2007/05/04 by Yury Arlinskii̇̆, Yury Arlinskii, Arlinskii, Yury
Computer Science · Engineering · Mathematics · #47A48 #47A56 #47A63 #47A64 #93B28 #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stability and Control of Uncertain Systems #math.FA #math.SP #msc:47A48 #msc:47A56 #msc:47A63 #msc:47A64 #msc:93B28
paper · pdf · doi:10.48550/arxiv.0705.0653
arxiv created 2007/05/04 · openalex publication_date 2007/05/04 · arxiv updated 2009/12/01 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We consider the Kalman - Yakubovich - Popov (KYP) inequality \beginpmatrix X-A^* XA-C^*C -A^*X B- C^*D\cr -B^*X A-D^* C I- B^*X B-D^*D \endpmatrix ≥ 0 for contractive operator matrices \beginpmatrix A&B\cr C &D \endpmatrix:\beginpmatrix\mathfrakH\cr\mathfrakM \endpmatrix→\beginpmatrix\mathfrakH\cr\mathfrakN \endpmatrix, where \mathfrakH, \mathfrakM, and \mathfrakN are separable Hilbert spaces. We restrict ourselves to the solutions X from the operator interval [0, I_\mathfrakH]. Several equivalent forms of KYP are obtained. Using the parametrization of the blocks of contractive operator matrices, the Kre\uın shorted operator, and the Möbius representation of the Schur class operator-valued function we find several equivalent forms of the KYP inequality. Properties of solutions are established and it is proved that the minimal solution of the KYP inequality satisfies the corresponding algebraic Riccati equation and can be obtained by the iterative procedure with the special choice of the initial point. In terms of the Kre\uın shorted operators a necessary condition and some sufficient conditions for uniqueness of the solution are established.