Approximation methods in optimal control problems for nonlinear infinite-dimensional systems

Serovaiskii, S. Ya.

doi:10.1134/s0001434613090277

Cited by 1 publication

(3 citation statements)

References 11 publications

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“…To estimate the second term, we introduce each term of the Laplace operator under the differential sign, after which we apply the boundary conditions (5). As a result, we have:…”

Section: Analysis Of the Differential Problemmentioning

confidence: 99%

“…Approximation of optimization problems is considered by many researchers. An important work is Serovaiskii (2013) [5], from which methods for obtaining estimates of the boundedness of the target functional are used.…”

Section: Introductionmentioning

confidence: 99%

“…(3)-(5) taking into account the discretization of the original problem. The grid function ℎ ℎ ℎ = ℎ ℎ ℎ ( ℎ ℎ ℎ ) will be the difference analogue of thefunction ( , , , ; ).…”

mentioning

confidence: 99%

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Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))

Kostikov¹,

Romanenkov²

2020

Civ Eng J

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This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.

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“…To estimate the second term, we introduce each term of the Laplace operator under the differential sign, after which we apply the boundary conditions (5). As a result, we have:…”

Section: Analysis Of the Differential Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%