholm Boundary Integral Equations of the First Kind for the Three-Dimensional Transmission Problem on the Spectrum","authors":"A. A. Kashirin, S. I. Smagin","doi":"10.1134/s0012266124020058","DOIUrl":"https://doi.org/10.1134/s0012266124020058","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers two weakly singular Fredholm boundary integral equations of the first\u0000kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The\u0000properties of these equations are studied on the spectra, where they are ill posed. For the first\u0000equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the\u0000transmission problem. The second equation in this case always has infinitely many solutions, with\u0000only one of them giving a solution of the transmission problem. The interpolation method for\u0000finding approximate solutions of the integral equations and the transmission problem in question\u0000is discussed.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"28 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553244","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 : 2024-06-05DOI: 10.1134/s0012266124020022
A. S. Makin
Abstract
We consider the spectral problem for the Dirac operator with arbitrary two-point boundary conditions and any square integrable potential (V). Necessary and sufficient conditions for an entire function to be the characteristic determinant of such an operator are established. In the case of irregular boundary conditions, conditions are found under which a set of complex numbers is the spectrum of the problem under consideration.
{"title":"On the Spectrum of Nonself-Adjoint Dirac Operators with Two-Point Boundary Conditions","authors":"A. S. Makin","doi":"10.1134/s0012266124020022","DOIUrl":"https://doi.org/10.1134/s0012266124020022","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the spectral problem for the Dirac operator with arbitrary two-point boundary\u0000conditions and any square integrable potential <span>(V)</span>. Necessary and\u0000sufficient conditions for an entire function to be the characteristic determinant of such an operator\u0000are established. In the case of irregular boundary conditions, conditions are found under which a\u0000set of complex numbers is the spectrum of the problem under consideration.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"71 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550812","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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