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

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

Solution of one optimum control problem regarding the depletion of gas reservoir

“…However, in this case, the question of the convergence of an approximate solution of a boundary value problem is not always confirmed. Without this, it is impossible to prove the convergence of the approximate solution of the approximating optimal problem, at least in terms of the functional [3].…”

“…By introducing an additional variable, the problem is reduced to the problem of controlling processes described by a set of equations in partial and ordinary derivatives. In [3], one class of problems on processes' optimal control, described by a set of nonlinear equations in ordinary and partial derivatives, is studied by the method of straight lines. Convergence in functional is proved and a constructive scheme for constructing a minimizing sequence of controls is proposed.…”

“…The results of the numerical solution of one variational problem related to thermal processes are presented. However, due to the nonlinearity of boundary value problems in [2] and [3], it was not possible to prove the maximum principle, which gives the necessary optimality conditions.…”

“…The paper [1][2][3] developed the concept of fuzzy numbers and arithmetic operations on them, which was expanded in [4]. Later, the work [5] contributed significantly by developing the main thought of LR fuzzy numbers and then presented a computational formula toward fuzzy number operations.…”

Методи Розрахунку Параметрів Технічних Систем

“…Розроблені методи раціонального розбиття дискретно-неперервних точкових множин на підмножини [2,3]. В публікації [4] вдосконалені математичні моделі і методи для оптимізації систем з розподіленими параметрами. Побудові математичних моделей і вдосконаленню існуючих методів для розрахунку і оптимізації процесу термічної дії на листовий метал задля збільшення ефекивності розв'язання прикладних задач у металургії та машинобудуванні присвячені результати публікацій [5,6].…”