Pub Date : 2023-06-27DOI: 10.37069/1683-4720-2023-37-5
Ihor Skrypnik, Maria Savchenko, Yevgeniia Yevgenieva
The study of the regularity of solutions to the elliptic equations with non-standard growth has been initiated by Zhikov, Marcellini, and Lieberman, and in the last thirty years, the qualitative theory of second-order elliptic and parabolic equations has been actively developed. Equations of this type and systems of such equations arise in various problems of mathematical physics and engineering (e.g. in describing electrorheological fluids, or in image recognition and data denoising). There are two cases of the type of growth. The simple so-called ''logarithmic'' case is studied very well and there are a lot of classical results in this regard. But the so-called ''non-logarithmic'' growth differs substantially from the logarithmic case. The non-logarithmic condition introduced by Zhikov turned out to be a precise condition for the smoothness of finite functions in the corresponding Sobolev space, which makes it extremely interesting to study. But to our knowledge, there are only a few results in this direction. Zhikov and Pastukhova proved higher integrability of the gradient of solutions to the $p(x)$-Laplace equation under the non-logarithmic condition. Interior continuity, continuity up to the boundary, and Harnack's inequality to the $p(x)$-Laplace equation were proved by Alkhutov, Krasheninnikova, and Surnachev. These results were generalized by Skrypnik and Voitovich. The qualitative properties of bounded solutions of $p(x)$-Laplace equation under the non-logarithmic condition were established by Skrypnik and Yevgenieva. As for unbounded solutions, there are just a few results. Ok has proved the boundedness of minimizers of elliptic functionals of the double-phase type under some assumptions on the growth parameters. The obtained condition gives a possibility to improve the regularity results for unbounded minimizers. The weak Harnack inequality for unbounded supersolutions of the corresponding elliptic equations with generalized Orlicz growth under the so-called logarithmic conditions was proved by Benyaiche, Harjulehto, H"{a}st"{o} and Karppinen. In the current paper, the weak Harnack inequality for unbounded solutions to the $p(x)$-Laplace equation has been proved under the precise non-logarithmic condition on the function $p(x)$.
{"title":"Weak Harnack inequality for unbounded solutions to the p(x)-Laplace equation under the precise non-logarithmic conditions","authors":"Ihor Skrypnik, Maria Savchenko, Yevgeniia Yevgenieva","doi":"10.37069/1683-4720-2023-37-5","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-5","url":null,"abstract":"The study of the regularity of solutions to the elliptic equations with non-standard growth has been initiated by Zhikov, Marcellini, and Lieberman, and in the last thirty years, the qualitative theory of second-order elliptic and parabolic equations has been actively developed. Equations of this type and systems of such equations arise in various problems of mathematical physics and engineering (e.g. in describing electrorheological fluids, or in image recognition and data denoising). There are two cases of the type of growth. The simple so-called ''logarithmic'' case is studied very well and there are a lot of classical results in this regard. But the so-called ''non-logarithmic'' growth differs substantially from the logarithmic case. The non-logarithmic condition introduced by Zhikov turned out to be a precise condition for the smoothness of finite functions in the corresponding Sobolev space, which makes it extremely interesting to study. But to our knowledge, there are only a few results in this direction. Zhikov and Pastukhova proved higher integrability of the gradient of solutions to the $p(x)$-Laplace equation under the non-logarithmic condition. Interior continuity, continuity up to the boundary, and Harnack's inequality to the $p(x)$-Laplace equation were proved by Alkhutov, Krasheninnikova, and Surnachev. These results were generalized by Skrypnik and Voitovich. The qualitative properties of bounded solutions of $p(x)$-Laplace equation under the non-logarithmic condition were established by Skrypnik and Yevgenieva. As for unbounded solutions, there are just a few results. Ok has proved the boundedness of minimizers of elliptic functionals of the double-phase type under some assumptions on the growth parameters. The obtained condition gives a possibility to improve the regularity results for unbounded minimizers. The weak Harnack inequality for unbounded supersolutions of the corresponding elliptic equations with generalized Orlicz growth under the so-called logarithmic conditions was proved by Benyaiche, Harjulehto, H\"{a}st\"{o} and Karppinen. In the current paper, the weak Harnack inequality for unbounded solutions to the $p(x)$-Laplace equation has been proved under the precise non-logarithmic condition on the function $p(x)$.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"283 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-27DOI: 10.37069/1683-4720-2023-37-4
Volodymyr Ryazanov, Evgeny Sevost'yanov
