Pub Date : 2024-02-26DOI: 10.1134/s0012266123120108
E. B. Afanas’eva, V. B. Vasil’ev, A. B. Kamanda Bongay
Abstract
We study the solvability of discrete elliptic pseudodifferential equations in a sector of the plane. Using special factorization of the symbol, the problem is reduced to a boundary value problem for analytic functions of two variables. A periodic analog of one integral transformation is obtained that was used to construct solutions of elliptic pseudodifferential equations in conical domains. The formula for the general solution of the discrete equation under consideration and some boundary value problems are described in terms of this transformation.
{"title":"Discrete Equations, Discrete Transformations, and Discrete Boundary Value Problems","authors":"E. B. Afanas’eva, V. B. Vasil’ev, A. B. Kamanda Bongay","doi":"10.1134/s0012266123120108","DOIUrl":"https://doi.org/10.1134/s0012266123120108","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the solvability of discrete elliptic pseudodifferential equations in a sector of the\u0000plane. Using special factorization of the symbol, the problem is reduced to a boundary value\u0000problem for analytic functions of two variables. A periodic analog of one integral transformation is\u0000obtained that was used to construct solutions of elliptic pseudodifferential equations in conical\u0000domains. The formula for the general solution of the discrete equation under consideration and\u0000some boundary value problems are described in terms of this transformation.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979174","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 : 2024-02-26DOI: 10.1134/s0012266123120157
Yu. L. Sachkov
Abstract
Sufficient conditions for the existence of optimal trajectories in general optimal control problems with free terminal time as well as in sub-Lorentzian problems are obtained.
摘要 在具有自由终点时间的一般最优控制问题以及亚洛伦兹问题中，获得了最优轨迹存在的充分条件。
{"title":"Existence of Sub-Lorentzian Longest Curves","authors":"Yu. L. Sachkov","doi":"10.1134/s0012266123120157","DOIUrl":"https://doi.org/10.1134/s0012266123120157","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Sufficient conditions for the existence of optimal trajectories in general optimal control\u0000problems with free terminal time as well as in sub-Lorentzian problems are obtained.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979786","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 : 2024-02-26DOI: 10.1134/s0012266123120182
N. V. Zaitseva
Abstract
We provide a formulation of the Cauchy problem in a strip for a two-dimensional hyperbolic equation containing a superposition of a differential operator and a shift operator with respect to the spatial variable varying along the entire real axis. The solution of the problem using integral Fourier transforms is constructed in explicit form.
{"title":"On One Cauchy Problem for a Hyperbolic Differential-Difference Equation","authors":"N. V. Zaitseva","doi":"10.1134/s0012266123120182","DOIUrl":"https://doi.org/10.1134/s0012266123120182","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We provide a formulation of the Cauchy problem in a strip for a two-dimensional\u0000hyperbolic equation containing a superposition of a differential operator and a shift operator with\u0000respect to the spatial variable varying along the entire real axis. The solution of the problem using\u0000integral Fourier transforms is constructed in explicit form.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979289","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 : 2024-02-26DOI: 10.1134/s0012266123120145
M. A. Velishchanskiy, V. N. Chetverikov
Abstract
We study the problem of optimal choice of model parameters with respect to any functional. Locally controllable affine systems and integral functionals depending on the program control are considered. Local controllability of affine systems with nonnegative inputs is proved in the case where the columns multiplying the controls form a positive basis. For such systems, we introduce the local controllability coefficient and pose the problem of its maximization depending on the choice of model parameters. As an example, we consider a very simplified model of an underwater vehicle and study the problem of finding an arrangement of its control propellers in which the energy consumption of the vehicle is minimal.
{"title":"Searching for Parameters of a Model with the Best Local Controllability","authors":"M. A. Velishchanskiy, V. N. Chetverikov","doi":"10.1134/s0012266123120145","DOIUrl":"https://doi.org/10.1134/s0012266123120145","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the problem of optimal choice of model parameters with respect to any\u0000functional. Locally controllable affine systems and integral functionals depending on the program\u0000control are considered. Local controllability of affine systems with nonnegative inputs is proved in\u0000the case where the columns multiplying the controls form a positive basis. For such systems, we\u0000introduce the local controllability coefficient and pose the problem of its maximization depending\u0000on the choice of model parameters. As an example, we consider a very simplified model of an\u0000underwater vehicle and study the problem of finding an arrangement of its control propellers in\u0000which the energy consumption of the vehicle is minimal.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979294","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 : 2024-02-26DOI: 10.1134/s0012266123120091
V. B. Vasil’ev
Abstract
We consider a special boundary value problem in the Sobolev–Slobodetskii space for a model elliptic pseudodifferential equation in a multidimensional cone. Taking into account the special factorization of the elliptic symbol, we write the general solution of the pseudodifferential equation that contains an arbitrary function. To determine it unambiguously, some integral condition is added to the equation, which makes it possible to write the solution in Fourier transforms.
