On a new class of realization formulas and their application
This paper is devoted to derive and develop the theory of a new set of realization formulas for a class of matrix-valued meromorphic functions \(W(\lambda)\), which includes the class of rational matrix-valued functions. The realization formulas for a general class of \(p\times q\) matrix-valued functions meromorphic in a connected nonempty open set \(\Omega\) of the complex plane are first established together with a number of auxiliary formulas and results. Then the authors discuss the concept of observability, controllability and minimality, and give a number of equivalent characterizations of each. The similarity of two minimal realizations is also discussed. Next, they develop the necessary and sufficient conditions in terms of Lyapunov-Stein-type equations for the existence of the realizations of this type, which are either isometric or coisometric (or both) matrix-valued functions with respect to a pair (or just one) signature matrices on the boundary of an appropriately defined region. Subsequently, they discuss pairings wherein a given \(W(\lambda)\) is ``paired'' with a partner \(W_\pi(\lambda)\), which is often an effective substitute for \(W(\lambda)^*\) when the conditions for \(W(\lambda)\) to be either isometric or coisometric (with respect to two given signature matrices) on the boundary fail. Finally, they devote to embedding the basic Lyapunov-Stein-type equations into a larger array of equations and related implications, and besides, factorization formulas for the pair space considered and minimal \(J\) unitary factorizations of \(W (\lambda)\) are treated.
scientific article; zbMATH DE number 931881
scientific article