J-Symmetric Factorizations and Algebraic Riccati Equations

Abstract

This paper discusses two interrelated topics: minimal J-symmetric factorizations of rational matrix functions and the algebraic Riccati equation. In particular, necessary and sufficient conditions are presented for the existence of a complete set of minimal J-symmetric factorizations of a selfadjoint rational matrix function with constant signature. For the algebraic Riccati equation the selfadjoint function which is of vital importance is the Popov function. Our first result for the algebraic Riccati equation describes the connection between the hermitian solutions, J-symmetric factorizations of the Popov function and generalized Bezoutians. Then, necessary and sufficient conditions are given for the algebraic Riccati equation to have a complete set of solutions. Both the continuous and discrete algebraic Riccati equation are treated.KeywordsConstant SignatureAlgebraic Riccati EquationMinimal RealizationImaginary LineLagrangian SubspaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.