Analysis of one class of optimal control problems for distributed-parameter systems

In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

Solving the problem of optimal control of fog diffusion

Solving the problem of optimal control of fog diffusion / Klyomin V.V., Suvorov S.S. // Hydrometeorological Research and Forecasting, 2021, no. 2 (380), pp. 52-65. The paper discusses a possibility of applying one of the fundamental modern optimization methods, namely, the Pontryagin’s method for solving process control problems, whose behavior is described by the diffusion equation. The parabolic diffusion equation is discretized by the method of straight lines and comes to a closed system of ordinary differential equations, which allow finding an optimal control impact in terms of operating speed. The existence of a solution to the problem of optimal control of fog diffusion is proved for the mentioned sampling. The methodology for finding control action switching points is substantiated, the calculations for the revealed two and three switching moments are performed. Keywords: Pontryagin’s method, fog diffusion control, diffusion equation

EFFICIENT ALGORITHMS FOR PARALLELIZING TRIDIAGONAL SYSTEMS OF EQUATIONS

The article is devoted to the development of the maximal parallel forms of mathematical models with a tridiagonal structure. The example of solving the Dirichlet and Neumann problems by the method of straight lines and the sweep method for the heat equation illustrates the direct fundamental features of constructing parallel algorithms. It is noted that the study of the heat and mass transfer processes is run through their numerical modeling based on modern computer technology.It is shown that with the multiprocessor computing systems’ development, there disappear the problems of increasing their peak performance. On the other hand, building such systems, as a rule, requires standard network technologies, mass-produced processors, and free software. The noted circumstances aim at solving the so-called big problems.It should be borne in mind that the classical approach to solving the tridiagonal structure models based on multiprocessor computing systems is far more time-consuming compared to single-processor computing facilities. That is explained by the recurrence relations that make the basis of classical methods. Therefore, the proposed studies are relevant and aim at the distributed algorithms development for solving applied problems.The proposed research aims to construct the maximal parallel forms of mathematical models with a tridiagonal structure.The paper proposes the schemes to implement parallelization algorithms for applied problems and their mapping to parallel computing systems.Parallelization of tridiagonal mathematical models by the method of straight lines and the sweeping method allows designing absolutely stable algorithms with the maximum parallel form and, therefore, the minimum possible time for their implementation on parallel computing devices. It is noteworthy that in the proposed algorithms, the computational errors of the input data are separated from the round-off errors inherent in a PC.The proposed approach can be used in various branches of metallurgical, thermal physics, economics, and ecology problems in the metallurgical industry.