Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

The behavior of the generalized solution of the initial-boundary value problem for the nonlinear parabolic equation

Daryna Shevchuk — 2024-05-23

Within the framework of this work, we study the behavior of the generalized solution (or the so-called energy solution) of an interesting initial-boundary value problem (namely, the Cauchy-Dirichlet problem is considered) for nonlinear parabolic equation. The research is carried out in a cylindrical area. A structural condition is imposed on the parameters of the equation corresponding to the slow diffusion process. So, in the article we are dealing with the distribution of the substance concentration in space and time, taking into account the initial and boundary conditions. This process has a practical aspect and is used in physics and engineering, for example, to study the diffusion of matter in environments with variable concentration or chemical influence. Solving such problems allows obtaining important data on the evolution of the system and predicting its behavior in various conditions. In the work, as a result of our research, several integral ratios, various estimates and inequalities were established, which lead to the need to analyze the behavior of the differential system, which in turn makes it possible to establish the presence of the solution localization property. So, relying on the well-known results regarding the behavior of the solution of the resulting differential system, it is possible to find a condition that guarantees the localization of the solution carrier for the Cauchy-Dirichlet problem under study. The main result of the work is a theorem that is proved for an arbitrary finite initial function and under the condition of fulfilling a certain restriction on the limit mode. The article has a fairly standard structure and, in addition to annotations and literature, contains the following structural elements: introduction; Formulation of the problem; basic definitions; formulation of the main result; auxiliary inequalities for proving the main result; auxiliary statements for proving the theorem; proving the main result; conclusions.

Adaptive dynamic programming for the optimal liver regeneration control

Valeriia Karieva — 2024-06-10

Every living organism interacts with an environment and uses that interaction for an improvement of its own adaptability, and, as a result, one’s survival and overall development. The process of evolution shows us that different species change methods of interaction with an environment with passage of time, which leads to natural selection and survival of the most adaptive ones. This learning, which based on actions, or reinforcement learning may embrace the idea of optimal behavior occurring in environmental systems. We describe mathematical formulas for reinforcement learning and the practical integration method also known as adaptive dynamic programming. That gives us the overall concept of controllers for artificial biological systems that both learn and show the optimal behavior.

This paper reviews the formulation of the upper optimality problem, for which the optimal regulation strategy is guaranteed to be better or equivalent to objective regulation rules that can be observed in natural biological systems.

In cases of optimal reinforcement learning algorithms the learning process itself moves from the analysis of the item take on system dynamics to the much higher level. The object of interest now is not the details of the system dynamics, but the quantity efficiency index, which clearly represents how optimally the control system works. Such scheme of reinforcement learning is learning technique of optimal behavior in order to monitor the response to non-optimal control strategies.

The purpose of this article is to show the possibility of using of reinforcement learning methods, the adaptive dynamic programming (ADP) in particular, to control biological systems using feedback. This article shows the on-line methods for solving the problem of searching the upper optimality estimate with adaptive dynamic programming.

Homogeneous approximations of nonlinear control systems with output and weak algebraic equivalence

Daria Andreieva — 2024-05-24

In the paper, we consider nonlinear control systems that are linear with respect to controls with output; vector fields defining the system and the output are supposed to be real analytic. Following the algebraic approach, we consider series $S$ of iterated integrals corresponding to such systems. Iterated integrals form a free associative algebra, and all our constructions use its properties. First, we consider the set of all (formal) functions of such series $f(S)$ and define the set $N_S$ of terms of minimal order for all such functions. We introduce the definition of the maximal graded Lie generated left ideal ${\mathcal J}_S^{\rm max}$ which is orthogonal to the set $N_S$. We describe the relation between this maximal left ideal and the left ideal ${\mathcal J}_S$ generated by the core Lie subalgebra of the system which realizes the series. Namely, we show that ${\mathcal J}_S\subset {\mathcal J}_S^{\rm max}$. In particular, this implies that the graded Lie subalgebra that generates the left ideal ${\mathcal J}_S^{\rm max}$ has a finite codimension. Also, we give the algorithm which reduces the series $S$ to the triangular form and propose the definition of the homogeneous approximation for the series $S$. Namely, homogeneous approximation is a homogeneous series with components that are terms of minimal order in each component of this triangular form. We prove that the set $N_S$ coincides with the set of all shuffle polynomials of components of a homogeneous approximation. Unlike the case when the output is identical, the homogeneous approximation is not completely defined by the ideal ${\mathcal J}_S^{\rm max}$. In order to describe this property, we introduce two different concepts of equivalence of series: algebraic equivalence (when two series have the same homogeneous approximation) and weak algebraic equivalence (when two series have the same maximal left ideal and therefore have the same minimal realizing system). We prove that if two series are algebraically equivalent, then they are weakly algebraically equivalent. The examples show that in general the converse is not true.

Well-posedness and parabolicity of the boundary-value problem for systems of partial differential equations

Alexander Makarov — 2024-06-03

The paper examines two-point boundary value problem for systems of linear partial differential equations. Not every system has a well posed two-point boundary value problem. For example, for equation

{$$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial x_1}+i \frac{\partial u(x_1,x_2,t)}{\partial x_2}$$

there are no boundary conditions of the form $au(x_1,x_1,0)+bu(x_1,x_2,T)=\varphi(x_1,x_2)$, under which this boundary value problem will be of a well posed in the Schwartz spaces.

In the work, the conditions for the matrix of the system under which exist well posed boundary-value problem are found in the Schwartz spaces, and the form of these boundary conditions is also indicated. So, for systems with a Hermitian matrix, the boundary value problem with conditions of form

$$u(x,0)+u(x,T)=\varphi(x)$$

will always be of a well posed in the Schwartz spaces, as well as in the spaces of functions of finite smoothness of power growth. For systems with one spatial variable, it is proved that well posed boundary value problems with the condition $u(x,0)+bu(x,T)=\varphi(x)$ ith positive b always exist. In addition, parabolic boundary value problems with property of increasing smoothness of the solutions are investigated.

Similar results hold for linear partial differential equations. The Helmholtz equation

$$\displaystyle\frac{\partial^2 u(x,t)}{\partial t^2}+\Delta u(x,t)=ku(x,t),$$

is considered as an example. It is not well posed according to Petrovsky, but for it exist a boundary value problem that is parabolic. For an equation with one spatial variable, sufficient conditions for the second-order equation in time are given, under which parabolic boundary value problems exist.

Anatoliy Vasylyovych Lutsenko (obituary)

Tetiana Smortsova — 2024-05-15