In this article we continue to develop the theory of several moduli of families of surfaces, in particular, strings (open surfaces) of various dimensions in Euclidean spaces. Since the surfaces in question can be extremely fractal (wild), the natural basis for studying them is the so-called Hausdorff measures. As is known, these moduli are the main geometric tool in the mo-dern mapping theory and related topics in geometry, topology and the theory of partial differential equations with appropriate applications to the boundary-value problems of mathematical physics in anisotropic and inhomogeneous media. In addition, this theory can also find its further applications in many other fields, including mathematics itself (nonlinear dynamics, minimal surfaces), theoretical physics (conformal field theory, string theory), and engineering (mathematical models of the filtration of gases and fluids in underground mines of water, gas and oil seams, crystal growth and others). On the basis of the proof of Lemma~1 about the connections between moduli and the Lebesgue measures, we have proved the corresponding analogue of the Fubini theorem in the terms of the moduli that extends the known V"ais"al"a theorem for families of curves to families of surfaces of arbitrary dimensions. It is necessary to note specially here that the most refined place in the proof of Lemma~1 is Proposition~1 on measurable (Borel) hulls of sets in Euclidean spaces. We also prove here the corresponding Lemma~2 and Proposition~2 on families of centered spheres. Finally, in a similar way, suitable results can be also obtained for families of several spheroids.
{"title":"On the surfaces moduli theory","authors":"Volodymyr Ryazanov, Evgeny Sevost'yanov","doi":"10.37069/1683-4720-2023-37-4","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-4","url":null,"abstract":"In this article we continue to develop the theory of several moduli of families of surfaces, in particular, strings (open surfaces) of various dimensions in Euclidean spaces. Since the surfaces in question can be extremely fractal (wild), the natural basis for studying them is the so-called Hausdorff measures. As is known, these moduli are the main geometric tool in the mo-dern mapping theory and related topics in geometry, topology and the theory of partial differential equations with appropriate applications to the boundary-value problems of mathematical physics in anisotropic and inhomogeneous media. In addition, this theory can also find its further applications in many other fields, including mathematics itself (nonlinear dynamics, minimal surfaces), theoretical physics (conformal field theory, string theory), and engineering (mathematical models of the filtration of gases and fluids in underground mines of water, gas and oil seams, crystal growth and others). On the basis of the proof of Lemma~1 about the connections between moduli and the Lebesgue measures, we have proved the corresponding analogue of the Fubini theorem in the terms of the moduli that extends the known V\"ais\"al\"a theorem for families of curves to families of surfaces of arbitrary dimensions. It is necessary to note specially here that the most refined place in the proof of Lemma~1 is Proposition~1 on measurable (Borel) hulls of sets in Euclidean spaces. We also prove here the corresponding Lemma~2 and Proposition~2 on families of centered spheres. Finally, in a similar way, suitable results can be also obtained for families of several spheroids.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-27DOI: 10.37069/1683-4720-2023-37-6
Sergey Chuiko, Olga Nesmelova, Mykyta Popov
The problem of solvability of nonlinear boundary value problems originates from the classical theory of periodic boundary value problems for systems of ordinary differential equations, developed in the works of A. Poincare, O.M. Lyapunov, I.G. Malkin, Yu.O. Mitropolsky, A.M. Samoilenko, O.A. Boichuk and others. In the classical works of R. Bellman, J. Hale, Y.O. Mitropolsky, A.M. Samoilenko and O.A.~Boichuk, the conditions for solvability of nonlinear boundary value problems for systems of differential equations in critical cases were obtained. To find solutions to nonlinear boundary value problems for systems of differential equations in critical cases, iterative schemes using the method of simple iterations were constructed in the monographs of A.M. Samoilenko and O.A.~Boichuk. In the works of O.A. Boichuk and S.M. Chuiko, iterative schemes based on the Newton--Kantorovich scheme with quadratic convergence were constructed to find solutions to nonlinear boundary value problems, and constructive conditions for convergence were obtained. The technique for constructing approximations to solutions of weakly nonlinear boundary value problems using the Adomian de-com-po-si-tion method investigated in this paper differs from the authors' previous results in that the boundary condition, the number of components of which, in general, does not coincide with the dimension of the solution. The results obtained can be transferred to weakly nonlinear boundary value problems with a boundary condition using nonlinear bounded vector functions. The article obtains constructive conditions for solvability and a scheme for constructing solutions to a weakly nonlinear boundary value problem for an ordinary differential equation in the critical case using the Adomian decomposition method.
