Kalman–Popov–Yakubovich lemma for ordered fields

The Kalman–Popov–Yakubovich lemma was generalized to the case where the field of scalars is an ordered field that possesses the following property: if each value of the polynomial of one variable is the sum of squares, then the polynomial itself is the sum of squares of polynomials. The field with this property was named the sum of squares (SOS) field. The SOS-fields are, for instance, those of rational numbers, algebraic numbers, real numbers, or rational fractions of several variables, with the coefficients from the aforementioned fields. It was proved that the statement of the Kalman–Popov–Yakubovich lemma about the equivalence of the frequency domain inequality and the linear matrix inequality holds true if the SOS-field is considered as a that of scalars. An example was presented which shows that in the SOS-field the fulfillment of the frequency domain inequality does not imply solvability of the corresponding algebraic Riccati equation.