In this paper, on the base of Pontryagin maximum principle, the optimal control problem with concentrated parameters for a degenerate differential equation with the Caputo operator and with coefficients from the Lebesgue space is studied. The efficiency indicator of the considered optimal control problem has an integral form of fractional order. A new version of the method of increments is applied and the concept of a conjugate equation with an integral form is essentially used.

Introduction. The article presents the results of a comprehensive assessment of disease incidence among male workers of shipbuilding and ship repair industries in the Astrakhan Region for creation of an organizational and management model of a comprehensive assessment of their disease incidence at workplaces in order to develop adequate preventive measures that exclude adverse health effects of occupational hazards in male shipbuilders. Materials and methods. Data for analysis were extracted from medical histories of inpatients and medical records of outpatients of the Russian Southern District Medical Center for the period 2015–2017. We applied methods of an opinion poll, expert examinations, correlation and regression analysis, mathematical modeling, as well as software products. Results. We built analytical models to assess health effects of determinants in shipbuilders. We also established the main occupational dangers, risks of occupational diseases, and the correlation between the number of patients and adverse industrial factors. We found that the risk of occupational diseases (0.8) was the highest in men over 45 (31.9 %) exposed to occupational hazards for more than 8.5 years. The lowest risk (0.2) was estimated for men aged 27–39 (68.1 %) with up to 6.6 years of occupational exposure. Conclusion. The organizational and management model was developed for the comprehensive assessment of causal relationships between the duration of exposure of shipbuilders to occupational hazards, the age of the employee, the risk of developing an occupational disease, and adequate preventive measures eliminating health risks and reducing effects of adverse industrial determinants.

Aim. The purpose of the study: the analysis of morbidity of workers of the ship-building and ship-repair enterprises of the city of Astrakhan taking into account a quantitative estimation of influence of determinants. Material and methods. Sociological survey, methods of nonparametric statistics and mathematical modeling. For a detailed assessment of the influence of risk factors on the health of workers in shipbuilding and ship repair enterprises, an analysis of risk factors was conducted. As a tool for assessing the effectiveness of a determinant on the health of shipbuilders using a predicative analysis and predictive model of a mathematical assessment of the development of medical, social and production determinants on the health of workers. Results and its discussion. According to the results of the study, the most significant production hazards were revealed, as well as the statistically significant effect of the duration of work in conditions of industrial hazards on the incidence of male workers in shipbuilding and ship repair enterprises in the city of Astrakhan.The results of the study testified to the high level of medical and social activity of workers in shipbuilding and ship repair enterprises in contact with industrial hazards and identified the reserves for its further growth in the framework of the continuous introduction of prevention programs. The revealed correlation between the frequency of registration of the pathology of shipbuilders and ship repairmen and their work experience has made it necessary to construct a predictive model with the identification of the determinants of morbidity. And the construction of a predictive model on the basis of actual data provided an opportunity to manage these determinants without further monitoring and forecasting. The conclusion. The reduction of the risk of occurrence of industrial hazards associated with industrial hazards in shipbuilding and ship repair workers is promoted by annual medical and preventive measures, timely diagnostics and qualitative medical examination, as well as improvement of working conditions for employees of the enterprise.

In this paper, a necessary and sufficient condition, such as the Pontryagin’s maxi-mum principle for a fractional optimal control problem with concentrated parameters, is given by the ordinary fractional differential equation with a coefficient in weighted Lebesgue spaces. We discuss a formulation of fractional optimal control problems by a fractional differential equation in the sense of Caputo fractional derivative. The statement of the fractional optimal control problem is studied by using a new version of the increment method that essentially uses the concept of an adjoint equation of the integral form.

For a fourth-order generalized Mangeron equation with nonsmooth coefficients defined on a rectangular domain, we consider the Neumann problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions to classical boundary conditions for the case in which the solution to the problem is sought in an isotropic Sobolev space. The problem is solved by reduction to a system of integral equations whose well-posed solvability is established based on the method of integral representations. The well-posed solvability of the Neumann problem for the generalized Mangeron equation is proved by the method of operator equations.