非线性边值问题的可解性问题起源于常微分方程系统周期边值问题的经典理论，该理论在A. Poincare, O.M. Lyapunov, I.G. Malkin, yuu . o .等人的著作中得到发展。Mitropolsky,点Samoilenko, O.A. Boichuk和其他人。在R. Bellman, J. Hale, Y.O. Mitropolsky, A.M.Samoilenko和O.A.~Boichuk给出了微分方程组非线性边值问题在临界情况下可解的条件。为了求临界情况下微分方程组非线性边值问题的解，在A.M.的专著中采用简单迭代法构造了迭代格式萨莫伊连科和O.A.~博伊丘克。在O.A. Boichuk和S.M. Chuiko的著作中，构造了基于二次收敛的Newton—Kantorovich格式的迭代格式来求非线性边值问题的解，并得到了收敛的构造条件。本文研究的用Adomian de -com -po -si -tion方法构造弱非线性边值问题近似解的方法与作者以往的结果不同，它的边界条件，其分量的个数，通常与解的维数不一致。所得结果可转化为具有非线性有界向量函数边界条件的弱非线性边值问题。本文利用Adomian分解方法，得到了一类常微分方程弱非线性边值问题在临界情况下的可解性的构造条件和构造方案。
{"title":"Adomian decomposition method in the theory of weakly nonlinear boundary value problems","authors":"Sergey Chuiko, Olga Nesmelova, Mykyta Popov","doi":"10.37069/1683-4720-2023-37-6","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-6","url":null,"abstract":"The problem of solvability of nonlinear boundary value problems originates from the classical theory of periodic boundary value problems for systems of ordinary differential equations, developed in the works of A. Poincare, O.M. Lyapunov, I.G. Malkin, Yu.O. Mitropolsky, A.M. Samoilenko, O.A. Boichuk and others. In the classical works of R. Bellman, J. Hale, Y.O. Mitropolsky, A.M. Samoilenko and O.A.~Boichuk, the conditions for solvability of nonlinear boundary value problems for systems of differential equations in critical cases were obtained. To find solutions to nonlinear boundary value problems for systems of differential equations in critical cases, iterative schemes using the method of simple iterations were constructed in the monographs of A.M. Samoilenko and O.A.~Boichuk. In the works of O.A. Boichuk and S.M. Chuiko, iterative schemes based on the Newton--Kantorovich scheme with quadratic convergence were constructed to find solutions to nonlinear boundary value problems, and constructive conditions for convergence were obtained. The technique for constructing approximations to solutions of weakly nonlinear boundary value problems using the Adomian de-com-po-si-tion method investigated in this paper differs from the authors' previous results in that the boundary condition, the number of components of which, in general, does not coincide with the dimension of the solution. The results obtained can be transferred to weakly nonlinear boundary value problems with a boundary condition using nonlinear bounded vector functions. The article obtains constructive conditions for solvability and a scheme for constructing solutions to a weakly nonlinear boundary value problem for an ordinary differential equation in the critical case using the Adomian decomposition method.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-27DOI: 10.37069/1683-4720-2023-37-2
Volodymyr Shcherbak, Nadiya Zhogoleva
The problem of determining the external force, which is given by the harmonic function of time and acts on a self-oscillating system of general type (Lienard oscillator) is considered. A general method of asymptotic estimation of oscillator velocity and force unknown parameters is proposed. Such problems of estimating the frequency, amplitude, and phase of an external force acting on a mechanical system are reflected in a sufficient number of publications both in past and present times. The reason for this interest lies in the use of appropriate techniques in various theoretical and engineering disciplines, for example, for mechanical systems for converting the kinetic energy of vibrations, in the problems of vibration isolation of periodic components of noise through rotating mechanisms, to compensate for harmonic disturbances in automatic control algorithms, in adaptive filtering during signal processing, and so on. In principle, the least squares method, Fourier analysis, and Laplace Transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. Despite the relative simplicity of the problem of determining the frequency, amplitude and phase