Factorization of the Popov Function of a Multivariable Linear Distributed Parameter System in the Non-coercive Case: a Penalization Approach

where ∗ denotes both the conjugate of a complex number and the conjugate of a matrix or an operator. The first result of this type is the well-known Fejer-Riesz theorem (Riesz and SzNagy, 1955), which states that if Π(iω) is a (scalar) polynomial, then we can choose a polynomial M(z) which, additionally, does not have zeros in the right half-plane. Extensions to polynomial, hence also to rational matrices are known and have been widely used in systems theory (Anderson 1967; Balakrishnan, 1995; Francis, 1987; Kalman, 1963; Yakubovich, 1973; Youla, 1961). The fact that we want to stress is the following: let us assume for a moment that Π(iω) is a rational (square) matrix function, and that R = lim|ω|→+∞Π(iω) is an invertible matrix, and hence a positive definite matrix. In this case the factorization