Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation

HL Trentelman*, P Rapisarda

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review


    We study the existence of positive and negative semidefinite solutions of algebraic Riccati equations (ARE) corresponding to linear quadratic problems with an indefinite cost functional. The formulation of reasonable necessary and sufficient conditions for the existence of such solutions is a long-standing open problem. A central role is played by certain two-variable polynomial matrices associated with the ARE. Our main result characterizes all unmixed solutions of the ARE in terms of the Pick matrices associated with these two-variable polynomial matrices. As a corollary of this result, we nd that the signatures of the extremal solutions of the ARE are determined by the signatures of particular Pick matrices.

    Original languageEnglish
    Pages (from-to)969-991
    Number of pages23
    JournalSiam Journal on Control and Optimization
    Issue number3
    Publication statusPublished - 19 Nov 2001


    • algebraic Riccati equation
    • existence of semidefinite solutions
    • two-variable polynomial matrices
    • Pick matrices
    • dissipative systems

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