Analysis of morbidity of shipbuilders and ship repairers exposed to harmful industrial factors allows to assess their health problems, the degree of the industrial impact o on the level of health, to detect groups of people with an increased risk of developing occupational pathology, to work out effective prevention strategies that ensure the preservation of health of the population.

Aim. The purpose of the study: the analysis of morbidity of workers of the ship-building and ship-repair enterprises of the city of Astrakhan taking into account a quantitative estimation of influence of determinants. Material and methods. Sociological survey, methods of nonparametric statistics and mathematical modeling. For a detailed assessment of the influence of risk factors on the health of workers in shipbuilding and ship repair enterprises, an analysis of risk factors was conducted. As a tool for assessing the effectiveness of a determinant on the health of shipbuilders using a predicative analysis and predictive model of a mathematical assessment of the development of medical, social and production determinants on the health of workers. Results and its discussion. According to the results of the study, the most significant production hazards were revealed, as well as the statistically significant effect of the duration of work in conditions of industrial hazards on the incidence of male workers in shipbuilding and ship repair enterprises in the city of Astrakhan.The results of the study testified to the high level of medical and social activity of workers in shipbuilding and ship repair enterprises in contact with industrial hazards and identified the reserves for its further growth in the framework of the continuous introduction of prevention programs. The revealed correlation between the frequency of registration of the pathology of shipbuilders and ship repairmen and their work experience has made it necessary to construct a predictive model with the identification of the determinants of morbidity. And the construction of a predictive model on the basis of actual data provided an opportunity to manage these determinants without further monitoring and forecasting. The conclusion. The reduction of the risk of occurrence of industrial hazards associated with industrial hazards in shipbuilding and ship repair workers is promoted by annual medical and preventive measures, timely diagnostics and qualitative medical examination, as well as improvement of working conditions for employees of the enterprise.

In this paper a necessary and sufficient condition, such as the Pontryagin's maximum principle for an optimal control problem with distributed parameters, is given by a hyperbolic equation of the second order with L p ( x ) -coefficients. The results can be used in the theory of optimal processes for distribution Pontryagin maximum principle for various controlled processes described by hyperbolic equations of second order with discontinuous coefficients in variable exponent Sobolev spaces with dominant mixed derivatives.

В данной статье выявлен гомеоморфизм между определенными парами банаховых пространств при исследовании четырехмерной задачи Гурса для одного дифференциального уравнения со старшей частной производной шестого порядка $D_{1} D_{2} D_{3}^{2} _{4}^{2} u(x)$ с разрывными коэффициентами $L_p$-коэффициентами) путем сведения этой задачи к эквивалентному интегральному уравнению.

В работе рассматривается краевая задача для системы псевдопараболических уравнений четвертого порядка с разрывными коэффициентами и условиями Бицадзе - Самарского и Самарского - Ионкина. Найдено интегральное представление функции в пространстве Соболева, которое позволяет однозначно восстановить ее посредством значений некоторых (определяющих) операторов, принимаемых на этой функции. Также, дается постановка задачи Гурса с неклассическими краевыми условиями.

For a fourth-order pseudoparabolic equation with nonsmooth coefficients in a rectangular domain, we consider the Dirichlet problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions and the classical boundary conditions for the case in which the solution of the problem is sought in a Sobolev space.

In the present paper, we study the Goursat problem for a three-dimensional equation with highest derivative of fifth order with L p -coefficients and establish a homeomorphism between certain pairs of Banach spaces by reducing this problem to the equivalent Volterra integral equation.

In this paper we consider a hyperbolic-type differential equation with L p -coefficients in a three-dimensional space. For this equation we study the Goursat problem with nonclassical boundary constraints not requiringmatched conditions. We prove the equivalence of these boundary conditions to classical ones in the case when one seeks for a solution to the stated problem in an anisotropic space introduced by S. L. Sobolev. In addition, we prove the correct solvability of the Goursat problem by the method of integral equations.

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.

It is known that different conditions can be found for correct solvability of the non-local problem. In this sense, in the paper we find correct solvability conditions of the non-local problem with loaded boundary conditions in the integral form for a fourth order pseudoparabolic equation.