of vibrations, approaches to solving them use a rather complex apparatus of modern methods of Applied Mathematics. The aim of this paper is to extend the method of synthesis of invariant relations to the problem of determining the parameters of external influence on a mechanical system. To obtain asymptotic estimates of the coefficients of external force, the method of invariant relations developed in analytical mechanics is used. Method was intended, in particular, to search for partial solutions (dependencies between variables) in problems of dynamics of rigid body with a fixed point. Modification of this method to the problems of observation theory made it possible to synthesize additional connections between known and unknown quantities of the original system that arise during the movement of its extended dynamic model. The asymptotic convergence of estimates of unknowns to their true value is proved. The results of numerical modeling of the asymptotic estimation process of oscillator velocity and external force parameters for the mathematical pendulum model are presented.
{"title":"Identification of external harmonic force parameters","authors":"Volodymyr Shcherbak, Nadiya Zhogoleva","doi":"10.37069/1683-4720-2023-37-2","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-2","url":null,"abstract":"The problem of determining the external force, which is given by the harmonic function of time and acts on a self-oscillating system of general type (Lienard oscillator) is considered. A general method of asymptotic estimation of oscillator velocity and force unknown parameters is proposed. Such problems of estimating the frequency, amplitude, and phase of an external force acting on a mechanical system are reflected in a sufficient number of publications both in past and present times. The reason for this interest lies in the use of appropriate techniques in various theoretical and engineering disciplines, for example, for mechanical systems for converting the kinetic energy of vibrations, in the problems of vibration isolation of periodic components of noise through rotating mechanisms, to compensate for harmonic disturbances in automatic control algorithms, in adaptive filtering during signal processing, and so on. In principle, the least squares method, Fourier analysis, and Laplace Transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. Despite the relative simplicity of the problem of determining the frequency, amplitude and phase of vibrations, approaches to solving them use a rather complex apparatus of modern methods of Applied Mathematics. The aim of this paper is to extend the method of synthesis of invariant relations to the problem of determining the parameters of external influence on a mechanical system. To obtain asymptotic estimates of the coefficients of external force, the method of invariant relations developed in analytical mechanics is used. Method was intended, in particular, to search for partial solutions (dependencies between variables) in problems of dynamics of rigid body with a fixed point. Modification of this method to the problems of observation theory made it possible to synthesize additional connections between known and unknown quantities of the original system that arise during the movement of its extended dynamic model. The asymptotic convergence of estimates of unknowns to their true value is proved. The results of numerical modeling of the asymptotic estimation process of oscillator velocity and external force parameters for the mathematical pendulum model are presented.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"53 31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-27DOI: 10.37069/1683-4720-2023-37-3
Mykola Krasnoshchok
We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_mu(s_theta$ under some conditions on weight $mu$, higher order of derivatives $k$ and the opening of the angle $theta$.
