Hamiltonian Systems in Dynamic Reconstruction Problems

We consider dynamic reconstruction (DR) problems for controlled dynamical systems linear in controls and nonlinear in state variables as inaccurate current information about real motions is known. A solution of this on-line inverse problem is obtained with the help of auxiliary problems of calculus of variations (CV) for integral discrepancy functionals. Key elements of the constructions are solutions of hamiltonian systems obtained with the help of optimality conditions for the CV problems. An illustrating example is exposed. © 201817-01-00074, 18-01-00221The work is supported by Russin Foundation for Basic Research (project no. 17-01-00074 and no. 18-01-00221)

Construction of the Viability Set in a Problem of Chemotherapy of a Malignant Tumor Growing According to the Gompertz law

The problem of chemotherapy of a malignant tumor growing according to the Gompertz law is considered. The mathematical model is a system of two ordinary differential equations. We study a problem of optimal control (optimal therapy) aiming at the minimization of the malignant cells in the body at a given terminal time T. The viability set of this problem, i.e., the set of initial states of the model (the volume of the tumor and the amount of the drug in the body) for which an optimal control guarantees that the dynamics of the system up to the time T is compatible with life in terms of the volume of the tumor, is constructed analytically. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00362)

Generalized solutions of Hamilton-Jacobi equation to a molecular genetic model

A boundary value problem with state constraints is under consideration for a nonlinear noncoercive Hamilton-Jacobi equation. The problem arises in molecular biology for the Crow-Kimura model of genetic evolution. A new notion of continuous generalized solution to the problem is suggested. Connections with viscosity and minimax generalized solutions are discussed. In this paper the problem is studied for the case of additional requirements to structure of solutions. Constructions of the solutions with prescribed properties are provided and justified via dynamic programming and calculus of variations. Results of simulations are exposed. © IFIP International Federation for Information Processing 2016

The value function in a problem of chemotherapy of a malignant tumor growing according to the Gompertz law

In this paper a construction of the value function is obtained in the problem of chemotherapy of a malignant tumor growing according to the Gompertz law, when a therapy function has two maxima. The aim of therapy is to minimize the number of tumor cells at the given final instance. © 2018Russian Foundation for Basic Research, RFBR: 17-01-00074Russian Academy of Sciences, RAS: 18-1-1-10Russian Foundation for Basic Research, RFBR17-01-00074, PRAS-18-01This work was supported by the Russian Foundation for Basic Research (project no. 17-01-00074) and by the Program of the Presidium of the Russian Academy of Sciences 01 ’Fundamental Mathematics and its Applications’ under grant PRAS-18-01

Weak*approximations for the solution of a dynamic reconstruction problem

We consider the problem of the dynamic reconstruction of an observed state trajectory x() of an affine deterministic dynamic system and the control that has generated this trajectory. The reconstruction is based on current information about inaccurate discrete measurements of x(). A correct statement of the problem on the construction of approximations ul() of the normal control ul() generating x∗() is refined. The solution of this problem obtained using the variational approach proposed by the authors is discussed. Conditions on the input data and matching conditions for the approximation parameters (parameters of the accuracy and frequency of measurements of the trajectory and an auxiliary regularizing parameter) are given. Under these conditions, the reconstructed trajectories xl() of the dynamical system converge uniformly to the observed trajectory x∗() in the space of continuous functions C as l → ∞. It is proved that the proposed controls ul() converge weakly∗to u∗() in the space of summable functions L1. © 2021 Krasovskii Institute of Mathematics and Mechanics. All rights reserved

Derivational slots of parametric adjectives: Cognitive aspect

This paper realized a new, modern approach to the study of a substantive aspect of the studied linguistic structures: correlative derivational slots are considered in the cognitive aspect, and also from the point of view of a language consciousness of certain linguistic culture carriers - Russian or English one. The current actual interest to the content of the word-formation processes, the analysis of the derivational-semantic space of a language allow to obtain the information about some features of nominative-cognitive activity of a man, namely to specify the structure of knowledge embedded in a derivative word. The comparative analysis ofe Russian and English derivational slots of parametric adjectives allows you to determine the ways of development and modification concerning the dimensional characteristics of objects in the acts of word formation, as well as to identify the national specifics of the derivationalnominative space of each of the slots

Optimal Control Theory and Calculus of Variations in Mathematical Models of Chemotherapy of Malignant Tumors

This paper is devoted to the analysis of mathematical models of chemotherapy for malignant tumors growing according to the Gompertz law or the generalized logistic law. The influence of the therapeutic agent on the tumor dynamics is determined by a therapy function depending on the time-varying concentration of the drug in the patient’s body. The case of a non-monotonic therapy function with two maxima is studied. It reflects the use of two different therapeutic agents. The state variables of the dynamics are the tumor volume and the amount of the therapeutic agent able to suppress malignant cells (concentration of the drug in the body). The treatment protocol (the rate of administration of the therapeutic agent) is the control in the dynamics. The optimal control problem for this models is considered. It is the problem of the construction of treatment protocols that provide the minimal tumor volume at the end of the treatment. The solution of this problem was obtained by the authors in previous works via the optimal control theory. The form of the considered therapy functions provides a specific structure for the optimal controls. The managerial insights of this structure are discussed. In this paper, the structure of the viability set is described for the model according to the generalized logistic law. It is the set of the initial states of the model for which one can find a treatment protocol that guarantees that the tumor volume remains within the prescribed limits throughout the treatment. The description of the viability set’s structure is based on the optimal control theory and the theory of Hamilton–Jacobi equations. An inverse problem of therapy is also considered, namely the problem of reconstruction of the treatment protocol and identification of the unknown parameter of the intensity of the tumor growth. Reconstruction is carried out by processing information about the observations of the tumor volume dynamics and the measurements of the drug concentration in the body. A solution to this problem is obtained through the use of a method based on the calculus of variations. The results of the numerical simulations are presented herein. © 2023 by the authors.Ministry of Education and Science of the Russian Federation, Minobrnauka: 075-02-2023-913The contribution of Nina Subbotina and Evgenii Krupennikov was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2023-913)

Optimal synthesis to inverse problems of dynamics

Perturbed inverse problems are under consideration for dynamical systems linear relative controls. It is assumed that sampling history and sampling error estimate are known. Auxiliary optimal control problems are introduced to minimize a regularized integral discrepancy functional. The trajectories of the system are constructed with the help of Optimal Synthesis in the domain of admissible motions. It is proven that realizations of Optimal Synthesis generating trajectories of the system and minimizing the discrepancy functional in the domain are solutions of the perturbed inverse problem of dynamics. © IFAC.Russian Foundation for Basic Research, РФФИ: 14–01– 00486, 14–01–00168This work was supported by the Russian Foundation for Basic Researches (projects No. 14–01–00168, 14–01– 00486).Boje E.Xia X.International Federation of Automatic Contro

SAFE EDUCATIONAL ENVIRONMENT IN THE SYSTEM OFEDUCATION OF A. S. MAKARENKO

В статье выполнен анализ процесса формирования безопасной образовательной среды в системе воспитания А.С. Макаренко. Проблема безопасной образовательной среды становится востребованной в связи с происходящими изменениями в системе образования. Указывается важность внедрения программы безопасной образовательной среды в школеThe article analyzes the formation of a safe educational environment in the system of education of A.S. Makarenko. The problem of safe educational environment is becoming popular due to the ongoing changes in the education system. The importance of introduction of the program of safe educational environment at school is specifie