Continuity properties of one-parameter families of linear-quadratic problems without stability

A.H.W. Geerts

    Research output: Book/ReportReportAcademic

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    Abstract

    In a recent paper ([1]) given one-parameter families of linear-quadratic control problems (with side condition that the state trajectory should vanish at infinity) have been investigated. It was proven there that, under two rather acceptable assumptions, the optimal cost depends continuously on the parameter. Moreover, optimal inputs (whenever they exist), state trajectories and outputs are continuous w.r.t. the parameter if the underlying systems are left invertible. However, in contrast with these problems with stability there generally proved to be no such continuity properties for problems without stability (i.e. the free end-point problems). In the present paper we will explain why. We will demonstrate that the definition of a new type of control problem with "partial" stability is necessary. The optimal cost for the "perturbed" problem then turns out to converge to the cost for this new problem. Additional results are found for inputs, states and outputs in case of left-invertibility. These results are established only by applying the assumptions made in the article mentioned above. Actually even less.
    Original languageEnglish
    Place of PublicationEindhoven
    PublisherTechnische Universiteit Eindhoven
    Number of pages25
    Publication statusPublished - 1988

    Publication series

    NameMemorandum COSOR
    Volume8817
    ISSN (Print)0926-4493

    Fingerprint

    Linear Quadratic Problem
    Control Problem
    Costs
    Trajectory
    Linear Quadratic Control
    Output
    Invertibility
    End point
    Invertible
    Vanish
    Infinity
    Family
    Converge
    Partial
    Necessary
    Demonstrate

    Cite this

    Geerts, A. H. W. (1988). Continuity properties of one-parameter families of linear-quadratic problems without stability. (Memorandum COSOR; Vol. 8817). Eindhoven: Technische Universiteit Eindhoven.
    Geerts, A.H.W. / Continuity properties of one-parameter families of linear-quadratic problems without stability. Eindhoven : Technische Universiteit Eindhoven, 1988. 25 p. (Memorandum COSOR).
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    Geerts, AHW 1988, Continuity properties of one-parameter families of linear-quadratic problems without stability. Memorandum COSOR, vol. 8817, Technische Universiteit Eindhoven, Eindhoven.

    Continuity properties of one-parameter families of linear-quadratic problems without stability. / Geerts, A.H.W.

    Eindhoven : Technische Universiteit Eindhoven, 1988. 25 p. (Memorandum COSOR; Vol. 8817).

    Research output: Book/ReportReportAcademic

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    T1 - Continuity properties of one-parameter families of linear-quadratic problems without stability

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    AB - In a recent paper ([1]) given one-parameter families of linear-quadratic control problems (with side condition that the state trajectory should vanish at infinity) have been investigated. It was proven there that, under two rather acceptable assumptions, the optimal cost depends continuously on the parameter. Moreover, optimal inputs (whenever they exist), state trajectories and outputs are continuous w.r.t. the parameter if the underlying systems are left invertible. However, in contrast with these problems with stability there generally proved to be no such continuity properties for problems without stability (i.e. the free end-point problems). In the present paper we will explain why. We will demonstrate that the definition of a new type of control problem with "partial" stability is necessary. The optimal cost for the "perturbed" problem then turns out to converge to the cost for this new problem. Additional results are found for inputs, states and outputs in case of left-invertibility. These results are established only by applying the assumptions made in the article mentioned above. Actually even less.

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    ER -

    Geerts AHW. Continuity properties of one-parameter families of linear-quadratic problems without stability. Eindhoven: Technische Universiteit Eindhoven, 1988. 25 p. (Memorandum COSOR).