{"title":"Dirihlet-Ventcel bounsdary problem for Laplace equation in an unbounded sector","authors":"Mykola Krasnoshchok","doi":"10.37069/1683-4720-2023-37-3","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-3","url":null,"abstract":"We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_mu(s_theta$ under some conditions on weight $mu$, higher order of derivatives $k$ and the opening of the angle $theta$.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-27DOI: 10.37069/10.37069/1683-4720-2023-37-1
Oleksandr Dovhopiaty
This article is devoted to the possibility of joining of two pairs of points of a convex domains by curves. We are interested in the case when these curves are not farther from each other than the distance between their end points, possibly, up to some absolute multiplicative constant. We have obtained some upper and lower bounds for the modulus of families of paths joining curves mentioned above.
{"title":"On the possibility of joining two pairs of points in convex domains using paths","authors":"Oleksandr Dovhopiaty","doi":"10.37069/10.37069/1683-4720-2023-37-1","DOIUrl":"https://doi.org/10.37069/10.37069/1683-4720-2023-37-1","url":null,"abstract":"This article is devoted to the possibility of joining of two pairs of points of a convex domains by curves. We are interested in the case when these curves are not farther from each other than the distance between their end points, possibly, up to some absolute multiplicative constant. We have obtained some upper and lower bounds for the modulus of families of paths joining curves mentioned above.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-27DOI: 10.37069/1683-4720-2022-36-07
Yevhen Zozulia
For the weighted parabolic equation vleft(x right)u_{t} -{hbox{div}({w(x)| nabla u |^{p-2}}} nabla u) = f , p >{2} we prove the local boundedness for weak solutions in terms of the weighted Wolff potential of the right-hand side of equation.
对于加权抛物方程vleft (x right){u＿t} - {hbox{div} (w(x)|{nabla u |^p{-2}}}nabla u) = f, p ＆gt;{2}，用方程右侧的加权Wolff势证明了弱解的局部有界性。
{"title":"Pointwise estimates of solutions to weighted parabolic p-Laplacian equation via Wolff potential","authors":"Yevhen Zozulia","doi":"10.37069/1683-4720-2022-36-07","DOIUrl":"https://doi.org/10.37069/1683-4720-2022-36-07","url":null,"abstract":"For the weighted parabolic equation vleft(x right)u_{t} -{hbox{div}({w(x)| nabla u |^{p-2}}} nabla u) = f , p >{2} we prove the local boundedness for weak solutions in terms of the weighted Wolff potential of the right-hand side of equation.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-27DOI: 10.37069/1683-4720-2022-36-06
Nadiya Zhogoleva, Volodymyr Shcherbak
Many applied control problems are characterized by a situation where some or all parameters of the initial dynamic system are unknown. In such cases, the problem of identification arises, which consists in determining the unknown parameters of the system based on information about its output - known information about movement. The ability to solve the problem of identification is an essential property of identifiability depends on the analytical structure of the right-hand sides of the dynamics equations and available information [1]. To solve the identification problem itself, this work uses the method of invariant relations [2], which was developed in analytical mechanics and is intended, in particular, for finding partial solutions (dependencies between variables) in problems of the dynamics of a rigid body with a fixed point. The modification of this method to the problems of the theory of control, observation made it possible to synthesize additional connections between the known and unknown quantities of the original system that arise during the movement of its extended model [3 - 5]. It is worth noting that a some more general approach, which forms a suitable method for solving observation problems for nonlinear dynamic systems due to the synthesis of an invariant manifold in the space of an extended system, was proposed in the works [6], [7] as a certain modification of the method stabilization of nonlinear systems I&I (Input and Invariance). The purpose of this work is to spread the method of synthesis of invariant relations in control problems to the problem of identifying parameters of pendulum systems. A general scheme for constructing asymptotically accurate estimates of the parmeters of a two-dimensional dynamical system is proposed. A relatively simple case of the identification problem will be considered, namely: 1) the output of the original system is the complete phase vector and 2) the system depends linearly on the unknown parameters. Generalizations to more general designs of input-output systems, including with the involvement of information about the output obtained on several trajectories, can be carried out using the approach described below and is the subject of a separate study. The computational experiment on the estimation of the parameters of the mathematical pendulum confirms the efficiency of the proposed identification scheme.
