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The Bulletin of Irkutsk State University. Series Mathematics, 2017, Volume 19, Pages 164–177 (Mi iigum295)  

Impulsive control systems with trajectories of bounded $p$-variation

O. N. Samsonyuk, M. V. Staritsyn

Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033

Abstract: The paper deals with impulse-trajectory relaxations of control-affine systems under the assumption that $L_1$-norms of the control functions are not uniformly bounded. Generalized solutions of such control systems may be of infinite total variation. Most of the known results related to impulse-trajectory relaxations of control-affine systems are mainly devoted to the case of state with bounded variation and controls of the type of bounded Borel measures and do not concern the considered case. The main issue of the study are constrictive techniques for impulse-trajectory relaxations within the class of functions of bounded $p$-variation ($p>1$) (in the sense N. Wiener) and explicit representations of relaxed systems.
Based on a sort of discontinuous time reparametrization, we propose a new approach to trajectory extension of systems with generalized solutions of bounded $p$-variation, $p>1$. This approach includes some space-time extension of the original system and its transformation to an auxiliary one with continuous solutions of bounded $p$-variation. In this paper, we dwell on the case of scalar control inputs, while a similar space-time transformation can be applied for control-affine systems with vector-valued inputs, even, in the absence of the involution property (or a more conventional commutativity assumption) of the vector fields.
For the case $p\in[1,2)$ and scalar control input, we obtain an explicit representation of the extended control system by a specific discrete-continuous integral equation involving Young integral.

Keywords: trajectory relaxations of control systems, solutions of bounded $p$-variation, impulsive control.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-31-60030_мол_а_дк
Ministry of Education and Science of the Russian Federation НШ-8081.2016.9


DOI: https://doi.org/10.26516/1997-7670.2017.19.164

Full text: PDF file (394 kB)
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Document Type: Article
UDC: 517.977.5
MSC: 93C10, 93C23

Citation: O. N. Samsonyuk, M. V. Staritsyn, “Impulsive control systems with trajectories of bounded $p$-variation”, The Bulletin of Irkutsk State University. Series Mathematics, 19 (2017), 164–177

Citation in format AMSBIB
\Bibitem{SamSta17}
\by O.~N.~Samsonyuk, M.~V.~Staritsyn
\paper Impulsive control systems with trajectories of bounded $p$-variation
\jour The Bulletin of Irkutsk State University. Series Mathematics
\yr 2017
\vol 19
\pages 164--177
\mathnet{http://mi.mathnet.ru/iigum295}
\crossref{https://doi.org/10.26516/1997-7670.2017.19.164}


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