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Mathematics of Control, Signals, and Systems

, Volume 31, Issue 3, pp 333–362 | Cite as

Approximately reachable directions for piecewise linear switched systems

  • Dan GoreacEmail author
Original Article
  • 42 Downloads

Abstract

This paper deals with some reachability issues for piecewise linear switched systems with time-dependent coefficients and multiplicative noise. Namely, it aims at characterizing data that are almost reachable at some fixed time \(T>0\) (belong to the closure of the reachable set in a suitable \({\mathbb {L}}^2\)-sense). From a mathematical point of view, this provides the missing link between approximate controllability toward 0 and approximate controllability toward given targets. The methods rely on linear–quadratic control and Riccati equations. The main novelty is that we consider an LQ problem with controlled backward stochastic dynamics and, since the coefficients are not deterministic (unlike some of the cited references), neither is the backward stochastic Riccati equation. Existence and uniqueness of the solution of such equations rely on structure arguments [inspired by Confortola (Ann Appl Probab 26(3):1743–1773, 2016)]. Besides solvability, Riccati representation of the resulting control problem is provided as is the synthesis of optimal (non-Markovian) control. Several examples are discussed.

Keywords

Reachability Approximate controllability Controlled switch process Linear–quadratic control Backward stochastic Riccati equation Stochastic gene networks 

Mathematics Subject Classification

93B05 93B25 60J75 

Notes

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShandong University at WeihaiWeihaiP.R. China
  2. 2.LAMA (UMR 8050), UPEMLV, UPEC, CNRSUniversité Paris-EstMarne-la-ValléeFrance

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