Approximate averaged bounded synthesis for a parabolic process with two switching points
DOI:
https://doi.org/10.17721/1812-5409.2024/1.19Keywords:
optimal control, parabolic process, semi-definite quality criterion, switching point, feedback control, approximate averaged synthesisAbstract
In this paper we analyze the problem of the optimal bounded control with semi-definite quality criterion for parabolic process with rapidly oscillating coefficients on finite time interval. The case when optimal control goes out both on lower, and on upper restriction, that is has two switch-points is considered. Our aim is to build a synthesized control which is practically preferable to program one. But after Fourier method application such a control will contain coefficients which are expressed by series and, in addition, it irregularly depends on small parameter ε. Thus, for practical application it is natural to restrict infinite series by finite sums and to get rid of dependence on a small parameter using average procedure. So, besides the law of optimal synthesis also approximate averaged feedback control, which provides control system behavior that is close to optimal one and thus has a series of advantages from the practical application point of view, is proposed and proved. The efficiency of approximate averaged control construction procedure consisted of cutting to finite sums of series in optimal control and replacing of Fourier coefficients nonregular depended on small parameter by corresponding average values is illustrated by an example of controlled system for parabolic process. We compare the properties of the averaged control and the sequence of optimal controls calculated at the different values of the small parameter. For comparison, we use the switching points of optimal control and averaged control, deviation between optimal control and averaged control, the difference in the values of the quality criterion on optimal control and averaged control. In considered example the precision of quality criteria value on approximate control has ε-order and for sufficiently small ε the precision is one order better then ε.
Pages of the article in the issue: 96 - 100
Language of the article: English
References
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Ladyzhenskaya, O. A. (2013). The Boundary Value Problems of Mathematical Physics. Springer-Verlag.
Lions, J.-L. (1971). Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag.
Temam, R. (1997). Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer.
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Copyright (c) 2024 Yuriy Loveikin, Anna Sukretna
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