Numerical solution of problems of optimal processes with distributed parameters

Leonchuk, M.P.

doi:10.1016/0041-5553(64)90091-6

Cited by 7 publications

(4 citation statements)

References 3 publications

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“…As can be seen, to obtain the gradient of functional (12) for given Q(t) and α, we have to solve two boundary value problems. First, from ( 9)-( 11) it is necessary to determine the function p(x, t) then put the resulting p(x, t) into ( 16)- (18), and from ( 16)-(18) find ψ(x, t) and, finally, ψ(0, t).…”

Section: T L P X T P X P X T X T P P P X T a P P X T T Xmentioning

confidence: 99%

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Numerical solution of the control problem on the depletion of gas reservoirs with weakly permeable top

Mamtiyev

Rzayeva

Abdulova

2023

EEJET

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The current stage in the development of mathematical and software support for the processes of designing the development of hydrocarbon fields is characterized not only by the improvement of the means of geological and hydrodynamic modeling of reservoir fluid filtration but also by the use of algorithms for optimizing the development of gas deposits. The paper considers the problem of optimal control of the depletion of a gas reservoir with a low-permeability top. Using the so-called Myatiev-Girinsky hydraulic scheme, a two-dimensional equation describing the unsteady gas flow in a reservoir with a jumper is averaged over the capacity of the productive reservoir. This comes down to a one-dimensional equation with an additional term, taking into account gas-dynamic relationships between the reservoir and the jumper. For the numerical solution of process control problems, a formula for the gradient of the functional characterizing the reservoir depletion is found, and the method of successive approximations based on Pontryagin’s maximum principle is applied. In this case, the direct and conjugate boundary value problems are solved by the method of straight lines, and the required flow rate, without taking it beyond the maximum and minimum possible, is found by the gradient projection method with a special choice of step. A brief block diagram of the algorithm for solving the problem is shown; on its basis, a computer program was compiled. The results of calculations are presented to identify the influence of the values of the complex communication parameter not only on the state of the object but also on the operating mode of the well. The expediency of using the presented optimization tool is dictated by an increase in the share of deposits

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Section: T L P X T P X P X T X T P P P X T a P P X T T Xmentioning

confidence: 99%

“…where k is an iteration number, and the parameter λ>0 is chosen depending on the change in the sign of the functions 1 ( ) k y t during iterations [18]. That is, if ψ 1 (t) does not change sign during iterations, then can be increased to speed up convergence.…”

Section: Developing An Algorithm and Carrying Out Software Implementa...mentioning

confidence: 99%

Numerical solution of the control problem on the depletion of gas reservoirs with weakly permeable top

Mamtiyev

Rzayeva

Abdulova

2023

EEJET

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“…In this regard, when solving practically important abovementioned types of optimal control problems using computational tools, it is very effective to use various approximate methods, in particular, the method of straight lines. This approach was used in [11], where, when approximating the heat conduction equations in phase variables, the problem associated with the choice of lumped, starting and distributed controls was reduced to solving a variational problem for systems of ordinary differential equations. In this paper and in the papers [6,8,10], the authors restricted themselves to indicating a method of approximate realization of the optimal control, while the convergence of the approximate solution was not proved.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Note that in the process of solving the problem (28)-(30), it turned out that ∂ ∂ £ H β 0, therefore, to avoid looping, as ∂ ∂ H 0 β in formula (36), the minimum element of the array ∂ ∂ H β was taken by its absolute value for 0 £ £ t T [11].…”

Section: Fig 2 Approximately Optimal Controlmentioning

confidence: 99%

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

Mamtiyev

Aliyeva

Rzayeva

2021

EEJET

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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

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A direct method for synthesis of optimum distributed systems

El-Fattah

Ghonaimy

1973

Calcolo

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Numerical solution of problems of optimal processes with distributed parameters

Cited by 7 publications

References 3 publications

Numerical solution of the control problem on the depletion of gas reservoirs with weakly permeable top

Numerical solution of the control problem on the depletion of gas reservoirs with weakly permeable top

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

A direct method for synthesis of optimum distributed systems

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