Pub Date : 2023-06-08DOI: 10.26565/2221-5646-2023-97-02
V. I. Korobov, T. V. Andriienko
This article is devoted to the controllability function method in admissible synthesis problems for linear canonical systems. The work considers methods of constructing such control so that the controllability function is time of motion of an arbitrary point to the origin. A canonical controlled system of linear equations $dot{x}_i=x_{i+1}, i=overline{1,n-1}, dot{x}_n=u$ with control constraints $|u| le d$ is considered. The controllability function $Theta$ can be found as the only positive solution of the implicit equation $2a_0Theta=(D(Theta)FD(Theta)x,x)$, where $D(Theta)= diag(Theta^ {-frac{-2n-2i+1}{2}})_{i=1}^n$. Matrix $F={f_{ij}}_{i,j=1}^n$ is positive definite and $a_0>0$ is chosen so that the control constraints are satisfied. The controllability function is motion time if $dot{Theta}= -1$. From this condition, an equation is obtained, the solution of which is considered in this work. Unlike previous works on this topic, no additional restrictions are imposed on the appearance of matrix $F$. The task of this article is to find the parameters set of the matrix $F$ and the column vector $a$, which satisfy the obtained equation and for which the controllability function is the time of movement from the point $x$ to the origin. In this way, we get a family of controls depending on this parameters such that the trajectory of system steers the origin in finite time. In general case, difficulties may arise when finding the solution of Cauchy problem of the corresponding system. Canonical system can be reduced to Euler's equation, for which a characteristic equation can be found, and therefore a trajectory in an explicit form. Two-dimensional, three-dimensional and four-dimensional canonical systems are considered. In each case, the matrix equation is solved and sets of parameters for which the controllability functions value will be the time of movement of an arbitrary point to the origin are found. Conditions on parameters are obtained from positive definiteness of the matrix $F$. Some parameters and an arbitrary initial point are chosen and the solution of Cauchy problem in analytical form is found.
本文研究了线性正则系统可容许综合问题的可控性函数方法。本文考虑了构造这种控制的方法，使可控性函数为任意点到原点的运动时间。考虑了一类具有控制约束$|u| le d$的正则控制线性方程组$dot{x}_i=x_{i+1}, i=overline{1,n-1}, dot{x}_n=u$。可控性函数$Theta$可以作为隐式方程$2a_0Theta=(D(Theta)FD(Theta)x,x)$的唯一正解，其中$D(Theta)= diag(Theta^ {-frac{-2n-2i+1}{2}})_{i=1}^n$。矩阵$F={f_{ij}}_{i,j=1}^n$为正定矩阵，选择$a_0>0$以满足控制约束。可控性功能是运动时间，如果$dot{Theta}= -1$。在此条件下，得到了一个方程，本文研究了该方程的解。与此主题的先前作品不同，没有对矩阵$F$的外观施加额外的限制。本文的任务是求出矩阵$F$和列向量$a$的参数集，满足得到的方程，其可控性函数为从点$x$到原点的运动时间。通过这种方式，我们得到了一系列依赖于这些参数的控制，使得系统的轨迹在有限时间内转向原点。一般情况下，在求解相应系统的柯西问题时会遇到困难。正则系统可以简化为欧拉方程，欧拉方程可以找到特征方程，因此可以找到显式轨迹。二维、三维和四维正则系统被考虑。在每种情况下，都求解矩阵方程，并找到一组参数，其中可控性函数的值将是任意点到原点的运动时间。由矩阵$F$的正定性得到参数的条件。选取一些参数和任意起始点，得到柯西问题的解析解。
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Pub Date : 2023-06-07DOI: 10.26565/2221-5646-2023-97-04
V. V. Karieva, S.V. Lvov
This publication investigates one of the fundamental problems of mathematical biology, specifically the development of mathematical models for the dynamics of complex biosystems that have a satisfactory explanatory and predictable power. A necessary condition for the development of such models is to find a solution for the problem of identifying the objective principles and rules of regulation of the "cellular system", which determines among all the possibilities exactly the "real path" of its dynamics observed in the experiment. One of the promising approaches to solving this problem is based on the hypothesis that the regulation of processes for support/restoration of the dynamic homeostasis of tissues and organs of the body occurs according to certain principles, and criteria of optimality, which have developed due to the natural selection of the body during its previous evolution. It is quite difficult to solve this problem at the current time due to the many uncertainties in the paths of the previous evolution of the organism, the dynamics of changes in external conditions, as well as the high computational complexity of solving such a problem. Instead of this, we have proposed a simplified formulation of the problem of searching for regulation control strategies, which gives us an upper estimate of optimality for the processes of maintaining/restoring dynamic homeostasis of the liver. The upper estimate of the optimality of regulation and testing of hypotheses for the model of liver regeneration was considered in the case of partial hepatectomy and was solved by Python software methods. The result shows that in the case of partial hepatectomy, the liver regeneration strategies obtained in numerous experiments for the problem of the upper optimality estimate qualitatively coincide with the processes of liver regeneration that can be observed during biological experiments. In plenty of experiments following hypotheses were also tested: how significant is the contribution of the process of controlled apoptosis, and how other processes (polyploidy, division, and formation of binuclear hepatocytes) affect the strategy of liver regeneration.
{"title":"Liver regeneration after partial hepatectomy: the upper optimality estimate","authors":"V. V. Karieva, S.V. Lvov","doi":"10.26565/2221-5646-2023-97-04","DOIUrl":"https://doi.org/10.26565/2221-5646-2023-97-04","url":null,"abstract":"This publication investigates one of the fundamental problems of mathematical biology, specifically the development of mathematical models for the dynamics of complex biosystems that have a satisfactory explanatory and predictable power. A necessary condition for the development of such models is to find a solution for the problem of identifying the objective principles and rules of regulation of the \"cellular system\", which determines among all the possibilities exactly the \"real path\" of its dynamics observed in the experiment. One of the promising approaches to solving this problem is based on the hypothesis that the regulation of processes for support/restoration of the dynamic homeostasis of tissues and organs of the body occurs according to certain principles, and criteria of optimality, which have developed due to the natural selection of the body during its previous evolution. It is quite difficult to solve this problem at the current time due to the many uncertainties in the paths of the previous evolution of the organism, the dynamics of changes in external conditions, as well as the high computational complexity of solving such a problem. Instead of this, we have proposed a simplified formulation of the problem of searching for regulation control strategies, which gives us an upper estimate of optimality for the processes of maintaining/restoring dynamic homeostasis of the liver. The upper estimate of the optimality of regulation and testing of hypotheses for the model of liver regeneration was considered in the case of partial hepatectomy and was solved by Python software methods. The result shows that in the case of partial hepatectomy, the liver regeneration strategies obtained in numerous experiments for the problem of the upper optimality estimate qualitatively coincide with the processes of liver regeneration that can be observed during biological experiments. In plenty of experiments following hypotheses were also tested: how significant is the contribution of the process of controlled apoptosis, and how other processes (polyploidy, division, and formation of binuclear hepatocytes) affect the strategy of liver regeneration.","PeriodicalId":53016,"journal":{"name":"Visnik Harkivs''kogo Nacional''nogo Universitetu im VN Karazina Cepia Matematika Prikladna Matematika i Mehanika","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135493076","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 : 2023-05-16DOI: 10.26565/2221-5646-2023-97-01
O. G. Rovenska
The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.
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