{"title":"Pseudodifferential Equations and Boundary Value Problems in a Multidimensional Cone","authors":"V. B. Vasil’ev","doi":"10.1134/s0012266123120091","DOIUrl":"https://doi.org/10.1134/s0012266123120091","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider a special boundary value problem in the Sobolev–Slobodetskii space for a\u0000model elliptic pseudodifferential equation in a multidimensional cone. Taking into account the\u0000special factorization of the elliptic symbol, we write the general solution of the pseudodifferential\u0000equation that contains an arbitrary function. To determine it unambiguously, some integral\u0000condition is added to the equation, which makes it possible to write the solution in Fourier\u0000transforms.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979316","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 : 2024-02-26DOI: 10.1134/s001226612312008x
A. B. Khasanov, T. G. Khasanov
Abstract
The inverse spectral problem method is applied to finding a solution of the Cauchy problem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions. A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded terms and a derivation of an analog of Dubrovin’s system of differential equations are proposed. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and the first trace formula actually satisfies the loaded nonlinear Korteweg–de Vries equation. In addition, we prove that if the initial function is a real (pi )-periodic analytic function, then the solution of the Cauchy problem is a real analytic function in the variable (x ) as well, and also that if the number ( {pi }/{n} ), (nin mathbb {N}), (nge 2 ), is the period of the initial function, then the number ({pi }/{n} ) is the period for solving the Cauchy problem with respect to the variable (x).
{"title":"Cauchy Problem for the Loaded Korteweg–de Vries Equation in the Class of Periodic Functions","authors":"A. B. Khasanov, T. G. Khasanov","doi":"10.1134/s001226612312008x","DOIUrl":"https://doi.org/10.1134/s001226612312008x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The inverse spectral problem method is applied to finding a solution of the Cauchy\u0000problem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.\u0000A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded terms\u0000and a derivation of an analog of Dubrovin’s system of differential equations are proposed. It is\u0000shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin\u0000system of equations and the first trace formula actually satisfies the loaded nonlinear\u0000Korteweg–de Vries equation. In addition, we prove that if the initial function is a real\u0000<span>(pi )</span>-periodic analytic function, then the solution of the\u0000Cauchy problem is a real analytic function in the variable <span>(x )</span> as well, and also that if the number\u0000<span>( {pi }/{n} )</span>, <span>(nin mathbb {N})</span>,\u0000<span>(nge 2 )</span>, is the period of the initial function, then the\u0000number <span>({pi }/{n} )</span> is the period for solving the Cauchy problem with\u0000respect to the variable <span>(x)</span>.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979286","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 : 2024-02-26DOI: 10.1134/s0012266123120029
N. A. Izobov, A. V. Il’in
Abstract
We prove the existence of a two-dimensional linear system (dot {x}=A(t)x ), (tgeq t_0), with bounded infinitely differentiable coefficients and all positive characteristic exponents, as well as an infinitely differentiable (m)-perturbation (f(t,y) ) having an order (m>1 ) of smallness in a neighborhood of the origin (y=0 ) and an order of growth not exceeding (m ) outside it, such that the perturbed system (dot {y}=A( t)y+thinspace f(t,y)), (yin mathbb {R}^2 ), (tgeq t_0), has a solution (y(t) ) with a negative Lyapunov exponent.
Abstract We prove existence of a two-dimensional linear system (dot {x}=A(t)x ), (tgeq t_0), withbounded infinitely differentiable coefficients and all positive characteristic exponents, as well as aninfinitely differentiable (m)-perturbation(f(t,y) ) having an order (m>. 1) in the neighborhood of origin(y=0) with an smallness and an order growth not exceed(m) outside it；在原点（y=0）的邻域内有一个小的增长阶次，而在它之外有一个不超过（m）的增长阶次、such that the perturbed system(dot {y}=A( t)y+thinspace f(t,y)),(yin mathbb {R}^2 ), (tgeq t_0), has asolution (y(t) ) with a negative Lyapunov exponent.
