Abstract
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
Highlights
The foundations of theoretical researches and practical development in the field of distributed-parameter systems were first laid down in [1] more than half a century ago
The main goal of this study is the approximate solution of the optimal control problem for systems, the processes in which are described by rather general nonlinear boundary value problems of parabolic type in combination with the Cauchy problem for ordinary differential equations
The main difference between the considered problem and the previously presented ones is that in order to prove the convergence of the approximate solution, at least in terms of the functional, this paper shows, first of all, the convergence of the approximating boundary value problem to the solution of the original one
Summary
The foundations of theoretical researches and practical development in the field of distributed-parameter systems were first laid down in [1] more than half a century ago. Over the years, control theory for distributed-parameter systems has been enriched with new ideas and results. Many important questions of the theory are not fully developed for optimal control problems for systems containing links with distributed parameters, the processes in which are described by boundary value problems for partial differential equations. Optimal control theory is one of the main areas of practical use of mathematics. The rapid development of control theory for lumped-parameter systems is largely associated with the use of Pontryagin’s maximum principle, Bellman’s optimality principle, and Krasovsky’s method of moments. Many real control objects have to be considered as distributed-parameter systems. The variety of spheres of application of the theory of distributed systems control, its methods and results is evidenced by its close connection with technical problems, with game theory and problems of positional control, with inverse problems of the dynamics of controlled systems
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