
Izv. IMI UdGU, 2017, Volume 50, Pages 110–120
(Mi iimi350)




Asymptotic expansion of a solution for the singularly perturbed optimal control problem with a convex integral quality index and smooth control constraints
A. A. Shaburov^{} ^{} Ural Federal University, ul. Mira, 19, Yekaterinburg, 620002, Russia
Abstract:
The paper deals with the problem of optimal control with a convex integral quality index for a linear steadystate control system in the class of piecewise continuous controls with smooth control constraints. In a general case, to solve such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. The main difference from the preceding article [5] is that the terminal part of the convex integral quality index depends not only on slow, but also on fast variables. In a particular case, we derive an equation that is satisfied by an initial vector of the conjugate system. Then this equation is extended to the optimal control problem with the convex integral quality index for a linear system with the fast and slow variables. It is shown that the solution of the corresponding equation as $\varepsilon\to0$ tends to the solution of an equation corresponding to the limit problem. The results obtained are applied to study a problem which describes the motion of a material point in $\mathbb{R}^n$ for a fixed interval of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion.
Keywords:
optimal control, singularly perturbed problems, asymptotic expansion, small parameter.
DOI:
https://doi.org/10.20537/2226359420175009
Full text:
PDF file (252 kB)
References:
PDF file
HTML file
Bibliographic databases:
Document Type:
Article
UDC:
517.977
MSC: 49N05, 93C70 Received: 04.08.2017
Citation:
A. A. Shaburov, “Asymptotic expansion of a solution for the singularly perturbed optimal control problem with a convex integral quality index and smooth control constraints”, Izv. IMI UdGU, 50 (2017), 110–120
Citation in format AMSBIB
\Bibitem{Sha17}
\by A.~A.~Shaburov
\paper Asymptotic expansion of a solution for the singularly perturbed optimal control problem with a convex integral quality index and smooth control constraints
\jour Izv. IMI UdGU
\yr 2017
\vol 50
\pages 110120
\mathnet{http://mi.mathnet.ru/iimi350}
\crossref{https://doi.org/10.20537/2226359420175009}
\elib{http://elibrary.ru/item.asp?id=32260611}
Linking options:
http://mi.mathnet.ru/eng/iimi350 http://mi.mathnet.ru/eng/iimi/v50/p110
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Translation

Number of views: 
This page:  92  Full text:  37  References:  16 