许多应用控制问题的特点是初始动态系统的部分或全部参数未知。在这种情况下，识别问题就出现了，这包括根据关于其输出的信息(关于运动的已知信息)确定系统的未知参数。求解辨识问题的能力是可辨识性的基本属性，这取决于动力学方程右侧的解析结构和可用信息[1]。为了解决识别问题本身，这项工作使用了不变关系方法[2]，该方法是在分析力学中发展起来的，特别是用于寻找具有固定点的刚体动力学问题的部分解(变量之间的依赖关系)。这种方法对控制理论问题的修正，观察使得在扩展模型运动过程中产生的原始系统的已知和未知量之间的附加联系成为可能[3 - 5]。值得注意的是，文献[6]、[7]中提出了一种更一般的方法，它是对非线性系统稳定化方法I&I (Input and Invariance)的某种修正，它形成了求解扩展系统空间中不变量流形的非线性动态系统观测问题的合适方法。本文的目的是将控制问题中不变量关系的综合方法推广到摆系统参数辨识问题中。提出了构造二维动力系统参数渐近精确估计的一般格式。本文将考虑一种相对简单的辨识问题，即:1)原始系统的输出是完整的相位矢量，2)系统线性依赖于未知参数。推广到更一般的输入输出系统的设计，包括涉及在几个轨迹上获得的输出信息，可以使用下面描述的方法进行，这是一个单独研究的主题。数学摆参数估计的计算实验验证了所提辨识方案的有效性。
{"title":"Identification of parameters of non-linear oscillators","authors":"Nadiya Zhogoleva, Volodymyr Shcherbak","doi":"10.37069/1683-4720-2022-36-06","DOIUrl":"https://doi.org/10.37069/1683-4720-2022-36-06","url":null,"abstract":"Many applied control problems are characterized by a situation where some or all parameters of the initial dynamic system are unknown. In such cases, the problem of identification arises, which consists in determining the unknown parameters of the system based on information about its output - known information about movement. The ability to solve the problem of identification is an essential property of identifiability depends on the analytical structure of the right-hand sides of the dynamics equations and available information [1]. To solve the identification problem itself, this work uses the method of invariant relations [2], which was developed in analytical mechanics and is intended, in particular, for finding partial solutions (dependencies between variables) in problems of the dynamics of a rigid body with a fixed point. The modification of this method to the problems of the theory of control, observation made it possible to synthesize additional connections between the known and unknown quantities of the original system that arise during the movement of its extended model [3 - 5]. It is worth noting that a some more general approach, which forms a suitable method for solving observation problems for nonlinear dynamic systems due to the synthesis of an invariant manifold in the space of an extended system, was proposed in the works [6], [7] as a certain modification of the method stabilization of nonlinear systems I&I (Input and Invariance). The purpose of this work is to spread the method of synthesis of invariant relations in control problems to the problem of identifying parameters of pendulum systems. A general scheme for constructing asymptotically accurate estimates of the parmeters of a two-dimensional dynamical system is proposed. A relatively simple case of the identification problem will be considered, namely: 1) the output of the original system is the complete phase vector and 2) the system depends linearly on the unknown parameters. Generalizations to more general designs of input-output systems, including with the involvement of information about the output obtained on several trajectories, can be carried out using the approach described below and is the subject of a separate study. The computational experiment on the estimation of the parameters of the mathematical pendulum confirms the efficiency of the proposed identification scheme.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"205 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-27DOI: 10.37069/1683-4720-2022-36-08
Yuriy Kononov, Akram Cheib
Assuming that the center of mass of an asymmetric rigid body is located on the third main axis of inertia of a rigid body, the influence of dissipative asymmetry on the stability of uniform rotation in a medium with resistance of a dynamically asymmetric rigid body is estimated. A rigid body rotates around a fixed point, is under the action of gravity, dissipative moment and constant moment in an inertial frame of reference. The stability conditions are represented by a system of three inequalities. The first and second inequalities have the first degree with respect to the dissipative asymmetry, and the third inequality has the third degree. The third inequality is the most difficult to study. Analytical studies of the influence of small and large dissipative asymmetries, restoring, overturning and constant moments on the stability of rotation of a rigid body are carried out. Conditions for asymptotic stability are obtained for sufficiently small values of the dissipative asymmetry and conditions for instability for sufficiently large values of the asymmetry. The stability conditions are written down to the second order of smallness with respect to the constant moment and the first - with respect to the restoring or overturning moments. Stability conditions for the rotation of a rigid body around the center of mass are studied.