{"title":"Existence of an Anti-Perron Effect of Change of Positive Exponents of the Linear Approximation System to Negative Ones under Perturbations of a Higher Order of Smallness","authors":"N. A. Izobov, A. V. Il’in","doi":"10.1134/s0012266123120029","DOIUrl":"https://doi.org/10.1134/s0012266123120029","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove the existence of a two-dimensional linear system <span>(dot {x}=A(t)x )</span>, <span>(tgeq t_0)</span>, with\u0000bounded infinitely differentiable coefficients and all positive characteristic exponents, as well as an\u0000infinitely differentiable <span>(m)</span>-perturbation\u0000<span>(f(t,y) )</span> having an order <span>(m>1 )</span> of smallness in a neighborhood of the origin\u0000<span>(y=0 )</span> and an order of growth not exceeding\u0000<span>(m )</span> outside it, such that the perturbed system\u0000<span>(dot {y}=A( t)y+thinspace f(t,y))</span>, <span>(yin mathbb {R}^2 )</span>, <span>(tgeq t_0)</span>, has a\u0000solution <span>(y(t) )</span> with a negative Lyapunov exponent.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979181","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 : 2024-02-26DOI: 10.1134/s0012266123120042
E. A. Baderko, K. D. Fedorov
Abstract
We consider the solution of the Cauchy problem in a strip on the plane for a homogeneous second-order parabolic system. The coefficients of the system satisfy the double Dini condition. The initial function is continuous and bounded along with its first and second derivatives. Using the Poisson potential, the nature of the smoothness of this solution is studied and the corresponding estimates are proved.
摘要 我们考虑了一个均质二阶抛物线系统在平面条带中的考奇问题的求解。该系统的系数满足双 Dini 条件，初始函数及其一阶导数和二阶导数是连续和有界的。利用泊松势研究了该解的平滑性，并证明了相应的估计值。
{"title":"On the Smoothness of the Poisson Potential for Second-Order Parabolic Systems on the Plane","authors":"E. A. Baderko, K. D. Fedorov","doi":"10.1134/s0012266123120042","DOIUrl":"https://doi.org/10.1134/s0012266123120042","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the solution of the Cauchy problem in a strip on the plane for a homogeneous\u0000second-order parabolic system. The coefficients of the system satisfy the double Dini condition.\u0000The initial function is continuous and bounded along with its first and second derivatives. Using\u0000the Poisson potential, the nature of the smoothness of this solution is studied and the\u0000corresponding estimates are proved.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979180","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 : 2024-02-26DOI: 10.1134/s0012266123120170
B. Yu. Irgashev
Abstract
A solution of the Cauchy problem is obtained for one degenerate equation with the Dzhrbashyan–Nersesyan fractional derivative, particular solutions of which are represented using the Kilbas–Saigo function.
{"title":"Solution of the Cauchy Problem for One Degenerate Equation with the Dzhrbashyan–Nersesyan Fractional Derivative","authors":"B. Yu. Irgashev","doi":"10.1134/s0012266123120170","DOIUrl":"https://doi.org/10.1134/s0012266123120170","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A solution of the Cauchy problem is obtained for one degenerate equation with the\u0000Dzhrbashyan–Nersesyan fractional derivative, particular solutions of which are represented using\u0000the Kilbas–Saigo function.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979998","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 : 2024-02-26DOI: 10.1134/s0012266123120078
M. S. Smirnov
Abstract
A necessary and sufficient condition is established for the closedness of the range or surjectivity of a differential operator acting on smooth sections of vector bundles. For connected noncompact manifolds it is shown that these conditions are derived from the regularity conditions and the unique continuation property of solutions. An application of these results to elliptic operators (more precisely, to operators with a surjective principal symbol) with analytic coefficients, to second-order elliptic operators on line bundles with a real leading part, and to the Hodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively, Dolbeault) cohomology group on a connected noncompact smooth (respectively, complex-analytic) manifold vanishes. For elliptic operators, we prove that solvability in smooth sections implies solvability in generalized sections.
{"title":"On the Solvability of Linear Differential Operators on Vector Bundles over a Manifold","authors":"M. S. Smirnov","doi":"10.1134/s0012266123120078","DOIUrl":"https://doi.org/10.1134/s0012266123120078","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A necessary and sufficient condition is established for the closedness of the range or\u0000surjectivity of a differential operator acting on smooth sections of vector bundles. For connected\u0000noncompact manifolds it is shown that these conditions are derived from the regularity conditions\u0000and the unique continuation property of solutions. An application of these results to elliptic\u0000operators (more precisely, to operators with a surjective principal symbol) with analytic\u0000coefficients, to second-order elliptic operators on line bundles with a real leading part, and to the\u0000Hodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively,\u0000Dolbeault) cohomology group on a connected noncompact smooth (respectively, complex-analytic)\u0000manifold vanishes. For elliptic operators, we prove that solvability in smooth sections implies\u0000solvability in generalized sections.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979177","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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