Characterization of well-posedness of piecewise linear systems

J.I. Imura, Arjan van der Schaft

Research output: Book/ReportReportOther research output

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Abstract

One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. This paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Carath\'eodory. The concepts of jump solutions or a sliding mode are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multi-modal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-posed.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Publication statusPublished - 1998

Publication series

NameMemorandum / Department of Mathematics
PublisherDepartment of Applied Mathematics, University of Twente
No.1475
ISSN (Print)0169-2690

Keywords

  • MSC-90C33
  • MSC-93A30
  • EWI-3295
  • IR-65664
  • MSC-93B99

Cite this

Imura, J. I., & van der Schaft, A. (1998). Characterization of well-posedness of piecewise linear systems. (Memorandum / Department of Mathematics; No. 1475). Enschede: University of Twente, Department of Applied Mathematics.
Imura, J.I. ; van der Schaft, Arjan. / Characterization of well-posedness of piecewise linear systems. Enschede : University of Twente, Department of Applied Mathematics, 1998. (Memorandum / Department of Mathematics; 1475).
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Imura, JI & van der Schaft, A 1998, Characterization of well-posedness of piecewise linear systems. Memorandum / Department of Mathematics, no. 1475, University of Twente, Department of Applied Mathematics, Enschede.

Characterization of well-posedness of piecewise linear systems. / Imura, J.I.; van der Schaft, Arjan.

Enschede : University of Twente, Department of Applied Mathematics, 1998. (Memorandum / Department of Mathematics; No. 1475).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - Characterization of well-posedness of piecewise linear systems

AU - Imura, J.I.

AU - van der Schaft, Arjan

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N2 - One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. This paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Carath\'eodory. The concepts of jump solutions or a sliding mode are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multi-modal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-posed.

AB - One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. This paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Carath\'eodory. The concepts of jump solutions or a sliding mode are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multi-modal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-posed.

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KW - IR-65664

KW - MSC-93B99

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Imura JI, van der Schaft A. Characterization of well-posedness of piecewise linear systems. Enschede: University of Twente, Department of Applied Mathematics, 1998. (Memorandum / Department of Mathematics; 1475).