{"title":"Influence of dissipative asymmetry on the of rotation stability in a resisting medium of a asymmetric rigid body under the action of a constant moment in inertial reference frame","authors":"Yuriy Kononov, Akram Cheib","doi":"10.37069/1683-4720-2022-36-08","DOIUrl":"https://doi.org/10.37069/1683-4720-2022-36-08","url":null,"abstract":"Assuming that the center of mass of an asymmetric rigid body is located on the third main axis of inertia of a rigid body, the influence of dissipative asymmetry on the stability of uniform rotation in a medium with resistance of a dynamically asymmetric rigid body is estimated. A rigid body rotates around a fixed point, is under the action of gravity, dissipative moment and constant moment in an inertial frame of reference. The stability conditions are represented by a system of three inequalities. The first and second inequalities have the first degree with respect to the dissipative asymmetry, and the third inequality has the third degree. The third inequality is the most difficult to study. Analytical studies of the influence of small and large dissipative asymmetries, restoring, overturning and constant moments on the stability of rotation of a rigid body are carried out. Conditions for asymptotic stability are obtained for sufficiently small values of the dissipative asymmetry and conditions for instability for sufficiently large values of the asymmetry. The stability conditions are written down to the second order of smallness with respect to the constant moment and the first - with respect to the restoring or overturning moments. Stability conditions for the rotation of a rigid body around the center of mass are studied.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-27DOI: 10.37069/1683-4720-2022-36-11
Volodymyr Shcherbak
This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9], [10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].
本文采用解析力学[8]中提出的不变关系理论，并在[9]、[10]中进一步研究(参见[11])，研究了正弦信号的幅、频、相位渐近估计问题。正弦信号的频率、幅度和相位估计问题在过去和现在的文献中都引起了极大的研究关注。这种兴趣的原因依赖于几个工程应用，在这些应用中，对该问题的有效和健壮的解决方案至关重要。值得一提的是，自动控制中的谐波干扰补偿、通信中的视相环路设计、信号处理中的自适应滤波等问题。原则上，最小二乘法、傅立叶分析、拉普拉斯变换为相应问题提供了一种潜在的解决方案。然而，这些方法可能不适合，例如，具有实时数据处理的控制算法。本文的目标是通过展示如何通过观察者的理论解决手头的问题，为这项任务提出进一步的贡献。采用不变关系法设计渐近观测方案。该方法基于对原系统的动态扩展和构造适当的不变关系，将未知变量表示为扩展系统轨迹上已知量的函数。最后在得到未知参数渐近估计的条件下进行综合。结果表明，在扩展状态空间中，通过适当选取一个吸引的不变流形，可以得到未知状态的渐近估计。证明了所寻相矢量分量的估计对其真值的渐近收敛性。仿真结果验证了该方法解决谐振子状态观测问题的有效性。值得注意的是，在文献[12,13]中提出的I&I方法(Input and Invariance)的改进，形成了一种更一般的方法，该方法由于不变量流形的综合而形成了求解非线性动力系统观测问题的合适方法。
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