Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices
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The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: R^n; R and its intervals [a,b], [0,∞); and the strips [a,b] ×R⊂R^2. A method for numerically computing such representations is also presented. The methodology is divided into two stages: (S1) diagonalizing the initial polynomial matrix based on the Schmüdgen's procedure Schmudgen09; (S2) for each diagonal element of the resulting matrix, find its low rank sosrf-representation satisfying the Artin's theorem solving the Hilbert's 17th problem. Some numerical tests and illustrations with OCTAVE are also presented for each type of polynomial matrices.
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